arXiv:2011.00845v1 [math.AG] 2 Nov 2020 eiwof Review 1 Introduction 0 Contents eiwo coeso ufcswt mhsson emphasis with surfaces on schobers of Review 3 etlna on fve n h ore rnfr 40 transform Fourier the and view of point Rectilinear 2 . coeso ufcs...... gluing . . . and . . . decompositions . semi-orthogonal . . on . . arro . Reminder Picard-Lefschetz . . . and . . triangles . 3.3 Picard-Lefschetz . . surfaces . . on 3.2 . Schobers . . . . . 3.1 . . . Infrared . . the . structur . of Stokes Algebra . . sheaves: baby . perverse The . of . transform topology Fourier 2.5 . sheaves: perverse . of approac . transform 2.4 rectilinear Fourier . the in . deformations 2.3 . Isomonodromic approach rectilinear 2.2 The 2.1 . utetb lcly osatsevs...... sheaves constant . . (locally) . . . by Quotient forms . . bilinear . . with theory. sheaves 1.4 Picard-Lefschetz . Perverse classical . the to . 1.3 Relation theory Picard-Lefschetz 1.2 Abstract 1.1 evreshbr n h ler fteInfrared the of Algebra the and schobers Perverse aycntutoso h ler fteIfae a edeve be can Infrared the of Algebra suc on schober the between (tunnelling) of interaction constructions 2-dimens the many a as of well D-branes as of theory, categories various encode can that hae.Apres coe nacmlxplane complex a ca on general) schober in perverse (conjectural, A are sheaves. which schobers perverse of sheaves. o coes ftegoercFuirtasomwl known well transform Fourier geometric the of schobers, for erlt h ler fteIfae fGaiotto-Moore-Witt of Infrared the of Algebra the relate We Perv ihi arnv a obla,LvSoukhanov Lev Soibelman, Yan Kapranov, Mikhail C hs osrcin a ese sgvn aiu features various giving as seen be can constructions These . p C A , q oebr3 2020 3, November Abstract 1 C a ese sa leri structure algebraic an as seen be can eoia nlg fperverse of analogs tegorical aeois eso that show We categories. h for C s...... 75 ...... ws oa uesmercfield supersymmetric ional 53 ...... e D 43 ...... h 22 ...... mdlsadperverse and -modules oe new aea have we once loped ucos...... 85 ...... functors nwt h theory the with en 48 ...... fteanalog, the of 27 ...... 38 ...... 13 ...... 40 . . . 60 . . . 70 . 70 13 2 3.4 Unitriangular (co)monads and glued N-termSOD’s ...... 90 3.5 The Fukaya-Seidel category of a schober on C ...... 100 3.6 The Lefschetz schober of a Landau-Ginzburg potential ...... 106

4 Algebra of the Infrared associated to a schober on C, I 110 4.1 Rectilinear approach and the Fourier transform of a schober...... 111 4.2 Theinfraredcomplexofaschober...... 113 4.3 The infrared monad and the Fukaya-Seidel monad ...... 121

5 Algebra of the Infrared associated to a schober on C, II 126 5.1 Backgroundonsecondarypolytopes ...... 126 5.2 Variation of secondary polytopes and exceptional walls ...... 129 5.3 The L8-algebra associated to a schober on C ...... 136 5.4 Deforming the infrared monad R ...... 142 5.5 Maurer-Cartan elements in the infrared L8-algebra...... 148

A Conventions on (enhanced) triangulated categories 152

0 Introduction

A. The goal of the paper. The term “Algebra of the Infrared” was coined by D. Gaiotto, G. Moore and E. Witten [36] to describe a remarkable new chapter of homological algebra dealing with triangulated categories of D-branes in quantum field theory. In the infrared limit, a physical theory reduces to the data of vacua and of tunnelling between the vacua. For a massive 2-dimensional theory with 2, 2 supersymmetry, they introduced a p q remarkable algebraic structure: a homotopy Lie (L8-) algebra g acting by higher deriva- tions on a homotopy associative (A8-)algebra R and a Maurer-Cartan element in g giving a deformation of R describing the global category of D-branes (Fukaya-Seidel category). Re- markably, these structures make appeal to rather deep aspects of convex geometry. In [51] a part of the constructions of [36] was interpreted using the language of secondary polytopes. The goal of this paper is to relate the Algebra of the Infrared with the theory of perverse schobers, which are (conjectural, in general) categorical analogs of perverse sheaves intro- duced in [54] and studied further in [16, 27, 28, 29]. More precisely, we show that perverse schobers form the natural framework for this type of algebra: many of its constructions can be developed each time we have a schober on C.

B. Tunnelling between vacua and perverse sheaves. Let us summarize (with some formal generalizations) the approach of [36]. A theory of the above kind has a discrete set A of vacua; for simplicity we assume A finite. Further, A w1, ,wN C is embedded into the complex plane parametrizing charges of the supersymmet» t ¨ry ¨ ¨ algebrauĂ [96]. For a Landau- Ginzburg (LG) theory with superpotential W : Y C the vacua correspond to critical Ñ

2 points yi of W (which we assume isolated but not necessarily Morse), and wi W yi are the corresponding critical values (assumed distinct). “ p q In the IR limit the vacua become separated (cease to interact) and so one can associate to the ith vacuum its own “category of infrared D-branes” Φi. If the vacuum is massive (in the b LG example this means that yi is Morse), then Φi D Vectk is “the triangulated category freely generated by one object” (the geometric vanishing“ sphere in the LG Morse example). More generally, if yi is an isolated singular point of W , then Φi is the local Fukaya-Seidel category of W at yi (see §3.6 below for a discussion). This is a categorical analog of the space of vanishing cycles of W at yi, in particular, the rank of its Grothendieck group is µi, the multiplicity, or Milnor number of yi. Now, switching the interaction, i.e., the tunnelling between the vacua, gives rise to trans- port functors Mij : Φi Φj. ÝÑ b b In the LG example, when both critical points yi and yj are Morse, Mij : D Vectk D Vectk is the functor of tensor multiplication with the Floer complex spanned by ζ-solitons,Ñ i.e., ´1 by gradient trajectories of Re ζ W joining yi and yj, see [36]. Here ζ is the slope of the p q interval wi,wj . r s We observe that data Φi, mij (formed by vector spaces and linear operators instead of categories and functors) appearp inq the mathematical problem of classification of perverse sheaves [41]. Originally [9], perverse sheaves were introduced as a language for intersec- tion cohomology theory, which is a cohomology theory for singular varieties that satisfies Poincaré duality [46]. They are “self-dual” local objects, situated half-way between sheaves and cosheaves, much like half-densities, so important in quantum mechanics (they form a natural L2-space), are situated between functions and volume forms. One manifestation of this is the relation of intersection cohomologies and L2-cohomologies, see [45] for a review. An additional remarkable property of perverse sheaves is their intimate connection with the concept of vanishing cycles: forming vanishing cycles in the most general situation pre- serves perversity [9, 60]. In particular, S. Gelfand, R. MacPherson and K. Vilonen [41] used vanishing cycles to identify the category of perverse sheaves on C or on a disk (modulo constant sheaves, see below), with the category of data Φi, mij subject to a mild non- degeneracy property. p q This parallelism leads us to suggest that perverse schobers should be a natural conceptual tool to analyze infrared behavior of theories as above.

C. The concept of a perverse schober. Intuitively, perverse schobers should be categorical analogs of perverse sheaves, where “vector spaces are replaced by triangulated categories” [54]. The original observation of [54] was that a datum of a spherical functor a and its right adjoint a˚ a Φ o / Ψ, a˚

3 see [4], can be viewed as a categorical analog of the diagram of [38] describing a perverse sheaf on C with one singularity at 0. The term Schober is one of the German analogs of the English word sheaf: it has the same root, also present in the German schieben (to push), Verschiebung (shift), Scheibe (moving disk), as well as in the English shove (as in “when push comes to shove”). For economy, we often speak simply about “schobers”, omitting the adjective “perverse”. However, a general theory of perverse schobers has not yet been developed. While per- verse sheaves can be defined in terms of the derived category formed by complexes of sheaves of vector spaces [9], there is no analogous pre-existing concept of “complexes of sheaves of categories”. Nor it is clear what such a concept should look like (although some important hints as to categorified homological algebra are provided by [31]). Already in the case of Riemann surfaces, a flexible topological treatment of perverse schobers is quite non-trivial, and is being developed in [32]. In the present paper our focus is not on the general theory but on things one can do with perverse schobers in a physically motivated context. This provides an additional motivation for future foundational studies. Accordingly, we adopt the most naive approach to perverse schobers on C, obtained by categorifying the description of [41] for perverse sheaves. That is, we consider a schober S with singularities at A w1, ,wN as represented by a collection N “ t ¨ ¨ ¨ u ai : Φi Ψ of spherical functors with common target. This description proceeds in the p Ñ qi“1 presence of a “spider” K which is a system of cuts joining some outside point v with the wi. The consideration of such spiders goes back to the Arnold school of singularity theory in the 1960-70’s, see [5]. Already with this minimal background, it is possible to capture a lot of beautiful phe- nomena. Some further features are not fully captured but it becomes clear that they would follow from a more systematic theory. We indicate them and leave them for future work.

D. The role of the Fourier transform. To be precise, the physical picture corresponds to looking at perverse sheaves and schobers with the intent of making the Fourier transform. That is, there is an important difference between the above two approaches. In the physical tunnelling picture, it is the rectilinear intervals wi,wj that play a distinguished role, so the transport happens along such intervals. But ther approachs of Gelfand-MacPherson-Vilonen involves the transport along curved paths passing through some common faraway point (“Vladivostok”, see Remark 1.1.18). In fact, the concept of a perverse sheaf being purely topological, the mathematical reason to consider rectilinear intervals may not be clear. The Fourier transform point of view makes this reason apparent because of the dominance order of various exponentials w ezw in the complex domain. The relevance of the FourierÑ transform for the physical picture is clear by looking at the Landau-Ginzburg example: a typical oscillatory integral

i (0.1) I ~ e ~ W pyqΩ y dy, p q “ p q żΓĂY

4 fundamental in the theory, can be seen as the result of, first, integrating over the fibers of W and then taking the single variable Fourier transform. To formulate our point of view more clearly, a large part of the Algebra of the Infrared can be interpreted as the study of the Fourier transform for perverse schobers. This last concept needs, of course, to be defined, and we can proceed by analogy with the case of perverse sheaves. Here we have the concept of the Geometric Fourier Transform [20, 76, 24] which corresponds to the formal Fourier transform (z w, z w) applied to a holonomic regular D-module M on C. The latter procedureÞÑ produces B B Ñ a holonomic, ´ typically irregular D-module M on the dual complex plane with the only possible singularity at 0. So ignoring the Stokes phenomena, the most basic datum provided by M, is a local system L on C 0 . zt u In the categorical| case the analog of L is provided by the Fukaya-Seidel category with its dependence on the direction at infinity [86, 36]. We show that| this data can be defined for an arbitrary schober S on C and define the further features that one can expect from the Fourier transform S, see §4.1. Many of the remarkable constructions of the Algebra of the Infrared can be interpreted as tools for finding these features in terms of the data defining S. q In fact, these constructions can be found already in their “baby version” at the level of perverse sheaves, more specifically in the (seemingly new) formulas expressing various curvilinear data in terms of rectilinear ones, using convex geometry. For example, we give a formula (Proposition 2.5.12) for the Stokes matrix of the Fourier transform of a perverse sheaf which can be seen as the de-categorifcation of the infrared A8-algebra R, involving alternating sums over convex paths.

E. Schobers vs. localized schobers. It is worth stressing that a “tunnelling datum” Φi, mij : Φi Φj of vector spaces and linear operators represents not exactly a perverse psheaf, but a localizedÑ q perverse sheaf, i.e., an object of the category Perv C, A obtained by quotienting Perv C, A , the category of all perverse sheaves with singularitiesp q in A, by the subcategory of constantp q sheaves [41]. Any F Perv C, A gives rise to a localized perverse sheaf F, i.e., a tunnelling datum as above. ToP lift Fp toq an F in terms of tunnelling data ai amounts to giving an extra vector space Ψ (the generic stalk) and maps Φi o / Ψ so that bi (the Vladivostok version of) mij factors as bjai. Accordingly, it is “localized schobers” that are, strictly speaking, relevant for our problem. In this paper we do not attempt to define the corresponding ( -)category, as its meaningful study would require more foundational work. Instead, we sta8rt with a schober represented ai by a diagram Φi Ψ of spherical functors and use some but not all information contained in such a diagram.p Ñ Theq physical meaning of the category Ψ for a schober seems less clear but possibly important as one can see from the following example.

A LG superpotential W : Y C gives rise to the Lefschetz perverse sheaf LW , see §1.2 Ñ and its categorification, Lefschetz schober LW , see §3.6. The vector space Ψ for LW is the ´1 middle cohomology of a generic fiber W b ; the category Ψ for LW is the Fukaya category p q

5 of W ´1 b . As such, they are sensitive to the way W is (or is not) compactified, i.e., made into a properp q map. Note that traditionally, the superpotential mirror dual to a Fano variety V is written in a bare-bones, non-proper form such as a Laurent polynomial C˚ C, e.g., p qÑ 1 W z1, , zm z1 zm p ¨ ¨ ¨ q “ `¨¨¨` ` z1 zm ¨ ¨ ¨ for V Pm, so there is in principle some ambiguity as to its compactification. This ambiguity has a“ counterpart in the discussion of the quantum differential equation (QDE) of V which we write in the 1-variable form as

dΨ µΨ ˚ ‚ (0.2) K K Ψ, Ψ: C H X, C . dq “ q ` ˚q Ñ p q Here K H2 X, Z is the canonical class of V and µ is the grading operator equal to i dim VP 2 onp Hiq. A part of the mirror symmetry package for V can be expressed by ´ p q{ saying that the QDE “is” the Fourier transform LW of LW (in particular, its solutions are expressed via oscillatory integrals for W ). But LW is a D-module on C, while (0.2) gives, a priori, a D-module only on C˚. Extension of it to allq of C corresponds precisely to specifying LW as a perverse sheaf (and not just a localizedq one), i.e., to specifying the above vector space Ψ precisely, i.e., to the question of compactification of W . From this point of view, the Dubrovin Conjecture ([30] §4.2.2), involving categorical lifts of the Stokes matrices of the QDE, should likely admit a formulation in terms of schobers.

F. Webs vs. secondary polytopes. Our approach to the “Algebra of the Infrared in the presense of a schober” is, following [51], based on the concept of secondary polytopes [39]. In that, it is dual to the original approach of Gaiotto-Moore-Witten [36, 37] which uses certain planar graphs called webs. For convenience of the reader, let us recall (cf. also [51] §13) this duality, which is made very natural by the Fourier transform point of view.

We view elements of A w1, ,wN as points of the complex plane C Cw on which a perverse schober S is defined,“ t and¨ consider ¨ ¨ u the convex polygon Q Conv A “. The secondary “ p q 1 polytope Σ A has faces corresponding to decompositions P of Q into a union Q Qi of (suitably marked,p q see §5.1) convex subpolygons with vertices in A. These decompositions“ must be regular, i.e., admit convex piecewise-affine-linear functions f : Q R thatŤ do 1 Ñ break along every possible juncture of the Qi. Given a schober S with singularities in A, each subpolygon Q1 Q gives rise to a natural “Clebsch-Gordan space” I Q1 , see §5.3. It Ă p q consists of natural transformations between the compositions of the transport functors Mij 1 along two complementary arcs in the boundary of Q . The infrared L8-algebra g gS is formed out of such I Q1 using the factorization properties of secondary polytopes. “ p q In comparison, the original approach of [36] proceeds in the dual complex plane Cz (on which, in our language, the Fourier transform S is supposed to live). Points of Cz represent R-linear functionals Cw R. Given a convex piecewise-affine-linear f : Q R as above, Ñ Ñ q 6 1 C the linear parts of f on different Qi give points pi z. Joining pi with pj by an edge when 1 1 P C Qi and Qj have an edge in common, gives a graph w in z which is precisely a web in the sense of [36, 37]. Topologically, w is a graph Poincaré dual to the subdivision P, see Fig. 1. So w encodes both the decomposition P Q1 and the function f. “ t iu

Cw Cz

1 Q2 1 1 p2 Q1 Q4 p3 p4 1 p1 Q3 P w

Figure 1: Webs w are dual to subdivisions P together with convex PL-functions.

Thus the two approaches are completely equivalent and one can easily translate any argument from one language to the other.

G. The role of higher coherences. An intrinsic difficulty in any upgrade from vector space to triangulated categories is that of higher coherences. First, “classical” triangulated categories cannot be used as primary objects since their Hom-spaces are not fundamental: they appear as the cohomology of certain cochain com- plexes or the sets of connected components of some topological spaces. Keeping track of these complexes or spaces leads to enhancements: pre-triangulated dg-categories [15] (the approach adopted here), or more fundamental stable -categories [74]. 8 Second, once such an approach is adopted, the meaning of any identity among the mor- phisms changes: it should now hold up to some “homotopy”, and there should be a coherence condition for these homotopies, which is itself a next level homotopy and so on ad infinitum. This of course complicates the study since every single step is now, so to say, replaced by an infinite routine. Some of these routines are by now standard and do not need special emphasis. Such are, for example, A8 and L8-structures (higher coherence data for associativity or the Jacobi identity); we use´ them in several places throughout the paper, mostly in the same context as [36, 37]. Beyond that, in this paper we keep such higher homotopy techniques to a minimum, in line with our “elementary” approach to schobers. That is, we work at the triangulated category level or, implicitly, both at the triangulated and the enhanced levels, when the transition between the two is easy. Hopefully, this makes the paper more accessible and helps to keep its size under control.

7 At the very end, we highlight one situation where higher coherences are essential and which will be studied in a future paper. This is the Maurer-Cartan element of g which gives, by deformation, the Fukaya-Seidel category. In [36], such an element was defined using the count of solutions of a -equation. From our point of view, such an element should be present for any schober andB appear as a collection of “higher coherence data for Picard-Lefschetz identities” in the following sense. For a perverse sheaf F on C with singularities in A, every “empty triangle” (i.e., a rectilinear triangle ∆ijk Conv wi,wj,wk not containing other wl) gives rise to a Picard- Lefschetz (“wall-crossing”)“ identityt amongu three linear transport operators:

1 (0.3) m mik mjkmij, ik “ ´ see Proposition 1.1.12 for details. For a perverse schober S on C with singularities in A, each such identity upgrades to an exact triangle involving the transport functors

1 uijk vijk M Mik Mjk Mij Mik 1 , ik ÝÑ ÝÑ ˝ ÝÑ r s cf. Proposition 3.2.1. It involves new data: the Picard-Lefschetz arrows (natural trans- formations of functors) uijk, vijk. Up to easy modifications (dualization etc.) there is just 1 one Picard-Lefschetz arrow for each geometric empty triangle ∆ijk. Any subpolygon Q Q with vertices in A, triangulated into empty triangles, gives then a 2-dimensional compositionĂ (pasting) of the Picard-Lefschetz arrows corresponding to the triangles. An important fact is that at the level of triangulated categories this pasting is independent of the triangulation1, see Propositions 3.2.14 and 3.2.16 for the effect of 2 2 and 3 1 moves. But at the dg-level we have a higher coherence problem for suchÑ independence.Ñ The condition of the corresponding higher coherences are precisely those of being a Maurer-Cartan element in gS. We expect that such an element is canonically, up to gauge equivalence, determined by the schober S. This should follow from a more “advanced” definition of a schober which would include a sufficient supply of coherence data at the very outset. We leave this for future work.

H. Overview of the paper. The paper consists of five chapters. Chapter 1 is devoted to an overview of perverse sheaves on topological surfaces, espe- cially on the complex plane C. This subject is elementary but our presentation emphasizes several subtler points important for the sequel. In §1.1 we develop, following [54], an “abstract” version of the classical Picard-Lefschetz theory, in the context of a perverse sheaf instead of a singular point of a Lefschetz pencil, as is usually done, see, e.g., [5] Ch.1. The key concept for this is the transport map mij γ : p q 1We have here a clash of two uses of the term “triangulated”: one for the algebraic concept of a triangulated category, the other for a geometric triangulation of a plane polygon.

8 Φi Φj between spaces of vanishing cycles associated to a path γ. Remarkably, the concept of aÑ perverse sheaf appears to be intrinsically designed to produce Picard-Lefschetz identities such as (0.3), see Proposition 1.1.12. One can further interpret (Proposition 1.1.15) these identities in terms of “Vassiliev derivatives” of the monodromy living on various strata in the space of paths. We also recall from [41] the explicit quiver-like description of perverse sheaves on C which we later use as a blueprint for defining schobers. In §1.2 we connect the abstract Picard-Lefschetz theory for perverse sheaves with the classical theory for Lefschetz pencils W : Y X where X is a Riemann surface. In fact, we consider a slightly more general situation whenÑ W has isolated but not necessarily Morse critical points. The connection goes via a particular perverse sheaf LW on X associated with W which we call the Lefschetz perverse sheaf. It seems to be an object of some importance. In particular, it has a natural categorification, the Lefschetz schober, which underlies many features of Fukaya-Seidel theory see §3.6. In §1.3 we extend the classical Φ, Ψ -description of perverse sheaves on the disk [38] to a description of perverse sheaves equippedp q with a nondegenerate, not necessarily symmetric, bilinear form. Recall that a triangulated category can be seen as a “higher” analog not just of a vector space but of a vector spaces with a bilinear form, whole role is played by the spaces Hom x, y . This form is, in general, non-symmetric, hence the difference between left and right adjointp q functors. Our description is, in agreement with the original proposal in [54], completely analogous to the data defining a spherical functor. In particular, we give, in this decategorified context, analogs of the entire spherical functor package [4] and of the alternative definition of spherical functors due to Kuznetsov [70]. In §1.4 we recall the description, due to [41], of the category Perv C, A of localized p q perverse sheaves (discussed above) in terms of tunnelling diagrams Φi, mij . p q Chapter 2 is devoted to a study, in the form convenient for us, of the Fourier transform of a perverse sheaf F on C (that is, of the holonomic regular D-module M corresponding to F). This problem was studied before, first of all by Malgrange [76] in much greater generality (M is allowed to be irregular). The work of Malgrange was revisited by D’Agnolo-Kashiwara [23] from the point of view of their approach via enhanced ind-sheaves [22]. It was recognized in [63, 21] that focusing specially on the case of regular M leads to important insights and further developments. We continue on this path, bringing out the connections with convex geometry and with the Algebra of the Infrared. In §2.1 we introduce, besides motivation and terminology, the rectilinear version of trans- port maps mij. We formulate Proposition 2.1.7 which says that these maps give an alternative description of the category Perv C, A . p q In §2.2 we prove Proposition 2.1.7. Our method is based on isomonodromic deformations of perverse sheaves, when we move the allowed singular points wi while keeping the local behavior near each wi unchanged. This concept is usually considered for linear ODE (the Schlessinger system), not perverse sheaves. In the perverse sheaf context, isomonodromic deformations have a transparent “wall-crossing” effect on the rectilinear transports. The

9 resulting wall-crossing formula is another manifestation of the Picard-Lefschetz identity. The corresponding wall of collinearity (called wall of marginal stability in [36]) consists of N configurations w1, ,wN C such that three points wi,wj,wk lie on a real line. We also discuss configurationp ¨ ¨ ¨chambersq P (connected components of the complement to the walls) and their relation with the combinatorial concept of an oriented matroid [11, 12]. This relation is surprisingly nontrivial. In §2.3, we describe, following [76, 21], the Fourier transform of (the holonomic regular D-module associated to a) perverse sheaf F on C, at the topological level, as a perverse sheaf

F on the dual plane Cz. We present a description of the Φ, Ψ -diagram of F (Proposition p q 2.3.10) as well as the representation of the section of F outside 0 as actual Fourier integrals (Propositionq 2.3.16). q In §2.4, we recall the concept of a Stokes structureq (a local system on S1 with a fitration indexed by the sheaf of exponential germs) due to Deligne and Malgrage [25] and focus specifically on the case of exponential Stokes structures where the exponential germs are given by a collection of simple exponentials z ewz, w A. We emphasize, in Proposition 2.4.8 and Remark 2.4.9, the little noticed factÞÑ that theP combinatorial data encoded by an exponential Hodge structure are related to the concept of oriented matroid. We also recall the description of the (exponential) Stokes structure on F due to [76, 21]. In §2.5 we present a “baby”, or decategorified, version of the Algebra of the Infrared, by which we mean a class of formulas related to perverse sheavesq (not schobers) and involving summation over convex polygonal paths. Recall that the actual Algebra of the Infrared of [36, 37] involves complexes whose summands are labelled by such paths. We give two examples of such formulas. One, in Proposition 2.5.3, expresses the transport along a curvilinear path in terms of iterated transport along convex polygonal paths. The other, in Proposition 2.5.12, gives a similar expression for the Stokes matrix of F. These results support our main thesis that the right context for the Algebra of the Infrared is provided by perverse schobers. Chapter 3 begins an upgrade, to perverse schobers,q of the study of perverse sheaves presented in Chapters 1 and 2. In §3.1 we present, following [54], our working definition of schobers on surfaces in terms of spherical functors at singular points complemented by a local system on the complement. We are especially interested in the case when the surface is the disk or the complex plane. In this case a schober can be defined in terms of a diagram of spherical functors, once we choose a “spider”. This is analogous to the Gelfand-MacPherson-Vilonen description [41] for perverse sheaves. In §3.2 we upgrade the Picard-Lefschetz identities of Proposition 1.1.12 to Picard-Lefschetz triangles [54]. These are exact triangles in the category of (dg-)functors connecting three transport functors which generalize the transport maps of perverse sheaves. The arrows (nat- ural transformations) in Picard-Lefschetz triangles, which we call Picard-Lefschetz arrows, will be fundamental objects of study in the rest of the paper. We prove, in Propositions 3.2.14 and 3.2.16, geometric identities on the “pasting” of these arrows which have the mean-

10 ing of invariance under elementary moves of plane triangulations. These identities can be seen as strating points of the analysis leading to the Algebra of the Infrared. In §3.3 we review the theory of semi-orthogonal decompositions (SODs) in triangulated categories and their mutations [14]. We also recall the construction of 2-term SODs via gluing (dg-)functors [71]. In §3.4 we develop a generalization of the technique of gluing functors to construct N- term SODs. Instead of a single functor, we need a uni-triangular monad, i.e., an associative algebra in the category of functors. We study mutations of uni-triangular monads, the Koszul duality as well as the straightening procedure for A8-monads. In the case of exceptional b collections (which are SOD’s into copies of the category D Vectk), monads reduce to usual associative dg-algebras and our analysis reduces to that of [17]. In §3.5 we define the Fukaya-Seidel category associated to a schober S on C in terms of a diagram of spherical functors associated to S via a spider. Our approach is a modification of the classical Barr-Beck theory [7] that studies a monad associated to a pair of adjoint functors. In our case we have not one but several pairs of adjoint functors with common target and we associate to them a uni-triangular monad which we call the Fukaya-Seidel monad. The Fukaya-Seidel category is defined as the category of algebras over this monad. In §3.6 we discuss a particular example of a schober on C, which we call the Lefschetz schober LW associated to a generalized Lefschetz pencil. i.e., to a superpotential W : Y C with isolated critical points. It can be seen as a categorification of the Lefschetz perverseÑ sheaf LW of §1.2. As with any Fukaya-style object, the construction of LW is bound to involve non-trivial technical questions of symplectic geometry. We give references to the literature where such questions have been addressed at various level of generality and detail, some of the work still ongoing. In the more “classical” case of a Lefschetz pencil (all critical points non-degenerate) we can rely on the techniques developed by Seidel [86]. Chapter 4 is the first of two chapters dedicated to the analysis of schobers on C from the “infrared” perspective In it, we collected the aspects involving more elementary convex geometry, i.e., making use of convex hulls, convex paths etc. but not of the concept of secondary polytopes. In §4.1, we explain the rectlinear approach to schobers and spell out the schober analog of the geometric Fourier transform for perverse sheaves. This analog features the Fukaya-Seidel category. In §4.2, we develop a categorical analog of the Baby Infrared Algebra of §2.5. It appears as a cochain complex in the derived category whose terms are direct sums of iterated rectilinear transports. We include it into and a larger diagram of exact triangles called the infrared Postnikov system which categorifies the individual identities used to obtain the formulas of §4.2. In §4.3, we realize the components of the Fukaya-Seidel monad M as “circumnavigating” transports of §4.2, and define another monad R, which we call the infrared monad, formed

11 by iterated rectilinar transports. This monad is the schober analog of the A8-algebra of half-plane webs from [36, 37], see also [51]. We formulate a conjecture that each component Mij of M can be obtained by adding lower order terms to the differential in Rij. Note that this is somewhat different from the main trust of [36, 37], where A8-deformations are considered. Chapter 5 is devoted to the schober analogs of some of the main constructions of [36, 37]: the L8-algebra g formed by “closed polygons” and its action on the monad R. Following [51], we use the language of secondary polytopes. In §5.1 we give a brief overview of secondary polytopes, the main reference being [39]. In §5.2 we discuss the variation of the secondary polytope Σ A under the change of A. Here we have a subtle phenomenon: the combinatorics of Σ Ap canq change without the naive convex geometry of A (what lies in the convex hull of what)p q undergoing any visible change. This phenomenon has been noticed in [36] at the level of webs and called exceptional wall-crossing. We develop a quantified version of this phenomenon in terms of jumps of the cohomology of a certain complex. We prove that the locus of “exceptional” configurations A (for which we do have a jump) has real codimension 1 so that we can indeed speak about walls. We also prove that for A outside of suchě walls, Σ A has the factorization property: each face of it is itself a product of other secondary polytopes.p q In this, we correct the treatment of [51] where the condition of exceptionally general position was overlooked.

In §5.3 we define the L8-algebra g gA S associated to a schober S Schob C, A . It consists of certain Clebsch-Gordan type“ spacesp q (or spaces of intertwiners) IPQ1 associatedp q to closed convex polygons Q1 with vertices in A. An element of I Q1 can bep seenq as a natural transformation between two iterated transport functors correspondingp q to two polygonal paths 1 forming the boundary of Q . The L8-operations in g are induced by pasting of natural transformations.

In §5.4 we define a L8-morphism from g to the deformation complex of the infrared 1 operad R. In particular, any Maurer-Cartan element β g gives rise to an A8-deformation R β of R. P p q In §5.5 we analyze Maurer-Cartan elements β in g more explicitly. We conjecture that for any schober S there is a canonical (up to gauge equivalence) β such that R β is quasi- isomorphic to the Fukaya-Seidel monad M. We indicate an approach to provingp q this con- jecture by deforming A from the “maximally concave” position, deforming S accordingly (isomonodromically), thereby crossing some walls and tracing the effect of the wall-crossings on Σ A and g gA S . p q “ p q Finally, Appendix A collects notations and conventions related to triangulated cate- gories, their dg-categorical enhancements, as well as adjunctions, Serre functors and other related features.

12 I. Acknowledgements. We would like to thank M. Abouzaid and K. Hori for useful dicussions and correspondence. The research of M.K. was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The research of Y.S. was partially supported by the Munson-Simu award at Kansas State University and NSF grant. He also thanks to IHES where part of the project was carried out.

1 Review of Perv C, A p q 1.1 Abstract Picard-Lefschetz theory A. Setup and definitions. We fix a field k of characteristic 0. Let X be a smooth (C8) surface, possibly non-compact and with boundary X. Let A w1, wN X X be a finite set of interior points. We refer to elements ofB A as marked“ t points¨ ¨ ¨. TheuĂ set A´Bdefines a stratification of X with strata being X A and 1-element subsets wi , i 1, , N. We denote by Db X, A the derived categoryz formed by bounded complexest u of“k-vector¨ ¨ ¨ spaces on X which arep constructibleq (in particular, have stalks with finite-dimensional cohomology) with respect to this stratification. Let Perv X, A Db X, A be the abelian category of perverse sheaves with respect to the middle perversity.p q Ă Explicitly,p q F Perv X, A iff: P p q (P `) Hi F is supported on a set of real codimension 2i. That is, H1 is supported on A andp Hqě2 0. ě “ ´ i (P ) For any stratum S the sheaves H F vanish for i 2 codimR S . Sp q ă p q

This implies that F XzA is a local system in degree 0 and any local system on X in degree 0 is perverse. Note that| our normalization differs by shift from that of [9], in which a local system in degree 1 is perverse. p´ q The category Perv X, A has a perfect duality p q (1.1.1) F F ˚ : D F 2 , ÞÑ “ p qr s where D is Verdier duality [60]. The shift by 2 comes from our normalization of perversity so that on local systems in degree 0 we get the usual (unshifted) duality. Further, Perv X, A is the heart of the perverse t-structure on Db X, A , in particular, we have the functorsp ofq perverse cohomology p q

(1.1.2) Hi : Db X, A Perv X, A . perv p q ÝÑ p q Let Diffeo X, A be the topological group of all self-diffeomorphisms of the pair X, A p q p q which preserve X pointwise and A as a set. Let also Diffeo X, IdA Diffeo X, A be the subgroup formedB by self-diffeomorphisms which preserve A pointwise.p qĂ If X isp oriented,q we

13 ` ` denote by Diffeo X, A Diffeo X, A , Diffeo X, IdA Diffeo X, IdA the subgroups of orientation preservingp diffeomorphisms.qĂ p q We denotep by Ă p qq

` ` ΓX,A π0 Diffeo X, A , Γ π0 Diffeo X, A , “ p q X,A “ p q ` ` ΓX,Id π0 Diffeo X, IdA , Γ π0 Diffeo X, IdA A “ p q X,IdA “ p q the corresponding mapping class groups of X, A . p q Remark 1.1.3. Alternatively, we can choose one new marked point on each component of X, denote the set of these new points by B and consider the group of diffeomorphisms (resp.B orientation preserving diffeomorphisms) of X, A which preserve not the full X but just the points from B. This will give the same groupsp q of connected components. B

The topological nature of the perversity conditions implies:

Proposition 1.1.4. The group ΓX,A acts on the category Perv X, A . p q

Example 1.1.5. If X D z 1 be the closed unit disk in C and A w1 “ “ t| | ď u ` “ t 㨨¨ă wn 1, 1 be a set of N points on the real line inside D. Then ΓD,A is canonically u Ă p´ q ` identified with BrN , and Γ D, IdA with PBrN , the braid group, resp. the pure braid group p q on N strands, cf [35] §9.1.3. Thus BrN acts on Perv D, A . p q B. The Classical Φ, Ψ description. Let X D z 1 be the unit disk in C and A 0 . We recallp theq classical [8, 38] description“ of“ the t| category| ď u Perv D, 0 . “ t u p q Proposition 1.1.6. Perv D, 0 is equivalent to the category Q of diagrams p q a Φ o / Ψ b

of finite-dimensional k-vector spaces are linear maps such that TΨ : IdΨ ab is an isomor- “ ´ phism (or, what is equivalent, such that TΦ IdΦ ba is an isomorphism). “ ´ For future use, we recall from [38] the explicit construction of the equivalence

aF (1.1.7) Perv D, 0 Q, F Φ F o / Ψ F . p q ÝÑ ÞÑ p q bF p q ( Consider the radial “cut” K 0, 1 D (one can choose any other simple curve starting at 0 and ending on the boundary“r of DqĂ), see Fig. 2, where D is depicted as an ellipse. Hi Consider the sheaves K F of hypercohomology with support in K. The assumption p q Hi H1 that F is perverse, implies that K F 0 for i 1. The sheaf K F on K is locally constant (hence constant) on K 0p .q Let “ ε 0,‰1 be a small nonzerop realq number. We define: ´ t u P p q H1 H1 Φ F : K F 0 K ∆, F (the space of vanishing cycles), p q “ p q »1 p q Ψ F : H F ε (the space of nearby cycles). p q “ Kp q 14 D

V U V` 0K 1 ε

V´

Figure 2: The spaces Φ F and Ψ F . p q p q

Since F Dzt0u is a local system in degree 0, after taking a small neighborhood V of ε and denoting by| V the two components of V K, we have ˘ ´ F V` F V´ Fε Fε Ψ F F V K F V p q‘ p q ‘ Fε p q “ p ´ q{ p q “ F V “ Fε » p q is just the stalk of F at ǫ, which explains the name “nearby cycles”. As for Φ F , it includes into the exact sequence (the standard sequence relating sheaves of hypercohomologyp q with and without support) β (1.1.8) 0 H0 U, F F U K Φ F H1 U, F 0, Ñ p qÑ p ´ q Ñ p qÑ p qÑ where U is any small contractible neighborhood of 0 in D (in fact, U D is also allowed). “ We now explain the construction of the maps aF and bF . H1 The map aF : It is just the generalization map of the constructible sheaf K F from the stalk at 0 to the stalk at a nearby point ε. More explicitly, restrictions from Dp toq U and V give the diagram Φ F H1 U, F rU H1 D, F rV H1 V, F Ψ F , p q“ UXK p q ÐÝ Kp q ÝÑ V XKp q“ p q ´1 with rU being an isomorphism, and we define a rV r . F “ ˝ U The map b : Let ψ Ψ F Fε. Continue ψ to a section of F in V . After this, extend F P p q “ ` it to a section ψ of F in D K, so ψ V` coincides with ψ. Then ψ U´K F U´K, and we define ´ | | P | r b ψ r β ψ U K Φ F . r F p q “ | ´ P p q In other words, we continue ψ along the dotted` contour˘ in Fig. 2 and look at the “multival- uedness” of this continuation near 0. The mapr bF is known as the variation map. The equivalence (1.1.7) depends on the choice of the cut K. In particular, Φ F and Ψ F are, intrinsically, not vector spaces but local systems on the (infinite-dimensional)p q spacep Kq formed by all possible cuts K (i.e., all simple closed curves joining 0 with the boundary of D). This space is homotopy equivalent to the circle S1 formed by directions at 0. So we can think of Φ F and Ψ F as being local systems on this circle and denote these local systems p q p q Φ F and Ψ F Their monodromies are the operators TΦ and TΨ above. p q p q 15 C. The transport maps mij γ and the Picard-Lefschetz identities. Let X, A ˝ p q “ w1, ,wN be as in n A. For any w Y we have the natural circle of directions t ¨ ¨ ¨ u P 1 ˚ S TwY 0 R . w “ ´ t u { ą0 ` ˘ Let F Perv Y, A . For any i 1, , N we can restrict F to a small disk around wi and P p q “ ¨ ¨ ¨ 1 ˝ associate to it local systems Φi Φw F , Ψi Ψw F on S , as in n B. “ i p q “ i p q wi Let α be a simple, piecewise smooth arc in X joining two marked points wi and wj and dir 1 not passing through any other marked points, see Fig. 3. Denote by i α Swi and 1 p q P dirj α S the tangent directions to α at the end points and equip α with the orientation p q P wj going from wi to wj. We want to study F in a neighborhood of α. Considering α as a closed 1 subset in Σ, we have the sheaf H F on α which is constant on the open arc α wi,wj . αp q ´ t u

Φi,α Ψα Φ w j,α w i p j α

wk

Figure 3: The transport map.

The stalks 1 1 Φi,α H F w , Φj,α H F w “ αp q i “ αp q j are just the stalks of the local systems Φi, Φj at the tangent directions diri α and dirj α respectively. The stalk p q p q 1 Ψα Γ α wi,wj , H F “ p ´ t u αp qq is identified with the stalk Fp at any intermediate point p α. Note that this identification depends on the chosen orientations of X and α which giveP a co-orientation of α, i.e., a particular numeration of the 2-element set of “sides” of Σ α near p (change of either orientation incurs a minus sign in the identification). These´ spaces are connected by the maps ai,α aj,α Φi,α o / Ψα o / Φj,α, bi,α bj,α

obtained from the description of F on small disks near wi and wj and using γ as the choice for a cut K. We define the transport map along α as

(1.1.9) mij α bj,α ai,α : Φi,α Φj,α. p q“ ˝ ÝÑ

By an admissible isotopy of paths from wi to wj we mean a continuous 1-parameter family of simple arcs αt t 0,1 , each αt joining wi with wj and not passing through any other wk. The p q Pr s

16 maps mij α are “unchanged” under such an admissible isotopy. More precisely, we have a commutativep q diagram mij pα0q Φi,α0 / Φj,α0

ti tj

 mij pα1q  Φi,α1 / Φj,α1 , 1 where ti is the monodromy of the local system Φi on S from diri α0 to diri α1 , and wi p q p q similarly for tj. We now describe what happens when a path crosses a marked point. That is, we consider 1 a situation as in Fig. 4, where a path γ from wi to wk approaches the composite path formed 1 by β from wi to wj and α from wj to wk. After crossing wj, the path γ is changed to γ.

γ

wj β α

wi wk γ1

Figure 4: The Picard-Lefschetz situation.

In this case we have identifications

(1.1.10) Φi,γ Φi,β Φi,γ1 , Φk,γ1 Φk,α Φk,γ, Φj,β Φj,α, Ñ Ñ Ñ Ñ Ñ given by clockwise monodromies of the local systems Φ around the corresponding arcs in the circles of directions. So after these identications we can assume that we deal with single vector spaces denoted by Φi, Φk and Φj respectively. Convention 1.1.11. In this paper we will often consider compositions of several transport maps (and later, transport functors) corresponding to composable sequences of several ori- ented paths. In all such cases our default convention will be to identify the intermediate Φ-vector spaces (and later, categories) using the clockwise monodromies from the incom- ing direction to the outgoing one. This extends the last identification in (1.1.10), which is indicated by an arc around wj in Fig. 4. Proposition 1.1.12 (Abstract Picard-Lefschetz identity). We have the equality of linear operators Φi Φk: Ñ 1 mik γ mik γ mjk α mij β . p q“ p q´ p q p q 17 This was proved in [54, Prop. 1.8] but with slightly different sign conventions. It can be seen as a disguised version of the identity TΨ IdΨ ab for the monodromy in the Φ, Ψ -description of Perv D, 0 . For convenience of“ the reader´ we give a detailed argument. p q p q Proof: Let us deform the situation of Fig. 4 by pinching the directions of the three paths at wi into a single direction, so that they all go together from wi to a nearby point wi and then diverge. Let us do the same at wk, see Fig. 5. r γ

wj r ‚ β α

wk wi r r ‚ wk wi ‚ 1 r ‚ ‚ r γ Figure 5: The Picard-Lefschetz situation, pinched. r It is enough to prove the statement for this deformed situation which we now assume. Thus, γ now consists of a short segment wi, wi (not necessarily straight) followed by a path r s γ followed by a similar segment wk,wk . Similarly, r s 1 1 r γ wi, wi γ wk,wk , α α wk,wk , β wi, wi β. r “ r sY Yr r s “ Yr s “r sY We view w and w as the midpoints to define the transport maps. Let Ψ , Ψ be the stalks i k r r r r r ri k r of F at wi, wk respectively. Then, we have maps r r ai : Φi Ψj, bk :Ψk Φk r r ÝÑ ÝÑ

corresponding to the intervals wi, wi and wk, wk , the monodromy maps r s r s

Tγ , Tγ1 :Ψi Ψk r rÝÑ corresponding to γ, γ1 and the maps r r

a : Φj,β Ψi, b :Ψi Φj,β r r j,β ÝÑ j,β ÝÑ

aj,αr : Φj,α Ψk, bj,αr :Ψk Φj,α ÝÑ ÝÑ corresponding to the segmentsr near wj. So by definition,r

1 mik γ bk Tγ ai, mik γ bk Tγ1 ai, (1.1.13) p q “ ˝ ˝ p q “ ˝ ˝ mij β b ai, mjk α bk aj,α. p q “ j,rβ ˝ p q “ ˝ r r 18 r ´1 Note that the transformation T T Tγ1 : Ψi Ψi is a full counterclockwise mon- “ γ ˝ Ñ odromy of Ψi around wj. Hence, T r 1 ra b “ ´ j,β j,β Now, we also have r r Tγ a aj,α j,β “ due to the clockwise identification Φj,β Φj,α. Combining previous equalities, we obtain »r r r

Tγ1 Tγ T Tγ 1 a b Tγ aj,αb “ “ p ´ j,β j,βq“ ´ j,β r r r Multiplying this equalityr byr bk onr the left and ai onr ther right we obtain the desired statement.

D. The iterated transport map as the “Vassiliev derivative” of the monodromy. Consider the following interated transport situation depicted in Fig. 6. That is, let x, x1 X A and γ be a (piecewise smooth) simple path joining x with x1 and passing, possibly,P z through some wi , ,wi A, in this order. 1 ¨ ¨ ¨ k P

wi1 wi2 wik x γ0 γ1 γk x1 ‚ ¨ ¨ ¨ ‚ ‚ ‚ ‚ ‚ ε1 0 ε2 1 εk 0 “ “ “ Figure 6: Iterated transport situation.

Let γ0 be the part of γ between x and wi1 , let γ1 be the part of γ between wi1 and wi2 , 1 and so on, until γk which is defined to be the part of γ between wik and x . Let us define the iterated transport map

(1.1.14) m γ ai ,γ mi ,i γk 1 mi ,i γ1 bi ,γ : Fx Fx1 . p q “ k k ˝ k´1 k p ´ q˝¨¨¨˝ 1 2 p q˝ 1 0 ÝÑ Here the identification of the Φ-categories at the intermediate points is made using the upward (clockwise) half-monodromies understood with respect to the direction of γ from x to x1 and to the chosen orientation of X, see Convention 1.1.11. If γ does not pass through any wi, then k 0 and m γ is defined to be the monodromy of F along γ. “ p q

Let ε1, ,εk 0, 1 . Denote by γε , ,ε the deformation of γ obtained by bending it ¨ ¨ ¨ P t u 1 ¨¨¨ k around wi upwards (in the above sense), if εν 0 and downwards, if εν 1, see Fig. 6. ν “ “ Thus γε1,¨¨¨ ,εk does not pass through any wi. The following is obtained by iterating the argument in the proof of Proposition 1.1.12.

Corollary 1.1.15. We have an equality of maps Fx Fx1 : Ñ

ε1`¨¨¨`εk m γ 1 m γε , ,ε . p q “ p´ q p 1 ¨¨¨ k q ε ,¨¨¨ ,ε Pt0,1u 1 ÿk 19 This has an attractive interpretation in terms of Vassiliev’s approach to invariants of knots, cf. [94] Ch. V. Let Π be the (infinite-dimensional) space of all simple paths joining x 1 with x . For each i 1, , N we have a hypersurface Hi Π consisting of paths passing “ ¨ ¨ ¨ N Ă through wi. The hypersurface H i“1 Hi (“discriminant”) cuts Π into various chambers (defined as connected components“ of Π H). For each chamber C we have a well defined monodromy Ť z m C m γ : Fx Fx1 , γ C. p q “ p q ÝÑ @ P The correspondence C m C can be seen as an “invariant” in the spirit of [94]. Further, the discriminant itself is furtherÞÑ p stratifiedq into “faces” by multiplicity of intersections of different components Hi.

Corollary 1.1.15 means that m γ for γ lying on such a face F Hi1 Hik is obtained as the alternating sum of the m Cp ,q where C runs over the 2k chambersĂ X¨¨¨ surrounding F . In other words, m γ is realized asp theq kth Vassiliev derivative of the invariant C m C . We will takep upq this point of view in a slightly different form in §4.2. ÞÑ p q

E. The spider descriptions of Perv D, A and Perv C, A . Let D be the unit disk and p q p q A w1, ,wN D D. A description of the category Perv D, A was given by Gelfand- MacPherson-Vilonen“ t ¨ ¨ ¨ uĂ [41].´B To recall it, it is convenient to introducep theq following terminology. Fix a point v D. Bya v-spider for D, A we will mean a system K γ1, ,γN of P B p q “ t ¨ ¨ ¨ u simple closed piecewise smoorth arcs in D so that γ joins v with wi and different γi do not meet except at v. See the left part of Fig. 7. Note that a v-spider for D, A defines a total order on A, by clockwise ordering of the p q slopes of the γi at v (assumed distinct). Let us assume that the numbering A w1, ,wN is chosen in this order, see again the left part of Fig. 7. “ t ¨ ¨ ¨ u

C w3 w1 ¨ ¨ ¨ wN γ1 γ3 γN γN v v wN γ1 γ2 γ2 ¨ ¨w ¨2 w1 w2

D

Figure 7: Equivalence depending on a spider.

Definition 1.1.16. Let QN be the category formed by diagrams (quivers) of finite-dimensional

20 k-vector spaces ΦN ❄❄ _ ❄❄❄ a ❄❄❄❄ N ❄❄❄❄ bN ❄❄❄ . . ? Ψ ⑧⑧⑧ a1 ⑧⑧ ⑧⑧⑧⑧ ⑧⑧⑧⑧ ⑧⑧ b1 ⑧⑧⑧ Φ1 such that TΦi : IdΦi biai is invertible for each i. This implies that each Ti,Ψ IdΨ aibi is invertible as“ well. ´ “ ´ Let now K be a v-spider for D, A and F Perv D, A . For any i 1, , N we denote p q P p q “ 1¨ ¨ ¨ Φi,K F Φi,γ F the stalk of the local system Φi F at the point in S represented by p q “ i p q p q wi the direction of γi. Let us also identify the stalk Ψγi F with the stalk Fv, as in §1.1 C. This gives a diagram p q

ΘK F Ψ F , Φi,K F , ai,γ , bi,γ QN . p q“p p q p q i i q P

Here ai,γi and bi,γi are the canonical maps along γi as in (1.1.9).

Proposition 1.1.17 ([41]). Let K be a v-spider for D, A . The functor ΘK : Perv D, A p q p qÑ QN is an equivalence. Remarks 1.1.18. (a) We can think of the point v D as being being far away at the infinity (“Vladivostok”). For this reason we will sometimesPB refer to the description of Perv D, A given by Proposition 1.1.17 as the Vladivostok description. p q (b) The significance of the choice of position of v becomes clearer when, instead of the disk D, we can consider the complex plane C so that the boundary of C is not given. Then we can take for v any point in C A and consider spiders spreading out from that point, see z the right part of Fig. 7. This will also lead to an identification Perv C, A QN . p qÑ We further recall the dependence of the equivalence ΘK on the choice of a spider K. Note that isotopic spiders give equivalences which are canonically isomorphic as functors because the space of spiders in a given isotopy class (i.e., each connected component of the space of spiders) is contractible. Let K D, v, A be the set of isotopy classes of v-spiders for D, A . p q p q It is classical, see, e.g., [5] §2.6, that K D, v, A forms a torsor over the braid group BrN . p q ` One way of seeing this is to use the realization BrN Γ D, A as the mapping class group, see Example 1.1.5. That is, every diffeomorphism “f : Dp Dq preserving A as a set and Ñ D pointwise, takes a spider K to a new spider f K . More explicitly, recall that BrN is B p q generated by elementary twists τ1, , τN 1 subject to the relations ¨ ¨ ¨ ´

τiτj τjτi, i j 2, τiτi 1τi τi 1τiτi 1. “ | ´ |ě ` “ ` `

In this presentation, the action of τi on a spider K γ1, ,γN leaves γj, j i 1, 1 “ t ¨ ¨ ¨ u ‰ ` unchanged and replaces γj`1 by the path γj`1 obtained by moving γj`1 counterclockwise

21 around γj, see Fig. 8. Thus the homomorphism BrN SN comes from the change of the Ñ order of the slopes of the γj at v.

wN wN ‚ ‚ γN γN . . wi`1 . γi`1 wi`1 . γ ‚ γi v ‚ i v ‚ wi ‚ w‚ . ‚ i . γ1 γ1 . i`1 γ1 . ‚w1 ‚w1

Figure 8: Action of τi BrN on spiders. P

Proposition 1.1.19 ([41], Prop. 1.3). (a) Replacing K by τi K changes a diagram Ψ, Φj, aj, bj p q p q to Ψ, τi Φj , τi aj , τi bj , where: p p q p q p qq

τi Φ1, , ΦN Φ1, , Φi 1, Φi 1, Φi, Φi 2, , ΦN , p ¨ ¨ ¨ q“p ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ q τi a1, , aN a1, , ai´1, Ti,Ψ ai`1, ai, ai`2, , aN , p ¨ ¨ ¨ q“p ¨ ¨ ¨ ´1 ¨ ¨ ¨ q τi b1, , bN b1, , bi 1, bi 1 T , bi, bi 2, , bN . p ¨ ¨ ¨ q“p ¨ ¨ ¨ ´ ` i,Ψ ` ¨ ¨ ¨ q

(b) Put differently, the formulas in (a) define an action of BrN on the category QN . For any spider K K D, v, A (and the corresponding numbering of A by slopes of the γi at v), P p q ` the equivalence ΘK takes the geometric action of BrN Γ on Perv D, A (Proposition “ D,A p q 1.1.4) to the above action of BrN on QN .

1.2 Relation to the classical Picard-Lefschetz theory. A. The Lefschetz perverse sheaf. When speaking of complex manifolds, we allow them to have a boundary. By a Riemann surface we mean a 1-dimensional complex manifold (possibly with boundary).

Definition 1.2.1. Let X be a Riemann surface. An X-valued generalized Lefschetz pencil of relative dimension n is a datum Y, W , where: p q (1) Y is an n 1 -dimensional complex manifold. p ` q (2) W : Y X is a holomorphic, proper (fibers are compact without boundary) map with only finitelyÑ many isolated singular points, lying in the interior of Y . We denote the critical points y1, ,yN , and put wi W yi to be the corresponding critical values. ¨ ¨ ¨ “ p q We assume that all the wi are distinct and put A w1, ,wN X. “ t ¨ ¨ ¨ uĂ

22 A Lefschetz pencil is a generalized Lefschetz pencil with only quadratic (Morse) singular points. k Given a generalized Lefschetz pencil Y, W , we can form a constructible complex RW˚ Y Db X, A . It gives rise to: p q P p q • Constructible sheaves RqW k Hq RW k . ˚ Y “ p ˚ Y q q k q k • Perverse sheaves RpervW˚ Y Hperv RW˚ Y Perv Σ, A . In particular, we have the Lefschetz perverse sheaf “ p q P p q

n L LW R W k , “ “ perv ˚ Y corresponding to the middle dimension.

Proposition 1.2.2 (Picard-Lefschetz theory I). Let Y, W be a generalized Lefschetz pencil. Then: p q

q k (a) For q n the sheaf R W˚ Y is locally constant (i.e., it does not have a singularity at ‰ q k any point of A) and is identifed with RpervW˚ Y . n k ((b) For q n, the perverse sheaf L is identified, outside A, with R W˚ Y , the local system of middle“ cohomology of the smooth fibers W ´1 b . p q

(c) Suppose that Y, W is a Lefschetz pencil. Then, for any wi A, the local system of p q 1 P n`1 vanishing cycles Φw L on S is 1-dimensional, with monodromy 1 . i p q wi p´ q

W ´1 ε p q

∆i W

wi ε

Figure 9: The vanishing cycle.

Further, in the classical theory, we can think of a radial direction ε S1 as a nearby P wi point of Y connected to wi by a path, see Fig. 9. Such a datum gives rise to the geometric ´1 (Lefschetz) vanishing cycle ∆i ∆i ε W ǫ which is a submanifold homeomorphic to “ p qĂ p q 23 the n-sphere Sn and defined canonically up to isotopy. See Fig. 9 and [5], Ch. 1. The stalk of Φw L at ε can be thought of (after identifying cohomology with homology by Poincaré i p q duality) as spanned by ∆i , the homology class of ∆i. Note that there are two choices of ∆i r s n r s corresponding to two orientations of ∆i S , of which none is preferred. The monodromy »˝ n around wi can be seen geometrically (see n B.) as induced by the antipodal map of S which preserves orientation for n 1 even and changes it for n 1 odd, which explains (b). ` ` Proposition 1.2.3 (Picard-Lefschetz theory II). Let Y, W be a Lefschetz pencil. The maps p q a Φ i / Ψ n ´1 k wi L ε o wi L ε H W ε , p q bi p q “ p p q q

describing L on a small disk near wi, have the following form:

(a) The map ai considers (any choice of) ∆i as an element of the middle (co)homology of W ´1 ε . r s p q n ´1 (b) The map bi takes any ξ H W ε , k into ∆i ,ξ ∆i . Here ∆i ,ξ k is P p p q q pr s q¨r s r s q P the intersection index (or, if we consider ∆i as a homology class, just the pairing of homology and cohomology). r s

Combining Proposition 1.2.3 with the identity TΨ Id ab gives the classical Picard- n ´1 “ ´ Lefschetz theorem: the monodromy of H W ε around wi is the transvection associated to the vanishing cycle. p p qq ˝ Let now α be a path joining wi and wj as in n 1.1 C. After choosing orientations of the two vanishing cycles ∆i and ∆j, we identify the vector spaces Φi and Φj with k, so the transport map mij α : Φi Φj becomes an element of k, in fact, an integer, the intersection index in W ´1 b : p q Ñ p q (1.2.4) mij ∆i b, ∆j b Z k “ r s r s W ´1pbq P Ă ` ˘ n ´1 Here b α is an intermediate point, as in Fig. 4, while ∆i b, ∆j b H W b , k are the P r s r s P p p q q parallel transports of ∆i and ∆j along α (with respect to the Gauss-Manin connection) r s r s from the vicinity of wi or wj to b.

More generally, let Y, W be a generalized Lefschetz pencil. Then the space Φwi LW ε, with its action of the monodromyp q operator, is the classical space of vanishing cycliespof theq isolated singularity yi of W , see [5] §2.2.

B. The symplectic connection. In the situation of n˝ A., assume that Y is equipped with a Kähler metric h g ? 1 ω so that g is a Riemannian metric and ω is a symplectic form on Y (considered“ as a`C8´-manifold¨ of dimension 2n 2). ` ´1 Let y Y y1, ,yN be a non-critical point for W . Then the fiber W W y is, near y, a smoothP ´ t complex¨ ¨ ¨ manifoldu of dimension n. We define p p qq

´1 K (1.2.5) Ξy Ty W W y TyY “ p p p qqq h Ă ` ˘ 24 to be the orthogonal complement (with respect to h) to its tangent space. Thus Ξy is a 1-dimensional complex submanifold in TyY with the property that the differential dyW : 8 Ξy TW y X is an isomorphism. We get a C complex rank 1 subbundle Ξ of TY y , ,y . Ñ p q ´t 1 ¨¨¨ N u Let X0 X A be the set of non-critical values. The map “ z N ´1 ´1 W Y0 : W X0 Y W wi X0 “ p q “ ´ i“1 p q ÝÑ ď is a C8-locally trivial fibration, and Ξ can be considered as a nonlinear connection in this fibration, known as the symplectic connection [79] §6.3. Denote this connection ∇W . Since we assumed W to be proper, any (piecewise) smooth path γ X0, joining two points x1 Ă and x2, gives a well defined diffeomorphism

´1 ´1 1 Tγ : W x W x . p qÑ p q

It is further known (see [79] Lemma 6.3.5) that Tγ is a symplectomorphism with respect to the symplectic forms on W ´1 x and W ´1 x1 obtained as the restrictions of ω. p q p q The connection ∇W is not flat but it can be seen as a geometrization of the Gauss- Manin connection on the cohomology of the W ´1 x , which it induces. In particular, the p qn “monodromy” of the geometric vanishing cycle ∆i S with respect to an appropriate closed » contour γ around wi (and appropriate Kähler metric) can be seen as a self-diffeomorphism of ∆i isotopic to the antipodal map ([79] Ex. 6.3.7).

C. mij α as the number of curved gradient trajectories. We keep the notations of n˝ B. p q

Definition 1.2.6. Let γ be a piecewise smooth path in Xgen. By γ-gradient trajectories we will mean integral trajectories of Ξ lifting γ.

Thus, if γ begins at some b Xgen, then there is a unique γ-gradient trajectory passing through any y W ´1 b . The nameP “γ-gradient trajectories is explained as follows. P p q Proposition 1.2.7. Let X C, so W is a “holomorphic Morse function”. Let p C be a non-critical value. “ P (a) Let γ be the horizontal half-line p ,p going from p to the left. We assume that this half-line consists of non-criticalp vales.´ 8 Thens γ-gradient trajectories coincide (as unparametrized curves) with the gradient trajectories of the real function Re W : Y R with respect to the Riemannian metric g, beginning in W ´1 p . p q Ñ p q (b) More generally, if ζ 1 and γ p ζ ,p is the half-line of slope ζ emanating from p, then γ-gradient trajectories| |“ coincide“p ´ with¨8 gradients trajectories of the real function Re ζ´1W . p q Proof: Part (b) follows from (a). Part (a) follows from the next well known fact.

25 Proposition 1.2.8. The gradient flow of Re W with respect to g coincides with the Hamil- tonian flow of Im W with respect to ω. Inp particular,q gradient trajectories of Re W are mapped, by W , top horizontalq lines Im W const in C. p q t p q“ u

Recall that Re W is a Morse function with the critical points yi (same as for W ), each of the same indexp q (number of negative eigenvalues) equal to n 1. The same holds ´1 ` for Re ζ W , ζ 1. Thus any critical point yi gives rise to the unstable manifold, or p q | | “ n`1 Lefschetz thimble Thζ yi R which is the union of all descending gradient trajectories ´1 p q » of Re ζ W originating from yi. By Proposition 1.2.7, the function W : Y C projects p q Ñ Thζ yi to the half-line wi ζ with slope ζ originating at wi. p q p ´ 8q Let now α be a piecewise smooth path in X joining wi and wj and not passing through any other wk, as in §1.1 C. Let ζ be the slope of α at wi, so that the half-line wi ζ is tangent p ´ 8q to α at wi. We can compare α-gradient trajectories with the wi ζ -gradient trajectories, i.e., with gradient trajectories of Re ζ´1W . This shows thatp γ´-gradient8q trajectories which p q approach yi over wi, also form an n 1 -dimensional manifold which we call the α-Lefschetz p ` q thimble originating at yi and denote Thα yi , see Fig. 10. By construction, W projects p q Thα yi to α, so that the preimage of any intermediate point p α is an n-sphere p q P ´1 n ∆i,α p W p Thα yi S . p q “ p qX p q » This is nothing but the translation to p (along α, with respect to the symplectic connection) of the geometric vanishing cycle ∆i. Starting from the other end, we have a similar Lefschetz thimble Thα yj originating at ´1 p q yj and the sphere ∆j,α p W p . p qĂ p q We now make the following

´1 Assumption 1.2.9. The submanifolds ∆i,α p , ∆j,α p in W p intersect transversely. In particular, being of middle dimension, theyp intersectq p inq finitelyp manyq points.

If we choose orientations of the n 1 -dimensional manifolds Thα yi , Thα yj , then we p ` q p q p q get orientations of the n-spheres ∆iα p and ∆j,α p . In particular, such a choice identifies p q p q the 1-dimensional vector spces Φi,α and Φj,α with k, and (1.2.4) implies:

Proposition 1.2.10. mij α equals the signed number of intersection points of ∆iα p and p q p q ∆j,α p (with the signs given by the orientations). p q

Since points of, say ∆i,α p correspond to α-gradient trajectories beginning at yi and ending in W ´1 p , we can reformulatep q the above as follows. p q

Corollary 1.2.11. mij α equals the signed number of α-gradient trajectories joining yi and p q yj.

26 W ´1 p p q

∆i,α b Thγ yi p q p q

Thγ yj yi p q yj

∆j,α b p q

wi α wj p

Figure 10: Curved Lefschetz thimbles over a path α.

Example 1.2.12. Let X C and α wi,wj be the straight line interval joining wi and “ “ r s wj. We assume, as before, that there are no other critical values on this interval. Let

wi wj ζij ´ “ wi wj | ´ | be the slope of α wi,wj . Corollary 1.2.11 implies that mij α is the number of gradient “r´1 s p q trajectories of Re ζ W joining yi and yj. We will return to this point of view in §2.1. p ij q ´1 Note that yi and yj are critical points of Re ζij W of the same index. As well known, for a generic Morse function f there are no gradientp trajectoriesq joining two critical points of the same index. In fact, functions that do have such trajectories, form a subvariety of real codimension 1 in the space of all Morse functions, see [53] and references therein. So a generic 1-parameter family of Morse functions will contain finitely many such “exceptional” ´1 functions. This is what happens in our situation: the 1-parameter family is Re ζ W ζ 1, t p qu| |“ and exceptional functions correspond to ζ ζij for all pairs i, j . “ p q 1.3 Perverse sheaves with bilinear forms A. Seifert duality for local systems on S1. Let LS S1 be the category of local systems of finite-dimensional k-vector spaces on S1. We introducep q the Seifert duality functor

27 ‹ on this category by defining, for a local system L,

‹ ˚ L ζ L ζ . p q “ p ´ q That is, the stalk of L‹ at the point ζ S1 is set to be the dual to the stalk of L at the opposite point ζ . P It is clear thatp´ q‹ is a perfect duality on LS S1 . For a local system L LS S1 , we will call a homomorphism q : L L‹ a Seifertp bilinearq form on L, called non-degenerateP p q , if it is an isomorphism. A localÝÑ system equipped with a nondegenerate symmetric (resp. antisymmetric) Seifert form, will be called a Seifert symmetric resp. Seifert antisymmetric local system.

Example 1.3.1. Let Sgn be the 1-dimensional local system on S1 with monodromy 1 . Then Sgn carries a natural antisymmetric scalar product q : Sgn Sgn‹. To describep´ it,q let us view S1 as the unit circle z 1 in C, and let p : S1 SÑ1 be the 2-fold covering Z | | “ 2 k Ñ Z (with Galois group 2) taking ζ to ζ . Then Sgn p˚ S1 consists of 2-anti-invariant sections. Given ζ S1{, the preimage p´1 ζ consistsĂ of two opposite points{ which generate a Z-sublattice Z Pp´1 ζ C of rank 1.p Anti-invariantq functions p´1 ζ k are the same as group homomorphisms¨ p q ĂZ p´1 ζ k, that is, p q Ñ ¨ p qÑ ´1 Sgn HomZ Z p ζ , k . ζ “ p ¨ p q q Now consider the standard symplectic form ω dx dy on C. Given two opposite points ζ, ζ S1, any ξ p´1 ζ and η p´1 ζ are“ orthogonal^ unit vectors and so ω ξ, η 1. Therefore´ P ω definesP a nondegeneratep q P pairingp´ q p q“˘

´1 ´1 Z p ζ Z Z p ζ Z p ¨ p qqb p ¨ p´ qq ÝÑ k which by the above, induces a pairing qζ : Sgnζ Sgn´ζ . Since ω is anti-symmetric, t b Ñ q ζ q . ´ “´ ζ Let now L, q be a local system on S1 with a Seifert bilinear form (not necessarily p q 1 symmetric or antisymmetric). Choose a point ζ S and let V Lζ be the stalk at ζ. By P “ the monodromy M along the positive (counterclockwise) half-circle we can identify V Lζ “ with L´ζ. Then q gives a bilinear form

t qζ ˚ M ˚ ˚ (1.3.2) B Bq : V Lζ L L V “ “ ÝÑ ´ζ ÝÑ ζ “ which we call the Seifert form on V associated to q. Compare with the definition of the Seifert form for an isolated singularity in [5] §2.3. Note that if q is symmetric (as a Seifert form on L), then Bq does not have to be symmetric as a bilinear form on a vector space. In fact, we have the following fact that is proved by direct comparison of definitions.

28 Proposition 1.3.3. (a) The category of Seifert symmetric local systems on S1 is equivalent to the category of pairs V, B : V V ˚ where V is a finite-dimensional k-vector space (stalk of L at some point)p and B isÑ a (notq necessarily symmetric) nondegenerate bilinear form on V . Under this equivalence, the monodromy of L corresponds to the Serre operator S B´1 Bt. “ ˝ (b) Tensoring with Sgn identifies the categories of Seifert symmetric and Seifert antisym- metric local systems on S1. In particular, the category of Seifert antisymmetric local systems is again equivalent to the category of pairs V, B; V V ˚ as in (a). Under this equivalence, the monodromy of L corresponds to the operatorp SÑ qB´1 Bt. ´ “´ ˝ B. Duality for vanishing cycles. Recall the functor of vanishing cycles Φ : Perv D, 0 LS S1 . p qÑ p q Proposition 1.3.4. The functor Φ takes (shifted) Verdier duality on perverse sheaves to Seifert duality of local systems, i.e., we have natural isomorphisms

Φ F ˚ Φ F ‹. p q»p p qq Proof: For simplicity we assume that F is given on the entire complex plane C, i.e., F Perv C, 0 . Let ζ S1. We have the closed half-line R ζ C and the embeddings P p q P ` ¨ Ă k i 0 ζ R ζ ζ C. t u ÝÑ ` ¨ ÝÑ By definition, ˚ ! Φ F ζ k i F 1 . p q “ ζ ζ p qr s Since the Verdier duality functor D interchanges i! with i˚, we have

˚ ! ˚ ˚ Φ F ζ k i F 1 , p q » p ζ ζ p qr sq where the last in the RHS is the usual duality of vector spaces. We now consider the commutative diagram˚ kζ 0 / R ζ t u ` ¨ k´ζ iζ   R` ζ / C. ¨ p´ q i´ζ It is a Cartesian square of closed embeddings, and so gives a natural morphism

(1.3.5) k! i˚ F k˚ i! F ´ζ ´ζ ÝÑ ζ ζ and it is immediate that in our case (F Perv D, 0 ) it is a quasi-isomorphism. This gives an identification Φ F ˚ Φ F ‹. P p q p q»p p qq

29 Remark 1.3.6. The identification (1.3.5) would of course hold if we take, instead of ζ , 1 Φ ˚ Φ ˚ p´ q any ζ ζ and so we have an identification F ζ F ζ1 . The role of the exact opposite‰ point ζ is that it provides a consistentp wayq » to choosep q such a ζ1 for any ζ. Any other consistentp´ wayq would do just as well, and would be homotopy equivalent to ζ ζ . ÞÑ p´ q The identification induced by (1.3.5) can be rewritten as a non-degenerate pairing (note that ζ R ζ R 0 ) p ¨ `qXp´ ¨ `q “ t u

˚ 1 1 ˚ βζ 2 (1.3.7) Φ F ζ Φ F H R C, F H R C, F H C, k k. p q b p ´ζq “ ζ¨ ` p qb ´ζ¨ ` p q ÝÑ t0up q“ 1 1 2 We note that this pairing, having the form H H H , is anti-symmetric, i.e., b ζ b Ñ ´ becomes, after interchanging the arguments, equal to the negative of bζ . This implies the following fact.

Proposition 1.3.8. Let F Perv D, 0 and e : F F ˚˚ and ε : Φ F Φ F ‹‹ be the canonical isomorphisms. ThenP thep identificationq ofÑ Proposition 1.3.4p takesq ÝÑe top q ε . p´ q C. Duality in the Φ, Ψ -description. p q Proposition 1.3.9. Let F Perv D, 0 be represented, via Proposition 1.1.6, by a diagram a P p q a1 ˚ ˚ ˚ Φ o / Ψ . Then F is represented by the diagram Φ o / Ψ , where b b1

a1 bt 1 atbt ´1 “ p ´ q b1 at. # “´ Remarks 1.3.10. (a) Sometimes a simpler formula is asserted, namely a1 bt, b1 at. However, this cannot be true for the following reason. The monodromy of“F on D“ 0 (where it is a local system) is T 1 ab. Now, for any local system L on any space X ztandu any path γ joining points x and “y, the´ monodromies of L and L˚ along γ are related by

˚ y y t ´1 ˚ ˚ y M L M L : L L , M L : Lz Ly p qx “ p p qxq x ÝÑ y p qx ÝÑ (this is the only possible expression` that˘ makes sense for any x and y). So the monodromy of F ˚ should be T t ´1 1 btat ´1, while the “simpler formula” above would give 1 btat T t. The formulap in Propositionq “p ´ 1.3.9q does give the correct monodromy due to the´ “Jacobson“ identity”

(1.3.11) 1 uv ´1 1 u 1 vu ´1v. p ´ q “ ` p ´ q See the discussion around Eq. (1.1.6) in [56] for the significance of this identity for the Φ, Ψ -description. p q

30 (b) Applying Proposition 1.3.9 twice, we get the Φ, Ψ description of F ˚˚ as the diagram a2 p q Φ o / Ψ , with b2

b2 a1 t 1 ba ´1b “p q “ ´p ´ q a2 b1 t 1 a1 t b1 t ´1 a 1 1 ba ´1ba ´1 a 1 ba ´1 # “ ´p q p ´p q p q q “´ p `p ´ q q “´ p ´ q a This diagram is isomorphic to Φ o / Ψ , via the isomorphisms b

´1 eΨ Id: Ψ Ψ, eΦ 1 ba : Φ Φ. “ ÝÑ “ ´p ´ q ÝÑ

The equality eΨ Id reflects the fact that dualization commutes with taking general fiber. “ The appearance of the minus sign in the formula for eΦ is a manifestation of Proposition 1.3.8. (c) The formula of Proposition 1.3.9 can be compared with the formula given in [10] (Prop. 4.5) for the effect of the Fourier-Sato transform on the Φ, Ψ -description of perverse sheaves on C, 0 . Both formulas are remindful of cluster transformations.p q p q

D. Duality for transport maps. Given F Perv X, A and a path α joining wi with F P p q wj. Then the transport map m α : Φi F α Φj F α acts between the stalks of Φi F ij p q p q Ñ p q p q and Φj F in the directions given by the path α, see (1.1.9). By Proposition 1.3.9, the dual p q F t Φ ˚ Φ ˚ map can be seen as acting can be seen as a map mij α : j F ´α i F ´α acting between the stalks in the directions, opposite to thep directq ionsp givenq ÝÑ by α.p Onq the other ˚ F ˚ ´1 hand, we have the dual perverse sheaf F and its own transport map mji α associated to the opposite path α´1, i.e., to α run in the opposite direction. Propositionp 1.3.9q implies at once: Corollary 1.3.12. We have an equality

F ˚ ´1 F t ´1 m α T ,i m α T , ji p q“´ ` ijp q `,j

where Ti,` is the counterclockwise half-monodromy from the opposite to direction of α to the direction of α, and similarly for T`,j.

E. Proof of Proposition 1.3.9. For simplicity we assume that F is given on the entire complex plane C, i.e., F Perv C, 0 . P p q The Φ, Ψ -diagram describing F is the stalk at ζ 1 of a local system of similar diagrams on the unitp circleq S1: “

aζ,F pIq pIIq H1 C F Φ F / Ψ F F H0 C R F (1.3.13) ζ¨R` , ζ o ζ ζ ζ ` , . p q“ p q bζ,F p q » » p zp ¨ q q

1 Here Fζ is the stalk of F at ζ S , and the identifications (I) and (II) go as follows: P 31 (I) (Local Poincaré duality) As in Fig. 2, choose a small disk V around ζ and write V ζ R` V` V´ with V` following ζ R` in the counterclockwise direction and V´ precedingz ¨ “ζ R\. Then by definition ¨ ¨ `

1 F V` F V´ Ψζ F H R V, F p q‘ p q, p q“ V Xpζ¨ `qp q“ F V p q and we send the class s ,s of a pair of sections s F V , to the image of s s r ` ´s ˘ P p ˘q ` ´ ´ in Fζ.

(II) Given an element s of Fζ , we extend it to a section in C ζ R` by continuing counterclockwise from ζ. zp ¨ q

With these identification, the map bζ,F is identified with the coboundary map

0 1 δ δζ : H C ζ R , F H R C, F . “ p zp ¨ `q q ÝÑ ζ.¨ ` p q

We focus in particular on ζ 1 and write Φ˘ F for Φ˘1 F etc. We use similar notations for F ˚. “˘ p q p q ˚ ˚ By Proposition 1.3.4, Φ´ F is identified with Φ` F . The non-degenerate pairing that Proposition 1.3.4 gives, can bep writtenq as the cup-productp q pairing (note that 0 R R ) t u“ ` X ´ ˚ 1 1 ˚ β 2 (1.3.14) Φ F Φ F HR C, F HR C, F H C, k k. `p qb ´p q “ ` p qb ´ p q ÝÑ t0up q“ At the same time, Ψ F ˚ is identified with Ψ F ˚. `p q `p q σ H0 C R , F ˚ x H1 C, F β x, δ σ Lemma 1.3.15. Let ´ and R` . Then the pairing ´ can be found as follows: P p z q P p q p p qq

1 • Choose a small disk V around 1, and let γ x H R V, F be the restriction of x p q P V X ` p q H1 F (i.e., the result of applying the generalization map of the sheaf R` to the image of 1 p q x in HR F 0). ` p q 0 • Represent γ x as the class s`,s´ of a pair s˘ H V˘, F (as in the identification (I) above). p q r s P p q

• Then β x, δ´ σ s`, σ s´, σ , where s˘, σ is the canonical pairing of sections p p qq“p q´p˚ q p q of the local systems F and F on V˘ .

32 Proof: Consider the diagram

α“Y H1 C, F H0 C R , F ˚ / H1 C R , k R` ´ pCzR´qXR` ´ p qb p z q ✈ p z q ✈ ✈ “ ✈ ✈  ✈ H0 C R H1 k Idbδ ✈ ´, R ✈ p z ` p qq ✈ ✈ ✈ » δ  ✈  1 1 ˚ ✈ 1 1 HR C, F HR C, F ✈ HR HR k ` ´ ✈ ´ ` p qb p ✈ q ❣❣❣❣ p p qq ✈ “ ❣❣❣❣❣ β“Y ✈ ❣❣❣❣❣ ✈ ❣❣❣❣❣  {✈ s❣❣❣❣ 2 HR R C, k k `X ´ p q“ whose outer rim is commutative by naturality of the coboundary maps δ (Note that both α and β are given by the cup-product.) Now, after the identification of the target of α with k given by the maps on the right side (the dotted arrow), the map α is immediately found to have the form claimed in the lemma. We now describe the standard Φ, Ψ -diagram (i.e., the stalk at ζ 1 of the local system (1.3.13) of diagrams) but for F ˚ insteadp q of F. We use the isomorphisms“

T t ´1 ˚ β ˚ p `q ˚ Φ` F Φ´ F Φ` F , p q ÝÑ p q ÝÑt p q pI ˚ q pI q Ψ F ˚ F F ˚ F Ψ F ˚, `p q ÝÑ 1 ÝÑ `p q where:

• T` : Φ` F Φ´ F is the anti-clockwise half-monodromy (along the upper semi- circle); p q Ñ p q

˚ • β β ˚ is the pairing (1.3.14) for F ; “ F ˚ • I and I ˚ are the identifications in (1.3.13) for F and F . p F q p F q After this, we define a1 and b1 by commutativity of the diagram

a`,F˚ ˚ ˚ Φ` F o / Ψ` F p q b`,F˚ p q t ´1 t pT`q ˝β pIF q ˝pIF˚ q  a1  ˚ ˚ Φ` F o / Ψ` F , p q b1 p q that is, by 1 ˚ ´1 t a IF IF ˚ a`,F ˚ β T`, 1 “t p ´1q ˝p q˝ ˚ ´˝1 ˝ t ´1 b T β b , ˚ I I . “ p `q ˝ ˝ ` F ˝p F q ˝ pp F q q

33 1 ˚ 1 To find a , let ϕ Φ F and ψ Ψ F F1 and find a ϕ, ψ . P `p q P `p q“ p q Let us denote the counterclockwise half-monodromies from 1 to 1 in all the local ` p´ q systems involved, by the same symbol T`, the counterclockwise half-monodromies from 1 to 1 by T , and the full monodromies from 1 to 1 by T T T . p´ q ´ “ ´ ˝ ` In this notation set

ϕ1 T ϕ Φ F ˚, f β´1 ϕ1 Φ F ˚ . “ `p q P ´p q “ p q P `p q By Lemma 1.3.15 applied to F ˚ instead of F,

a , ˚ f , ψ β f, δ s ψ , p ` F p q q “ p ´ ´p qq

where s´ ψ is the section of F on C R´ equal to ψ at 1 and β is viewed as a pairing. p q ˚ z ˚ Viewing β as an isomorphism Φ` F Φ´ F , we can rewrite the RHS of the previous equality as p q Ñ p q β f , δ s ψ T ϕ , δ s ψ . p p q ´ ´p qq “ p `p q ´ ´p qq Now, recall that b ψ Φ F is obtained as δ s1 ψ , where s1 ψ is the section of F on F p q P `p q ` p q p q C R` obtained by extending ψ counterclockwose from R`. z ´1 1 Note that s´ ψ T´ s ψ . So we write the RHS of the previous displayed equality as p q “ p p qq T ϕ , δ T ´1s1 ψ T ϕ , T ´1δ s1 ψ p `p q ´ ´ p qq “ p `p q ´ ` p qq “ T T ϕ , δ s1 ψ T ϕ , b ψ . “p ´ `p q ` p qq “ p p q F p qq This means that a1 bt T , where T is the monodromy of Φ F ˚, i.e., “ ˝ p q T 1 ba t ´1 1 atbt ´1 “ pp ´ q q “p ´ q as claimed. Let us now find b1. For this, fix x Ψ F ˚ F ˚ and y Φ F and find P `p q “ 1 P `p q 1 t ´1 b x ,y T β b , ˚ x ,y β b , ˚ x , T y . p p q q “ p `q p p ` F p qqq “ p ` F p qq `p q 1 ` 1 ˘ ` ˚ ˘ Now, b , ˚ x δ s x , where s x is the section of F in C R obtained by continuing ` F p q “ ´ p q p q z ` x counterclockwise from R`. So by Lemma 1.3.15 in which the roles of R` and R´ are interchanged, the RHS of the previous displayed equality is equal to minus the pairing of 1 the values of s x and a , T y at the point 1 . The minus sign comes from the fact p q ´ F p `p qq p´ q that interchanging roles of R` and R´ amounts to change of orientation or, put differently, it comes from the anticommutativity of the product H1 H1 H2. But the value of s1 x at 1 is equal to T x , we we get b Ñ p q p´ q `p q

T x , a , T y T x , T a , y x, a , y . p `p q ´ F p `p qqq “ p `p q `p ` F p qq “ p ` F p qq This means that b1 at, as claimed. “´

34 F. Perverse sheaves with bilinear forms. The baby spherical functor package. Denote by Perv D, 0 BF the category formed by pairs F, q , where F Perv D, 0 is a perverse sheaf andpq is aq nondegenerate bilinear form on Fp, i.e.,q an isomorphismP q :p F q F ˚. No symmetry conditions on q are assumed. Morphisms are morphisms of perverse sheavesÑ compatible with bilinear forms. We will describe Perv D, 0 BF in terms of Φ, Ψ -diagrams with additional structure. The answer is completely analogp qous to (is a “decategorificationp q of”) the concept of a spherical functor [4]. We start with some generalities. Let V be a finite-dimensional k-vector space equipped with a non-degenerate bilinear form B. We view B alternately as an isomorphism B : V V ˚ and as a pairing Ñ v, v1 B v, v1 : v, B v1 , p q ÞÑ p q “ x p qy where on the right hand side we have the canonical pairing of V ˚ and V . The change of arguments in the pairing B v, v1 corresponds to passing to the dual map t ˚ ˚ ˚ p q ´1 t B : V V V . The form B gives rise to the Serre operator SB B B : V V , so that“p we haveq Ñ “ ˝ Ñ B v1, v2 B v2,SB v1 . p q“ p p qq Further, let V, W be two finite-dimensional k-vector spaces equipped with non-degenerate bilinear forms BV , BW and f : V W be a linear map. Then the right and left adjoints to f are the maps f ˚, ˚f : W V definedÑ by Ñ ˚ ˚ BW f v ,w BV v, f w , BV f w , v BW v, f w . p p q q“ p p qq p p q q“ p p qq ˚ ˚ Viewing BV , BW as isomorphisms BV : V V and BW : W W , we have Ñ Ñ ˚ ´1 t ˚ t ´1 t t (1.3.16) f B f BW , f B f B “ V “p V q W where f t : W ˚ V ˚ is the map dual to f. One can further define iterated right and left adjoints such asÑf ˚˚ f ˚ ˚, ˚˚f etc. “p q Specializing to W, BW V, BV , let us note the following. p q“p q Lemma 1.3.17. Let T : V V be an endomorphism such that T ˚ T 1. Then ˚T ˚ ´1 Ñ 1 1 ˝ “ “ T T and T is an isometry, i.e., BV v, v BV Tv,Tv . “ p q“ p q Definition 1.3.18. Let Φ, BΦ and Ψ, BΨ be finite-dimensional k-vector spaces equipped with non-degenerate bilinearp forms.q p A linearq map a : Φ Ψ is called spherical, if the following conditions are satisfied: Ñ

˚ (S1) The map TΦ 1 a a : Φ Φ is invertible. “ ´p q Ñ (S2) We have a˚ ˚a ˚a aa˚ 0. `p q´p q “ Compare with [4]. The condition (S2) is analogous to Kuznetsov’s definition of a spherical functor, see [70], Def. 2.12 . Futher, the next proposition is a de-categorified version of the full spherical functor package, see [4], Th. 1.1.

35 Proposition 1.3.19. If a is spherical, then:

˚ (a) The map TΨ 1 aa :Ψ Ψ is invertible. “ ´ Ñ

(b) Both TΦ and TΨ are isometries (for BΦ and BΨ respectively).

´1 (c) The left multiplication by TΦ takes the right adjoint to minus the left adjoint: ˚a 1 a˚ a ´1 a˚ . p q“´p ´p q q p q (d) We have ˚a aa˚ a˚ a ˚a . p q “p q p q

Note that the requirement that TΦ and TΨ are isometries is strictly weaker than the requirement that a is a spherical map, while in categorical version it is enough to require that TΦ and TΨ are autoequivalences. Proof: Multiplying (S2) by a on the left and adding 1 we get

˚ ˚ ˚ ˚ ˚ ˚ ˚ TΨ TΨ 1 a a 1 aa 1 a a a a aa 1, p q˝ “p ´ p qqp ´ q“ ´ p `p q´p q q“

so TΨ is invertible, giving (a) and is an isometry by Lemma 1.3.17. Similarly, multiplying (S2) by a on the right and adding 1, we get

˚ ˚ ˚ ˚ ˚ ˚ ˚ TΦ TΦ 1 a a 1 a a 1 a a a aa a 1 p q˝ “p ´p q qp ´ q“ ´p `p q`p q q “ ´1 ˚ so TΦ is an isometry, giving (b). Next, (c) is reduced to (S2) by substituting TΦ 1 a a. Finally, let us rewrite the equality claimed in (d) by using (c). It will read: “ ´p q

˚a a 1 a˚ a ˚a ? 1 a˚ a ˚a a ˚a . p q p ´p q qp q “ p ´p q qp q p q But this is obviously true since ˚a a commutes with 1 ˚a a ´1 1 a˚a . p q p ´p q q “p ´ q Proposition 1.3.20. The category Perv D, 0 BF is equivalent to the category formed by data consisting of: p q • Finite-dimensional k-vector spaces Φ, Ψ equipped with non-degenerate bilinear forms BΦ, BΨ; • A spherical map a : Φ Ψ. Ñ a Proof: Suppose F Perv is represented by a diagram Φ o / Ψ . Note that the diagram P b a: ˚ ˚ ˚ representing, by Proposition 1.3.9, the dual F is naturally isomorphic to Φ o / Ψ with b:

a: bt 1 atbt ´1 “´ p ´ q b: at. # “ 36 The isomorphism is given by 1 on Φ˚ and 1 on Ψ˚. Let us use this diagram for F ˚. The bilinear form is then a map q`: F F ˚, whichp´ q we represent as the following commutative diagram: Ñ a Φ o / Ψ b BΦ BΨ :  ˚ a ˚ Φ o / Ψ , b: Using (1.3.16), we write

˚ ´1 t ´1 1 a BΦ a BΨ BΦ b BΨ b ˚ “ ´“1 t 1 t “ ´1 ˚ ´1 ˚ ´1 ˚ a BΨaB a 1 ba b 1 a a a T a , #p q“p Φ q “p q “ ´p ´ q “ ´p ´p q q “´ Φ p q whence the statement.

G. Calabi-Yau perverse sheaves. We now focus on symmetry conditions for q : F F ˚. Ñ Definition 1.3.21. We call an even (resp. odd) Calabi-Yau perverse sheaf a perverse sheaf F with a symmetric (resp. antisymmetric) nondegenerate bilinear form q : F F ˚. Ñ Let Perv D, 0 CY` and Perv D, 0 CY´ be the categories formed by even and odd Calabi- Yau perversep sheavesq on D, 0 .p Theyq are full subcategories in Perv D, 0 BF. p q p q Proposition 1.3.22. (a) A spherical map a : Φ Ψ corresponds to an even Calabi-Yau perverse sheaf, if and only if: Ñ

(a1) BΨ is symmetric. ˚ (a2) TΦ 1 a a is equal to SB , the negative of the Serre operator for BΦ. “ ´p q p´ Φ q (b) A spherical map a : Φ Ψ corresponds to an odd Calabi-Yau perverse sheaf, if and only if: Ñ

(b1) BΨ is anti-symmetric. ˚ b2) TΦ 1 a a is equal to SB , the Serre operator for BΦ. “ ´p q Φ The proof is an immediate consequence of the next corollary, which, in turn, follows at once from Proposition 1.3.8. Corollary 1.3.23. If F is even (resp. odd) Calabi-Yau perverse sheaf on D, 0 , then Φ F is a Seifert anti-symmetric (resp. symmetric) local system. p qq p q Example 1.3.24. Let W : X Σ be a generalized Lefschetz pencil (Definition 1.2.1) of Ñ relative dimension n and LW Perv Σ, A be the corresponding Lefschetz perverse sheaf. P p q Poincaré duality, formulated in the language of perverse sheaves, implies that LW is Calabi- Yau.

37 k More precisely, since the constant sheaf X is Verdier self-dual (up to a shift by n 1), k n `k the complex RW˚ X is Verdier self-dual up to a shift by n and so LW Hperv RW˚ X p q “ p˚ p qq is self-dual, i.e., equipped with a non-degenerate bilinear form q : LW LW which is symmetric for even n and anti-symmetric for odd n. Ñ In particular, let us restrict LW to a small disk around a singular value wi W xi . Corollary 1.3.22 gives then the following classical result: the monodromy of an“ isolatedp q singularity transposes (up to a sign) the Seifert form. See [5] Th. 2.5, where the operator Var differs from the Seifert form by a sign.

1.4 Quotient by (locally) constant sheaves A. The category Perv X, A . Let A be an abelian category. We recall that a full subcategory B A is calledp a Serreq subcategory, if it is closed under subobjects, quotient objects and extensions.Ă In particular, B is abelian itself and a strictly full subcategory in A (i.e., any object of A isomorphic to an object in B, is in B). In this case we have a new abelian category A B, called the quotient category of A by B. By definition, Ob A B Ob A , and { p { q “ p q 1 1 Hom A, A Hom A, A IA,A1 , A{Bp q“ Ap q{ 1 where IA,A1 consists of all morphisms A A which factor through an object of B. See [41] and references therein for more details.Ñ An alternative definition of A B is as follows. { 1 Proposition 1.4.1. Let QB Mor A consist of morphisms f : A A such that both Ker f and Coker f lie in BĂ. Thenp Aq B is identified with A Q´1 ,Ñ the localization of A p q p q { r B s with respect to QB. We now specialize to A Perv X, A . Let LS X Perv X, Perv X, A be the full subcategory formed by local“ systemsp q on X. Followingp q “ [41],p we noteHq Ă that Bp LSqX is a Serre subcategory in the abelian category Perv X, A , The corresponding quotient“ categoryp q will be denoted p q (1.4.2) Perv X, A Perv X, A LS X . p q “ p q p q Objects of Perv X, A will be called localized perverseL sheaves. p q Denoting, as before, A w1, ,wN , we note that the vanishing cycle functors Φi descend to the quotient category,“ t giving¨ ¨ ¨ theu functors that we still denote

1 Φi : Perv X, A LS S . p q ÝÑ p wi q Further, for any path α joining wi and wj and not passing through any other wk, we have the natural transformation of these new functors

mij α : Φi,α Φj,α, p q ÝÑ where Φi,α takes, as in (1.1.9), a perverse sheaf F to the stalk of Φi F at the direction of p q α at wi. These transformations satisfy the formal Picard-Lefschetz identities in Proposition 1.1.12.

38 B. The spider description of Perv D, A . We now specialize to X D being a disk. In this case LS D consists of constantp sheaves.q The spider description of“ Perv D, A (Proposition 1.1.17)p implies,q as pointed out in [41], a description of Perv D, A as well.p Toq p q formulate it, let K γi, ,γN be a v-spider for D, A , see again Fig. 7. For any “ t ¨ ¨ ¨ u p q i, j 1, , N let αij be the path obtained by going first from wi to v via γi and then back “ ¨ ¨ ¨ to wj via the reverse of γj. Let also Ti F : Φi,γ F Φi,γ F be the monodromy of the p q i p q Ñ i p q local system Φi F . p q Definition 1.4.3. Let MN be the category whose objects are diagrams consisting of:

(0) Vector spaces Φi, i 1, , N. “ ¨ ¨ ¨ (1) Linear operators mij : Φi Φj given for all i, j and such that IdΦ mii is invertible. Ñ i ´

Proposition 1.4.4 ([41]). Let K be a v-spider for D, A . The functor ΞK : Perv D, A p q p qÑ MN taking F to Φi Φi,γ F and “ i p q

mij αij , if i j; mij p q ‰ “ Id Ti F , if i j, # ´ p q “ is an equivalence of categories. In other words, we have the following commutative diagram of functors ΘK Perv D, A / QN p q „ λ µ  „  Perv D, A / MN p q ΞK with horizontal arrows being equivalences, λ being the canonical localization functor and µ given explicitly by Ψ, Φi, ai, bi Φi, mij bjai . p q ÞÑ p “ q

Remark 1.4.5. In fact, Perv D, A can be canonically embedded back into Perv D, A as the full subcategory of perversep sheavesq F s.t. H‚ D, F 0, see [41], Prop. 3.1p and [63],q p q “ §2.3.2. At the level of quivers, this corresponds to the full embedding t : MN ã QN given by Ñ N m N N j ij proj.i t Φi, mij Ψ Φj, Φi Φi, ai : Φi Φj, bi : Φj Φi , p q “ “ “ ÀÝÑ ÝÑ ˆ j“1 j“1 j“1 ˙ à à à and which is right adjoint to µ, see [41], proof of Prop. 2.3. Given a diagram Φi, mij MN , p q P we will refer to the perverse sheaf F Perv D, A corresponding to the quiver t Φ, mij QN P p q p q P as the Gelfand-MacPherson-Vilonen representative of Φi, mij . p q Let us also note the relation of the GMV description with the Katzarkov-Kontsevich- Pantev description of nc Betti structures, see Sect. 2.3.2 of [63].

39 C. Dependence on the spider. For later use let us recall, following [41], the effect of the change of spider K on the equivalence ΞK : Perv D, A MN . We use the action p q Ñ of the braid group BrN on the set K D,v,A of spiders, see Fig. 8. Proposition 1.1.19 is complemented by the following fact whichp is itsq direct consequence (it is also a consequence of the Picard-Lefschetz identities).

Proposition 1.4.6 ([41], Prop. 2.4). (a) Replacing K by τi K changes a diagram Φj, mν,j to the diagram Φ1 , m1 , where; p q p q p j ν,jq 1 1 Φ , , Φ Φ1, , Φi 1, Φi 1, Φi, Φi 2, , ΦN p 1 ¨ ¨ ¨ N q“p ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ qu

mν,j, if i j, j 1,ν,ν 1 ; R t ` ` u $mν,i, if j i 1, ν i, i 1 ; ’ “ ` R t ` u ’mi,j, if ν i 1, j i, i 1 ; ’ “ ` R t ` u ’m m 1 m ´1m , if j i, ν i, i 1 ; ’ ν,i`1 i,i`1 ii ν,i 1 ’ ` p ´ q “ R t ` u mν,j ’mi`1,j mijmi`1,i, if ν i, j i, i 1 ; “ ’ ´ “ R t ` u &’ 1 mi,i mi`1,i, if ν i, j i 1; p ´ q ´ “ ` ’mi,i`1 1 mi,i , if ν i 1, j i; ’ p ´ q “ ` “ ’mi`1,i`1, if ν j i; ’ “ “ ’m , if ν j i 1. ’ i,i ’ “ “ ` ’ Br M (b) Put differently,%’ the formulas in (a) define an action of N on the category N . For any spider K K D,v,A (and the corresponding numbering of A by slopes of the γi at v), P p q ` the equivalence ΞK takes the geometric action of BrN Γ on Perv D, A (coming from “ D,A p q Proposition 1.1.4) to the above action of BrN on MN .

2 Rectilinear point of view and the Fourier transform

2.1 The rectilinear approach

A. Notation and terminology. Let A w1, ,wN C. The following notation and terminology, pertaining to convex geometry“ tof C ¨ ¨ ¨R2, willu Ă be used throughout the rest of the paper. “ For any points x, y C we denote by x, y the straight line interval between x and y. We denote by Q ConvP A the convex hullr ofsA, a convex polygon in C. “ p q Definition 2.1.1. Let ζ C, ζ 1. We say that A is: P | |“ (a) in convex position, if each element of A is a vertex of Q.

(b) in linearly general position, if no three elements of A lie on a real line in C.

(c) in strong linearly general position, if no two intervals wi,wj and wk,wl are parallel. r s r s 40 (d) in linearly general position including ζ-infinity, if A is in linearly general position and ´1 all the Im ζ wi are distinct. When ζ 1, we simply speak about linear position including infinityp q. “

Condition (d) can be understood as A Rζ being in linearly general position, where R 0 is a very large positive number. Y t u " The rectilinear approach to the study of perverse sheaves F Perv C, A , which we develop in this section, consists in using the tools from convex geometry,P p in particular,q the above concepts, to analyze F.

B. Motivation: Fourier transform brings convex geometry. The rectilinear approach may seem unnatural, since the concept of a perverse sheaf is purely topological. However, convex geometry appears naturally, if we want to do the Fourier transform. In other words, assume that our perverse sheaves are formed by some analytic functions (solutions of linear differential equations) and we are interested in Fourier transforms of these functions.

More precisely, let w be the coordinate in C, and write C Cw to signify that we deal with the plane with this coordinate. Let z be the “dual variable”“ to w, so that the complex plane Cz with coordinate z is the dual C- vector space to Cw via the pairing z,w z w. The naive formula for the Fourier (Laplace) transform p q“ ¨

(2.1.2) f z f w e´zwdw p q “ p q żγ needs, in the context of complex analyticq functions, several well known qualifications. First, f being possibly multi-valued, the countour γ must be a 1-cycle with coefficients in the local system of determinations of f. Further, γ must be an unbounded (Borel-Moore) cycle such that the integrand decays appropriately at its infinity. The case relevant to our situation is when (each branch of) f has at most polynomial growth: this means that the differential equation has regular ´z0w singularities. Then, for f z0 to be defined, the function w e must decay at the p q ÞÑ infinity of γ, i.e., γ can go to infinity only in the half-plane Re z0w 0. p qą q ´z0w In this way, studying the relative growth of exponentials e for different z0, leads nat- urally to questions of convexity, to emphasis on straight lines etc. The following elementary example (interchanging the roles of z and w) may be useful.

Example 2.1.3. Let A w1, ,wN Cw be as before and consider an exponential sum “ t ¨ ¨ ¨ uĂ N wiz g z cie , p q“ i“1 ÿ with all ci 0. The asymptotic geometry of the distribution of zeroes of g z is governed by the convex‰ polygon Q Conv A . More precisely, the zeroes concentrate, asymptotically,p q on “ p q 41 the rays which are normal to the edges of Q. Indeed, if we approach the infinity in any other direction, there will be a single summand in g z that will dominate all the other summands, so g z 0. This observation, due to H. Weylp [95],q is relevant for understanding of integrals likep (2.1.2)q‰ and (0.1) by the stationary phase approximation.

C. Rectilinear transport maps. Let us assume that A is in linearly general position 1 1 including infinity, i.e., ζ in Definition 2.1.1(d). For any wi A, we identify Swi , the “ 1 P circle of directions at wi, with the unit circle S ζ 1 C. As in Example 1.2.12, denote by “ t| | “ u Ă

wi wj 1 (2.1.4) ζij ´ S “ wi wj P | ´ | the slope of the interval wi,wj . Thus ζij R by our assumption. r s R Let F Perv C, A . We denote by Φi F the stalk of the local system Φi F at the horizontalP directionp 1 q S1 S1. For anyp iq j we define the rectilinear transportp q map P wi » ‰ mij mij F : Φi F Φj F as the composition “ p q p qÑ p q

T1,ζji mij pri,jsq Tζji,1 (2.1.5) Φi F Φi F 1 Φi F ζ Φj F ζ Φj F 1 Φj F , p q“ p q ÝÑ p q ji ÝÑ p q ij ÝÑ p q “ p q where:

• mij wi,wj is the transport map along the rectilinear interval wi,wj . pr sq r s

• T1,ζji , resp. Tζij ,1 is the monodromy from 1 to ζji, resp. from ζij to 1, taken in the counterclockwise direction, if Im wi Im wj , and in the clockwise direction, if p q ă p q Im wi Im wj , see Fig. 11. p qą p q

wj ‚

Tζji,1

T wi 1,ζji ‚ Figure 11: Rectilinear transport.

We also define mii biai : Φi F Φi F , “ p q ÝÑ p q ai where ai and bi are the map in the standard Φ, Ψ -diagram Φi F o / Ψi F , represent- p q p q bi p q ing F near wi in the horizontal direction. Thus 1 mii Ti F is the counterclockwise ´ “ p q monodromy of Φi F , in particular, it is invertible. p q 42 Thus the data Φi F , mij form an object of the category MN , see Definition 1.4.3 and we have the functorp ofp rectilinearq q transport data

k k (2.1.6) Φ Φ : Perv C, A MN , F Φi F , mij . “ A p q ÝÑ ÞÑ p p q q Proposition 2.1.7. The functor Φk is an equivalence of categories.

Remark 2.1.8. This statement is similar but not identical to Proposition 1.4.4, because the wi,wj do not all pass through a single point v (“Vladivostok”). Moreover, they cannot, in r s general, be deformed (without crossing other wk) to paths which have this property. However, if A is in convex position, then we can take v inside Q and form a spider K consisting of the intervals v,wi (dotted lines in Fig. 12). Then each interval wi,wj will be isotopic to r s r s the path wi, v v,wj used to define the Gelfand-MacPherson-Vilonen (GMV) transport map, so inr thissYr case thes statement does reduce to Proposition 1.4.4.

‚ wi wj ‚ ‚ v ‚ K

‚ ‚ Figure 12: For a convex configuration, rectilinear = GMV.

We postpone the proof of Proposition 2.1.7 until the discussion of isomonodromic defor- mations in the rectilinear approach.

2.2 Isomonodromic deformations in the rectilinear approach A. Isomonodromic deformations of perverse sheaves. Denote by CN CN the ‰ Ă space of all numbered tuples (“configurations”) of N distinct points A w1, ,wN . We Cn “ p ¨ ¨ ¨ q refer to ‰ as the configuration space. It is a classifying space of the pure braid group PBrN . CN Let A t w1 t , ,wN t 0ďtď1 be a continuous path in ‰ joining two configurations A 0 andpAq“p1 . Sincep q the¨ ¨ ¨ conceptp qq of a perverse sheaf is purely topological, we have a canonical (definedp q uniquelyp q up to a unique isomorphism) equivalence of categories

IA t : Perv C, A 0 Perv C, A 1 , p qq p p qq ÝÑ p p qq which we call the isomonodromic deformation along the path A t . This elementary but important concept can be understood in either of three folllowingp ways.q

43 (1) Intuitively, we “move the singularities” wi t of a perverse sheaf F while keeping the p q a Φ i / Ψ 1 1 data near the singularities, i.e., the local systems of diagrams i o i on Swi S , the bi “ same. (2) Note that A t gives a unique isotopy class of diffeomorphisms f : C, A 0 p q p p qq Ñ C, A 1 sending wi 0 to wi 1 and identical near infinity. The equivalence IA t is just p p qq p q p q p q the direct image f˚. It is clear that IAptq depends (up to a unique isomorphism) only on CN the homotopy class of the path A t in ‰ . Therefore the IAptq for all possible A t (with arbitrary beginning and end configurations),p q define a local system of abelian categoriesp q on CN CN C ‰ . The corresponding action of π1 ‰ , A 0 PBrN on Perv , A 0 is the one described in Example 1.1.5. p p qq “ p p qq r CN CN CN (3) Let p : ‰ ‰ be the universal covering. Explicitly, we take ‰ to consist of homotopy classes ofÑ paths originating at some given point A 0 , so (the constant path at) p q A 0 is a distinguishedr point of CN . Consider the trivial bundle r p q ‰ r π : CN C CN ‰ ˆ ÝÑ ‰ C with fiber and let wi, i 1, , N rbe its tautologicalr section whose value at B with “ ¨ ¨ ¨ N p B w1, ,wN , is wi. Denote by Γi C C be the graph of wi. p q“p ¨ ¨ ¨ q Ă ‰ ˆ CN C r CN C Let A ‰ be the union of the Γi and Perv ‰ , A be the category of perverse Ă ˆ r p ˆ q r sheaves on CN C whose microsupport [60] is contained in the union of the conormal bundles ‰ ˆ to the Γir. Sincer such sheaves are smooth in directionsr transversalr to the fibers of π and since CN CN ´1 ‰ is homeomorphicr to a ball, for any B ‰ with p B A, the restriction on π B gives an equivalence P p q “ p q N r rB : Perv C C,rA Perv C, A . p ‰ ˆ q ÝÑ p q CN In particular, if a path A t joins A 0 with some A 1 , it lifts to a path A t in ‰ joining p q p rq r p q ´1 p q A 0 with some point B over A 1 , and we define IA t rB r . p q p q p q “ ˝ Ap0q Our goal is to analyze the effect of isomonodromic deformations on rectilinearr r transport maps.

B. Walls of collinearity/marginal stability and configuration chambers. Let CN Cn gen ‰ consist of A in linearly general position. Explicitly, for A w1, ,wn consider the realĂ matrix “p ¨ ¨ ¨ q 1 1 1 ¨ ¨ ¨ P A Re w1 Re w2 Re wN . p q “ » p q p q ¨¨¨ p qfi Im w1 Im w2 Im wN p q p q ¨¨¨ p q – fl For 1 i j k N let pijk A be the 3 3-minor of P A on the columns numbered ď ă CNă ďCN p q ˆ p q i, j, k. Then gen ‰ i,j,k Dijk, where Dijk is the real (of real codimension 1) quadratic hypersurface “ z Ť Dijk pijk A 0 wi,wj,wk lie on a straight line “ p q“ p q ( 44 which we call the wall of collinearity. (In [36] it is called a wall of marginal stability.) CN CN The connected components of gen (into which the walls Dijk subdivide ‰ ) are open sets which we call configuration chambers. Describing these chambers in a more combinatorial fashion is a classical but very difficult problem of geometry. An obvious discrete invariant is as follows.

N Definition 2.2.1. (a) The orientation datum of A w1, ,wN C is the collection “ p ¨ ¨ ¨ q P gen O A eijk A of signs p q“p p qq

eijk A sgn pijk A 1 , 1 i j k N. p q “ p q P t˘ u ď ă ă ď (b) The orientation datum O `C of a configuration˘ chamber C is defined as O A for any A C. p q p q P

C. Oriented matroids. Note that the signs eijk A are subject to restrictions coming p q from the Plücker relations among the minors pijk. To write them down, let us extend the pijk A (and eijk A , in a unique way, to alternating functions defined for all triples of i, j,p k q 1, , Nqp (inq particular, make them equal to 0 when two of the indexes coincide). ThenP the t relations¨ ¨ ¨ u are as follows.

Proposition 2.2.2. For any i1, , i4 and j1, j2 between 1 and N, we have ¨ ¨ ¨ 4 1 νp p 0. i1,¨¨¨ ,iν ,¨¨¨ ,i4 j1,j2,iν ν“1p´ q ¨ “ ÿ p This leads to the following definition, see [11, 12] for more information.

Definition 2.2.3. (a) A structure of an oriented matroid on 1, , N is a collection O of O t ¨ ¨ ¨ u signs εijk εijk 1 given for any triple i, j, k 1, , N of distinct integers, which satisfy the“ axioms:P t˘ u P t ¨ ¨ ¨ u

(1) εijk depends on i, j, k in an alternating way: εijk εjik etc. “´

(2) Let us extend εijk to all triples by putting εijk 0 if i, j, k are not all distinct. Then, “ for any i1, , i4 and j1, j2 between 1 and N, the set ¨ ¨ ¨ ν 1 ε εj ,j ,i , ν 1 , 4 0, 1 p´ q i1,¨¨¨ ,iν ,¨¨¨ ,i4 ¨ 1 2 ν “ ¨ ¨ ¨ Ă t ˘ u p ˇ ( either contains 1 or is equal to 0 . ˇ t˘ u t u O CN CN (b) For an oriented matroid εijk let O gen be the set of A such that eijk A “p q NĂ p q“ εijk for all i, j, k. We say that O is realizable, if CO . ‰ H Thus, the orientation datum O A for any A CN is an oriented matroid. p q P gen

45 Remarks 2.2.4. (a) More precisely, the concept we defined is known as a uniform oriented matroid (or simplicial chirotope) of rank 3, see [11, 12]. Since in this paper we consider only this specific type of oriented matroids, we use the simplified terminology. N (b) By construction, each CO is an open and closed subset, i.e., a union of possibly several CN CN connected components (configuration chambers) in gen. In fact, O can be disconnected. In other words, it is possible that more than one configuration chamber has the same orientation datum. For an example (with N 14) and estimates for large N see, respectively, [89] and [12], Cor. 6.3. “ (c) For large N, most oriented matroids are not realizable, and it is a difficult problem to decide whether a given oriented matroid is realizable or not. See [12], Fig. 1-1 for an example of non-realizable oriented matroid with N 9. “ CN CN D. Walls of horizontality and refined configuration chambers. Let gen,8 gen be the open part formed by confugurations in linearly general position including infinity.Ă It is obtained by removing the additional walls

Dij Dij w1, ,wN Im wi Im wj , 1 i j N, “ 8 “ p ¨ ¨ ¨ q p q“ p q ď ă ď ˇ ( CN which we call the walls of horizontality. Theˇ connected components of gen,8 will be called refined configuration chambers. They give rise to uniform oriented matroids on 1, ,N, . t ¨ ¨ ¨ 8u E. Isomonodromic deformations and rectilinear transport. Let A t be a path in CN p q ‰ whose end points A 0 and A 1 are in linearly general position including infinity, so the k k p q p q functors ΦAp0q and ΦAp1q are defined, see (2.1.6). We would like to describe the effect of the isomonodromic deformation IAptq on these functors by constructing an explicit equivalence JAptq fitting into the commutative diagram

Φk Ap0q (2.2.5) Perv C, A 0 / MN p p qq IAptq „ „ JAptq Φk  Ap1q  Perv C, A 1 / MN p p qq and compatible with composition of paths. For this, it suffices to consider the elementary situation of “wall-crossing”. That is, we assume that:

• A 0 and A 1 lie in neighboring refined configuration chambers C0 and C1, separated p q p q by a wall of collinearity Dijk or of horizontality Dij. • A t crosses the wall once, transversally at a smooth point not lying on other walls. p q The corresponding wall-crossing formulas for the change of Φk can be easily extracted from the general Picard-Lefschetz formula of Proposition 1.1.12.

46 First of all, crossing any wall does not change the vector spaces Φi themselves, as Φi F p q is determined by the behavious of F on a small neighborhood of wi. So the effect of JAptq will be the change of the maps mij. We will denote by mij the “old” map, corresponding to 1 A 0 and by m the “new” map corresponding to A 1 . We write Ti 1 mii, the invertible p q ij p q ´ ´ transformation corresponding, under Φk, to the counterclockwise monodromy of F around wi.

Consider first the crossing of a wall of collinearity Dij. That is, at some t0 0, 1 , we P p q have Im wi t0 Im wj t0 . The shape of JA t will depend on whether wj moves above p p qq “ p p qq p q wi or vice versa and also on the relative position of Re wi and Re wj (at t t0). Following through various monodromies, we get: p q p q “

Proposition 2.2.6 (Wall crossing: horizontality). Crossing Dij leaves unchanged all mpq other than mij and mji. Further, we have

mijTi, if wj moves above wi and Re wj Re wi ; p qă p q Tjmij, if wj moves above wi and Re wj Re wi ; m1 $ p qą p q ij “ ’m T ´1, if w moves below w and Re w Re w ; ’ ij i j i j i & ´1 p qă p q T mij, if wj moves below wi and Re wj Re wi . j p qą p q 1 ’ and similarly for mij%’(interchange the roles of i and j above). Consider now the crossing of a wall of collinearity Dijk. We choose the order of indices i, j, k so that for some t t0 0, 1 the point wj t crosses the interval wi t ,wk t in its “ Pp q p q r p q p qs interior point (so not necessarily i j k). See Fig. 13. We denote by εijk 1 the sign ă ă P t˘ u of Definition 2.2.1 for A 0 , i.e., the orientation of the triangle △ wi 0 ,wj 0 ,wk 0 and by 1 p q 1 r p q p q p qs εijk the similar sign for A 1 . Clearly, εijk εijk: the orientation changes as we cross the wall. p q “´

Proposition 2.2.7 (Wall-crossing:collinearity). Crossing Dijk (so that wj crosses wi,wk ) r s does not change any mpq other than mik and mki. Further, 1 m mik εijkmjk mij, ik “ ` ˝ and similarly for mki (interchange the roles of i and k above).

wj 0 p q wk ‚ ‚

‚

wi ‚ ‚wj 1 p q Figure 13: Crossing a wall of collinearity.

47 Remark 2.2.8. Propositions 2.2.6 and 2.2.7 can be seen as giving an action, on the category CN MN , of the fundamental groupoid of ‰,8, in which we take one base point in each refined configuration chamber. It is a convex geometry counterpart of the (purely topological) action N of the fundamental group π1 C PBrN on MN described in [41], Prop. 2.4. p ‰ q“ F. Proof of Proposition 2.1.7. We now go back to the proof of the Proposition 2.1.7. k We have already seen in Remark 2.1.8 that for A in convex position, ΦA is an equivalence. Now, any configuration A CN can be connected to a configuration in convex position P gen,8 by a chain of wall-crossings. For each wall-crossing we have a diagram (2.2.5) with IAptq and k k JAptq being equivalences. Therefore, if ΦAp0q is an equivalence, then ΦAp1q is an equivalence CN as well. In this way we inductively deduce our statement for arbitrary A gen,8 from the case of convex position. P

2.3 Fourier transform of perverse sheaves: topology A. Reminder on D-modules and the Riemann-Hilbert correspondence. We recall some basic material, see [49, 76] for more detail. Let X be a smooth algebraic variety over C We denote by DX the sheaf of algebraic differential operators on X in Zariski topology. Let DX - Mod be the category of sheaves of left DX -modules M on X which are coherent (=locally finitely presented) over DX . Such an M can be regarded, in a standard way (see loc. cit.) as a system of algebraic linear partial differential equations on X. an an Let X the complex analytic manifold corresponding to X, with its sheaf OX of holo- morphic functions. Clearly, OXan is a left DX -module (not coherent). The solution complex of M DX - Mod is the complex of sheaves P (2.3.1) Sol M RHom M, Oan p q “ DX p X q an 0 Hom M Oan on X . Its th cohomology sheaf, i.e., DX , X , is the sheaf of holomorphic so- lutions, in the usual sense, of the system of PDEp correspondiq ng to M. Note that Sol is a contravariant functor. Let hol h. r . DX - Mod DX - Mod Ą be the full subcategories in DX - Mod formed by modules which are holonomic (“maximally overdetermined”) and holonomic regular (“maximally overdetermined with regular singular- ities”). We refer to [49, 76] for precise definitions. Let us just mention that the condition of being regular involves, for a non-compact variety X, also the behavior at infinity, i.e., on a compactification of X. Let Perv X be the category of perverse sheaves on Xan (constructible with respect to some algebraicp q stratification). The base field k is assumed to be C. The following is a summary of the basic results of the theory; part (b) is known as the Riemann-Hilbert correspondence (for D-modules).

48 Theorem 2.3.2. (a) If M is any holonomic DX -module (regular or not), then Sol M is a perverse sheaf. p q h. r . (b) The restriction of the functor Sol to DX - Mod is an equivalence of categories

h. r . „ op DX - Mod Perv X . ÝÑ p q Remark 2.3.3. The theorem implies that for any holonomic (possibly irregular) M, there is a canonical holonomic regular module Mreg (the “regularization” of M) such that Sol Mreg Sol M . See [59] for the general discussion and [76] (I.4.6) for an example which showsp thatq» Mregp cannot,q in general, be defined in a purely algebraic way.

B. Fourier transform for D-modules and perverse sheaves. Let V be a finite- dimensional C-vector space and V ˚ is the dual space. Considering V as an affine algebraic variety, we see that DV - Mod is equivalent to the category of finitely generated modules M over the Weyl algebra D V of global polynomial differential operators on E, via the localizattion p q 0 M M M C V OV , M H V, M . ÞÑ “ b r s “ p q Here C V is the algebra of polynomial functions on V . Explicitly, D V is generated by r s ˚ p q linear functions l V and constant vector fields v, v V , subject to v,l l v . This leads to the identificationP B P rB s “ p q

˚ ˚ ˚ ˚ f : D V D V , v v, f f, v V V , f V . p q ÝÑ p q ÞÑ ´B B ÞÑ P “p q P Given a D V -module M, its Fourier transform is the D V ˚ -module M which, as a vector space, is thep q same as M but with action of D V ˚ givenp viaq f. Note that the sheaves M p q and M of DV and DV ˚ -modules associated to M and M, can look quite| different, since it is the action of the v on M that is localized in forming M. B Let| now F Perv V be a perverse sheaf. By Theorem| 2.3.2(b), F Sol M for a P p q | “ p q unique holonomic regular sheaf M of DV -modules, which is encoded by the D V -module 0 p q M H V, M . Let us the Fourier transform M and the corresponding sheaf M of DV ˚ - “ p q modules. Note that M may be irregular, but by Theorem 2.3.2(a), its solution complex is perverse, so we have a new perverse sheaf, now on| V ˚, |

| an (2.3.4) F Sol M RHom ˚ M, O ˚ , “ p q “ DpV qp V q which we call the Fourier transform of F. q | |

C. Description of F. In this paper we will be interested in the 1-dimensional case ˚ only. That is, we take V Cw, V Cz to be the complex lines with “dual” coordinates w “ “ and z. As before, we fix qA w1, ,wN and take F Perv Cw, A . “ t ¨ ¨ ¨ u P p q A description of F in this case be deduced from results of Malgrange [76]. We present this description in an explicit form which highlights the analogy with the Fukaya-Seidel category. First, we have: q

49 Proposition 2.3.5. The only possible singularity of M in the finite part of Cz is at z 0, “ and this singularity is regular. In particular, F Perv Cz, 0 . P p q | Proof: This is a particular case of more generalq statements of [76] involving not necessarily regular holonomic DC-modules. More precisely, the determination of singularities follows from [76] (XII.2.2). In fact, both this determination and the claim that the singularity at 0 is regular follow from the “stationary phase theorem” of [76] (see [23] Th. 1.3 for a concise formulation) which describes the exponential Puiseux factors of M as the Legendre transforms of those of M. | Next, as F lies in Perv Cz, 0 , it is completely determined by its Φ, Ψ -diagram (Propo- sition 1.1.6) p q p q q a“aζ (2.3.6) Φ: Φ F ζ o / Ψ F ζ : Ψ “ p q qb“qbζ p q “

q q q q q q for some direction ζ. To describe it, we introduce some notation.

The circles of directions around wi in Cw and around 0 or in Cz are both parametrized by the standard unit circle ζ 1 but we will use the following8 orientation reversing iden- tification: | |“

1 1 1 (2.3.7) σi : S Cz S Cz S Cz , ζ ζ. 0 p q“ 8p q ÝÑ wi p q ÞÑ ´ For any ζ S1 and any r R we consider the following half-plane, half-rays of slope ζ P P ´ originating from the wi and their union:

Hr ζ w C Re ζ w r , (2.3.8) p q “ P p´ ¨ qě R Ki ζ wi ζ `ˇ, K ζ K(i ζ . p q “ ´ ¨ ˇ p q “ p q ď see Fig. 14 which corresponds to ζ 1. The choice of ζ vs. ζ in these definitions is explained by the following obvious“ fact. ´ p´ q p´ q

ζ¨w Proposition 2.3.9. (a) The function w e decays in Hr ζ . ÞÑ p q ζ¨w (b) If r wi for all i, then K ζ H r ζ . In particular, w e decays on each ě | | p q Ă ´ p q ÞÑ Ki ζ . p q We assume that elements of A are in linearly general position including ζ -infinity and p´ q numbered so that Im ζw1 Im ζwN . Then the rays Ki ζ do not intersect. The union K ζ can be viewedp´ asq㨨¨ă a “spider”p´ in whichq the “Vladivostok”p pointq v has been moved to ζ p,q see §1.1 E. p´ 8q

50 Cw

H r ζ Hr ζ ´ p q p q KN ζ p q wN ‚ . . ‚ K2 ζ w2 ‚ p q K1 ζ w1 ‚ p q s

Figure 14: The plane Cw with the cuts Ki ζ at wi. Here ζ 1. p q “´ Φ 1 Ψ 1 Let i F LS Swi be the local system of vanishing cycles of F at wi and F LS S8 be thep localq P systemp q of nearby cycles of F at . Let us denote for short p q P p q 8 1 0 Φi Φi F H F w , Ψ Ψ F i H Hr ζ , F , r 0. “ p q´ζ “ Kipζqp q i “ p q ´ζ “ p p q q "

The restriction of sections from Hr ζ to the sufficiently far part of Ki ζ identifies Ψ with q p q q p q the space Ψi F ´ζ of the standard Φ, Ψ -description of F near wi given by Ki ζ . Thus we have, for eachp iq, the diagram p q p q

a i / Φi o Ψ , ai ai,´ζ, bi bi,´ζ, bi “ “ describing F near wi in the direction ζ . In particular, Ti,Φ 1 biai is the full counter- p´ q “ ´ clockwise monodromy on Φi. We also denote Ti,Ψ 1 aibi the monodromy on Ψ obtained “ ´ by going around wi along Ki ζ (i.e., from to near wi along Ki ζ , then around wi, then p q 8 p q back to along Ki ζ ). 8 p q Proposition 2.3.10. We have identifications of the spaces in the diagram (2.3.6) as

N N Φ Ψ Ψ Φi i F ´ζ, Φ Ψ F ´ζ, » i“1 “ i“1 p q » “ p q à à so that the maps areq identified as q

N ´1 ´1 ´1 b aiTi,Φ , a a1, , aN , ai bi Ti´1,Ψ...T1,Ψ. “´ i“1 “p ¨ ¨ ¨ q “ ÿ q Proof: This is a reformulation of [21],q Prop.q 6.1.4.q Alternativelyq , the statements are partic- ular cases of more general results (involving not necessarily regular D Cw -modules) given in [76] §XIII.2, see especially formulas (XIII.2.4) and (XIII.2.5). In comparingp q with [21, 76],

51 one need to be mindful of different conventions. For instance, in [76] p. 199, the plane Cz is equipped with the orientation opposite to the one given by the complex structure, which replaces various monodromies by their inverses. This corresponds to our identifications (2.3.7).

Remarks 2.3.11. (a) A one-shot description of the map a is as follows. Take a section 0 s H Hr, F and look at its values in the area below K Ki (i.e., in the lower right P p q 1 “ C corner of Fig. 14). Extend s from that area to a section s qin K. Then a s is obtained 1 1 Ťz p q by taking the image of s in each H F w under the coboundary map. Ki p q i (b) Proposition 2.3.10 implies that the monodromy operator on Φ has theq form

T T ´1 T ´1 T ´1 Φ “ N,Ψ N´1,Ψ ¨ ¨ ¨ 1,Ψ q the full clockwise monodromy on ΨqF , cf. [76] (XIII.2.6). For example, for N 2 we have, applying the Jacobson identity (1.3.11),p q “

´1 ´1 ´1 T 1 ba 1 T b1 a2T b2T Ψ “ ´ “ ` 1,Φ ` 2,Φ 1,Φ “ 1 1 ´1 1 ´1 1 ´1 a2q b1a1 b1 a2 b2a2 b2 b1a1 “ `´1 p ´ qq ´1` p ´ q p ´´1 q “ ´1 1 a1b1 1 a1b1 1 a2 1 b2a2 b2 1 a1b1 “p ´ q ´p ´ q `p´1` p ´ ´1 q qp ´ q “ 1 a2b2 1 a1b1 , “p ´ q p ´ q and similarly for any N. This relation is clear from the first principles since moving ζ counterclockwise amounts to rotating the half-planes Hr ζ and the cuts Ki ζ clockwise. p q p q (c) Let A 0 . In this case the Dw-module M corresponding to F is monodromic, and “ t u so M is also regular, see [19, 20]. Proposition 2.3.10 in this case reduces to Proposition 4.5 of [10]. | D. F and Fourier integrals. The first identification of Proposition 2.3.10 which we write as q N N Φ „ Ψ Φ Ψ (2.3.12) gζ gi,ζ : i F ´ζ F ζ , gi,ζ : i F ´ζ F ζ , “ i“1 i“1 p q ÝÑ p q p q Ñ p q ÿ à q q is in fact given in terms of actual Fourier integrals of (distribution) solutions of M, the regular holonomic D-module corresponding to F. Here we present this interpretation, following [76] Ch. XII.

For this, we take M DC DC p to correspond to a single polynomial differential op- “ w { w ¨ erator p D Cw . This is known to be possible for any holonomic D-module in one variable, see [76] §V.1.P p (Itq is probably possible to adjust the argument to avoid this assumption, by working with matrix differential operators.) Thus F Sol M consists, outside of A, of holomorphic solutions of p f 0. “ p q p q“

52 Let Γ C be an arc, i.e., an 1-dimensional real analytic submanifold, possibly with boundary.Ă Recall [47, 80] that the sheaf of hyperfunctions on Γ is defined as

1 ‰1 BΓ H OC (and we have H OC 0). “ Γp q Γ p q“ Sometimes sections of BΓ near the endpoints of Γ are referred to as microfunctions. The sheaf BΓ is acted upon by DC via its action on OC. According to Sato, see loc. cit., sections of BΓ are interpreted as a version of distributions (generalized functions) supported on Γ. For F Sol M with M holonomic regular as above, we have identifications of sheaves “ p q 1 1 1 (2.3.13) H Sol M H RHom M, O Hom M,H O Hom M, BΓ Γ p q “ Γ Dp q » Dp Γp qq “ Dp q with the sheaf of solutions of p f 0 in BΓ. Such solutions can be viewed as distributions in any other classical sense. p q“

We take Γ Ki ζ , the half-line from (2.3.8), and let “ p q 1 1 ϕ Φi F H F w H C, F , P p q´ζ “ Kipζqp q i “ Kipζqp q the second identification coming from the fact that H1 F is (locally) constant on the Kipζqp q open ray Ki ζ wi . Let p qzt u (2.3.14) αi ϕ Hom M, BK ζ f BK ζ Ki ζ p f 0 p q P Dp ip qq “ P ip qp p qq p q“ be the global hyperfunction solution of M on Ki ζ correspondingˇ to ϕ (by (2.3.13). As. p q ˇ distribution, αi ϕ has at most polynomial growth at infinity (since M is regular). Therefore the Fourier integralp q

zw (2.3.15) Gi,ζ ϕ z αi ϕ w e dw p qp q “ p qp q żwPKipζq

converges for z ζ R 0 and extends to a holomorphic solution of M in an open sector P ¨ ą around ζ R 0. Thus we have a map Gi,ζ : Φi F Ψ F ζ. ¨ ą p q´ζ Ñ p q | Proposition 2.3.16. The map Gi,ζ coincides with gi,ζ, i.e., the latter is given by the integral (2.3.15). q Proof: This is [76] Thm. XII.1.2.

2.4 Fourier transform of perverse sheaves: Stokes structure

A. Setup. As before, let A Cw and F Perv Cw, A . Let M be the holonomic Ă P p q regular DC -module with Sol M F, let M be its Fourier transform and F Sol M . w p q “ “ p q By Proposition 2.3.5, M has only one possible singularity in the finite part of Cz, namely 0, which is regular. But the singularity at may| be irregular, so M contains more information 8 than F, encoded in its| Stokes structure near . This structure has been studied, rather 8 explicitly, in [76, 21]. The goal of this section and the next one is| to express it in a particularly explicitq and striking for rently, not noticed before) which can be seen as a de-categorification of the A8-Algebra of the Infrared.

53 B. Stokes structures: general case. We start with recalling the necessary concepts. The basic references are [6, 25, 63, 84]. We consider the case of only one irregular singular point, at . Let 8 C z´1 C z´1 pp qq Ą tt uu be the fields of formal Laurent series vs. of germs of meromorphic functions near . Both 8 fields are equipped with the action of the derivative z. A formal (resp. convergent) meromorphic connectionB near is a datum E, ∇ , where E is a finite-dimensional vector space over C z´1 (resp. C 8z´1 ) and ∇p: E q E is a C-linear map satisfying the Leibniz rule pp qq tt uu Ñ

´1 ´1 ∇ f e zf e f ∇ e , f C z or C z , e E. p ¨ q “ pB q ¨ ` ¨ p q P pp qq tt uu P DC N E, ∇ A holonomicp z -module gives a convergentp meromorphic connection near , ´1 p q 8 where E N C z and ∇ is induced by the action of z on N . “ bO tt uu B 1 1 Let S S8 be the circle of directions at , identified with the unit circle ζ 1 in “ 1 8 t| | “ u C. For an open U S let CU z arg z U be the corresponding domain in C (so it is an open sector if UĂis connected).“ t | p q P u The circle S1 carries a locally constant sheaf A called the sheaf of exponential germs. By definition, for an open arc U as above, A U consists of germs of 1-forms α defined near the p q infinity of CU and having a Puiseux expansion of the form (for some p Z depending on α) P k α akz dz. “ kP 1 Z, ką´1 p ÿ Further, A is made into a sheaf of partially ordered sets. That is, for any open U S1 we Ă have a partial order U on A U . It is defined by ď p q

α U β exp α β is of at most polynomial growth a the infinity of CU . ď ô p ´ q ż This partial order is compatible with restrictions and gluing of sections. In particular, it 1 gives an order ζ on the stalk Aζ at any ζ S . ď P Note that for two given sections α β on some U, there are finitely many points ‰ ζ1, ,ζn U at which the order ζ switches, so α and β become incomparable at ζ ζj, ¨ ¨ ¨ P ď “ while for all other ζ ζj we have either α ζ β or β ζ α. ‰ ă ă As A, is a sheaf of posets, one can speak about A-filtered and A-graded sheaves on S1 in ap straightforwardďq way, see [84] for precise definitions.

Thus, an A-filtration F on a sheaf L gives, for each ζ, a filtration FαLζ in the stalk Lζ labelled by elements α of the poset Aζ. An A-grading on L gives, for each ζ, a grading, i.e., a direct sum decomposition L L on the stalk. These local filtrations and ζ αPAζ ζ,α gradings should vary with ζ in an appropriately“ continuous way, see loc. cit. À 54 Any grading gives a tautological filtration called split:

FαLζ Lζ,α. “ βďα à Any filtration F in L gives the associated graded sheaf grF L with grading summands being the quotients of the filtration:

F (2.4.1) grα L FαL FβL. “ βăα N ÿ If F is the split filtration corresponding to a grading, then grF L recovers the grading.

Definition 2.4.2. Let L be a finite rank local system of C-vector spaces on S1.

(a) A Stokes structure on L is an A-filtration F which is locally split, i.e., locally isomorphic to the filtration coming from a grading. A local system with a Stokes structure is called a Stokes local system.

(b) The Stokes sheaf U L of a Stokes local system L, F is the sheaf of automorphisms of L preserving thep filtrationq and inducing the identityp q on grF L.

Thus U L is a constructible sheaf of (non-abelian, in general) unipotent groups on S1. In particular,p q any A-graded local system L is a Stokes local system via the split filtration, and we can speak about the sheaf U L . The following is then straightforward. p q Proposition 2.4.3. Let L be an A-graded local system on S1. The following are in natural bijection:

(i) The set of isomorphism classes of Stokes local systems L, F together with an isomor- phism grF L L of A-graded local systems. p q Ñ (ii) The first non-abelian cohomology set H1 S1, U L . p p qq Given a convergent meromorphic connection E, ∇ near , we have the local system L E∇ of flat sections on S1. That is, E∇ Up , forq an open8 U S1, consists of flat “ p q ∇Ă holomorphic sections of E near the infinity of CU . The sheaf E comes with a Stokes structure given by the filtrations

∇ ∇ FαE U s E U exp α s has at most polynomial growth at the infiity of CU , p q “ P p q ´ q¨ " ˇ ż * ˇ ` ˇ where α A U . The “irregularˇ Riemann-Hilbert correspondence” due to Deligne and Mal- grange, isP asp follows.q

55 Theorem 2.4.4. The above construction gives an equivalence of categories RH fitting into a commutative diagram whose lower row is also an equivalence:

RH Convergent meromorphic connections near o / Stokes local systems on S1 8 Formalization gr

 RH  Formal meromorphic connections near o / A-graded local systems on S1 8 y If L is a filtered local system corresponding, under RH, to a connection E, ∇ , then U L is identified with the sheaf of flat sections of End E whose asymptoticp expansionq at p isq identically 1. p q 8 Let G Perv C be a perverse sheaf of C-vector spaces (with some finite set of sin- gularities).P The behaviorp q of G near gives a local system on S1 S1 which we denote 8 8 “ Ψ8 G . pTheoremq 2.4.4 is complemented, for D-modules, as follows, see [84] Theorem 2.4.5. The following categories are equivalent:

(i) Holonomic DC-modules N which are regular everywhere except, possibly, . 8 (ii) Data G, F , where G Perv C is a perverse sheaf and F is a Stokes structure on the local systemp q Ψ G . P p q 8p q

C. Exponential Stokes structures. Let A w1, ,wN Cw. For simplicity we assume that A is strong linearly general position,“ see t Definition¨ ¨ ¨ 2.1.1(cuĂ ). One can weaken this assumption by more careful reasoning. 1 The constant sheaf A on S is embedded into A as consisting of constant forms αi widz. wiz “ Thus exp αi e is a simple exponential. While the“ sheaf A is constant, the partial order on it induced from A, changes with ş ζ the direction of ζ S1. Explcitly, ď P (2.4.6) wi ζ wj Re ζwi Re ζwj ď ô p qď p q (this means that epwi´wj qz does not grow exponentially on ζ R ). In other words, after ¨ ` the orthogonal projection to the line ζ R (note the complex conjugation!) the image of wi ¨ precedes the image of wj in the orientation given by ζ R , see Fig. 15. ¨ ` Remark 2.4.7. It is convenient, following Kontsevich, to depict a local system of finite posets on S1 by a “Lissajoux figure”, i.e., a curve L encircling 0 so that cutting L with an oriented line Rζ gives the order ζ on the finite set L Rζ (the stalk of the local system at ζ). The above implies the followingď version of this descriptiX on in our case. Note that the locus of orthogonal projections of wi to all the lines through 0 is Ci, the circle with diameter 0,wi . Thus i ζ j, if Rζ meets Ci before Cj or, equivalently, if Rζ meets Ci before Cj. r s ă This means that L in our case can be taken as the union of the circles Ci with diameters 0, wi , see Fig. 16. r s 56 ‚

Q ‚ wi w ‚ j ζ ‚ ď ‚ ‚ ζ

Figure 15: Convex geometry of exponential Stokes structures.

Clearly, the data of all the ζ on A contains the information about the convex geometry of A, in particular, about theď polygon Q Conv A . To illustrate this, let us assume for “ p q simplicity hat A is in linearly general position and recover, from the data of the ζ , the ď uniform oriented matroid structure on A, i.e., the orientation datum εijk , see Definition 2.2.1. p q

Proposition 2.4.8. Let i, j, k 1, , N and let us choose ζ0 such that wi ζ wj ζ wk. P t ¨ ¨ ¨ u ă 0 ă 0 After we rotate ζ clockwise from ζ0 to ζ0 , the order will change to the opposite, so we have three changes of order. p´ q

(a) If the sequence of orders is ijk ikj kij kji , then εijk 1, i.e., the p qÑp qÑp qÑp q “ ` triangle wi,wj,wk is positively oriented. r s

(b) If the sequence is ijk jik jki kji , then εijk 1, i.e., the triangle is negatively oriented.p qÑp qÑp qÑp q “ ´

More generally, the order ζ switches at N N 1 values ζ ? 1 ζij where ζij are the slopes defined in (2.1.4). Theseď slopes are distinctp ´ byq our assumption“ ´ on¨ A. We will call the ? 1 ζ the anti-Stokes directions. ´ ¨ ij Passing through each ? 1 ζ switches the order of just two elements, wi and wj, leaving ´ ¨ ij all the other order relations intact. If ζ is not anti-Stokes, then ζ is a total order on A. ď Starting from a non-anti-Stokes ζ0 and going counterclockwise to ζ0 , we reach the opposite order by a sequence of N N 1 2 elementary switches, i.e.,p´ weq get a reduced p ´ q{ decomposition of the maximal permutation smax SN into elementary tranpositions: P

smax si si , l N N 1 2, si i, i 1 , i 1, , N 1. “ 1 ¨ ¨ ¨ l “ p ´ q{ “p ` q “ ¨ ¨ ¨ ´

57 j ζ i ă

wi i ζ1 j ‚ ă

wj ‚

0‚

Figure 16: The Lissajoux figure for an exponential Stokes structure is a union of circles.

Remark 2.4.9. Not every reduced decomposition R of smax comes from some A. In fact, R gives rise to a uniform oriented matroid structure (Definition 2.2.3) on 1, , N . The correspondence goes through B N, 2 , the second higher Bruhat order of Manin-Schechtmant ¨ ¨ ¨ u [77]. More precisely, a datum ofp R,q in the form of a sequence of switches as above, is the N same an admissible, in the sense of [77, Def. 2.2] order on the set of 2 pairs 1 i j N. The set B N, 2 is the quotient of the set of such orders by a natural equivaleďnceă relationď ` ˘ (moves betweenp q reduced decompositions by interchanging neighboring pairs of elementary transpositions that commute). So R gives an element of B N, 2 and all elements of B N, 2 appear from some R. Further, any element of B N, 2 givesp aq uniform oriented matroidp ofq rank 3 and each such matroid, after renumberingp 1, q , N , appears from B N, 2 . This is shown in [57, 58, 97]. t ¨ ¨ ¨ u p q Now, decompositions that come from some A, give, in this process, realizable oriented matroids while most oriented matroids are not realizable, see Remark 2.2.4(b).

Definition 2.4.10. An A-Stokes structure on a local system L on S1 is a Stokes structure F F such that grα L 0 unless α wdz is a constant differential form and w A. An exponential Stokes structure“is an A-Stokes“ structure for some A C. P Ă Theorem 2.4.5 together with Proposition 2.3.5, specialize to the following.

Corollary 2.4.11. Passing from a perverse sheaf to a holonomic regular DCw -module and then to the Fourier transform, establishes an equivalence between the following categories:

(i) Perv Cw, A . p q

(ii) The category of pairs G, F , where G Perv Cz, 0 and F is an A-Stokes structure on Ψ G . p q P p q 8p q

58 D. The Stokes structure on the Fourier transform F. We now specialize to our original situation, that is: q A w1, ,wN Cw: a finite set, assumed in strong linearly general position; F “Perv t ¨C ¨ ¨, A : auĂ perverse sheaf; P p q 1 1 Φi F : the local system on S Cw S formed by vanishing cycles of F near wi; p q wi p q» M: the holonomic regular DC -module with Sol M F; w p q“ M: the Fourier transform of M; F Sol M Perv Cz, 0 (Proposition 2.3.5,); “ p q P p q F|: the A-Stokes structure on the local system Ψ8 F classifying M (Corollary 2.4.11). q | p q Our goal is to describe F explicitly. We will call Stokes directions the conjugates ζij, and Stokes intervals the N N 1 open arcs into which theq Stokes directions| cut S1. Thus the p ´ q ˝ anti-Stokes directions ? 1 ζij are the 90 rotations of the Stokes directions. We further assume for simplicity that´ no¨ Stokes direction is anti-Stokes and vice versa (this assumption can also be eliminated by more careful reasoning). Recall the union K ζ of the half-lines p q Ki ζ from (2.3.8). In terms of it, the role of Stokes vs. anti-Stokes directions is as follows: p q • At Stokes directions, the topology of K ζ (with respect to A) changes. p q

• An anti-Stokes directions, the order ζ of dominance of exponentials (but not the topology) changes. ď

We further recall the maps gζ from (2.3.12), which, by Proposition 2.3.16, are given by Fourier integrals (2.3.15). Recall also the identifications σi from (2.3.7).

Proposition 2.4.12. (a) When ζ varies within a Stokes interval, gζ is covariantly constant with respect to the local system structures on the source and target, and

FiΨ F ζ gζ Φj F 8p q “ p q´ζ ˆwj ďζ wi ˙ à q becomes the split filtration associated with the grading in the RHS.

(b) The gζ for non-Stokes ζ glue together into an identification of A-graded local systems

N ˚Φ „ F Ψ g : σi i F gr 8 F . i“1 p q ÝÑ p q à Proof: The “covariantly constant” part in (a) is clear, as theq topology of the picture with K ζ as well as A, does not change within a Stokes interval. p q The identification of Fi in (a) follows from Proposition 2.3.16 and the fact that the leading wiz term in the growth of gi,ζ ϕ z along ζ R is const e , the contribution to the integral p qp q ¨ ` ¨ from near the origin of the ray Ki ζ . p q

59 To show (b), consider ζ crossing a single Stokes direction ζij counterclockwise, starting from ζ ζ0 just before the crossing and arriving at ζ ζ1 just after. Then the Kν ζ , ν 1, “ , N, rotate clockwise, see Fig. 17. “ p q “ ¨ ¨ ¨

Ki ζ0 p q Kj ζ0 p q Ki ζ1 p q

‚wi Kj ζ1 p q

wj‚

Figure 17: Crossing a Stokes direction.

For each ν let us identify the Φν F , ζ ζ0,ζ1 with each other by monodromy and p q´ζ P r s denote the resulting single space by Φi. Now, for ν j, the integration contour Kν ζ2 is ‰ p q isotopic (relatively to A and to the allowable ways to approach ) to Kν ζ0 . Therefore for 8 p q ϕ Φν the Fourier integral gν,ζ1 ϕ z is the analytic continuation of gν,ζ0 ϕ z . We simply writeP p qp q p qp q gν,ζ ϕ z gν,ζ ϕ z , ν j 1 p qp q “ 0 p qp q ‰ (as analytic functions in a sector near ζ R ). ¨ ` But Kj ζ1 is not isotopic (rel. A and the allowable infinity) to Kj ζ0 : it differs by p q p q crossing wi. Therefore, instead of the equality, we have the Picard-Lefschetz relation

(2.4.13) gj,ζ ϕ z gj,ζ ϕ z gi,ζ mji ϕ z , ϕ Φj, 1 p qp q “ 0 p qp q´ 1 p q p q P

where mji : Φj Φi is the rectilinear transport. ` ˘ Ñ It remains to notice that for ζ ζ0,ζ1 we have wi ζ wj, so the second summand in the RHS of the above equality lies inPr the lowers level of filtrationď and so the filtration remains unchanged.

2.5 The baby Algebra of the Infrared A. Meaning of the term. By baby Algebra of the Infrared we mean a class of identities for perverse sheaves on C that express the transport or variation map along a curved path as a (possibly alternating) alternating sum of compositions of rectilinear transport maps corresponding to certain convex polygonal paths. These identities are elementary and are obtained by iterated application of the Picard-Lefschetz identities (Proposition 1.1.12). But the sums or alternating sums that appear, can be seen as de-categorifications (baby versions)

60 of filtrations or complexes2 whose summands are labelled by such convex polygonal paths. Such complexes constitute an essential part of the Algebra of the Infrared of [36]. In this section we collect several examples starting from the simplest one which establishes the pattern.

B. Circumnavigation around a polygon. Let A w1, ,wN C be in linearly “ t ¨ ¨ ¨ uĂ general position, and Q Conv A . Note that not every wk is a vertex of Q, some of the “ p q wk may lie inside. Let wi,wj be two vertices of Q and ξ be a path going around Q (say, clockwise) and joining wi and wj, see the left part of Fig. 18. Without loss of generality, we can assume that wi,wj is actually an edge of Q; if not, r s we can replace Q with one of the two parts on which wi,wj dissects it and replace A by its intersection with that part, see the right part of Fig.r 18. s

ξ ξ

‚ ‚ ‚ ‚ wi2 wi ‚ 2‚ wi3 wi ‚ ‚ 3 ‚ Q ‚ wi ‚ ‚ wj wi ‚ ‚ wj

Q ‚ ‚

Figure 18: Circumnavigation vs. inland convex paths.

Suppose F Perv C, A is given, so we have the local system of vanishing cycles Φk F S1 Pk p q Φ F p q on wk for each . Let us identify, using the monodromy, the stalks i ζ for all directions 1 p q ζ S between ζji (the direction towards wj) and the tangent direction to γ and denote P wi the resulting single space Φi. Let us define Φj similarly on the other end. Then we have the transport map mij ξ : Φi Φj. p q Ñ In this section we will consider polygonal paths

(2.5.1) γ w1 , ,wi : wi ,wi wi ,wi wi ,wi . “ r 1 ¨ ¨ ¨ n s “ r 1 2 sYr 2 3 sY¨¨¨Yr p´1 p s with vertices in A. Let Λ i, j be the set of such paths joining wi and wj (i.e., with i1 p q “ i, ip j) and satisfying the following convexity property: “

• γ wi,wj is the boundary of a convex polygon. Yr s 2There is no essential difference between the two concepts. A filtration can be seen as a complex in derived category whose terms are shifted quotients of the filtration.

61 See Fig. 18. For each γ Λ i, j we define the iterated transport map m γ : Φi Φj as the composition P p q p q Ñ

(2.5.2) m γ mi ,j mi ,i mi,i p q “ p´1 ˝¨¨¨˝ 2 3 ˝ 2 where mkl : Φk Φl is the rectilinear transport, and the identification of the stalks of the intermediate localÑ systems Φ is done using clockwise monodromies, per Convention 1.1.11.

Proposition 2.5.3. We have the identity

mij ξ m γ . p q “ p q γPΛpi,jq ÿ Proof: We view the path ξ as a cord and proceed by “tightening the cord”, straightening it first so that it contains the segment wi,wk , where wk is the next after wi vertex of Q on r s the way to wj, see Fig. 19. More precisely, we get a composite path consisting of wi,wk 1 r s and a curved path ξ going around Q and connecting wk with wj. Moving the cord further 2 to cross the vertex wk and no other vertex, we get a path ξ .

ξ ξ1 2 wk ξ ‚ ‚ ‚ ‚ Q ‚ wi ‚ ‚ wj

Figure 19: Tightening the cord.

The Picard-Lefschetz identity (Proposition 1.1.12) gives:

2 1 mij ξ mij ξ mkj ξ mik. p q “ p q` p q 1 2 Then we proceed to tighten similarly the paths ξ and ξ , keeping the straight part wi,wk r s intact (i.e., viewing wk as a fixed “eye” through which we pass the rope). Continuing like this, we represent mij ξ as the sum of the effects of polygonal paths consisting entirely of straight line segments. Thesep q polygonal paths will have precisely the convex nature claimed.

C. Counterclockwise circumnavigation: alternating sums. For future reference, we note a version of Proposition 2.5.3 which concerns the path ξ run counterclockwise, i.e., ´1 in our general notation, the path ξ running from wj to wi.

62 Let Λ j, i be the set of the same polygonal convex paths as in Λ i, j but oriented from p q p q wj to wi. As before, each γ Λ j, i gives the iterated transport map m γ : Φj Φi. Here, too, we identify the stalks ofP thep intermediateq local systems Φ using clockwisep q monodromies,Ñ per Convention 1.1.11.

Proposition 2.5.4. We have an equality in Hom Φj, Φi p q ´1 lpγq mji ξ 1 m γ . p q “ p´ q p q γPΛpj,iq ÿ

Here l γ γ A 2 is the number of points from A other than wi and wj lying on γ. p q “ | Y |´ Proof: The argument is similar to that of Proposition 2.5.3 using Picard-Lefschetz identities which in this case will lead to signs in the RHS due to difference in orientation. For example, the very first identity above will take the form

´1 2 ´1 1 ´1 mji ξ mji ξ mkimjk ξ . p q “ pp q q´ pp q q Therefore, in the process of tightening the cord, the sign of a new summand will change each time we acquire a new element from A on the path. Collecting the signs, we get the statement.

D. The Stokes matrices of an exponential Stokes structure. Next, we want to give a "baby infrared” formula for the Stokes matrices of the Fourier transform of a perverse sheaf. As a first step, we explain, following [26], the correspondence between “abstract” exponential Stokes structures and “concrete” Stokes matrices.

Let A w1, ,wN . Assume that A is in strong linearly general position. Note that an A-Stokes“ t structure¨ ¨ ¨ onu a local system L can be equivalently seen as a filtration indexed just by the sheaf of posets A, . That is, we have globally defined subsheaves FiL L p ďq Ă (corresponding to wi A) such that, for a given direction ζ, the inequality wi ζ wj implies P ď that FiL FjL at tle level of stalks at ζ. Note that FiL is not locally constant, in general. Ă Further, an A-grading on a local system L is the same as an A-grading, i.e., a direct sum N decomposition L Li. “ i“1 Proposition 2.5.5.ÀLet L be an A-graded local system and U L be the corresponding Stokes sheaf. Let I S1 is an open (connected) arc. p q Ă

(a) If I contains at least N N 1 2 consecutive anti-Stokes directions ? 1ζij, then we have H0 I, U L 1 .p ´ q{ ´ p p qq“t u (b) If I contains at most N N 1 2 consecutive. anti-Stokes directions, then we have H1 I, U L 1 . p ´ q{ p p qq“t u

63 Proof: We can trivialize each Li on I, that is, assume Li V i where Vi is some finite- dimensional vector space. Then U L is identified with the“ subsheaf in the constant sheaf p q with stalk i,j Hom Vi,Vj , consisting of block-matrices with 1’s on diagonals which are upper-triangular withp respectq to switching orders, see Fig. 20. The orders switch at the anti-Stokesś directions.

1 0 1 0 0 1 0 0 0 1 ˚ 1 1 0 »0 0 1˚fi »0˚ 0 1˚fi »˚ 0 1fi ˚ – fl – fl – fl 1 1 0 1 0 0 1 0 0 0 1˚ ˚ 1 ˚ 1 1 0 »0 0 1˚fi »0˚ 0 1˚fi »˚ 0 1˚fi »˚ 1fi ˚ ˚ ˚ – fl – fl – fl – fl 123 ‚ 213 ‚ 312 ‚ 321 p q p q p q p q Figure 20: A part of the Stokes sheaf in the matrix form for N 3. “

In the situation (a), since the orders corresponding to ζ and ζ are opposite, the only p´ q possible matrix which will be upper-triangular with respect to all the orders ζ , ζ I and with 1’s on diagonal, will be the identity, so H0 1 . ď P “ t u To prove (b), we proceed by induction. Let us write I as an open interval θ1, θ2 in the p q counterclockwise direction. Let ϕ I be the last anti-Stokes direction on the way from θ1 P to θ2. 1 1 2 Let I θ1,ϕ and I2 ψ, θ2 , where ψ is the direction just before ϕ. Then I I I form a covering“p ofq I and by“p the inductiveq assumption U L has no higher cohomologyY (i.e.,“ no H1) on I1,I2 and I1 I2. Therefore the first cohomologyp q in question is identified with the set of double cosets X

(2.5.6) H1 I, U L U L I1 U L I1 I2 U L I2 . p p qq “ p qp q p qp X q p qp q J N Since I1 I2 is a single arc which does not contain anti-Stokes directions, U L I1 I2 consists X p qp X q of block-unitriangular matrices with respect to some total order ζ. The two subgroups on the left and right in (2.5.6) consist of matrices unitriangular with respectď to two partial orders 1 2 , of which ζ is a minimal common refinement. So it is immediate to see that the any ď ď ď 1 2 matrix g uni-triangular with respect to ζ is the product g g of matrices unitriangular with respect to 1 and 2 respectively. ď ď ď Note that the cases (a) and (b) of Proposition 2.5.5 have nontrivial intersection: when I contains exactly N N 1 2 anti-Stokes directions (so the width of I is slightly greater than 180˝). Such arcsp will´ beq{ called balanced. Thus, on a balanced arc, U L has neither H0 nor H1. p q

64 Note further that we can form a covering of S1 by two balanced arcs (a balanced covering, for short) S1 I` I´ whose intersection must consist of two intervals: I` I´ J ` J ´, “ Y X “ \ containing no anti-Stokes directions and giving rise to opposite orders ζ , see Fig. 21. ď J `

‚ ‚ ζ0 I` I´ ˚ ‚ ‚

J ´ Figure 21: A balanced covering.

Corollary 2.5.7. For a balanced covering S1 I I we have a canonical identification “ ` Y ´ H1 S1, U L U L J ` U L J ´ . p p qq » p qp qˆ p qp q Proof: Indeed, because on the I˘ the sheaf U L does not have H1, we can write H1 S1, U L as the double coset space as in (2.5.6). We havep q p p qq

U L I` I´ U L J ` U L J ´ , p qp X q“ p qp qˆ p qp q while the two subgroups by which it is factorized, reduce to 1 . t u Definition 2.5.8. Let L, F be an A-Stokes structure and L grF L . Let S1 I` I´ be a balanced covering.p Theq Stokes matrices associated to F“(andp theq covering)“ areY the sections C U L J so that C ,C corresponds to F via Corollary 2.5.6. ˘ P p qp ˘q p ` ´q More explicitly, Proposition 2.5.5 means that there exist unique splittings, i.e., isomor- phisms g : L I˘ L I˘ ˘ | ÝÑ | of A-filtered sheaves. The Stokes matrices are defined as

` ´1 ` ` C g` J` g´ J` : L J L J , (2.5.9) ´ “ ´1| ˝ | p ´q ÝÑ p ´q C g J´ g J´ : L J L J . “ ` ` | ˘ ˝ ` ´| ˘ p q ÝÑ p q ˘ ` ˘ ` ˘ Note that each ζ J gives the same order ζ ˘ on A. Thes ˘ are total orders, opposite to each other. ByP construction, C˘ being sectionsď “ď of U L , are unitriangularď with respect to p q 65 ˘ ˘ ˘ ˘ these orders, i.e., if we write C Cij with Cij : Li Lj , then Cij 0 only for wi ˘ wj and C˘ 1. “} } Ñ ‰ ď ii “ Note further that C` and C´ act in different spaces. To bring them together, let

T L : L J ` L J ´ , T L : L J L J ` , `{´ p q ÝÑ p q ´{` p ´q ÝÑ p q T L T L T L : L J ` L J ` “ ´{` ˝ `{´ p q ÝÑ p q be the counterclockwise monodromies for L. As L is graded, they are represented by diagonal matrices. Put C´ T L ´1 C´ T L : L J ` L J ` . “ p `{´q ˝ ˝ `{´ p qÑ p q Let also T L : L J ` L J ` be the full counterclockwise monodromy of L. Using the p q Ñ`r p q ` L identification g` : L J L J , we can view T as an automorphism (normalized mon- odromy) p qÑ p q T L g´1T Lg : L J ` L J ` . norm “ ` ` p q ÝÑ p q L ` L ´ ´1 L Proposition 2.5.10. We have an equality Tnorm C T C which realizes Tnorm as the product of an upper triangular, diagonal and lower“ triangularp q matrices. r Proof: This is straightforward by analyzing the commutative diagram

res´1 res res´1 res L J ` / L I` / L J ´ / L I´ / L J ` ■ ❏ p O q p O qd ■■ g p O q p O q p O qd ❏❏ g ■■ ` ❏❏ ` g` g` ■■ g´ g´ g´ ❏❏ ■■ ❏❏ L J ` / L I` / L J ´ / L J ´ / L I´ / L J ` / L J ` , p q res´1 p q res p qpC´q´1 p q res´1 p q res p q C` p q where res means the restriction isomorphisms for the local systems L and L. The composition L ` L ´ ´1 L of the top row is T . The composition of the bottom row is C T´{` C T`{´, and our statement follows from this. p q

E. ζ-convex paths. Next, we introduce some terminology. Let ζ C be such that ζ 1. For a subset B A we call the ζ-convex hull and denote Convζ BP the set | |“ Ă p q

ζ (2.5.11) Conv B Conv w ζ Rě0 . p q “ wPBp ` ¨ q ˆ ď ˙ Thus Convζ B is an unbounded polygon in C with (one or) two infinite edges in the ζ- direction. p q A polygonal path γ as in (2.5.1) will be called ζ-convex, if γ is the finite part of the boundary of the polygon Convζ γ . For ζ 1 we will use the term left convex instead of “1-convex” and for ζ 1 we willp q use the term“ right convex instead of “ 1 -convex”. We also write Conv˘ for Conv“ ´ ˘1. See Fig. 22. p´ q

66 wip ‚ . . ‚ Conv` B p q wi2 ‚

‚ wi1

Figure 22: A left convex path.

F. The Stokes matrix of F. We now specialize to the situation and notations of §2.4 D. Let N Ψq F ˚Φ L 8 F , L gr L σi i F . “ p q “ p q“ i“1 p q à Given a balanced covering S1 qI` I´ as above, take a generic (non-Stokes, nor anti- “ `Y Stokes) point ζ0 in the middlle of I , so that ζ ? 1 ζ0 J , see Fig. 21, where ` “ ´ ´ ¨ P ` ζ0 1. Let ζ and let us number A w1, ,wN so that w1 wN . For “´ ď`“ď ` “ t ¨ ¨ ¨ u ą` ¨¨¨ą` the choice ζ0 1 this means ordering wi so that Im w1 Im wN , as in Fig. 14. The Stokes matrix“ ´ C` describing the Stokes structure pF onq㨨¨ăL, has componentsp q

` Φ Φ ` Cij : Li,ζ` i F j F Lj,ζ` , i j, Cii 1. “ p q´ζ` ÝÑ p q´ζ` “ ě “ For each i let us identify, by the quarter-circle monodromy,

„ Φi F Φi : Φi F . p q´ζ` ÝÑ “ p q´ζ0

For the choice ζ0 1 this corresponds to considering the stalks of the Φi in the real positive “´ direction, as in §2.1 C. So we have the rectilinear transport maps mij : Φi Φj, see (2.1.5). Ñ For i j let Λ i, j be the set of ζ -convex (left convex, for ζ0 1) paths starting ă p q p´ 0q “´ from wi and ending at wj. For any γ Λ i, j we have the iterated transport map m γ : P p q p q Φi Φj, as in (2.5.2). Ñ Proposition 2.5.12. For i j we have ă C` m γ . ij “ p q γPΛpi,jq ÿ ´ In a similar way, Cij is interpreted as a sum over ζ0 -convex (right convex, for ζ0 1) paths. p` q “´

67 Remark 2.5.13. Another formula for the Stokes matrices of F was given in [21] (Th. 5.2.2 and Rem. 5.3.3) using a different approach (enhanced perverse sheaves). This formula is essentially equivalent to Proposition 2.5.12. The “infrared” shapeq of our formula is due to the rectilinear approach adopted.

Proof of Proposition 2.5.12: Recall from (2.3.12) and Proposition 2.3.16 the isomorphism gζ : Lζ Lζ betweent the stalks of L and L over ζ. Here ζ is any non-Stokes direction. Let Ñ ˘ g : L L be the isomorphism of local systems over I which coincides with g ζ at the ˘ Ñ ˘ 0 point ζ0. ˘ Φ To analyze g˘, we extend the construction of solutions Gi,ζ ϕ z , ϕ i F ´ζ, see (2.3.15), to a more flexible setup which allows us to deform thep pathqp q of integration.P p q Let z0 Cz be given and γ be any simple path, starting at wi in direction ζ, avoiding other P ´ z0w wj and eventually going to infinity along a straight line on which the function w e ÞÑ decays. Then, as in (2.3.14), we construct, out of αi,ζ ϕ Ψi F , a distribution solution p q P p q´ζ αi,γ ϕ w of p f 0 defined on the path γ. We then put p qp q p q“

zw Gi,γ ϕ z αi,γ ϕ w e dw p qp q “ p qp q żγ

It is a solution of M, i.e., a section of F, defined for z with arg z close to arg z0 . Thus p q p q the Gi,ζ ϕ z of (2.3.14) corresponds to the case when γ Ki ζ is the straight path in the p qp q “ p q direction ζ and z0| ζ. q ´ “ Now, to analytically continue Gi,γ ϕ z to z with other values of arg z , all we need to do is to simply bend the eventual directionp qp q of the path γ so that w ezw pcontinuesq to decay on it. With these notation, we prove: ÞÑ

˘ Lemma 2.5.14. g˘ is compatible with the filtration everywhere on I (split filtration in the source, Stokes filtration in the target).

Proof of the Lemma: This an extension of the argument in the proof of Proposition 2.4.12(b). More precisely, look at the specialization of g at some ζ I`. By the above, it is given by ` P the collection of the Gi,γi , where γi is obtained by bending the eventual direction of the ray Ki ζ0 so that it approaches the infinity in the direction ζ , not ζ . p q p´ q p´ 0q To see that g` is compatible, over ζ, with the filtrations, we compare it with Gζ which is obtained by integration over straight rays Ki ζ which go in the direction ζ right away, p q p´ q without bend. As we move the direction from ζ0 to ζ, we may cross some Stokes directions and so the Gi,γi will be expressed through the Gi,ζ via a series of Picard-Lefschetz formulas, like (2.4.13). Now, by keeping track of the dominance of exponentials, we see the following:

˝ (!) As long as ζ does not deviate from ζ0 by more than 90 in either direction, the correc- tion term in each such Picard-Lefschetz formula will belong to the lower layer of the filtration!

68 This means that compatibilty with filtrations will hold all the way until the first Stokes ˝ ` ´ directions differing from ζ0 by more than 90 , i.e., everywhere on I . Similarly for I . ˘ ´ ´ ´ ` ` ` K1 K2 KN KN K2 K1

w ‚ . N .

w‚ 2

w‚ 1

Figure 23: The Stokes matrix: combing at 90˝. ˘ Now, by definition, C` g´1g : L L, computed anywhere on J `, for example at “ ` ´ Ñ ζ` ? 1 ζ0. That is “´ ´ ¨ ` g´ i g` j Cij . p q “ jěip q ÿ ` In other words, C expresses the solutions G ´ ϕ z , ϕ Φi, through the solutions i,Ki ` p qp q P G ` ϕ z . Here K is the path going from wi in the direction ζ “all the way out i,Ki i 0 p qp q ˝ ´ p´ q of Q Conv A ” and then turning 90 left. Similarly, Ki goes from wi in the opposite “ p q ˝ direction ζ0 all the way out and then turns 90 right, see Fig. 23.

KN

K1 w ‚ . Kr N . N r K2 ‚ K r w2 2

w‚ 1 K1

Figure 24: The Stokes matrix: straight paths vs. combing at 180˝.

` ˘ ˝ Now, to find Cij , it is enough to rotate the eventual directions of the Ki by 90 clockwise ` ´ so that K becomes unwound, i.e., becomes the straight path Ki Ki ζ0 . The paths K i “ p q i 69 ˝ will then become twisted by 180 and we denote the resulting paths Ki, see Fig. 24 which corresponds to the case ζ0 1. So we need to express the solutions g in terms of the gK . “´ Ki i Iterated application of the Picard-Lefschetz formulas gives, in the wayr completely similar to the proof of Proposition 2.5.3 (“tightening the a sum over left convexr paths as claimed. Proposition 2.5.12 is proved.

3 Review of schobers on surfaces with emphasis on C

We now turn to the categorical analogs of the above constructions. We use the conventions of Appendix A.

3.1 Schobers on surfaces A. Schobers on D, 0 : spherical functors. Let Φ, Ψ be (enhanced) triangulated categories and a : Φp Ψq be an exact (dg-) functor. Assume that a has left and right adjoints ˚a, a˚ :Ψ ΦÑwith the corresponding unit and counit maps Ñ ˚ ˚ e : IdΦ a a, η : a a IdΨ, 1 ÝÑ ˝˚ 1 ˝˚ ÝÑ e : IdΨ a a , η : a a IdΦ ÝÑ ˝p q p q˝ ÝÑ Using the dg-ehnancement, we define the functors

paq paq (3.1.1) TΦ T Cone e 1 , TΨ T Cone η . “ Φ “ p qr´ s “ Ψ “ p q They are called the spherical cotwist and spherical twist associated to a. The notation paq paq TΦ , TΨ will be helpful in situations when there is more than one spherical functor connect- ing Φ and Ψ. By construction, we have exact triangles of functors

λ e ˚ f g ˚ η µ TΦ IdΦ a a TΦ 1 , TΨ 1 a a IdΨ TΨ. Ñ Ñ ˝ Ñ r s r´ s Ñ ˝ Ñ Ñ and dual triangles

˚ 1 ˚ ˚ ˚ f ˚ η ˚λ ˚ ˚ µ e1 ˚ g ˚ TΦ 1 a a IdΦ TΦ, TΨ IdΨ a a TΨ 1 . r´ s Ñ ˝ Ñ Ñ Ñ Ñ ˝ Ñ r s Proposition 3.1.2 (Spherical functor package [4] Th. 5.1). In the following list, any two properties imply the two others:

(i) TΨ is an equivalence (quasi-equivalence of dg-categories).

(ii) TΦ is an equivalence. (iii) The composite map ˚ 1 ˚ ˚ p aq˝g ˚ ˚ η ˝pa q ˚ a TΨ 1 a a a a ˝ r´ s ÝÑ ˝ ˝ ÝÑ is an isomorphism (quasi-isomorphism of dg-functors).

70 (iv) The composite map ˚ 1 ˚ ˚ pa q˝e ˚ ˚ f˝p aq ˚ a a a a TΦ a 1 ÝÑ ˝ ˝ ÝÑ ˝ r s is an isomorphism.

The functor a is called spherical, if the conditions of Proposition 3.1.2 are satisfied.

Proposition 3.1.3 ([70] Prop. 2.13). A functor a is spherical if and only if the following two conditions are satisfied:

(1) The map a˚˝e1`e˝p˚aq a˚ ˚a a˚ a ˚a ‘p q ÝÑ p q˝ ˝p q is an isomoprhism.

(2) The map pp˚aq˝η,η1 ˝p˚aqq ˚a a a˚ ˚a a˚ ˝ ˝ ÝÑ ‘ is an isomorphism .

These conditions can be seen as categorical analogs of the conditions of Definition 1.3.18 and Proposition 1.3.19 which describe perverse sheaves F Perv D, 0 equipped with a non-degenerate bilinear form. This motivates the following [54].P p q

Definition 3.1.4. A perverse schober S on D, 0 is a datum of a spherical functor a : Φ Ψ between enhanced triangulated categories. p q Ñ

With a view towards future more intrinsic theory, we will say that a perverse schober S is represented by a spherical functor a : Φ Ψ and write Φ Φ S , Ψ Ψ S . Further, Ñ “ p q “ p q TΦ and TΨ being equivalences, we can use them to construct local systems of (emhanced) triangulated categories on S1. More precisely, we define Φ S to be the local system with p q stalk Φ and monodromy TΦ 2 and Ψ S to be the local system with stalk Ψ and monodromy r s p q TΨ. The introduction of the shift by 2 is explained as follows.

Proposition 3.1.5. (a) We have canonical identifications of functors

´1 ˚˚ ´1 a TΦ 1 a TΨ a 1 , ˝ r´ s »˚˚ » ˝ r s a TΦ 1 a TΨ a 1 , ˝ r s » » ˝ r´ s in particular, a˚˚ and ˚˚a exist. (b) The functor a gives a morphism of local systems

a a S : Φ S Ψ S . “ p q p q ÝÑ p q

71 Proof: (a) Let us write the identifications of Proposition 3.1.2 (iii)-(iv)

˚ ˚ ˚ a TΨ 1 a TΦ a 1 . ˝ r´ s » » ˝ r s Note that for an equivalence of categories, the inverse equivalence is both left and right adjoint. Therefore, taking the right adjoints of the above identifications, we get the first line of identifications in (a). The second line is proved similarly.

(b) The second line of identifications in (a) implies a TΦ 2 TΨ a which means that a is a morphism of local systems. ˝ r s» ˝ We will refer to a morphism of local systems a : Φ Ψ (or, put differently, a “local system of spherical functors”) obtained from a spherical fuÑnctor a : Φ Ψ, a spherical local system. This gives an equivalent, more geometric point of view on a sÑpherical functor. Corollary 3.1.6. For a spherical functor a : Φ Ψ, all iterated adjoints Ñ ,˚˚ a, ˚a, a, a˚, a˚˚, ¨ ¨ ¨ ¨ ¨ ¨ exist and are spherical functors. Remark 3.1.7. The twist and cotwist associated by (3.1.1) to an iterated adjoint of a will differ, in general, from the one for a. For example, for the spherical functor ˚a :Ψ Φ we have Ñ p˚aq e1 ˚ paq ´1 TΨ Cone IdΨ a a 1 TΨ , “ t ÝÑ ˝1 ur´ s“p q p˚aq ˚ η paq ´1 T Cone a a IdΦ T , Φ “ t ˝ ÝÑ u“p Φ q see [4] §1.

The following is a categorical analog of the concept of a perverse sheaf with a symmetric or antisymmetric bilinear form. Definition 3.1.8. Let n Z. A perverse schober S on D, 0 represented by a spherical functor a : Φ Ψ is calledP a Calabi-Yau schober of dimensionp q n (or, shortly, an n-CY- schober), if theÑ following conditions hold: (1) Ψ is a Calabi-Yau category of dimension n.

(2) The functor TΦ n 1 is the Serre functor for Φ. r ` s (3) The data in (1) and (2) are compatible in the following sense. For any objects ϕ Φ, ψ Ψ, the diagram of identifications P P

adj ˚ t ˚ ˚ p a,aq ˚ HomΦ a ψ ,ϕ o / HomΨ ψ, a ϕ p O p q q p O p qq Serre data CY data   ˚ HomΦ ϕ, TΦ a ψ n 1 HomΨ a ϕ , ψ n p p qqrj❱❱` s ❥5 p p q r sq ❱❱❱❱ ❥❥❥❥ ❱❱❱❱ ❥❥❥ ` ˚ ˚ ˘ ❱❱❱ ❥❥❥ ˚ a »TΦ˝ ar1s ❱❱❱ ❥❥ adjpa,a q ❱❱* u❥❥❥ ˚ HomΦ ϕ, a ψ n p p qr sq 72 is commutative.

Example 3.1.9 (Spherical objects). The simplest class of examples of spherical functors b b has Φ D Vectk. A (dg-)-functor a : D Vectk Ψ is uniquely determined by the datum “ Ñ ˚ b of a single object E a k Ψ, by a V V k E. The adjoint a :Ψ D Vectk is given ‚ “ p q P p q“ b Ñ by F HomΨ E, F . SupposeÞÑ Ψ phas aq Serre functor S. Recall, cf. [50, Ch.8] [87], that E is called a spherical object of dimension n, if:

(1) S E E n . p q» r s k, if j 0, n; (2) HjHom‚ E, E “ p q “ #0, otherwise.

b b In this case TΦ : D Vectk D Vectk is easily identified, using (2), with the shift by n 1 , so it is an equivalence. It wasÑ shown in [87], see also [50, Prop. 8.6] that condition (1)´p implies` q that TΨ is also an equivalence, i.e., a is a spherical functor. Note that if Ψ is a Calabi-Yau category of dimension n, then (1) is automatic, so if (2) holds, then a gives a Calabi-Yau schober.

B. Schobers on surfaces. Transport functors. Let X, A w1, ,wN be a 1 p “ t ¨ ¨ ¨ uq stratified surface and Swi be the circle of directions at wi. Let Di be a small disk around wi ˝ ˝ 1 and Di Di wi be the punctured disk. Thus we have a homotopy equvalence Di Swi which is“ “homotopyzt u canonical”, i.e., defined uniquely up to a contractible space of choices.Ñ In this paper we adopt the following naive definition [54, 28].

Definition 3.1.10. A perverse schober on X, A si a datum consisting of: p q

• A local system S0 S X A of pre-triangulated categories on X A. “ | z z a : Φ Ψ 1 • For each i, a spherical local system i i S i S on Swi (or, equivalently, on ˝ p q Ñ p q Di ).

• An identification S0 D˝ Ψi S for each i. | i » p q In fact, we will really need only the case when X D (a disk) or C (complex plane) in which an even easier definition is possible (see below).“ A more systematic definition, using the language of -categories, will be given in [32]. Such a definition leads naturally to a construction of an 8-category Schob X, A of perverse schobers for X, A . In this paper we will not use this8 -category butp willq write S Schob X, A justp toq signify that S is a schober for X, A . 8 P p q p q As in §1.1 C., let α be a piecewise smooth oriented path in X, joining wi with wj and avoiding other wk. For a schober S Schob X, A we have the category Φi,α Φi S dir α P p q “ p q ip q (the stalk of the local system Φi S in the direction of α) and the similar category Φj,α. p q

73 Denoting Ψα the stalk of the local system S0 at any intermediate point of α, we have two spherical functors ai,α aj,α Φi,α Ψα Φj,α ÝÑ ÐÝ and we define the transport functor

˚ (3.1.11) Mij α a ai,α : Φi,α Φj,α, p q “ j,α ˝ ÝÑ analogous to the transport map (1.1.9) for perverse sheaves. ´1 ´1 Let α be the path obtained by reversing the direction of α, so α goes from wj to wi. We have the transport functor

´1 ˚ (3.1.12) Mji α a aj,α : Φj,α Φi,α. p q “ i,α ˝ ÝÑ Proposition 3.1.13. We have identifications of functors

˚ ´1 ´1 ˚ Mij α TΦ Mji α 1 Mji α TΦ 1 . p q » j ˝ p qr s » p q ˝ i r s Proof: Taking the left adjoint of (3.1.12), we get

˚ ´1 ˚ ´1 ˚ ´1 Mji α aj,α ai,α T a ai,α 1 T Mij α 1 , p q » ˝ » Φj ˝ j,α ˝ r´ s “ Φj p qr´ s ˚ ˚ where we used the relation between aj,α and aj,α from Proposition 3.1.2(iv). This gives the first claim of the proposition. The second claim is proved similarly by considering the right ˚˚ ´1 adjoint, using the identification a ai T 1 , see Proposition 3.1.5. i,α » ˝ Φi,α r s Definition 3.1.14. A schober S Schob X, A is called an n-Calabi-Yau schober, if : P p q (1) S0 consists of n-Calabi-Yau categories and the monodromies preserve the Calabi-Yau structure.

(2) For each i, the spherical local system ai : Φi S Ψi S represents an n-Calabi-Yau p qÑ p q schober on the disk Di around wi.

C. Schobers on a disk via spiders. We now specialize to the case X D being a “ closed disk. As in §1.1 E., we choose v D and a v-spider K γ1, ,γN for , A . P B “ p ¨ ¨ ¨ q p q Choose the numbering A w1, ,wN according to the slopes of the γi at v, see Fig. 8. Given a schober S Schob“D, t A¨, ¨ ¨we associateu to it a diagram P p q (3.1.15) Φ Φ 1 ❇ N ❇❇ ¨ ¨ ¨ ④④ ❇❇ ④④ a1 ❇❇ ④④ aN ❇! ④④ Ψ }

of N spherical functors with common target as follows:

Φi Φi S dir pγ q is the stalk of Φi S in the direction of γi. “ p q wi i p q 74 Ψ S0 v is the stalk of S0 at v. “p q ai is the stalk at dirw γi of the spherical local system ai : Φi S Ψi S followed by i p q p qÑ p q the monodromy of S0 from wi to v along γi. It is cleat that the data (3.1.15) determine S completely, in analogy with Proposition 1.1.17. Indeed, a choice of a spider K identifies π1 D A, v with the free group is generators p z q γi, going from v to wi along γi, then making a counterclockwise loop around γi and then returning back along γi. Given a diagram (3.1.15), we define a local system S0 on D A to z correspond to the action of π1 D A, v on Ψ such that γi acts by p p z q ˚ η Ti,Ψ Cone ai a IdΨ . “ t ˝ i ÝÑ p u

The spherical local system ai : Φi S Ψi S comes from the spherical functor ai. p qÑ p q Definition 3.1.16. We say that S Schob D, A is represented by the diagram (3.1.15) in the presense of the spider K, if it correspondsP p to theq diagram by the procedure just described.

Given S, the diagram (3.1.15) representing it, depends on the choice of a spider K. The change of the diagram corresponding to the change of spider under an action of the braid group BrN , is described by the formulas identical to those of Proposition 1.1.19. More precisely:

Proposition 3.1.17. Changing K to τi K changes the diagram (3.1.15) to the diagram aν p q Φν Ψ ν 1, ,N , where p Ñ q “ ¨¨¨

Φ1, , ΦN Φ1, , Φi 1, Φi 1, Φi, Φi 2, , ΦN , p ¨ ¨ ¨ q“p ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ q a1, , aN a1, , ai 1, Ti,Ψ ai 1, ai, ai 2, , aN . p ¨ ¨ ¨ q“p ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ q (Note that the composition of a spherical functor and an equivalence is again spherical).

3.2 Picard-Lefschetz triangles and Picard-Lefschetz arrows A. Picard-Lefschetz triangles. Let X, A be a stratified surface, and S Schob X, A be a schober with singularities in A. Thep Picard-Lefschetzq identity of PropositionP 1.1.12p isq now categorified into a Picard-Lefschetz triangle ([54], Prop. 2.13). More precisely, consider the situation of Fig. 4, where a path γ, joining wi and wk, is moved across wj. Let us define the categories Φi, Φj and Φk using the use the same identification of the appropriate stalks of Φi S , Φj S and Φk S as described in (1.1.10) p q p q p q

Proposition 3.2.1. In the situation described we have an exact triangle of functors Φi Φk Ñ u 1 α,β,γ1 vα,β,γ 1 Mik γ 1 Mjk α Mij β Mik γ Mik γ . p qr´ s ÝÑ p q˝ p q ÝÑ p q ÝÑ p q Proof: we give a detailed argument by revisiting the proof of Proposition 1.1.12, now in the categorical context. As in that proof, we deform the situation of Fig. 4 to that of Fig. 5,

75 and prove the statement for that deformed situation. We use the same notations for the 1 intermediate points wi, wj, wk and paths α, β, γ, γ .

We view wi and wk as the midpoints to define the transport functors. Let Ψi, Ψk be the r stalks of S0 at wi, wkr respectively.r r We thenr haver r the spherical functors r r ai : Φi Ψj, ak : Φk Ψk r r ÝÑ ÝÑ corresponding to the intervals wi, wi and wk, wk , the monodromy functors r s r s

Tγ , Tγ1 :Ψi Ψk r rÝÑ 1 r r for S0 corresponding to γ, γ and the spherical functors

a : Φj,β Ψi, aj,α : Φj,α Ψk r rj,β ÝÑ ÝÑ r r corresponding to the segments near wj. So by definition we have (similar to (1.1.13))

˚ 1 ˚ Mik γ ak Tγ ai, Mik γ ak Tγ1 ai, (3.2.2) p q “ ˝ ˚ ˝ p q “ ˚ ˝ ˝ Mij β a ai, Mjk α a aj,α. p q “ j,rβ ˝ p q “ k ˝ r We proceed analogously. Consider ther area between γ and γ1 asr a disk with one marked ´1 point wj and the radial cut along β. The transformation T Tγ Tγ1 :Ψi Ψi is the full Ψ “ ˝ Ñ counterclockwise monodromy of S0, i.e., of i S , aroundr wj.r Thus T TΨj is the spherical p q r r“ twist for the spherical functor a r: Φj,β Ψi and we have the defining exact triangle j,β Ñ r ˚ η T 1 a a IdΨ T r´ s ÝÑ j,β ˝ j,β ÝÑ i ÝÑ r r of functors Ψi Ψi. Now, taking the composition Tγ is an exact functor Fun Ψi, Ψi Ñ ˝p´q p qÑ Fun Ψi, Ψk so it transforms the above triangle to p q r ˚ Tγ1 1 aj,α a Tγ Tγ1 . r´ s ÝÑ ˝ j,β ÝÑ ÝÑ r r r r r Note that here the equality aj,α Tγ a is used, which is due to the clockwise identification » ˝ j,β of Φj,β, Φj,α. Our desired triangle is obtainedr r fromr this by the exact functor

˚ a ai : Fun Ψi, Ψk Fun Φi, Φk , k ˝p´q˝ p q ÝÑ p q in virtue of (3.2.2).

B. The Picard-Lefschetz arrows: duality. We now focus on the arrows (natural transformations)

1 (3.2.3) uαβγ1 : Mik γ 1 Mjk α Mij β , vαβγ : Mjk α Mij β Mik γ p qr´ s ÝÑ p q˝ p q p q˝ p q ÝÑ p q

76 in the triangle of Proposition 3.2.1, which we call the Picard-Lefschetz arrows. Note that the 1 1 data of α,β,γ define γ and α,β,γ define γ (uniquely up to isotopy), so the notations uαβγ1 and vαβγ are unambiguous. We want to express the v-arrow in terms of the u-arrows corresponding to the paths running in the opposite direction. Recall (A.7) that a natural transformation k : F G between two functors gives rise to the left and right transposes tk : ˚G ˚F and kt : GÑ˚ F ˚ between the corresponding adjoints (when the adjoints exist). Ñ Ñ Proposition 3.2.4. With respect to the identifications of the adjoints of the transport func- tors given by Proposition 3.1.13, we have natural identifications

t t vαβγ u TΦ TΦ uβ,α,γ. » β,α,γ ˝ i » i ˝ Let us explain the meaning of these identifications more precisely. We have

´1 uβαγ : Mki γ 1 Mji β T 2 Mkj α . p qr´ s ÝÑ p q˝ Φj r´ s˝ p q Here, the appearance of T ´1 2 in the middle comes from the fact that we identify the Φj r´ s stalks of Φj S at the directions of β and α using the monodromy along the arc opposite p q to that used in defining uαβγ, and the counterclockwise monodromy of Φ S is TΦ 2 . This identification is different from that of Convention 1.1.11. p q r s So the right transpose acts as

t ˚ ˚ ˚ u : Mkj α TΦ 2 Mji β Mki γ 1 . β,α,γ p q ˝ j r s˝ p q ÝÑ p q r s Using Proposition 3.1.13 to transform the source and target of this, we write

t ´1 ´1 ´1 u : Mjk α T 1 TΦ 2 Mij β T 1 Mik γ T β,α,γ p q˝ Φj r´ s˝ j r s˝ p q˝ Φi r´ s ÝÑ p q˝ Φi so t ´1 u T : Mjk α Mij β Mik γ , β,α,γ ˝ Φi p q˝ p q ÝÑ p q t just as vαβγ . Similarly for the second identification involving u. The proof of Proposition 3.2.4 reduces to a general statement involving a single spherical functor a : Φ Ψ and the corresponding exact triangle of functors Ñ µ (3.2.5) Id / T Ψ ❍ Ψ c ❍❍ ✇✇ ❍❍ `1 ✇✇ η ❍❍ ✇✇g ❍ {✇✇ a a˚ ˝ Proposition 3.2.6. The triangle (3.2.5) is self-dual, i.e., we have natural identifications

ηt T ´1 η, tη g T ´1 1 , » Ψ ˝ » ˝ Ψ r s µt µ T ´1, tµ T ´1 µ. » ˝ Ψ » Ψ ˝ 77 t ˚ The meaning of these identifications is as follows. By definition, η : IdΨ a a. By ˚ ˚ ´1 t t ˚ Ñ ´˝1 Proposition 3.1.2(iii), we have a a TΨ 1 , so η acts as η : IdΨ a a TΨ 1 , and so does g T ´1 1 . The following» technical˝ propositionr s will be used inÑ the˝ pro˝of: r s ˝ Ψ r s Proposition 3.2.7. Suppose we are given a pair of functors F,G : C D, admitting ÝÑ adjoints, and the natural transform α : F G. Denote by ηF , ηG the canonical maps from ˚ ˚ ÝÑ F F and G G to IdC . Then, the following diagram is commutative: ˝ ˝ ˚F ˝α ˚G F / ˚G G ˝ ˝ tα˝F ηG ˚  ηF  F F / IdC ˝ ˚ ˚ Proof: Denote by eF and eG the canonical maps from IdD to F F and G G respectively. Then, the map tα : ˚G ˚F can be described as the following˝ composition,˝ cf. (A.7): ÝÑ ˚ ˚ ˚ ˚ ˚G G˝eF ˚G F ˚F G˝α˝ F ˚G G ˚F ηG˝ F ˚F ÝÑ ˝ ˝ ÝÑ ˝ ˝ ÝÑ The statement then follows reduces to the adjointness axiom which states that the following composition is identity: F F ˝eF F ˚F F ηF ˝F F ÝÑ ˝ ˝ ÝÑ and the obvious commutativity of the following square:

˚ G˝G˝ηF ˚G G F F / ˚G G ˝ ˝ ˚ ˝ ˝ ˚ ηG˝ F ˝F ηG ˚  ηF  F F / IdC ˝

Proof of Proposition 3.2.6: First of all, note that for any functor a with left and right adjoints ˚a, a˚ and units and counits e, η (for a, a˚ ) and e1, η1 (for ˚a, a ), we have identifications p q p q tη e1, te η1. » » This is because, on one hand, tη and te can serve as a unit and counit exhibiting ˚a, a as an adjoint pair and, on the other hand, the adjoint functor is defined uniquely upp to aq unique isomorphism (up to a contractible space of choices in the dg-setting), see Appendix A4. Therefore (Proposition A.10), the triangle for the cone of e1, which we denote

t t t µ e1 ˚ g ´1 µr1s ∆e1 ... IdΨ a a T 1 1 “ Ñ Ñ ˝ Ñ Ψ r s Ñ r su is identified with the left adjoint/transpose of the triangle for the cone of η, which we denote g ˚ η µ ∆η TΨ 1 a a IdΨ TΨ . “ r´ s Ñ ˝ Ñ Ñ u

78 So it is enough to construct an isomorphism of triangles ∆e1 TΨ 1 ∆η. We consider ˝ r´ s Ñ the vertical arrow given as the composition in Proposition 3.1.2(iii), composed with TΨ 1 and the other two arrows being Id with respect to our identifications: r s

˚ ˚ a a TΨ 1 t / IdΨ t / TΨ / a a TΨ ˝ ˝ r´ s g˝TΨr´1s µ˝TΨ ˝ ˝ a˝˚a˝g a˝˚a˝gr1s   a ˚a a a˚ Id Id a ˚a a a˚ 1 ˝ ˝ ˝ ˝ ˝ ˝ r s a˝η1˝a˚ a˝η1˝a˚r1s   ˚ η  µ  ˚ a a / IdΨ / TΨ / a a 1 ˝ ˝ r s First, the middle square is clearly commutative from the Proposition 3.2.7, applied to G TΨ, F Id. The leftmost square is commutative from the application of the same “ “ ˚ proposition to F TΨ 1 , G a a . The rightmost square is the application of the dual statement and also“ canr´ bes reduced“ ˝ to the case of the leftmost square by dualizing on the right and unraveling the a˚˚ T ´1 a 1 . » Ψ ˝ r s C. Picard-Lefschetz arrows: rotation invariance. Consider a curvilinear geometric 2 triangle ∆ R consisting of three distinguished points wi,wj,wk and unoriented simple Ă paths α (joining wj,wk), β (joining wi,wj) and γ (joining wi,wk) and containing no other distinguished points inside or on the boundary. We assume that the triple wi,wj,wk has the counterclockwise orientation. p q The triangle ∆ gives rise to several Picard-Lefschetz situations and therefore to several Picard-Lefschetz u- and v-arrows. For example (our earlier choice), moving γ past wj to a path γ1, we get the arrows (3.2.3). But we can also move any side of ∆ past the opposite vertex. In this way, for any cyclic rotation (i.e., even permutation) a, b, c of i, j, k we get arrows p q p q ´1 uc,b,a : Mca 1 McbT 2 Mba, va,b,c : MbcMab Mac, r´ s ÝÑ Φb r´ s ÝÑ where we omitted the notation for the path joining the points, this path being always one of the sides of ∆.

wk γ ‚

wi ‚ x ‚ α β

‚wj

Figure 25: A geometric triangle ∆.

79 The appearance of T ´1 2 in the first but not the second arrow above means the follow- Φb r´ s ing. In forming vabc we identify the fibers of the local system Φb using clockwise monodromy (going inside the triangle), in accordance with Convention 1.1.11. In forming uc,b,a we use the identification using the monodromy along the arc which still goes inside the triangle but which is now anticlockwise, thus making the choice opposite to Convention 1.1.11. This is done so that the target of ub,a,c can be more easily related to the source of va,b,c by adjunction. After simplifying the shifts, we can write the u-arrow as

´1 uc,b,a : Mca McbT 1 Mba. ÝÑ Φb r´ s In this way we get three u-arrows and three v-arrows. Proposition 3.2.8. (a) The tensor rotation isomorphisms (Corollary A.9) give canonical identifications

Hom MjkMij, Mik Hom MkiMjk, Mji Hom MijMki, Mkj ´1p q » p ´1 q » p q ´1 Hom Mki, MkjT Mji 1 Hom Mij, MikT Mkj 1 Hom Mjk, MjiT Mik 1 . p Φj r´ sq » p Φj r´ sq » p Φj r´ sq

(b) The isomorphisms in (a) take the vabc, resp. uabc, into each other. Remark 3.2.9. Note that the v-maps are dual to the u-maps by Proposition 3.2.4. Propo- sition 3.2.8 therefore means that ∆ gives rise to what is essentially a single natural transfor- mation (“Picard-Lefschetz plaquette”) whose behavior is similar to that of Clebsch-Gordan spaces in representation theory. Remark 3.2.10. The second line of identification is dual to the first up to the application ˚ ´1 of TΦ, using the formula M MbaT 1 from 3.2.4: ab “ Φa r´ s

˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ Hom M TΦ , M M TΦ Hom M TΦ , M M TΦ Hom M TΦ , M MkiTΦ p ik i jk ij i q » p ji j ki jk j q » p kj k ij k q Proof of Proposition 3.2.8: (a) We prove the very first identification, the others are simi- lar. Let A,B,C be three functors such that Hom AB,C makes sense. By composing two consecutive tensor rotation isomorphisms of Corollaryp A.9q, we get an identification

˚ ˚ (3.2.11) ρABC : Hom AB,C Hom BC , A . p q ÝÑ p q C M , A M , B M . A˚ M T ´1 1 Let now ik jk ij By Proposition 3.1.13, kj Φi and ˚ ´“1 “ “ “ r´ s C MkiT 1 , so “ Φi r´ s Rijk : ρM ,M ,M TΦ 1 : Hom MjkMij, Mik Hom MijMki, Mkj “ jk ij ik ˝ i r s p q ÝÑ p q is the desired identification.

(b) Let us prove that Rijk vijk vjki, the other such statements being similar. Recalling the construction of the tensorp rotationq“ isomorphisms, cf. the proof of Proposition A.8, we see that for general A,B,C as above, ρABC is the composition (3.2.12) Hom AB,C Hom A˚ABC˚, A˚CC˚ Hom BC˚, A˚ p q ÝÑ p q ÝÑ p q 80 with the first arrow given by composition with A˚ on the left and C˚ on the right, and the ˚ ˚ second arrow given by insertion of the unit eA : Id A A and the counit ηC : CC Id. Ñ Ñ This means that for any v : AB C, the arrow ρABC v is the composition Ñ p q

˚ ˚ eA˝BC A˚˝u˝C˚ A ˝ηC (3.2.13) BC˚ / A˚ABC˚ / A˚CC˚ / A˚.

Now, specialize this to C Mik etc. as in (a). Since we are dealing with a single triangle ∆, “ we can represent our schober S in terms of three spherical functors ai : Φi Ψ, aj : Φj Ψ, Ñ Ñ ak : Φk Ψ, cf. (3.1.15) where Ψ corresponds to a point x positioned inside ∆. The transportÑ functors, which represent passing through the midpoints of the edges of ∆, can be just as well taken to represent passing through x and back, that is

˚ ˚ ˚ Mik a ai, Mjk a aj, Mij a ai, etc. “ k “ k “ j Then ˚ ˚ ˚ ˚ v vijk : a aja ai a ai, v a ηj ai “ k j ÝÑ k “ k ˝ ˝ ˚ is given simply by insertion of the counit ηj : ajaj IdΨ. Now, applying (3.2.12) to v, i.e., implementing (3.2.13), we get the transformationÑ which is the composition c (the dotted arrow below):

1 2 ˚ ˚ ˚˚ p q ˚ ˚˚ ˚ ˚ ˚ ˚˚ p q ˚ ˚˚ ˚ ˚ ˚˚ a ai a a / a a a aj a ai a a / a a a ai a a p j q p i qp k q p j qp k qp kq p j q p i qp k q p j qp k qp kq p i qp k q p3q c  - a˚ a˚˚ p j qp k q Here (1) is the insertion of two units on the left and (2) is the contraction via the counit ˚ aj aj Id in the middle of the 8-term composition, while (3) is the contraction via, first, p qÑ ˚ ˚ ˚˚ the counit ai ai Id and then via the counit ak ak Id. Now, thep axiomsqÑ of units and counits of adjunctionsp qp qÑ imply that c is the equal to just the ˚ ˚ ˚ ˚˚ result of contraction via the counit ai ai Id in the 4-term composition aj ai ai ak . ˚˚ ´1 p q Ñ p q p qp q At the same time, a ak T 1 , so after composing c on the right with TΦ 1 , we get k “p q Φk r´ s k r s the Picard-Lefschetz arrow vjki : MijMki Mkj. Ñ D. Picard-Lefschetz arrows: (co)associativity. Among the Picard-Lefschetz ar- rows (3.2.3), the arrow vα,β,γ looks vaguely like a multiplication, while uα,β,γ1 looks like a comultiplication. To pursue this analogy, consider a curved 4-gon formed by 4 points wi,wj,wk,wk A and paths α1,α2,α3,γ, as in the middle for Fig. 26. Let β1 and β2 be the two diagonals,P as in the left and right path of Fig. 26.

81 wj α2 wk wj α2 wk wj α2 wk ‚ ‚ ‚ ‚ ‚ ‚ ò ò β1 β2 α3 α1 α3 α1 α3 α1 ò ò ò wi‚ γ ‚wl wi‚ γ ‚wl wi‚ γ ‚wl

Figure 26: Coassociativity of Picard-Lefschetz arrows.

Proposition 3.2.14. (a) The u-arrows are coassociative, i.e., the diagram

uα1,β1,γ r1s Mil γ / Mkl α1 Mik β1 1 p q p q˝ p qr s

uβ2,α3,γ r1s Mklpα1q˝uα2,α3,β1 r2s   Mjl β2 Mij α3 1 / Mkl α1 Mjk α2 Mij α3 2 u ˝M pα qr2s p q˝ p qr s α1,α2,β2 ij 3 p q˝ p q˝ p qr s is commutative. (b) The v-arrows are associative, i.e., the diagram

vα1,α2,β2 ˝Mij pα3q Mkl α1 Mjk α2 Mij α3 / Mkl α1 Mik β1 p q˝ p q˝ p q p q˝ p q vβ ,α ,γ Mklpα1q˝vα2,α3,β1 2 3   Mjl β2 Mij α3 v / Mil γ p q˝ p q α1,β1,γ p q is commutative.

Here (as elsewhere) the composition of rectilinear transports is defined following Conven- tion 1.1.11. That is, the stalk of the local systems Φ at the intermediate points are identified via the clockwise monodromy from the incoming direction to the outgoing one. Proof: We prove (b), since (a) then follows by duality (Proposition 3.2.4). As in the con- struction of the Picard-Lefschetz triangle (Proposition 3.2.1), we “pinch” the diagram at wi ai and wl, after which all the functors in the diagram have a common initial part Φi Ψi and ˚ a l Ñ a common final part Ψl Ψl, see the left part of Fig. 27. Ñ So it suffices to prove the analogous statement for the “stripped” functors acting from Ψi to Ψl. Now, after such stripping off the common beginning and end, all that remains from Mil γ is the monodromy from Ψi to Ψl. So we can eliminate this monodromy also, p q contracting the part of the path γ joining the points carrying Ψi and Ψj, to a point and replacing the spaces Ψi and Ψj by a common space Ψ (identified with them both via the above monodromy). This gives a diagram in the right part of Fig. 27.

82 Φj Φk Φj Φk ‚ ‚ ˚ ‚ ‚ ˚ ak ak aj aj

Ψi Ψl Φ Φ i ‚monodromy ‚ l Ψ‚ ‚ ‚ Figure 27: Curved quadrilateral, pinched... and then contracted.

The composition Mjl β2 Mij α3 , stripped of its initial and final parts, now becomes ˚ p q ˝ p q aj aj : Ψ Ψ, while the composition Mkl α1 Mik β1 , similarly stripped, becomes ˝ ˚ Ñ p q ˝ p q ak a :Ψ Ψ. So we are reduced to the commutativity of the diagram ˝ k Ñ ˚ ˚ ˚ ak a aj a / ak a ˝ k ˝ ˝ j ˝ k

 ˚  aj a / IdΨ ˝ j obtained by two separate contractions of the counits for the adjunctions. This diagram clearly commutes.

E. Pasting of natural transformations. The compositions of the paths in the squares of Proposition 3.2.14 can be seen as instances of 2-dimensional composition, or pasting of natural transformations (or, more generally, of 2-morphisms in a 2-category), see [75, 82]. To perform pasting, one needs a pasting diagram which can be seen as a plane region decomposed into (curvilinear) polygons so that:

(0) To each vertex of each polygon there is associated a category.

(1) Each edge is oriented and carries a functor between the categories associated to its vertices.

(2) Each polygon is oriented, which (together with orientation of the edges) subdivides its edges into “source” edges and “target” edges. These two groups of edges should form two composable chains with common beginning and end vertices. Further, the polygon carries a natural transformation (indicated by a double arrow) between the functors given by these composable chains. It is indicated by a double arrow inside the polygon.

(3) Each internal edge appears once in a source chain of a 2-cell and once in the target chain of another 2-cell.

Under these conditions, one can compose the natural transformations associated to all the 2-cells to a single natural transformation (“pasting theorem”). See [82] for more details.

83 P ζ ζ ‚ ‚ Pi B` ó ‚ ‚ ‚ ‚ ‚ Pi p pζ “ ó ó ‚ ‚ ‚ ‚ R

Figure 28: A pasting diagram from a polygonal subdivision.

Examples 3.2.15. (a) The two sides of Fig. 26 are pasting diagrams whose pastings are the compositions of the two paths in the square in Proposition 3.2.14(a). The proposition says that these compositions are equal, which is represented by a single transformation in the middle of the figure.

(b) Let ζ C, ζ 1, be a direction. Then we have the projection p pζ : C R to the line orthogonalP to |ζ which|“ we identify with R and orient as in Fig. 28. Let“P ConvÑ B C, B A, B 3, be a convex polygon with vertices in A. Assume that A is in“ linearlyp generalqĂ positionĂ | including|ě to ζ-infinity. Then p x p y whenever x, y is an edge of P . Therefore every such edge x, y acquires an orientationp q‰ p fromq x to y, wherer s p x p y . In addition, r s p q ă p qζ the boundary P becomes decomposed into the upper boundary ` P ` P and the B ζ B p q“B p q lower boundary ´ P ´ P . By definition, ` P is “what is visible from the source of the projection”,B seep Fig.q“B 28,p andq P is the complementaryB p q arc (with the same beginning B´p q and end). This makes P into a pasting scheme with one 2-cell, in which the arc ` P consists of source edges and P of target edges. B p q B´p q Further, let P Pi i I be a polygonal decomposition of P , i.e., a finite set of convex “ t u P polygons Pi P , also with vertices in A, such that P Pi and any intersection Pi Pj Ă “ X is a common face (i.e., , or a vertex or an edge) of both. By the above, each Pi becomes a pasting scheme with oneH2-cell. This makes the whole subdivisionŤ P into a pasting scheme as well. See [52].

F. Picard-Lefschetz arrows: the 3 1 move. Geometrically, associativity and coassociativity of the Picard-Lefschetz arrowsÑ represent the elementary 2 2 move of plane triangulations: replacing one of the two possible triangulations of a 4-gonÑ into two triangles by the other triangulation. We have also a companion statement for the 3 1 move: passing from a triangulation of a triangle into 3 small triangles, to the “big” triangleÑ itself, see the left part of Fig. 29.

Proposition 3.2.16. In the situation of Fig. 29 , the pasting of the natural transformations uζ,ε,α, vε,β,δ and vζ,δ,γ is equal to 0.

84 Note that the structure of a schober does not give a non-trivial natural transformation associated to the “big” triangle.

wj wj ‚ wj ‚ ‚ r β ε α β ε α

ó ó ó ó wl r r wl r δ ‚ ζ δ ‚ ζ ó wi ó wk w w r i ‚γ ‚ k wi ‚r γ ‚ wk ‚ ‚ r r Figure 29: The 3 1 relation for PL arrows and proof by pinching. Ñ r

Proof: As in the proof of coassociativity, we pinch the picture at the points wi,wj,wk, see the right part of Fig. 29. So it suffices to prove that the pasting of the “inner” triangle on the right is zero. In this inner triangle, we have only one singular point, wl and two paths from wi to wk, one above and one below wl, i.e., we have a Picard-Lefschetz situation. The first of these is the composite path αβ and the second is γ. So we have a Picard-Lefschetz p q triangle,r denoter it ∆, corresponging to this Picard-Lefschetz situation. Note that the pasting of the twor smallr triangles on ther left and on the right (the lower triangle excluded) is just the Picard-Lefschetz arrow uζ,δ,pαβq involving the composite path. The the pasting of all three triangles is the composition of uζ,δ,pαβq with vζ,δ,γ, which is the composition of two successive morphisms in ∆, so it isr zero.r r r r r r r r r r 3.3 Reminder on semi-orthogonal decompositions and gluing func- tors A. 2-term SOD’s. We recall the material from [14], see also [50], Ch.1. Let V be a k-linear triangulated category. A full triangulated subcategory B V is called strictly full, if any object of V isomorphic to an object of B, is in B. In this section,Ă all subcategories of V will be assumed triangulated and strictly full. For a class of objects I Ob V we denote by I the smallest strictly full triangulated subcategory of V containingĂI. p q x y Given a subcategory B V, its left and right orthogonals are the subcategories defined as follows: Ă KB KB : V V Hom V, B 0, B B , “ V “ P p q“ @ P (3.3.1) K K B BV : V V ˇHom B,V 0, B B (. “ “ P ˇ p q“ @ P We can further form the iterated orthogonals ˇ BK2 BK K, BKp´2q ( K KB etc. General ˇ notation: BKn, n Z. “ p q “ p q P 85 Definition 3.3.2. (a) A pair B, C of subcategories in V is called a semi-orthogonal decom- position (SOD for short) of Vp, if Homq B, C 0 (i.e., C BK) and each object V V can be included into an exact triangle p q “ Ă P

BV V CV BV 1 , BV B, CV C. ÝÑ ÝÑ ÝÑ r s P P (b) A subcategory B V is called left admissible, if KB, B is an SOD, and right admissible, if B, BK is an SOD.Ă p q p q Remarks 3.3.3. (a) The condition of existence of a triangle in part (a) of Definition 3.3.2 is equivalent to V B, C , i.e., to V being generated by B and C as a triangulated category. So it is standard to“x speaky about a “semiorthogonal decomposition V B, C ”. “x y (b) For an SOD V B, C , we have C BK and B KC. “x y “ “ (c) The triangle in Definition 3.3.2 (a) in fact depends functorially on V , which is reflected in the following fact.

Proposition 3.3.4. B V is left admissible (resp. right admissible) if and only if the embedding functor ε : Bã ĂV admits a left adjoint ˚ε (resp. right adjoint ε˚). Ñ ˚ For example, for a right admissible B we have ε V BV , the B-component of the p q “ canonical triangle for V . Indeed, Hom ε B ,V Hom B, BV , since Hom B,CV 0. V p p q q“ Bp q V p q“ We further say that B is -admissible, if all iterated orthogonals BKn, n Z, are left admissible (they are then right8 admissible as well). We have the action of theP group Z on the set of -admissible subcategories in V generated by the transformation B BK with inverse being8 B KB. ÞÑ ÞÑ Proposition 3.3.5. Suppose V has a Serre functor S (in particular, all Hom-spaces in V are finite-dimensional over k). Then for any left and right admissible B V we have S B BKK and B is -admissible. Ă p q“ 8 For example, if B B and C BK, then Hom C,S B Hom B,C ˚ 0 so S B V KK. P P p p qq “ p q “ p q Ă

B. N-term SOD’s. Let V be as before.

Definition 3.3.6. A sequence B‚ B1, , BN of (strictly full, triangulated) subcategories in V is said to form a semi-orthogonal“p decomposition¨ ¨ ¨ q (SOD) of V, if:

(1) The Bi are semi-orthogonal, i.e., Hom Bi, Bj 0 for i j. p q“ ă

(2) V B1, , BN is generated by the Bi. “x ¨ ¨ ¨ y Thus, an N-term SOD can be seen, in several ways, as a nested sequence of 2-term SOD’s:

V B1, B2 , B3 , , Bn B1, B2, , BN 1, BN . “ x¨¨¨x y y ¨ ¨ ¨ y“x x ¨ ¨ ¨ x ´ y¨¨¨y 86 Of considerable interest for us will be the action of the braid group BrN on the set of appropriate N-term SOD’s. Given an SOD V B‚ B1, , Bn , and i 1, , N 1, we define its right and left mutation at the spot“xi toy“x be the sequences¨ ¨ ¨ y of subcategories“ ¨ ¨ ¨ ´

K τiB B1, , Bi 1, Bi 1, B , , Bi 2, , BN , ‚ “ ¨ ¨ ¨ ´ ` p i`1qxBi Bi`1y ` ¨ ¨ ¨ (3.3.7) ´1 K τ B B1, , Bi 1, Bi , , Bi 1, Bi 2, , BN . i ‚ “` ¨ ¨ ¨ ´ p qxBi Bi`1y ` ` ¨ ¨ ¨ ˘ These sequences are semi-orthogonal,` but may not generate V so may not˘ be SODs.

Definition 3.3.8. (a) A SOD V B‚ is called -admissible, if all the sequences obtained “x y ´18 from B by iterated application of the τi and τ , i 1, , N 1, are SOD’s. ‚ i “ ¨ ¨ ¨ ´ (b) An -admissible filtration on V is a sequence 8

F F1V F2V , FN V V “ à 訨 Ă “ of strictly full triangulated subcategories` such that ˘

FiV B1, , Bi “ x ¨ ¨ ¨ y for some -admissible SOD V B1, , BN . 8 “x ¨ ¨ ¨ y B V K Note that i in the definition is recovered uniquely as Fi´1 FiV . Thus considerations of -admissible filtrations and -admissible SOD’s are equivalent.p q The following is well known [14].8 8

Proposition 3.3.9. The group BrN acts on the set of -admissible N-term SOD’s of V 8 (or, equivalently, on the set of -admissible N-term filtrations) so that the generators τi and their inverse act by (3.3.7). 8

Further, recall ([61], Thm. 1.24) that BrN has a center which, for N 3, is freely generated by δ2, where δ is the “Garside discriminant” element ě

(3.3.10) δ τ1 τ2τ1 τ3τ2τ1 τN 1 τ1 . “ p qp q¨¨¨p ´ ¨ ¨ ¨ q The following is also well known.

Proposition 3.3.11. (a) We have

KK KK KK KK δ B1, , BN BN B , B , , B , , , , B . p ¨ ¨ ¨ q “ “p N qBN p N´1qxBN´1 BN y p N´2qxBN´2 BN´1 BN y ¨ ¨ ¨ 1 (b) We also have ` ˘ 2 KK KK KK δ B1, , BN B , B , , B . p ¨ ¨ ¨ q“p 1 2 ¨ ¨ ¨ N q (c) Assume that V has a Serre functor S. Then,

2 δ B1, , BN S B1 , ,S BN . p ¨ ¨ ¨ q“p p q ¨ ¨ ¨ p qq

87 Proof: Part (a) is proved directly by induction on N. To see (b), notice that (a) used 2 KK twice implies the statement for the first component: δ B 1 B . To deduce it for any p ‚q “ 1 other component, say the ith, let τp1i τi´1τi´2 τ1 be the minimal lift to BrN of the transposition 1i . Then, using the factq that“ δ2 is central,¨ ¨ ¨ we have p q 2 2 2 KK KK δ B i τ 1i δ B 1 δ τ 1i B 1 τ 1i B 1 B . p ‚q “p p q ‚q “p p q ‚q “ pp p q ‚q q “ i Finally, (c) follows from (b). Indeed, for any -admissible subcategory B V we have BKK S B , which is obvious from 8 Ă “ p q Hom B,C Hom C,S B ˚, C BK. p q “ p p qq P Convention 3.3.12. Note that the concept of an SOD can be developed purely in the framework of classical triangulated categories: no enhancements are needed. We will use it in the enhanced framework as well: by an SOD in a pre-triangulated category V we will mean an SOD in the triangulated category H0 V . Since V and H0 V have the same objects, this does not cause confusion. p q p q

C. Gluing 2-term SODs: gluing functors. One can ask: given triangulated cate- gories B1, , BN , what extra data are needed to construct an SOD V B1, , BN ? An obvious invariant¨ ¨ ¨ of such an SOD is the collection of functors “x ¨ ¨ ¨ y

op hij : B Bj Vectk, hij Bi, Bj Hom Bi, Bj , i j. i ˆ ÝÑ p q“ V p q ą

However, it does not seem possible to use just the data of hij as triangulated functors to recover (“glue”) V as a triangulated category. On the other hand, it is well known how to perform such gluing in the dg-enhanced setting. In this paragraph and the next subsection we recall this gluing procedure, starting from the case of 2-term SODs [71]. Let B, C be pre-triangulated dg-categories and f : C B is a dg-functor. Define a new Ñ dg-category B f C to have, as objects, triples V B,C,u , where B B, C C and u : f C B isˆ a degree 0 closed morphism in B. The“ pHom-complexq betweenP V PB,C,u and Vp 1 qÑB1,C1,u1 is defined, as a graded vector space to be “p q “p q (3.3.13) Hom‚ V,V 1 Hom‚ B, B1 Hom C,C1 Hom f C , B1 1 , Bˆf Cp q “ Bp q‘ Cp q‘ Bp p q qr´ s with the differential defined in such a way that we have an exact triangle

‚ 1 Hom B, B dis Hom‚ V,V 1 Bp q‘ Hom‚ f C , B1 Hom‚ V,V 1 1 . Bˆf Cp q ÝÑ Hom‚ C,C1 ÝÑ Bp p q q ÝÑ Bˆf Cp qr s ‘ Cp q Here dis is the obvious “discrepancy” map, measuring the failure of a pair of morphisms B B1, C C1 to be compatible with u and u1, see [71] §4.1 for more details. Informally, Ñ HomÑ‚ V,V 1 dis putting Bˆf C on the left of in the triangle, is the homotopy analog of taking its kernel, so we enforcep q compatibility at the homotopy level.

88 Proposition 3.3.14 ([71]). (a) B f C is pre-triangulated. ˆ (b) We have a semi-orthogonal decomposition B f C B, C , with B B f C consisting of the B, 0, 0 and C consisting of the 0,C, 0 . ˆ “x y Ă ˆ (c)p The functorq p q

‚ op ‚ ‚ h : C B dgVectk, h C, B Hom C, B ˆ ÝÑ p q “ Bˆf Cp q is given by h‚ C, B Hom‚ f C , B 1 . p q “ Bp p q qr´ s ˚ ˚ Assume now that f has a right adjoint f : B C, so we have the counit η : f f IdB. For an object B B consider the object Ñ ˝ Ñ P ˚ ˚ k B B, f B, ηB : f f B B B f C. p q“p p qÑ q P ˆ Proposition 3.3.15. (a) The correspondence B k B extends to a dg-functor k : B ÞÑ p q Ñ B f C which gives quasi-isomorphisms on Hom-complexes. ˆ K KK (b) The category k B is identified with C B B f C, and we have an SOD p q “ Ă ˆ B f C C,k B . ˆ “x p qy (c) This extends to an equivalence

C f ˚ B B f C ˆ ÝÑ ˆ which takes C C f ˚ B to C B f C by the identity functor, and B C f ˚ B to k B Ă ˆ Ă ˆ Ă ˆ p qĂ B f C by the functor k. ˆ Proof: (a) Notice that for any B, B1 we have a canonical embedding of complexes

Hom‚ B, B1 ã Hom‚ k B ,k B1 , Bp qÑ Bˆf Cp p q p qq which is, moreover, a quasi-isomorphism. Indeed, by (3.3.13),

Hom‚ k B ,k B1 Hom B, B1 Hom f ˚B, f ˚B1 Hom f ˚B, f ˚B1 1 Bˆf Cp p q p qq “ Bp q‘ Cp q‘ Cp qr´ sq as a graded vector space. Looking at the differentials, we see that the embedding of the LHS as the first summand in the RHS is a morphism (embedding) of complexes, and the quotient by the image is contractible (is the cone of the identity map of the second summand in the RHS). This defines a dg-functor k with the required property. (b) Let C C and B B. By definition (3.3.13), P P Hom C,k B Hom C, f ˚B Hom f C , B 1 Bˆf Cp p qq “ Cp q‘ Bp p q qr´ s as a graded vector space. Looking at the differential, we recognize this as the cone of the adjunction (quasi-) isomorphism) so it is acyclic. This shows an inclusion k B CK. To show that this inclusion is an equality and to establish an SOD, it suffices to showp q thatĂ every

89 object B,C,f C u B fits into an exact triangle with an object from C on the left and an objectp of k Bp qonÑ theq right. For this, consider the following closed degree 0 morphism in p q B f C, written vertically, with the third component, as in (3.3.13), equal to 0: ˆ u B, C, f C / B p q Id` adjpuq ˘   B, f ˚B, f f ˚B / B p q ηB ` ˘ Its target lies in k B and its cone is of the form 0,C1, 0 , so lies in C. p q p q (c) This follows from the identification

Hom‚ k B ,C Hom‚ f ˚ B ,C Bˆf Cp p q q “ Cp p q q which determines the gluing functor to be f ˚, cf. Proposition 3.3.14(c).

Corollary 3.3.16. If f admits iterated adjoints of all orders, then the subcategory B Ă B f C is -admissible. ˆ 8 3.4 Unitriangular (co)monads and glued N-term SOD’s A. Unitriangular (co)monads and their (co)algebras. We will use a version of the classical categorical concept of a monad [75].

Definition 3.4.1. Let V V1, , VN be a finite ordered sequence of pre-triangulated dg-categories. “ p ¨ ¨ ¨ q

(a) A unitriangular (dg-)monad on V is a collection M Mij : Vi Vj iăj of dg-functors together with closed, degree 0 natural transformations, called“p compositionÑ q maps

cijk : Mjk Mij Mik, i j k, ˝ ÝÑ ă ă such that for any i j k l the diagram below commutes (associativity condition): ă ă ă

Mkl˝cijk Mkl Mjk Mij / Mkl Mik ˝ ˝ ˝ cjkl˝Mij cikl   Mjk Mij / Mil. ˝ cijl

We put Mii Id . “ Vi (b) An M-algebra is a sequence of objects V V1, ,VN , Vi Vi, together with closed, degree 0 morphisms (action maps) “p ¨ ¨ ¨ q P

αij : Mij Vi Vj, i j, p q ÝÑ ă 90 such that for any i j k the diagram below commutes: ă ă

Mjkpαij q Mjk Mij Vi / Mjk Vj p p qq p q cijk,Vi αjk   Mik Vi / Vk. p q αik Remark 3.4.2. Note that any dg-functor commutes with direct sums. So, compared to the general concept of [75], our unitriangular dg-monads M are of “linear” nature and rather resemble associative dg-algebras (in fact, they can be viewed as “algebras in the category of functors”). Accordingly, it would be more natural to refer to M-algebras as “modules”, but we retain the classical terminology of [75].

Any collection of objects U U1, , UN , Ui Vi, gives rise to two M-algebras: the free M-algebra M U and the reduced“p free M-algebra¨ ¨ ¨ qMă UP, with p q p q ă (3.4.3) M U j Mij Ui , M U j Mij Ui , p q “ iďj p q p q “ iăj p q à à and the action maps given by the composition maps in M. Definition 3.4.4. (a) A unitriangular (dg-)comonad E in V is a unitriangular dg-monad in op op op V : VN , , V1 . Explicitly, E consists of dg-functors Eij : Vi Vj, i j, together with“ closed p degree¨ ¨ ¨ 0 dg-functorsq (cocomposition maps) Ñ ă

εijk : Eik Ejk Eij, i j k, ÝÑ ˝ ă ă satisfying the coassociativity condition dual to that in Definition 3.4.1(a). We put Eii IdVi . To keep notation straight, we will sometimes denote by Eop the unitriangular dg-monad“ in Vop corresponding to E.

(b) A coalgebra over E us a collection of objects Vi Vi together with closed degree 0 morphisms (coaction maps) P βij : Vj Eij Vi , i j, ÝÑ p q ă satisfying the compatibility condition dual to that in Definition 3.4.1(b).

Any collection of objects U U1, , UN , Ui Vi, gives rise to the free E-coalgebra E U and the restricted free E-coalgebra“ p ¨E ¨ă ¨ U onq U withP p q p q ă (3.4.5) E U j Eij Ui , E U j Eij Ui , p q “ iďj p q p q “ iăj p q à à and the coaction maps given by the cocomposition maps in E.

If M is a unitriangular monad in V1, , VN , then the collections of the right and left ˚ ˚ ˚ ˚ p ¨ ¨ ¨ q adjoints M Mij iăj and M Mij iăj (provided they exist), are unitriangular comonads “ p q “ p q ˚ in VN , , V1 . Further, M-algebras are the same as M -coalgebras. p ¨ ¨ ¨ q 91 B. Dg-categories of (co)algebras and SODs. Let V V1, , VN be as before. 1 1 “ p 1 ¨ ¨ ¨ 1 q For two sequences of objects V V1, ,VN and V V , ,V , Vi,V Vi, we define “p ¨ ¨ ¨ q “p 1 ¨ ¨ ¨ N q i P N Hom‚ 1 Hom‚ 1 V V,V Vi Vi,Vi . p q “ i“1 p q à In this way we view V as a dg-category (i.e., as the direct sum of the Vi). Let now M be a unitriangular dg-monad in V and V,V 1 be M-algebras. We define the naive complex of morphisms between V and V 1 as

Hom‚ V,V 1 ϕ Hom‚ V,V 1 ϕ commutes with the actions . naivep q “ P V p q ˇ ( This definition is not homotopy invariant and we takeˇ the derived functor of it in the following explicit way. For any M-algebra V we consider the reduced bar-resolution of V which is the complex of free M-algebras (and differentials being closed, degree 0 naive morphisms) (3.4.6) M M2 cMăpV q´ pcV q` pαq cV ´Mpαq Bar‚ V / M Mă Mă V / M Mă V / M V . p q “ ¨ ¨ ¨ p p p qqq p p qq p q " *

The differential d is the alternating sum of elementary contractions using c cijk or 2 “ p q α αij , so d 0 in virtue of the axioms. Note that this complex terminates since Mă“N p V q 0. We“ fix the grading of the complex so that M V is in degree 0. Let B V Totp qBarp ‚q“V be the total object. p q p q“ p p qq Proposition 3.4.7. The action map α : M V V gives rise to a quasi-isomorphisms B V V . p q Ñ p qÑ Proof: We need to show that the augmented complex

M Mă Mă V M Mă V M V α V, ¨¨¨Ñ p p p qqq Ñ p p qq Ñ p q ÝÑ ` ` denote it B , is exact. This means that for each j 1, , N, the jth component Bj of B ` “ ¨ ¨ ¨ is exact. Explicitly, Bj is the complex (in Vj)

Mi2,j Mi1,i2 Vi1 Mij Vi Vj. ¨¨¨Ñ i1ăi2ďj p p qÑ iďj p qÑ à à ` Thus, Bj as a graded object of Vj, consists of summands

S i1, , im Mi ,j Mi ,i Vi m , p ¨ ¨ ¨ q “ m ¨ ¨ ¨ 1 2 p 1 qr s ` labelled by sequences i1 im j. For p 0 let Fp B be the sum of the S i1, im 㨨¨ă ď ě Ă j p ¨ ¨ ¨ q such that # ν iν j p. It is immediate that Fp is closed under the differential, i.e., t | ă u ď ` is a subcomplex. So we have a finite increasing filtration F0 F1 Fj B by Ă Ă ¨¨¨ Ă “ j 92 subcomplexes. Now, each quotient Fp Fp´1 is a 2-term complex with both terms identified with {

Mip,j Mi1,i2 Vi1 i1㨨¨ăipăj ¨ ¨ ¨ p q à ` and the differential being the identity. So each Fp Fp 1 is exact and therefore B is exact. { ´ j

Definition 3.4.8. Let V V1, , VN be as before. “p ¨ ¨ ¨ q (a) For a unitriangular dg-monad M in V we define the dg-category AlgM to have, as objects, M-algebras and, as Hom-complexes,

Hom‚ V,V 1 Hom‚ B V , B V 1 . AlgM p q “ naivep p q p qq op (b) For a unitriangular dg-comonad E in V we define the dg-category CoalgE as AlgEop , where Eop is the dg-monad in Vop corresponding to E, see Definition 3.4.4(a). p q

Remarks 3.4.9. (a) Since B V is quasi-free (i.e., it becomes a free algebra, if we forget differentials in both B V andp qM), the quasi-isomorphism B V 1 V 1 induces a quasi- isomorphism p q p q Ñ

Hom‚ V,V 1 Hom‚ B V ,V 1 AlgM p q ÝÑ naivep p q q “ Tot Hom‚ V,V 1 δ0 Hom‚ Mă V ,V 1 δ1 Hom Mă Mă V ,V 1 δ2 . “ V p q ÝÑ V p p q q ÝÑ V p p p qq q ÝÑ ¨ ¨ ¨ " *

Note that the first differential δ0 in the last complex is the “discrepancy map”

1 ă 1 δ0 ϕ ϕ α α ϕ : M V V , p q “ ˝ ´ ˝ p qÑ q 1 1 ‚ 1 where α and α are the action maps for V and V . So Ker δ0 Hom V,V . p q“ naivep q (b) Explicitly, the category CoalgE has, as objects, E-coalgebras, and

Hom‚ V,V 1 Hom‚ C V ,C V 1 . CoalgE p q “ naivep p q p qq Here C V Tot E V E Eă V p q “ p qÑ p p qqѨ¨¨ is the total object of the reduced cobar-resolution of V , defined in( the way dual to (3.4.6), ‚ and Homnaive is the complex of morphisms commuting with the coalgebra structures.

Proposition 3.4.10. (a) Let M be a quasi-triangular dg-monad in V V1, , VN . The dg- “p ¨ ¨ ¨ q category AlgM is pre-triangulated. It has an SOD AlgM V1, , VN , where Vi is identified “x ¨ ¨ ¨ y with the full dg-subcategory in AlgM formed by algebras V with Vj 0 for j i. “ ‰ (b) Dually, let E be a quasi-triangular dg-comonad in V. The category CoalgE is pretrian- gulated. It has an SOD CoalgE V1, , VN , where Vi is identified with the full dg-category “x ¨ ¨ ¨ y of coalgebras V with Vj 0 for j i. “ ‰

93 Proof: We prove only (a), since (b) is obtained by duality. We start by showing that naive AlgM is pre-triangulated. Let AlgM be the dg-category with the same objects as AlgM ‚ 1 naive and Hom-complexes Homnaive V,V . Then AlgM is pre-triangulated: the total complex ‚ p ‚ q naive Tot U of a twisted complex U over AlgM is found componentwise, i.e., separately for p q ‚ ‚ each i 1, , N. The ith component of Tot U is Tot Ui , the total object in Vi of “ ¨ ¨ ¨ ‚ p q p q the twisted complex Ui over Vi. Now, AlgM is, by definition, the full dg-subcategory in naive AlgM on objects of the form B V . To prove that it is pre-triangulated, it suffices to show the following. If, in U ‚ above,p q each U p has the form B V p , then U Tot U ‚ is p q 0 naive“ p q homotopy equivalent to some B V (i.e., becomes isomorphic to it in H AlgM ). For this, we can take V U and look atp theq canonical quasi-isomorphism α : Bp U qU. By our assumptions, U “is quasi-free, and therefore α is a homotopy equivalence. p q Ñ i ďi Next, for i 1, , N, let AlgM (resp. AlgM ) be the full dg-subcategory in AlgM formed “ ¨ ¨ ¨ i by algebras V V1, ,VN such that Vj 0 only for j i (resp. for j i). Then AlgM “ p ¨ ¨ ¨ q ‰ ď“i ďi´1 ďi is isomorphic to Vi. Further, we have a 2-term SOD AlgM AlgM , AlgM . Indeed, for i 1 ďi´1 1 “ x y V AlgM and V AlgM we have HomAlgM V,V 0 by the construction. At the same P “ ď˚i q“ time, for any V V1, ,Vi, 0, , 0 AlgM we have an exact triangle “p ¨ ¨ ¨ ¨ ¨ ¨ q P

V1, ,Vi 1, 0, 0 V1, ,Vi, 0, , 0 0, , 0,Vi, 0, , 0 p ¨ ¨ ¨ ´ ¨ ¨ ¨ q ÝÑp ¨ ¨ ¨ ¨ ¨ ¨ q ÝÑp ¨ ¨ ¨ ¨ ¨ ¨ q so which shows that we have an SOD. In this way we show by induction that

1 N AlgM AlgM, , AlgM V1, , VN . “ x ¨ ¨ ¨ y“x ¨ ¨ ¨ y Example 3.4.11. Let N 2. The datum of a uni-triangular monad M reduces in this case “ to the datum of a single dg-functor f M12 : V1 V2 and nothing else. The dg-category “ Ñ AlgM for such an M is precisely the glued dg-category V1 f V2, see Proposition 3.3.14. ˆ C. Mutations of uni-triangular dg-monads. We now describe the analog, for mon- ads, of the concept of mutations of exceptional sequences and of their Ext-algebras, see [83, 86]. Note that the formulas for the mutations below (3.4.12) are similar to those of Proposition 1.4.6. This similarity is a clear indication of the role of perverse schobers in the theory of mutations.

Let V V1, , VN be a sequence of pretriangulated dg-categories. For any i 1, , N “1 we p denote¨ ¨ ¨ q “ ¨ ¨ ¨ ´

siV V1, , Vi 1, Vi 1, Vi, Vi 2, , VN “p ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ q the sequence obtained from V by permuting the ith and i 1 st terms. p ` q ˚ Let M be a uni-triangular dg-monad in V. Assume that each Mij has both adjoints Mij ˚ 1 2 and Mij. We construct two uni-triangular dg-monads M LiM and M RiM in siV which we call the left and right mutations of M. More precisely, define“ the functors“ M1 , 1 ν j N, ν,j ď ă ď

94 by

Mν,j, if ν j i 1; cν,i,i`1 ă ď ´ $Cone Mi,i`1 Mν,i Mν,i`1 1 , if ν i 1, j i. ˝ ÝÑ r´ s ď ´ “ ’M , if ν i 1, j i 1; ’ ν,i ( ’ ď ´ “ ` ’Mν,j, if ν i 1, j i 2; 1 ’ ď ´ ě ` (3.4.12) Mν,j ’˚ “ ’ Mi,i`1, if ν i, j i 1; &’ t “ “ ` ci,i`1,j ˚ Cone Mi`1,j Mij Mi,i`1 , if ν i, j i 2; ’ ÝÑ ˝p q “ ě ` ’Mi,j, if ν i 1, j i 2; ’ ( “ ` ě ` ’M , if i 2 ν j. ’ ν,j ’ ` ď ă tc ’ c : M M M M Here i,i`1,j corresponds%’ to the composition map i,i`1,j i`1,j i,i`1 i,j of by the cyclic rotation isomorphism Hom F G,H Hom F,H ˚G ,˝ see CorollaryÑ A.9. p ˝ q» p ˝p qq We now describe the compositions in M1. Leaving aside the cases when they reduce to the compositions in M, we consider the following possibilities. 1 1 1 For ν i the composition Mi,i`1 Mν,i Mν,i`1, which by our definitions, should be a map ă ˝ Ñ ˚ ˚ Cone Mi,i 1 Mi,i 1 Mν,i Mi,i 1 Mν,i 1 Mν,i, ` ˝ ` ˝ Ñ ` ˝ ` ÝÑ is induced by the following (vertical) map of a (horizontal)(2-term complex to a 1-term complex ˚ ˚ Mi,i 1 Mi,i 1 Mν,i / Mi,i 1 Mν,i 1 ` ˝ ` ˝ ` ˝ ` η˝Mν,i   Mν,i / 0, where η is the counit of the adjunction. 1 1 For ν i and j i 1 the composition Mi,j Mν,i Mν,j which, by our definitions, should be a mapă ą ` ˝ Ñ ˚ Cone Mi 1,j Mi,j Mi,i 1 Cone Mi,i 1 Mν,i Mν,i 1 1 Mν,j, t ` ˝ ˝ ` u˝ t ` ˝ Ñ ` ur´ s ÝÑ is obtained by considering the double complex

Mi 1,j Mi,i 1 Mν,i / Mi 1,j Mν,i 1 ` ˝ ` ˝ ` ˝ `

˚  ˚  Mi,j Mi,i 1 Mi,i 1 Mν,i / Mi,j Mi,i 1 Mν,i 1, ˝ ` ˝ ` ˝ ˝ ` ˝ ` then mapping it, using the counit of the adjunction and the compositions in M, to the double complex Id Mν,j / Mν,j

Id Id   M / M , ν,j Id ν,j

95 and then noticing that the total object of the latter double complex is canonically homotopy equivalent to Mν,j. Dually, we define the functors M2 for 1 ν j N by ν,j ď ă ď

Mν,j, if ν j i 1; ă ď ´ $Mν,i`1, if j i, ν i 1; t “ ď ´ ’ ci,i`1,j ’Cone M M˚ M 1 , if ν i 1, j i 1; ’ ν,i i,i`1 ν,i`1 ’ ÝÑ ˝ r´ s ď ´ “ ` 2 ’Mν,j, if ν i 1, j i 2; Mν,j ’ ( ď ´ ě ` “ ’M˚ , if ν i, j i 1; &’ i,i`1 “ “ ` Mi`1,j, if ν i, j i 2; ’ ci,i`1,j “ ě ` ’Cone Mi`1,j Mi,i`1 Mij , if ν i 1, j i 2; ’ ˝ ÝÑ “ ` ě ` ’ ’Mν,j, ( if i 2 ν j. ’ ` ď ă ’ 2 1 The composition maps%’ for M are similar to those for M and we leave them to the reader. Proposition 3.4.13. Assuming that the relevant (iterated) adjoints exist, the following hold: 1 2 (a) Both LiM M and RiM M are unitriangular dg-monads in siV. “ “ (b) We have canonical quasi-isomorphisms LiRiM RiLiM M. » » (c) We have canonical quasi-isomorphisms

LiLjM LjLiM, i j 2, LiLi 1LiM Li 1LiLi 1M, i 1, , N 2, » | ´ |ě ` » ` ` “ ¨ ¨ ¨ ´ and similarly for the Ri. 1 (d ) For any M-algebra V V1, ,VN in V the sequence “p ¨ ¨ ¨ q λi V V1, ,Vi 1, LV Vi 1 ,Vi,Vi 2, ,VN , where p q “ ¨ ¨ ¨ ´ i p ` q ` ¨ ¨ ¨ LV Vi 1 : Cone Mi,i 1 Vi Vi 1 1 (“left mutation”), i p ` q “` ` p q ÝÑ ` r´ s ˘ is naturally an LiM-algebra. This defines a quasi-equivalence( of pre-triangulated categories λi : AlgM Alg M. The pullback, under λi, of the standard SOD in Alg M is the SOD Ñ Li Li K τi V1, , VN V1, , Vi´1, V , Vi, Vi`2, , VN x ¨ ¨ ¨ y “ ¨ ¨ ¨ p i qxVi,Vi`1y ¨ ¨ ¨ see (3.3.7). @ D (d2) Dually, for any V as in (d1), the sequence

ρi V V1, ,Vi´1,Vi`1, RVi`1 Vi ,Vi`2, ,VN , where p q “ ¨ ¨ ¨ ˚ p q ¨ ¨ ¨ RV Vi : Cone Vi M Vi 1 1 (“right mutation”), i`1 p q “` ÝÑ i,i`1p ` q r´ s ˘ is naturally an RiM-algebra. This defines a quasi-equivalence( of pre-triangulated categories ρi : AlgM Alg M. The pullback, under ρi, of the standard SOD in Alg M is the SOD Ñ Ri Ri ´1 K τ V1, , VN V1, , Vi´1, Vi`1, Vi`1 , Vi`2, , VN i x ¨ ¨ ¨ y “ ¨ ¨ ¨ p qxVi,Vi`1y ¨ ¨ ¨ see again (3.3.7). @ D

96 Proof: It is completely similar to the (now classical) arguments in the case of exceptional 1 sequences [83, 86], and we leave the details to the reader. The complicated formulas for Mν,j 2 (resp. Mν,j) can be obtained at once by analyzing the Hom-complexes between elements of λi V (resp. ρi V ). p q p q Corollary 3.4.14. Suppose each Mij has left and right adjoints of all orders. Then:

(a) Iteration of the Li, Ri define an action σ Lσ of the braid group BrN on the category ÞÑ ´1 of unitriangular dg-monads, so that Lτi M LiM, Lτ M RiM. “ i “ (b) The SOD V1, , VN in AlgM is -admissible. x ¨ ¨ ¨ y 8 For future reference let us mention another, more minor operation on monads and alge- bras. For p1, ,pN Z and a unitriangular monad M on V as before, we have the shifted ¨ ¨ ¨ P monad M p1, ,pN on the same V defined by r ¨ ¨ ¨ s

(3.4.15) M p1, ,pn ij Mij pj pi , r ¨ ¨ ¨ s “ r ´ s with composition maps induced by those in M in an obvious way. Thus we have an equiva- lence of dg-categories

(3.4.16) AlgM AlgM , V1, ,VN V1 p1 , ,VN pn . ÝÑ rp1,¨¨¨ ,pns p ¨ ¨ ¨ q ÞÑ p r s ¨ ¨ ¨ r sq

D. Koszul duality for monads. Let V V1, , VN be as before, and G Gij : “ p ¨ ¨ ¨ q “ p Vi Vj iăj be any collection of dg-functors. We can form the free (unitriangular dg-) monad FMÑG andq the free (unitriangular dg-) comonad FC G generated by G. As dg-functors, p q p q

FM G ij FC G ij Gi,i , ,i ,j, p q “ p q “ 1 ¨¨¨ p (3.4.17) pě0 iăi1㨨¨ăipăj à à Gi,i , ,i ,j : Gi ,j Gi ,i Gi,i . 1 ¨¨¨ p “ p ˝ p´1 p ˝¨¨¨˝ 1

The composition in FM G is the formal concatenation. The cocomposition εijk : FC G ik p q p q Ñ FC G jk FC G ij, i j k, is given by “splitting”. This means that the summand Gi,i , ,i ,k p q ˝ p q ă ă 1 ¨¨¨ p is sent to 0, if no iν is equal to j and to Gi , ,i ,k Gi,i , ,i (by the identity map), if iν j. ν ¨¨¨ p ˝ 1 ¨¨¨ ν “ Let now M be a uni-triangular dg-monad in V. Consider the free comonad FC M 1 with p r sq its differential d. Let D be the differential in FC M 1 whose matrix element Mjk Mij Mik p r sq ˝ Ñ equals cijk for each i j k, and which is extended uniquely to the free coalgebra by ă ă 2 compatibility with cocomposition. Then D 0 by associativity of the cijk and dD Dd 0 “ ` “ by compatibility of the cijk with the differentials in M. So we have the uni-triangular dg- comonad M! FC M 1 ,d D “ p r sq ` which we call the cobar (or Koszul) dual` to M. Explicitly,˘ for i j we have ă

! D D D M Tot Mi ,j Mi ,i Mi,i Mi ,j Mi,i Mij , ij “ ¨ ¨ ¨ Ñ 2 ˝ 1 2 ˝ 1 Ñ 1 ˝ 1 Ñ " iăi1ăi2ăj iăi1ăj * à à 97 where the (horizontal) grading is such that deg Mij 1. The differential D is the alter- nating sum of adjacent composition maps. p q“´ Dually, let E be a uni-triangular dg-comonad on V. Consider the free monad FM E 1 with its differential d. Let D be the differential in FM E 1 which, on the generators,p r´ sq is p r´ sq given by the cocomposition maps εijk : Eik Ejk Eij and is extended to the free monad by the Leibniz rule. As before, D2 0 andÑdD ˝Dd 0, so we have the uni-triangular

“ ` “ dg-monad !

E FM E 1 ,dD , “ p r´ sq

called the bar- or Koszul dual to E. Explicitly,` ˘ ! D D D E Tot Eij Ei ,j Ei,i Ei ,j Ei ,i Ei,i , ij “ Ñ 1 ˝ 1 Ñ 2 ˝ 1 2 ˝ 1 Ѩ¨¨ " iăi1ăj iăi1ăi2ăj * à à with the horizontal grading such that deg Eij 1.

p q“` Proposition! 3.4.18. For any uni-triangular dg-monad M we have a canonical quasi-isomorphism

! M M! . Dually, for any uni-triangular dg-comonad E we have a canonical quasi-isomorphism Ñ E E !.

Ñ Proof: This is completely analogous to the proof of the corresponding statement! for the bar- ! cobar duality for operads [44], so we give just a bried sketch. Looking at Mij as just a graded

functor, we find that it is the direct sum of many shifted copies of summands of the form Mi,i , ,i ,j, see (3.4.17), and there is just one copy (which is! Mij) for p 0. The compositions 1 ¨¨¨ p “

: ! in M define, for each i j, the projection,! denote it πij Mij Mij, and these projections give

ă ! Ñ a morphism of dg-monads π! : M M. To prove that each πij is a quasi-isomorphism, we ! Ñ consider a filtration F in Mij with Fm being the sum of all summands corresponding to the

Mi,i1,¨¨¨ ,ip,j with p m. This is compatible with the differential, and each quotient Fm Fm´1 splits into the directď sum of dg-functors of the form {

Mi,i , ,i ,j k C i, i1, , im, j , 1 ¨¨¨ m b p ¨ ¨ ¨ q where C i, i1, , im, j is a combinatorial complex of k-vector spaces which is easily found out to bep exact¨ ¨ ¨for p q0. ą Next, let M be a quasi-triangulat dg-monad in V and V V1, ,VN be an M-algebra. Consider the free M!-coalgebra M! V with its differential d.“p As before,¨ ¨ ¨ itq has a unique dif- p q ferential D, compatible with the coalgebra structure, whose matrix elements Mij Vi Vj, 2 p q Ñ i jm are given by the action maps αij for V . It satisfies D 0, dD Dd 0, so we haveă the M !-coalgebra V ! M! V ,d D . Explicitly, “ “ “ “p p q ` q

! V Tot Mi ,i Mi ,j Vi Mij Vi Vj . j “ ¨¨¨ ÝÑ 1 2 p 1 p 1 qq ÝÑ p q ÝÑ " i1ăi2ăj iăj * à à

98 Proposition 3.4.19. The functor V V ! defines a quasi-equivalence of dg-categories ÞÑ AlgM CoalgM! . Ñ

Proof: Again, this is completely similar to the corresponding statement for algebras over an

! ! operad [44]. More precisely, we also have the dual construct! ion, which for any uni-triangular dg-comonad E and an E-coalgebra C defines an E -coalgebra C E C! ,d D . Then

“ p p! q ` q one proves easily that for any V AlgM we have a! quasi-isomorphism V V and for any P ! Ñ C CoalgE we have a quasi-isomorphism C C . P Ñ Note that Koszul duality can be expressed through mutations.

Proposition 3.4.20. Let M be a quasi-triangular dg-monad. The comonad M! is identified ˚ with LδM , where δ BrN is the Garside discriminant (3.3.10) p q P This is a direct generalization of a statement about mutations of exceptional collections (when instead of monads we have algebras) which was proved in [17].

Proof: Direct comparison: unraveling the definition of LδM, we find that it involves only the ˚ ! left adjoints Mij and after passing to the right adjoints we recover M . Corollary 3.4.21. Let M be a quasi-triangular dg-monad. Assuming the relevant double ˚˚ ˚˚ ˚˚ ˚˚ adjoints exist, the dg-monads M Mij and M M are identified with L 2 M and “ p q “ p ij q δ Lδ´2 M respectively.

E. A8-monads and their straightening. As dg-monads can be seen as associative dg-algebras in the category of functors, one can also consider the homotopy, or A8-version, in which associativity holds only up to a coherent system of higher homotopies. This version is similar to the classical concept of A8-algebras [64, 67] and we discuss it briefly in our context. 7 Let V V1, , VN be a sequence of pre-triangulated dg-categories. Denote V the “ p ¨ ¨ ¨ q i graded category obtained from Vi by forgetting the differentials in the Hom-complexes. We 7 can consider Vi as a dg-category with trivial differential.

Definition 3.4.22. A uni-triangular A8-monad M in V is a datum of:

7 7 7 (1) A collection M of dg-functors Mij : V V i j p i Ñ j q ă 7 2 (2) A differential Q QM in the free comonad FC M 1 of degree 1, satisfying Q 0 and compatible with“ the cocomposition. p r sq ` “

An A8-morphism of unitriangular A8-monads M L is, by definition, a morphism of 7 Ñ7 unitriangular dg-comonads FC M 1 , QM FC L 1 , QL . p p r s qÑp p r s q Explicitly, Q QM is determined by its matrix elements with values in the “space of cogenerators” M 7 “1 . After moving all the shifts to the right, these matrix elements can be r s

99 written as: dij : Mij Mij 1 , i j; ÝÑ r´ s ă cijk : Mjk Mij Mik, i j k; ˝ ÝÑ ă ă cijkl : Mkl Mjk Mij Mil 1 , i j k l; ˝ ˝ ÝÑ r s ă ă ă ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ ci , ,i : Mi ,i Mi ,i Mi ,i p 3 , i1 ip; 1 ¨¨¨ p p´1 p ˝¨¨¨˝ 1 2 ÝÑ 1 p r ´ s 㨨¨ă ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ The condition Q2 0 unravels, in a standard way, to a series of conditions: “ 2 (0) d 0, so each Mij becomes a dg-functor. We denote by d the differentials in all ij “ dg-functors obtained from the Mij by composition.

(1) d,cijk 0, i.e., cijk is a morphism of dg-functors. r s“ (2) d,cijkl is the associator for the cijk, i.e., the difference between the two paths in the squarer s of Definition 3.4.1(a). ( ) And so on. ¨ ¨ ¨

Thus the ci1,¨¨¨ ,ip form a system of higher homotopies for associativity of the cijk. Note that a uni-triangular A8-monad M in V gives a uni-triangular! (strictly coassociative) dg-comonad M ! FC M7 1 , Q . Therefore the Koszul dual M! is a strictly associative uni-

“p p r s q triangular dg-monad, which we call the straightening of M. Further,! in the same way as in Proposition 3.4.18, we have a canonical A -quasi-isomorphism M! M of A -monads. 8 Ñ 8 This procedure of straightening allows us to ignore the A8-issues in the sequel, even though many constructions of monads in the later sections produce, strictly speaking, A8- monads.

3.5 The Fukaya-Seidel category of a schober on C A. Uni-triangular Barr-Beck theory for spherical functors. Let

(3.5.1) Φ Φ 1 ❇ N ❇❇ ¨ ¨ ¨ ④④ ❇❇ ④④ a1 ❇❇ ④④ aN ❇! ④④ Ψ } be a diagram of N spherical dg-functors with common target, as in §3.1 C. It gives a uni- triangular dg-monad

˚ (3.5.2) M M a1, , aN Mij a ai : Φi Φj “ p ¨ ¨ ¨ q “ “ j ˝ ÝÑ iăj ` ˘ on Φ: Φ1, , ΦN . The composition “p ¨ ¨ ¨ q ˚ ˚ ˚ cijk : Mjk Mij a aj a ai a ai Mik ˝ “ k ˝ ˝ j ˝ ÝÑ k ˝ “ 100 ˚ ˚ is obtained from the counit ηj : aj aj IdΦj by composing with ak on the left and ai on the right. We will call M the Fukaya-Seidel˝ Ñ monad associated to the diagram (3.5.1).

Accordingly, we have a pre-triangulated category AlgM Φ1, , ΦN with its standard semi-orthogonal decomposition. “x ¨ ¨ ¨ y ˚ ˚ For any object U Ψ the collection b U a1 U , , aN U is naturally an M-algebra. The action map P p q“p p q ¨ ¨ ¨ p qq ˚ ˚ ˚ ˚ αij : Mij a U a ai a U a U p i p qq “ j i p q ÝÑ j p q ˚ is obtained from the counit ai a Id. This gives a dg-functor ˝ i Ñ

(3.5.3) b :Ψ AlgM ÝÑ which we call the Barr-Beck functor. ˚ Let us find the left adjoint b : AlgM Ψ. For an M-algebra V V1, ,VN , we define the Beck-Barr complex BB V to be theÑ following complex of objects“p of Ψ¨ ¨(with ¨ differentialsq being closed morphisms ofp degreeq 0):

˚ ˚ ˚ aj aj ai2 ai2 ai1 Vi1 aj aj ai Vi aj Vj . ¨¨¨ ÝÑ i1ăi2ăj p q ÝÑ iăj p q ÝÑ j p q à à à The grading is normalized so that the complex terminates in degree 0. The differential d on the p th term p´ q ˚ ˚ ˚ aj aj aip aip ai2 ai2 ai1 Vi1 i1㨨¨ăipăj ¨ ¨ ¨ p q à ˚ ˚ is the alternating sum of the results of applying the counits aj aj Id, , ai2 ai2 Id ˚ 2 ˝ Ñ ¨ ¨ ¨ ˝ Ñ and the action map a ai Vi Vi . It is clear that d 0. i2 1 p 1 qÑ 2 “ Proposition 3.5.4. The functor V Tot BB V is left adjoint to b. ÞÑ p p qq Proof: By definition of the Hom-complexes in AlgM, for any U Ψ we have P

‚ ‚ ˚ Hom V, b U Tot Hom Vj, a U AlgM p p qq “ Φj p j p qq Ñ " j à ‚ ˚ ˚ ‚ ˚ ˚ ˚ Hom a ai Vi , a U Hom a ai a ai Vi , a U Ñ Φj p j p q j p qq Ñ Φj p j 2 i2 1 p 1 q j p qqѨ¨¨ iăj i1ăi2ăj * à à ‚ Tot Hom aj Vj , U “ Ψp p q qÑ " j à ‚ ˚ ‚ ˚ ˚ Hom aj a ai Vi , U Hom aj a ai a ai Vi , U Ñ Ψp j p q qÑ Ψp j 2 i2 1 p 1 q qѨ¨¨ iăj i1ăi2ăj * à à Hom‚ Tot BB V , U . “ Ψp p p qq q

101 Remarks 3.5.5. (a) The complex BB V has some resemblance with B`, the augmented bar-complex of V (see the proof of Propositionp q 3.4.7) or, more precisely, with the direct sum ` aj B . In fact, the two complexes have the same terms but the differential in BB V j p j q p q has more summands than that in a B` . Of course, each B` and therefore a B` is À j j j j j exact while BB V is, in general, not. p q p q p q À À a (b) The classical Barr-Beck theory [7] concerns a single adjoint pair C o / D . In this a˚ ˚ case M a a : C C is a monad in the classical sense, and we have a functor b : D AlgM sending“U ˝ a˚ UÑ. The Barr-Beck monadicity theorem [7] gives a criterion for b toÑ be an ÞÑ p q equivalence. The complex BB V , V AlgM, can always be defined as a simplicial object in D. In our uni-triangular context,p q ourP interest is not in b being an equivalence but in a different property.

pbq Proposition 3.5.6. The functor b is spherical. The spherical cotwist TΨ Cone IdΨ ˚ ´1 ´1 paiq ˚“ t Ñ b b is identified with T T , where Ti,Ψ T Cone ai a IdΨ is the ˝ u N,Ψ ˝¨¨¨˝ 1,Ψ “ Ψ “ t ˝ i Ñ u spherical twist for ai. Proof: By Corollary 3.1.6 and Remark 3.1.7 it suffices to show that ˚b is spherical and to p˚bq show that T T1,Ψ TN,Ψ. So we prove the statements in this latter form. Ψ » ¨ ¨ ¨ p˚bq From the definition of b and from Proposition 3.5.4, we see that TΨ can be written as Tot C‚ , where p q ‚ ˚ ˚ ˚ C aj a ai a aj a Id “ ¨¨¨ ÝÑ j i ÝÑ j ÝÑ " iăj j * à à (the grading terminates in degree 0). This C‚ can be obtained as follows. Take the 2-term ˚ complexes of functors ai ai IdΨ , i 1, , N (with grading in degrees 1, 0) and compose them, gettingt an ˝N-foldÑ complexu “ (commutative¨ ¨ ¨ N-cube) ´

˚ ˚ a1 a IdΨ aN a IdΨ . t ˝ 1 Ñ u˝¨¨¨˝t ˝ N Ñ u Converting this cube, in a standard way, into a single complex gives C‚. But

‚ Tot C T1,Ψ TN,Ψ. p q » ˝¨¨¨ p˚bq This proves that TΨ is an equivalence and has the claimed form. ˚ p bq ˚ Now look at the functor T Cone IdAlg b b 1 . Let V V1, ,VN AlgM. AlgM “ t M Ñ ˝ ur´ s “p ¨ ¨ ¨ q P From Proposition 3.5.4, we find that TAlg V p is the total object of the double complex M p q

(3.5.7) Vp

e  ˚ ˚ ˚ ˚ ˚ ˚ / a aj a ai a ai Vi / a aja ai Vi / a aj Vj , ¨ ¨ ¨ i1ăi2ăj p j 2 i2 1 p 1 q iăj p j p q j p p q À ˚ À À where Vp is in bidegree 0, 0 and e takes Vp to a ap Vp by the unit of th adjunction. p q p p q 102 ˚ To analyze this, for any i, j 1, , N (not necessarily i j), let Mij aj ai : Φi Φj. lP t ¨ ¨ ¨ N u ă “ ˝ Ñ Then the matrix of functors M Mi,j is a “matrix monad”, i.e., we have compositions “p qi,j“1 cijk : Mjk Mij Mik for any i, j, k and the units ei : IdΦi Mii, satisfying the usual axioms. Let M˝ă, MěÑbe the matrices of functors given by Ñ

ă Mij, if i j; ě Mij, if i j; Mij ă Mij ě “ #0, otherwise; “ #0, otherwise.

So formally, Ml Mă Mě is the decomposition of Ml into the strict upper-triangular and non-strict lower-triangular“ ` parts. Our uni-triangular monad M can be formally written in this notation as M 1 Mă. “ ` p˚bq Preliminary step: Symbolic computation. To analyze TAlgM as a functor, we first do a symbolic computation with matrices, “at the level of Euler characteristics”, interpreting the total object of the double complex (3.5.7) for each p as the alternating sum of its terms. In p˚bq this symbolic way, TAlgM is represented by the terminating matrix series

1 Ml Ml Mă Ml Mă 2 ´ ` ¨ ´ ¨p q `¨¨¨ 1 Ml 1 Mă ´1 1 Mě 1 Mă ´1. “ ´ p ` q “ p ´ qp ` q 1 ě T ´1 Now, M is lower triangular with diagonal elements corresponding to the functors Φi ´ ˚ “ Cone IdΦi ai ai 1 . So, in our formal computation, these diagonal elements should be consideredt Ñ invertible˝ ur´ ands so 1 M ě is invertible, as is 1 Mă ´1. z p ` q Upgrade to functors. We now upgrade the above symbolic computation to an actual argument with functors. For this we first define the functors

ď ě T , T : AlgM AlgM. ÝÑ ď More precisely, define T to send V AlgM to P

Tot Mă Mă V Mă V α V . ¨¨¨ ÝÑ p p qq ÝÑ p q ÝÑ " * Here the complex in curly brackets is a subcomplex of the augmented bar-complex of V (see the proof of Proposition 3.4.7), but it is not exact. By construction, T ď is upper- unitriangular, i.e., it sends each Φi into the subcategory Φ1, , Φi . In this, it is an analog of the matrix 1 Mă ´1 in the above preliminary reasoning.x ¨ ¨ ¨ Notey further that T ď is an equivalence forp the` sameq reason a unitriangular matrix is invertible. That is, T ď preserves the admissible filtration

Φ1 Φ1, Φ2 Φ1, , ΦN AlgM x yĂx y訨Ăx ¨ ¨ ¨ y “ corresponding to our original SOD Φ1, , ΦN , and induces identity on each quotient p ¨ ¨ ¨ q Φ1, , Φi Φ1, , Φi 1 Φi. x ¨ ¨ ¨ y x ¨ ¨ ¨ ´ y“ L 103 ě Further, define the functor M : AlgM AlgM by Ñ ě M V i Mpi Vp . p p qq “ pěi p q à This is made into an M-algebra as follows. For i j the map ă

Mij Mpi Vp MijMpi Vp Mpj Vp p q “ p q ÝÑ p q ˆ pěi ˙ pěi pěj à à à sends the summand MijMpi Vp to Mpj Vp via cijp : MijMpi Mpj, if j p, and sends such a summand to 0, if j p. p q p q Ñ ě ą ε ě We have a natural transformation Id M given by the units of the adjunction IdΦi ˚ ě Ñ Ñ Mii a ai. Now define T Cone ε 1 , so “ i ˝ “ p qr´ s T ě V T ´1 V 1 V 1 . i Φi i Mpi p p p qq “ p qr´ s ‘ pąi p qr´ s à By construction, T ě preserves the admissible filtration

ΦN ΦN 1, ΦN Φ1, , ΦN AlgM x yĂx ´ y訨Ăx ¨ ¨ ¨ y “ corresponding to the mutated SOD Lδ Φ1, , ΦN , see Proposition 3.3.11, and induces on each quotient p ¨ ¨ ¨ q Φi, Φi 1, , ΦN Φi 1, , ΦN Φi x ` ¨ ¨ ¨ y x ` ¨ ¨ ¨ y » T ´1 T ě an equivalence, namely Φi . Therefore isL an equivalence as well. It is the functor upgrade of the matrix 1 Mě from the above preliminary step. ´ ˚ ˚ p bq ě ď p bq Finally, we notice that TAlgM T T and so it is an equivalence. Since both TΨ and ˚ “ ˝ p bq ˚ TAlgM are equivalences, the functor b is spherical and so is b.

˚ p bq ě ď L ` L ´ ´1 Remark 3.5.8. The splitting TAlgM T T is analogous to the formula Tnorm C T C representing the normalized monodromy» ˝ of the Fourier transform of a perverse“ sheaf in termsp q of the Stokes matrices, see Proposition 2.5.10. More precisely, T ě is analogous to the productr C`T L, the latter being non-strict lower triangular with isomorphisms (monodromies) on the diagonal. The functor T ď, whose very shape is suggestive of a geometric series, is analogous to C´ ´1 (strict upper-triangular with identities on the diagonal). p q

B.r The Fukaya-Seidel category. Let now A w1, ,wN C and S Schob D, A be a schober on C with singularities in A. Topologically,“ t C¨ ¨is ¨ the sameuĂ as the standardP p openq disk, so we use the approach of §3.1 C. That is, we choose a direction at infinity which we think of as a complex number ζ with ζ 1 and take a “Vladivostok” point v Rζ, R 0 far away in the direction ζ. Let K be| a |“v-spider for C, A . Then S is represented“ by a dia-" p q gram (3.1.15) or (3.5.1) in the presense of K. We denote the spherical functor ai : Φi Ψ by Ñ

104 ai,K to emphasize its dependence on K. In particular, we have the uni-triangular dg-monad M M K M a1,K , , aN,K as in (3.5.2). “Wep defineq“ p the Fukaya-Seidel¨ ¨ ¨ q category of S relatively to v and K to be

(3.5.9) FK C, v; S : AlgM . p q “ pKq We now study the dependence of this construction on K. Recall that the set K C, v, A p q of v-spiders for C, A is acted upon, simply transitively, by the group BrN , see §1.1 E. By p q Proposition 3.1.17, the effect of this action on the functors ai is found, on the generators τi BrN , by P

a1,τ K , , aN,τ K a1,K, , ai 1,K, Ti,Ψai 1,K, ai,K, ai 2,K , , aN,K . ip q ¨ ¨ ¨ ip q “ ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ ` ˘ ` ˘ Proposition 3.5.10. Let ai : Φi Ψ, i 1, , N, be spherical dg-functors as in (3.5.1). For i 1, , N 1 let Ñ “ ¨ ¨ ¨ “ ¨ ¨ ¨ ´

a1, , aN a1, , ai 1, Ti,Ψai 1, ai, ai 2, , aN . p ¨ ¨ ¨ q“p ¨ ¨ ¨ ´ ` ` ¨ ¨ ¨ q Then we have a natural quasi-isomorphism of dg-monads

M a1, , aN LiM a1, , aN δ1,i, , δN,i . p ¨ ¨ ¨ q»p p ¨ ¨ ¨ qqr ¨ ¨ ¨ s

Here δν,i is the Kronecker symbol and the square brackets mean the shift of the monad (3.4.15).

˚ Proof: This is a direct comparison. Let M M a1, , aN . We express the Mν,j aj aν ˚ “ p ¨ ¨ ¨ q “ through the Mij a aν using the definition of the aν to arrive at the formulas (3.4.12) for “ j the left mutation. The only new aν are

˚ ˚ ai Ti,Ψ ai 1 Cone Id ai ai 1 ai 1 Cone ai 1 ai ai ai 1 1 , “ ` “ t Ñ ˝ ur´ s ˝ ` “ t ` Ñ ˝ ˝ ` ur´ s

and ai 1 ai. ` “ So Mν,j Mν,j for ν, j i, i 1 , which matches the 1st, 4th and last lines of (3.4.12). “ R t ` u If ν i 1 and j i, then ď ´ “ 8 ˚ ˚ ˚ ˚ ˚ ˚ Mν,i ai aν Cone ai`1 ai ai ai`1 aν Cone ai`1 ai ai aν ai`1 aν “ “ t ˝ ˝ 1Ñ u ˝ “ t ˝ ˝ ˝ Ñ ˝ u Cone Mi,i 1 Mν,i Mν,i 1 M 1 , “ t ` ˝ Ñ ` u “ ν,ir s 1 where Mν,i is given by the second line of (3.4.12). If ν i 1, j i 1, then ď ´ “ ` ˚ ˚ 1 Mν,i 1 a aν a aν Mν,i M , ` “ i`1 “ i “ “ ν,i`1 as given by the third line of (3.4.12).

105 If ν i, j i 1, then “ “ ` ˚ ˚ ˚ Mi,i`1 ai`1ai ai Cone ai`1 ai ai ai`1 1 ˚ “ ˚ “˚ ˝ t Ñ ˝ ˚ ˝ ˚ ur´ s “˚ Cone a ai 1 a ai ai ai 1 1 Cone a a ai ai ai 1 1 . “ t i ˝ ` Ñ i ˝ ˝ ˝ ` ur´ s “ t i Ñ i ˝ ˝ u ˝ ` r´ s ˚ By Proposition 3.1.3(1), the last cone is identified with ai, so we get ˚ ˚ 1 aiai 1 1 Mi,i 1 1 M 1 , ` r´ s “ ` r´ s “ i,i`1r´ s see the 5th line of (3.4.12). 1 In this way we find that Mν,j M δi,j δi,ν as claimed. “ ν,jr ´ s Corollary 3.5.11. The categories FK C,ζ; S for different K K C, v, A , are canonically (uniquely up to a contractible space ofp higher homotopies)q identifiedP p with eachq other. Proof: Follows from Proposition 3.5.10, and from the identification (3.4.16). Definition 3.5.12. (a) The Fukaya-Seidel category of a schober S on C with respect to a direction ζ at infinity is defined as

F C,ζ; S : FK C, v; S , v Rζ, R 0 and K K C, v, A . p q “ p q @ “ " P p q (b) Similarly, the Fukaya-Seidel category of a schober S on a disk D with respect to a boundary point v D is defined as PB F D, v; S : FK D, v; S , K K D, v, A . p q “ p q @ P p q 3.6 The Lefschetz schober of a Landau-Ginzburg potential In this section we explain how various constructions related to the “classical” Fukaya-Seidel categories in symplectic geometry [86] can be interpreted in our schober language.

A. The idea of the Lefschetz schober. We consider the situation of §1.2 A. That is, X is a Riemann surface and W : Y X is an X-valued generalized Lefschetz pencil of Ñ relative dimension n (Definition 1.2.1). Let A w1, ,wN X be the set of singular values of W . In this case we have the Lefschetz“ perverse t ¨ ¨ ¨ sheafu Ă n LW R W k Perv X, A . “ perv ˚ Y P p q We will construct a natural categorification of L which we call the Lefschetz schober and denote LW Schob X, A . The data of Definition 3.1.10 defining LW will be as follows: P p q • The local system of categories LW 0 on X A will consist of the Fukaya categories of the W ´1 x , x X A, which arep consideredq z as compact real symplectic manifolds with the symplecticp q P formz given by a Kähler metric.

1 • The local system Φi LW on S will correspond to the local Fukaya-Seidel category p q wi ΦSi W with the natural monodromy action, see below. p q Thus, in our “upgrade” from perverse sheaves to perverse schobers, the Fukaya category categorifies the middle (co)homology while ΦSi W categorifies, following [86], the space of vanishing cycles. p q

106 B. Reminder on Fukaya categories. We first recall, following [86], the construction of the Fukaya category of a compact Kähler manifold. We start with recalling the concept of a quadratic differential, necessary to define a dg-category with Z-graded (not just Z 2-graded) Hom-complexes. {

Let V be a C-vector space of dimension n. Denote by GRR n, V the open part of the p q Grassmannian GR n, V of n-dimensional real subspaces W V formed by W which are totally real, i.e., Wp iWq V . Ă A quadratic differential‘ “on V is a C-linear map η : ΛnV b2 C. A nonzero quadratic differential η on V gives a map, called the phase functionp q Ñ

b2 1 η e1 ... en α : GRR n, V S , W p ^ ^ q , η b2 p q ÝÑ ÞÑ η e1 ... en `p ^ ^ q ˘ ˇ ˇ where e1, , en is any R- basis of W . ` ˘ t ¨ ¨ ¨ u ˇ ˇ Let now Z be a compact complex manifold of dimension n. A quadratic differential on Z is a holomorphic section η Γ Z, Ωn b2 . Thus, for each z Z, we have a quadratic P p p Zq q P differential ηz on the C-vector space TzZ. We say that Z is an Enriques manifold, if it admits a nowhere vanishing quadratic n b2 n differential, i..e, if ΩZ OZ . For example, any Calabi-Yau manifold (ΩZ OZ ) is obviously Enriques.p q » » In the sequel we will assume that Z is Enriques and fix a nowhere vanishing η. Assume further that Z is equipped with a Käbler metric h g iω, so ω is a symplectic form on “ ` Z. So we can speak about Lagrangan R-subspaces in the tangent spaces TzZ and about Lagrangian submanifolds in Z (considered as a 2n-dimensional real manifold).

Any Lagrangian subspace in any TzZ is totally real, so the phase functions αηz , z Z, assemble into a map (also called the phase function) P

1 αZ : LG TZ S , p q ÝÑ where LG TZ is the Lagrangian Grassmannian bundle associated to the tangent bundle TZ. p q The Fukaya category Fuk Z is a pre-triangulated dg-category defined as the pre-triangulated p q 1 envelope of (i.e., the category of twisted complexes over) the A8-category Fuk Z defined as follows. p q 1 5 Objects of Fuk Z are Lagrangian branes, i.e., data L L, αL,PL , where: p q “p q • L Z is a compact Lagrangian submanifold; Ă

• αL : L R is a smooth map (called a grading of L) such that ÝÑ

2πiαX pTz Lq αL z e , p q“

• PL is a Pin-structure on L.

107 So, a Lagrangian brane is a Lagrangian submanifold L decorated by some additional data. 5 5 Let L1, L2 be two Lagrangian branes. Let us suppose that L1 and L2 intersect trans- ‚ 5 5 ‚ 5 5 versely. One defines the Hom-complex C L1, L2 HomFuk1pZq L1, L2 as follows. As a k-vector space p q “ p q ‚ 5 5 k C L1, L2 : z p q “ zPL1XL2 ¨ à For each z L1 L2 we have two Lagrangian subspaces TzL1, TzL2 TzZ, and their index P X Ă i TzL1, TzL2 Z is defined as in [86] (11.25); to define it one uses the decorations. This p q P 5 5 defines a Z-grading on C L , L , by setting deg z i TzL1, TzL2 . p 1 2q p q“ p q 5 5 i 5 5 i`1 The Floer differential d dL ,L : C L , L C L , L is defined by its matrix “ 1 2 p 1 2q ÝÑ p 1 2q elements: for z L1 L2, deg z i P X p q“ dz n z,y y, “ p q y L L i`1 Pp 1ÿX 2q n z,y being the number, with appropriate signs, of pseudo-holomorphic (i.e., holomorphic withp respectq to a generic almost complex deformation of the complex structure) bigons in Z with vertices z and y. If L1 and L2 do not intersect transversely, one uses an appropriate Hamiltonian displace- ment of one of them, to make them transverse. Compositions are defined similarly, using the numbers of pseudo-holomorphic polygons, see [86]. We note the following:

Proposition 3.6.1 ([81]). For any Lagrangian brane L5 Ob Fuk Z we have a natural isomorphism P p p qq ˚ 5 5 ˚ H HomFuk Z L , L H L; k . p p qp qq “ p q C. The local system of Fukaya categories. Let now W : Y X be a generalized Ñ Lefschetz pencil of relative dimension n, with the set of singular values A X. Let X0 ´1 Ă ´1 “ X A and Y0 W X0 , so W : Y0 X0 is a smooth proper map. We write Yx W x forz the fibers“ of thisp map.q Ñ “ p q We assume that both X and Y are Enriques, with chosen nowhere vanishing quadratic differentials ηX and ηY . Then each Yx, x X0. is an Enriques manifold, with quadratic ηX P differential ηY ηY W Y . We fix a Kähler metric h on Y ; it induces a metric on each x “p { q| x Yx, x X0, so the category Fuk Yx is defined. P p q Recall (1.2.5) that the fibration W : Y0 X0 carries the symplectic connection ∇W . Ñ 1 In particular, any path γ in X0 joining two points x and x gives rise to a diffeomorphism Tγ : Yx Yx1 , preserving the symplectic structure. The connection ∇W is not flat, i.e., Ñ Tγ , as a diffeomorphism, can change under small deformations of γ, but the change is given by a Hamiltonian isotopy. Therefore ∇W gives rise to a local system Fuk Y0 X0 of pre- p { q triangulated dg-categories on X0 whose stalk at any x X0 is Fuk Yx . P p q

108 D. The case of a Lefschetz pencil. Assume further that W is a Lefschetz pencil, i.e., all the critical points yi Y are Morse. Let Di be a small closed disk around wi W yi , P “ p q choose a point εi Di and let γi a simple path in Di (“radius”) joining wi with εi. Let also ´1 PB Yi W wi , a compact Enriques Kähler manifold. As explained in §1.2, γi gives rise to “ p q ´1 the Lefschetz thimble Thi W Di whose projection to Di is γi. In particular, we have Ă p q n the geometric vanishing cycle ∆i Thi Yi, diffeomorphic to S , see Fig. 9. It is known “ X 5 that ∆i is Lagrangian in Yi and, moreover, has an upgrade to a Lagrangian brane ∆i, which is an object of Fuk Yi , see [86] §16f. Further, by Proposition 3.6.1, p q H‚ Hom‚ ∆5, ∆5 H‚ Sn, k . FukpYiqp i iq “ p q 5 whici, together with the fact that Fuk Yi is Calabi-Yau, means that ∆ is a spherical object p q i in Fuk Yi , see Example 3.1.9. This implies that the dg-functor p q b 5 ai : Φi : D Vectk Ψi : Fuk Yi , V V k ∆ “ ÝÑ “ p q ÞÑ b i is spherical.

The functor ai gives rise to a perverse schober LW,Di on Di with the only singularity at wi. In particular, on Ci Di, the schober LW,Di gives a local system of dg–categories coming from the self-equivalence“ B

˚ TΨ Cone ai a IdΨ : Ψi Ψi. i “ ˝ i ÝÑ i ÝÑ ( On the other hand, consider the local system LW 0 : Fuk Y0 X0 of dg-categories on X0 p q “ p { q “ X A. It has, as the stalk at εi, the same dg–category Ψi Fuk Yi , and the monodromy z k “ p q of L around Ci is, by construction, induced by the symplectomorphism TC : Yi Yi, the S i Ñ holonomy of the symplectic connection. It is known, see [86] §16c, that TCi is Hamiltonian isotopic to the Dehn twist around ∆i, which means that the induced self-equivalence on k Fuk Yi is identified with TΨ . Therefore, LW,D and L agree on Ci. p q i i W

Definition 3.6.2. We define the Lefschetz schober LW on X to be obtained by gluing the local system LW 0 on X0 with the schobers LW,D on each Di along the above identifications p q i on the Ci.

E. The case of a generalized Lefschetz pencil. Let now W : Y C be a generalized Ñ Lefschetz pencil, with yi Y being an isolated singular point with multiplicity µi. Let the P disk Di wi W yi , as well as εi Di and γi be as in § D. Q “ p q PB Note that µi is the dimension of the space of vanishing cycles of W at yi. The idea of categorifying this space, i.e., upgrading it to a Fukaya-style pre-triangulated (A8-)category ΦSi W , goes back to Kontsevich [66]. We will refer to ΦSi W as the local Fukaya-Seidel p q p q category of W at yi and note that there are two natural approaches to its construction. (1) Deformation approach. Deform W to an actual Lefschetz pencil W : Y C so that Ñ yi splits into µi Morse critical points yi,1, ,yi,µ , lying close to each other and projecting ¨ ¨ ¨ i Ă r 109 ´1 under W to distinct points wi,1, ,wi,µ Di. Denoting Yi W Di , we have a Lefschetz ¨ ¨ ¨ i P “ p q pencil W : Y D and so, by the above, the Lefschetz schober L on D . We then define i i i Wi i Ă Ñ r Ă ΦSi W ε F Di,εi; L , Ă r p q i “ p Wi q Ă to be the Fukaya-Seidel category of Di with coefficientsĂ in this schober or, more directly, as the Fukaya-Seidel category of Wi in the classical sense [86]. The result should, morally, not depend on the choice of the deformation, since, as far as local deformations (those in aĂ neighborhood of yi only) go, different choices are connected by the braid group action. Such an action translates into a mutation of SOD on the Fukaya- Seidel category while the category itself is unchanged, see Corollary 3.5.11. The difficulty with this approach is that one can guarantee the existence of a good deformation only locally (in a neighborhood of yi) and not as a proper map. For a non- proper map, however, the symplectic connection may be non-integrable, i.e., the parallel transport of a point can go to infinity. This makes it difficult to define the local system part of the schober L . Wi

(2) The directĂ approach. As in [66], in Def. 3.7 of [1] and in App. A of [2] we can take, as objects of ΦSi W ε , a certain clas of (possibly singular) Lagrangian branes which project p q i by W to the radial path εi and vanish to yi over wi. The implementation of this approach involves several subtle issues of symplectic geometry, for which we refer to §A.2 of [2] and references therein. Let us assume that we are in situation where this approach can be implemented. In this case Proposition A.4 and Theorem A.6 of [2] imply that we have a spherical functor

k ´1 ai : ΦSi W ε L ε Fuk W εi p q i ÝÑ p W q i “ p p qq whose associated spherical reflection is identified with the effect of the monodromy around k Di. This means that we can construct a schober LW by gluing L with the schobers on B W the Di associated to the ai, as before.

4 Algebra of the Infrared associated to a schober on C,I

In this and the next chapter we analyze schobers on C in the spirit of the Algebra of the Infrared, relating curvilinear and rectilinear transport functors. Such analysis involves natu- rally convex geometry of the set A, but splits naturally into two parts, which use respectively: (1) “Elementary” convex geometry related to the data of which elements of A lie in the convex hulls of which subsets. This data is formalized by the oriented matroid O A , see Definition 2.2.1. p q (2) “Advanced” convex geometry involving the concept of the secondary polytope Σ A . This polytope depends on more subtle aspects than just the data of O A . p q p q In the present Chapter 4 we concentrate on the first part, with the second part postponed till the next Chapter 5.

110 4.1 Rectilinear approach and the Fourier transform of a schober. A. Rectinilear approach to schobers on C. We now adopt the point of view of §2.1 and emphasize the rectilinear and convex-geometrical approach to the (purely topological) concept of a schober on C. We assume that A is in linearly general position and S Schob C, A . We distinguish 1 1 P Φ p q the horizontal direction (ζ ) in each circle Swi and denote i S the stalk of the local Φ 1 1 “ 1 p q system i S at Swi . For any distinct i, j , , N we have the rectilinear transport functor p q P P t ¨ ¨ ¨ u Mij : Mij wi,wj : Φi S Φj S , “ pr sq p q ÝÑ p q as in (2.1.5).

B. Fourier transform of a schober on C: topology. Let A w1, ,wN C. “ t ¨ ¨ ¨ u Ă Following the conventions of §2.3, we write C Cw and denote the dual complex line as “ Cz with coordinate z dual to w. If F Perv Cw, A is a perverse sheaf with singularities P p q in A, then we can define its Fourier transform F by (2.3.4), using the Riemann-Hilbert correspondence. Proposition 2.3.5 says that F Perv Cz, 0 . P q p q Let now S Schob Cw, A be a perverse schober with singularities in A. We would like P p q q to define its Fourier transform, which should be a new schober S Schob Cz, 0 . At present, it is not known how to extend the Riemann-Hilbert correspondenceP to schobers.p q However, the above constructions provide a working ad hoc definition. q First, we define two local systems of triangulated categories Φ S , Ψ S on the circle 1 p q p q 1 S ζ 1 which we think of as the circle of directions of Cz at 0. Given ζ S , let “ t| | “ u P vζ Rζ, R 0 be a “Vladivostok” point in Cw very far in the directionq qζ. We put “´ " ´

Φ S ζ Sv Ψ S p q “ ζ » 8p q´ζ

to be stalk of S at vζ which can beq thought as the stalk at ζ of the local system of nearby cycles of S near . Further, we put ´ 8

(4.1.1) Ψ S ζ F Cw, ζ; S FK C,ζ; S Alg p q “ p ´ q “ p q “ MK to be the Fukaya-Seidelq category of S in the direction ζ, see Definition 3.5.12(a). Here ´ K is any vζ -spider for A. Further, choosing some particular value of ζ, we define S to be represented by the Barr-Beck spherical functor q b : Φ S ζ Sv Alg Ψ S ζ p q “ ζ ÝÑ MK “ p q of (3.5.3). Comparison with Propositionq 2.3.10 shows that thisq is indeed a natural categorical analog. Study of Fukaya-Seidel categories can be therefore seen as study of the Fourier transform for schobers.

111 C. Categorical Stokes structures. A categorification of the concept of a Stokes struc- ture (i.e., data defining “irregular perverse schobers” in complex dimension 1) was introduced by Kuwagaki [69] (and also in an earlier unpublished note [62]). Here we recall this cate- gorified concept in a simple form convenient for us. Let Λ be a local system of finite posets on S1. Thus for each ζ S1 we have a finite set P Λζ with a partial order ζ . Let us assume the following two properties of Λ (the second one added for simplicity). ď

(Λ1) The order ζ is a total order for any ζ outside of a finite set ∆ Λ (the anti-Stokes directions).ď Ă

(Λ2) For ζ ∆ the order ζ has exactly two incomparable elements, whose order switches P ď as we cross ζ. That is, if ζ´,ζ` are nearby directions immediately before and after ζ in the counterclockwise order, then the two orders ζ , ζ (which can be considered ď ` ď ´ on the same set Λζ) have the form (for some k)

λ1 ζ λ2 ζ ζ λk ζ λk 1 ζ ζ λN , ă ´ ă ´ ¨¨¨ă ´ ă ´ ` ă ´ ¨¨¨ă ´ λ1 ζ λ2 ζ ζ λk 1 ζ λk ζ ζ λN ă ` ă ` ¨¨¨ă ` ` ă ` ă ` ¨¨¨ă ` Definition 4.1.2. Let Λ be a local system of posets on S1 satisfying (Λ1) and (Λ2). A categorical Λ-Stokes structure is a datum of:

(1) A local system L of triangulated categories on S1.

(2) For each ζ S1 ∆, an -admissible filtration (Definition 3.3.8) F ζ on the triangulated P z 8 category Lζ indexed by the totally ordered set Λζ, ζ , so that: p ď q (3) F ζ is covariantly constant on (each component of) S1 ∆. z ζ` ζ´ (4) If ζ ∆, then in the notation of (Λ2) we have F τkF (the result of mutation P “ corresponding to the generator τk BrN ). P The definition implies that the quotient categories

F K 1 1 gr Lζ FλLζ Fλ1 Lζ Fλ1 Lζ , λ maxµ λ µ, ζ § ∆, λ “ “ p qF λLζ “ ă P z L 1 i.e., the terms of the SOD’s associated to the -admissible filtrations in Lζ , ζ S ∆, unite into a Λ-graded local system of triangulated categories8 L grF L on all of S1.P Notez that in this categorical context, a splitting, i.e., an identification L“ L usually does not exist, even locally. Ñ

112 D. Fourier transform of a schober: Stokes structure. We now specialize to the case when Λ A is the constant sheaf corresponding to A w1, ,wN C in strong “ “ t ¨ ¨ ¨ u Ă linearly general position and ζ is given by the dominance order of the exponentials (2.4.6). Then A satisfies (Λ1) and (Λ2ď) and we will speak about A-Stokes structures. The set ∆ of

anti-Stokes directions for A consists of the directions ζij, where ζij are the slopes from of (2.1.4). Let us construct an A-Stokes structure on the local system L Ψ S formed by the Fukaya-Seidel categories (4.1.1). Let ζ S1 be non-anti-Stokes, so“ we havep q a picture as in P Fig. 14. Let vζ Rζ, R 0 be very far in the direction ζ. We canq then take the “ ´ " ´ vζ -spider for A of the form N

K vζ wi, vζ , p q “ i“1 r s ď the union of the straight intervals. As R 0, it approximates the union K ζ of straight rays " p q Ki ζ in the direction ζ from Fig. 14. By definition, Ψ S ζ is the Fukaya-Seidel category p q p´ q p q FK v Cw, vζ ; S , i.e., the category of algebras over the uni-triangular monad MK v . As p ζ qp q p ζ q such, it comes with a semi-orthogonal decomposition or, equq ivalently, see Definition 3.3.8, with an -admissible filtration F ζ. 8 ζ Proposition 4.1.3. The filtrations F at the non-anti-Stokes fibers Ψ S ζ form a categori- fied A-Stokes structure. p q q Proof: Let us look at the effect of crossing an anti-Stokes direction ζij. Let ζ´, ζ` be nearby v v´ directions immediately before and after ζij and K ζ´ , K ζ` be the corresponding v ´ p q p q v rectilinear spiders. Then K ζ` is isotopic to the spider obtained from K ζ´ by an elementary braiding of the ithp andqjth tentacles (these tentacles will have successivep numbersq , say k and k 1 in the numeration according to ζ ). So the effect of crossing ζ on the ` ă ´ ´ ij filtration is, by Propositions 3.1.17 and 3.5.10, is the mutation corresponding to τk.

4.2 The infrared complex of a schober. A. The setup. We now present categorical analogs of formulas of §2.5 expressing the transport along a curved path in terms of a sum of rectilinear ones. This sum will be categorified by a complex of functors while the proof, involving Picard-Lefschetz identities, will be categorifed by a diagram of exact triangles which generalize Postnikov systems familiar in homotopy theory and homological algebra [40].

We assume that A w1, ,wN C is in linearly general position, denote Q Conv A and fix a schober“ tS Schob¨ ¨ ¨ Cu, A Ă . Consider the situation depicted on the right of“ p q P p q the Fig. 18, reproduced below on the Fig. 30, so we have an edge wi,wj of Q and a path r s ξ from wi to wj which circumnavigates Q. We would like to express the transport functor M ξ : Φi Φj in terms of compositions of rectilinear transports associated to convex paths. pAccordingly,q Ñ we will categorify the sum formula of Proposition 2.5.3.

113 B. Paths as graphs of functions. We further assume that there is an affine coordinate system x, y on C R2 such that: p q “ • wi and wj lie on the horizontal axis y 0. “ • The projection of Q to the horizontal axis along the vertical one is the interval wi,wj and not a bigger set. Further, no edges of Q are vertical, see Fig. 30. r s This assumption will be automatically satisfied in our main application. In general it can always be ensured by peforming a real projective transformation of R2 (preserving straight lines and therefore preserving convex geometry).

y

ξ

‚ ‚ wi2 ‚ wi ‚3 ‚ Q x ‚ ‚ wi wj

Figure 30: Convex paths as graphs of convex functions.

Let P be the (infinite-dimensional) convex cone formed by all convex functions f : wi,wj R 0 such that f wi f wj 0. Each such f gives a path γf , the graph r s Ñ ě p q “ p q “ of f, joining wi and wj. We refer to paths γ of the form γ γf as convex paths. In particular, we can assume our circumnavigating path ξ to be convex.“

Each point wν A, ν i, j, gives an affine hyperplane Hν in P formed by all convex P ‰ paths γ passing through wν. If wν has coordinates xν,yν in our system, then Hν consists p q of all convex functions f such that f xν yν. This condition is given by vanishing of the affine-linear functional p q “

(4.2.1) lν : P R, lν f f xν yν. ÝÑ p q“ p q´

We get an affine hyperplane arrangement H Hν in P. In particular, we can speak about “ t u flats of H, i.e., intersections FI : Hν for vanious subsets I 1, , N i, j . This “ νPI Ă t ¨ ¨ ¨ uzt u includes I which corresponds to FH P. Each flat F has well-defined codimenion “ H Ş “ codim F , defined as the number of the Hν containing F . This definition is meaningful becausep ofq the following.

114 Proposition 4.2.2. The hyperplanes from H intersect transversely.

Proof: The statement means that for any I J 1, , N i, j such that I J and Ă Ă t ¨ ¨ ¨ uzt u ‰ FI , the inclusion FJ FI is proper. But this is clear, as among convex paths passing ‰ H Ă through all the wν, ν I (which exist by the assumption FI ) we can always find a path P ‰ H not passing through any other wµ,µ I. R Further, the arrangement H subdivides P into faces which are maximal subsets on which each equation lν has the same sign , , 0. Faces are locally closed convex sets. A face Γ ` ´ can be considered as an isotopy class of convex paths γ joining wi with wj, with isotopies understood relatively to the set A. The set of faces will be denoted by P . For a face Γ P we denote by Aff Γ its affine envelope which is the minimal flat of H containing Γ. ThusP Γ is open on Aff Γ p, andq we denote codim Γ : codim Aff Γ . In particular,p q open faces (i.e., facesp ofq codimension“ p 0p)qq correspond to empty paths, i.e., paths not containing any wν, ν i, j. ‰ C. The infrared Postnikov system. Each face Γ P gives a transport functor M Γ : P p q Φi Φj. Explicitly, we choose any path γ Γ and denote wk wi,wk , ,wk wj all Ñ P 1 “ 2 ¨ ¨ ¨ p “ the elements of A lying on γ, in order from wi to wj. Denote γν be the segment of γ between wkν and wkν`1 . Then

M Γ : Mk ,k γp 1 Mk ,k γ1 . p q “ p´1 p p ´ q˝¨¨¨˝ 1 2 p q Following Convention 1.1.11, the composition of the transport functors above is defined by identifying the stalks of the intermediate local systems Φ using clockwise monodromies, i.e., along the arcs going inside Q.

Definition 4.2.3. Three faces Γ , Γ0, Γ P are said to form a wall-crossing triple, if: ´ ` P

(1) Γ´ and Γ` are open in the same flat F Aff Γ´ Aff Γ` . In particular, they have the same codimension, say r. “ p q“ p q

(2) Γ0 has codimension r 1 and separates Γ and Γ in F . That is, Γ0 is contained in ` ´ ` the closures of both Γ and Γ , and Aff Γ0 is the intersection of F and one more ´ ` p q (unique, by Proposition 4.2.2) hyperplane Hν.

(3) The equation lν of Hν is negative on Γ´, zero on Γ0 and positive on Γ`. See Fig. 31.

Γ´ Γ0 Γ` lν increases ‚

Figure 31: A wall-crossing triple.

Thus, crossing the “wall” Γ0 in the direction from Γ´ to Γ` correspond, in terms of a path γ, to moving γ across one point wν, in the direction from below to above. Crossing just this

115 one wall and no others means that no other points on A leave γ or appear on γ. Therefore we have a Picard-Lefschetz situation and the corresponding Picard-Lefschetz triangle gives rise to the wall-crossing triangle of functors

vΓ0,Γ` zΓ`,Γ´ uΓ´,Γ0 (4.2.4) M Γ0 M Γ M Γ M Γ0 1 . p q ÝÑ p `q ÝÑ p ´q ÝÑ p qr s Definition 4.2.5. The diagram (in the homotopy category of dg-functors Φi Φj) formed Ñ by the functors M Γ , Γ P and the natural transformations vΓ ,Γ , zΓ ,Γ and uΓ ,Γ for p q P 0 ` ` ´ ` 0 all wall-crossing triples of faces Γ´, Γ0, Γ` , will be called the infrared Postnikov system and denoted P. p q Note that any arrow of P is contained in a unique exact triangle.

D. Examples of infrared Postnikov systems. Example 4.2.6 (The maximally non-convex position). Let A be as in Fig. 32 on the left. That is, Q is a triangle with one side w1,wN , and all the other points w2, ,wN´1 A are positioned very close to the perpendicularr tos that side (but not exactly on¨ ¨ ¨ the perpen-P dicular, to satisfy the assumption that A is in linearly general position). The face structure of the hyperplane arrangement P is depicted on the right of Fig. 32 in the form of a 1-dimensional transversal slice. Open faces correspond to empty paths, i.e., paths containing only w1 and wN ; there are N 1 such faces, corresponding to the paths ´ ξ1 w1,wN , ξ2 and so on up to the circumnavigating path ξ ξN 1. Faces of codimension “r s “ ´ 1 correspond to polygonal paths 12N w1,w2 w2,wN , 13N w1,w3 w3,wN and so on up to 1, N 1, N . r s“r sYr s r s“r sYr s r ´ s

ξN´1 ξ wN´1 “ ‚ P ξN´2 ‚ wN´2 . ξ1 ξ2 ξN´2 ξN´1 . ‚‚¨ ¨ ¨ ‚‚ ξ2 . r12Ns r12Ns r1, N ´ 1, Ns ‚ w2 ‚ ‚ w1 ξ1 w1,wN wN “r s Figure 32: The maximally nonconvex position.

The diagram P in this case is a Postnikov system in the usual sense, i.e., (see [40] or [55] §1A) a sequence of interlocking exact triangles of the form:

116 M2N M12 M3N M13 MN´1,N M1,N´1 > ❉❉ = ❉❉ 8 ▼ `1 ⑥⑥ ❉ `1 ③③ ❉ `1 qq ▼▼ ⑥⑥ ❉❉ ③③ ❉❉ qqq ▼▼▼ ⑥⑥ ❉❉ ③③ ❉❉ qqq ▼▼▼ ⑥⑥ ❉! ③③ ❉! qq ▼& M1N o Mpξ2q o Mpξ3q o ¨ ¨ ¨ o MpξN´2q o MpξN´1q.

As with any Postnikov system, P can be viewed in two different ways, which emphasize two different parts of the diagram. P as an analog of a cofiltration. The part of P corresponding to empty paths (open cells) M ξ M ξN 1 M ξ2 M ξ1 M1N p q“ p ´ q ÝѨ¨¨ ÝÑ p q ÝÑ p q“ can be seen as an analog of a cofiltration in M ξ . The polygonal transfers, i.e., M1N and p q M1iMiN , play the role of kernels of the cofiltration. So P represents M ξ as “assembled” out of these kernels. p q P as a complex. Composing the adjacent non-horizontal arrows in P, we get a diagram in the derived category involving only polygonal transfers:

‚ R M1N M2N M12 1 M3N M13 2 MN 1,N M1,N 1 N 2 . “ ÝÑ r s ÝÑ r s ÝѨ¨¨ ÝÑ ´ ´ r ´ s It is a cochain complex: the composition of any two consecutive arrows is zero.( So P represents M ξ as a “total object” of the complex I‚. p q Example 4.2.7 (The convex position). Now assume that A w1, ,wN consists of vertices of a convex N-gon, numbered clockwise, see Fig. 33 for “N t 4.¨ ¨ ¨ u “

r134s ξ

ζ1 η1 14 r s η1 r12sY ζ1 r1234s r124s ‚ ‚ ‚ w2 w3 ξ η2

‚ ‚ 2 34 w1 w4 ζ Y r s

Figure 33: Convex position: N 4. “ In this case H consists of N 2 hyperplanes intersecting transversely along a flat of codimension N 2, so we have 3´N´2 faces. For N 4 they are depicted on the right of ´ “ 117 Fig. 33. Here, along with the paths ξ (the circumnavigating path), η1 and ζ1 depicted on the left figure, we use the paths η2 and ζ2 which are symmetric to η1 and ζ1 with respect to the 2 vertical axis of symmetry of Q. For example, ζ joins w1 with w3, going over w2. We also write ij for wi,wj , as well as ijk for wi,wj wj,wk etc. r s r s r s r sYr s The infrared Postnikov system P is a commutative 3 3 3-diagram consisting of exact triangles (written linearly) in each of the N 2 directions.ˆ ˆ¨¨¨ˆ For example, for N 4 we get a 3 3 diagram with all rows and columns being´ exact triangles: “ ˆ 1 1 (4.2.8) M ξ / M η / M ζ M12 p q p q p q

 2   M η / M14 / M24M12 p q

 2   M34M ζ / M34M13 / M34M23M12. p q This is not a Postnikov system in the standard sense but it can be analyzed along the same lines as in Example 4.2.6: P as an analog of a filtration. The part of P formed by empty paths, corresponding to the upper left 2 2 square in (4.2.8), can be seen as an analog of a cofiltration in M ξ . This “cofiltration”,ˆ however, is labelled not by a totally ordered set but by a poset: the posetp q of all subsets in 2, , N 1 . Note that we are working in the derived category, and the naive abelian-categoricalt ¨ ¨ ¨ definition´ u (2.4.1) of the kernels of a cofiltration labelled by a poset, does not admit an immediate generalization to this case. P as a complex. The part of the diagram P “opposite” to the above, i.e., corresponding to the lower right 2 2 square in (4.2.8), is a commutative N 2 -dimensional hypercube formed by polygonalˆ transfers. This hypercube can be converp ted´ intoq a cochain complex R‚ in a standard way. The entire diagram can be seen as expressing M ξ as a total object of R‚ in the sense that we will make precise later. p q

E. The infrared complex. We now develop the cochain complex point of view on the infrared Postnikov system P in the general context of Definition 4.2.5. As in (2.5.1), let Λ Λ i, j P be the set of (isotopy classes of) polygonal convex “ p q Ă paths γ with vertices in A joining wi and wj. Note that each such isotopy class contains precisely one polygonal path, so we can think of L consisting of polygonal paths themselves. For such a γ we put l γ A γ wi,wj to be the set of the intermediate vertices of γ and l γ l γ the numberp q“ of suchX points,zt cf.u Proposition 2.5.4. p q “ | p q| Consider also the height function

h :Λ Z , h γ h γ , h γ : A Conv γ wi,wj , ÝÑ ` p q “ | p q| p q “ X p qzt u counting the number of “new” points (i.e., points other than wi,wj) on or under γ. Thus h γ l γ is the number of “new” points strictly under γ. Let also or γ max khpγq be thep q´1-dimensionalp q “orientation space” of Γ. p q“ p q Ź 118 Let Λm Λ be the set of paths with height m. Ă We now define a cochain complex of functors

R‚ 0 R0 q R1 q R2 q , “ Ñ Ñ Ñ Ñ¨¨¨ (4.2.9) m R : M γ k or γ dgFun Φ(i, Φj . “ γPΛm p qb p q P p q à The differential q : Rm Rm`1 is defined as follows. Ñ Let γ Λm 1 and w l γ be an intermediate vertex. The reduction of γ at w, denoted P ` P p q wγ, is a new convex path obtained by removing the vertex w from γ and taking the convex hullB of the remaining vertices, see Fig. 34. Thus, a 2-segment part γ w1,w,w2 of γ is replaced by a p 1-segment path “r s ` 1 1 2 γ w ,wj , ,wj ,w p 0, “ r 1 ¨ ¨ ¨ p s ě 1 of γ wγ so that there are no elements of A inside the nonconvex polygon bounded by γ and γ“B1. By construction,

h wγ h γ w , h wγ h γ 1, l wγ l γ p 1. pB q“ p qzt u pB q“ p q´ pB q“ p q` ´

w ‚

γ Ω

‚ ‚ wj wj w2 p 1 w1 ‚ 1 ¨ ¨ ¨ ‚ γ wγ “B

1 Figure 34: Reduction γ wγ of a path γ at a vertex w. “B Denote by Ω the (isotopy class of the) curved path in the above polygon that joins w1 2 and w avoiding all elements of A. Let Ω be the path joining wi and wj which extends Ω by coinciding with γ and γ1 before w1 and after w2. The curvilinear triangle formed by γ and Ω gives, by (4.2.4), the Picard-Lefschetz “co- multiplication” arrow v : M Ω M γ 1 . Note that we also have a curvilinear polygon Ω,γ p qÑ p qr s formed by Ω and the polygonal path γ1. Because of the coassociativity of the “multiplica- tion” Picard-Lefschetz u-arrows (Proposition 3.2.14), this latter triangle gives the iterated 1 multiplication arrow u 1 : M γ M Ω . Let γ ,Ω p qÑ p q v 1 u M γ1 γ ,Ω M Ω Ω,γ M γ 1 p q ÝÑ p q ÝÑ p qr s 119 be the natural transformations between the functors Φi Φj obtained by pre- and post- Ñ composing vγ,Ω and uΩ,γ1 by the identity natural transformations of the transport functors corresponding to the common parts of γ,γ1 and Ω, (unchanged under our move). 1 Let now γ Λm 1 and γ Λm. We define the matrix element P ` P 1 1 qγ1,γ : M γ or γ M γ or γ 1 p qb p q ÝÑ p qb p qr s 1 of q to be 0 unless γ wγ for some w l γ , in which case we put “B P p q

qγ1,γ uΩ,γ vγ1,Ω k w. “ p ˝ qb ^ Here max max w : or γ khpγ1q or γ khpγq ^ p q“ p q ÝÑ p q“ p q is the Grassman multiplication byľ an element of khpΓq correspondingľ to w. p q 2 Proposition 4.2.10. The differential q with the matrix elements qγ1,γ satisfies q 0 and so ‚ 0 “ R , q is a complex over H dgFun Φi, Φj , the homotopy category of dg-functors Φi Φj. p q p q Ñ Proof: By definition, each matrix element of q2 has the form

2 1 q γ2,γ qγ1,γqγ2,γ1 , γ Λm`2, γ Λm. p q “ γ1PΛ P P ÿm`1 1 For a summand in the RHS to be nonzero we must have γ wγ for some w l γ and 2 1 1 “ B P p q γ yγ for some y l γ . There are three possibilities: “B P p q

(1) y wj1 , ,wjp is one of the “new” vertices which become exposed after we remove w.P t ¨ ¨ ¨ u

(2) y is one of the “old” vertices of γ which remain as vertices of γ1. Here one can subdivide further:

(2a) r w,w1 is at one of the ends of the new segment γ1. P t u (2b) r w,w1 is separated from the new segment γ1. R t u 1 2 In the case (1) there is only one possible path γ , so q γ2,γ qγ1,γqγ2,γ1 . Since each of the two factors here is itself a composition of a u-arrowp andq a v“-arrow, the vanishing of their composition follows from the face that the composition of a v-arrow and a u-arrow in the same Picard-Lefschetz triangle is zero. 1 In the case (2a) or (2b) there are two possible paths γ , namely w γ and y γ . In this 2 B p q B p q case q γ2,γ will be the sum of two equal summands with opposite signs. p q

120 4.3 The infrared monad and the Fukaya-Seidel monad A. The Fukaya-Seidel monad as circumnavigation. The “circumnavigating” trans- port functors studied in §4.2 can be viewed as the components of the Fukaya-Seidel monad M from (3.5.2) defining the Fukaya-Seidel category of a schober S Schob C, A . P p q More precisely, fix a direction ζ, ζ 1 and suppose A w1, ,wN is in linearly | | “ “ t ¨ ¨ ¨ u general position including ζ-infinity, so that the ζ-direction rays wi ζR` do not intersect, ´1 ` ´1 see Definition 2.1.1. Let us number the wi so that Im ζ w1 Im ζ wN and consider the Fukaya-Seidel category F C, ζ; S correspondingp q toă ¨¨¨the ă oppositep directionq ζ . This category is obtained by takingp ´ the “Vladivostok”q point v ζR, R 0 very far p´ q “ " in the direction ζ and forming the v-spider K consisting of the straight intervals v,wi , almost parallet top´ eachq other, see the right of Fig. 35. This spider defines spherical functorsr s ai : Φi Φi S Sv and we have the Fukaya-Seidel monad M M K Mij iăj, where ˚“ p Ñ “ p q“p q Mij a ai : Φi Φj. The category F C, ζ; S is then the category of M K -algebras. “ j ˝ Ñ p ´ q p q The nature of K implies that Mij is nothing but the circumnavigating transport functor from wi to wi, see the left of Fig. 35.

wj wj ‚ K ‚ ξ v ‚ ‚ Qζ . Qζ . . ‚ ‚ . ‚ w‚ i w‚ i

Figure 35: Circumnavigation = Fukaya-Seidel monad.

B. The infrared monad. The iterated rectilinear transforms studied in §4.2, form a monad as well. This is in fact a replacement, in the general schober situation, of the A8-algebra R8 defined in [36, 37], see also [51] §10. More precisely, let Qζ Convζ A (the shaded region in Fig. 35). We adjust slightly the definitions of §4.2 E. That“ is, for anyp q i j we let Λ i, j be the set of ζ-convex polygonal ă p q paths γ joining wi and wj. For γ Λ i, j we have the iterated transport M γ : Φi Φj. Recall (Convention 1.1.11) that inP thep definitionq of M γ we identify the stalksp q of theÑ local systems Φ at the intermediate points using the clockwisep q monodromy: along the arcs going inside Qζ . We also denote by l γ the number of intermediate vertices of γ and h γ the set A ζ p q p q X Conv γ wi,wj and l γ l γ . Define the dg-functor Rij : Φi Φj by p zt u p q “ | p q| Ñ

Rij Rγ , “ (4.3.1) γPΛpi,jq max à hpγq Rγ M γ or γ , or γ : k . “ p qb p q p q “ p q ľ 121 This is similar to the second line of (4.2.9), except we do not consider the differential q yet. Let i j k and γ Λ j, k , γ2 Λ i, j . If γ γ1 is ζ-convex, then it belongs to Λ i, k , and we haveă ă a natural mapP p q P p q Y p q cγ,γ1 : Rγ Rγ1 Rγ γ1 . b ÝÑ Y It is defined as the tensor product of the composition map

M γ1 M γ M γ1 γ , p q˝ p q ÝÑ p Y q of the isomorphism or γ1 or γ or γ1 γ p qb p q ÝÑ p Y q obtained by contracting the two copies of the basis vector of kA. Let us define the map cijk : Rjk Rij Rik by b Ñ 1 cγ,γ1 , if γ γ is convex; R R cijk γ b γ1 Y | “ #0, otherwise.

ζ Proposition 4.3.2. The maps cijk make R R Rij i j into a uni-triangular monad. “ “p q ă Proof: Since the cijk are given by composition of functors, they are associative. We call R the infrared monad associated to S and the direction ζ. Example 4.3.3 (Maximally concave position). Suppose that A lies on a single ζ - convex polygonal path. Then from the point of view of ζ -convexity, A is “maximallyp´ q concave”, see Fig. 36. p` q

wj ‚

‚ Mij Rij

‚

‚ wi

Figure 36: Maximally concave position: Mij Rij Mij. “ “

In this case, any ζ-convex polygonal path consists of a single segment wi,wj . This segment is isotopic to the circumnavigating path passing through any farawayr points in the direction ζ . Therefore in this case the components of the monads M and R coincide: Mij Rij p´ q “ “ Mij is the rectilinear transport. But the composition maps in R are zero, while in M they do not have to be zero.

122 C. Two layers of infrared analysis. By the infrared analysis of a schober S we will mean, somewhat simplistically, the study of the relations between the infrared monad R and the Fukaya-Seidel monad M or of ways to obtain (the components of) M from (those of) R. In fact, there can be two layers of such analysis:

(1) Deforming the differential in each component Rij to get a dg-functor quasi-isomorphic to Mij.

(2) Deforming the compositions cijk in the monad R to get an A8-monad quasi-isomorphic to M. These two steps involve two levels of convex geometry mentioned in the beginning of this chapter. Here we consider the first step which is more elementary.

Similarly to (4.2.9), we have a differential q in each Rij, thus upgrading Rij to a cochain complex

R‚ 0 R0 q R1 q R2 q , ij “ Ñ ij Ñ ij Ñ ij Ѩ¨¨ (4.3.4) m R : M γ k or γ dgFun Φi, Φj . ij “ p qb p q P p( q γPΛmpi,jq à ‚ We would like to say that this complex “represents” the dg-functor Mij. However, Rij, as 0 we defined it, is a complex over the triangulated category H dgFun Φi, Φj and there is, a priori, no canonical way of associating to it a single object of this category.p q Here, a general discussion may be useful.

D. Totalization of complexes and Maurer-Cartan elements. Let C be a pre- triangulated dg-category and C‚, q be a bounded complex over the triangulated category H0C. We would like to associatep toq it a “total object” Tot C‚ C which would categorify the formal alternating sum 1 iCi. p q P p´ q Remark 4.3.5. In this sectionř we use slightly different notation - the differential q in a complex now has degree zero. The reason that it is not automatic, is as follows. By our assumptions, each Ci is an object of C and each q : Ci Ci`1 is an element of 0 ‚ i i`1 2 0 ‚ i i`2 Ñ H HomC C ,C so that q 0 in H HomC C ,C . This means that if we lift q to a p q “ p i q i`1 2 system q1 of actual closed degree 0 morphisms q1,i : C C , then q1 does not have to be Ñ i i`2 zero but has the form d, q2 where q2 is a system of degree 1 -morphisms q2,i : C C . r s p´ q Ñ To continue in a meaningful way, we need to extend q1, q2 to a full coherence data. Such data can be formulated in terms of Maurer-Cartan elements. Let g‚,d be a dg-Lie algebra over k. In particular, g0, the degree 0 component, is a Lie algebrap inq the usual sense. We recall that a Maurer-Cartan element in g‚ is an element 1 1 α g such that dα 2 α,α 0. The set of Maurer-Cartan elements in g is denoted MCP g‚ . ` r s “ p q 123 In particular, let E‚,d be an associative dg-algebra. Then E‚ can be considered as a dg-Lie algebra via thep gradedq commutator. In particular, we can speak about Maurer-Cartan elements in E‚. Explicitly, they are elements α E1 such that dα α2 0. The degree 0 part E0 is an associative algebra in the usual sense,P so we have the` group“E0,ˆ of invertible elements in E0. This group acts on MC E by conjugation Two elements of MC E‚ lying in the same orbit of this action, are calledp gaugeq equivalent. p q Returning to our situation of C‚, q , consider the associative dg-algebra p q ‚ ‚ i j E HomC C ,C j i 1 , “ iďj p qr ´ ´ s à ‚ with the differential d, . A Maurer-Cartan element q MC E is a sum q jě1 qj 1´jr i´s i`j P2 p q “ with qj i Hom C ,C and the condition d, q q 0 unravels into a system of P p q r 2 s` “ ř equations of which the first two are d, q1 0 and q1 rd, q2 0. r À r s“ `r s“ Definition 4.3.6. An MC-lifting of the complex C‚,r q isr a Maurer-Cartan element q E 1 p q P such that the cohomology class of q1 is q.

b r Example 4.3.7. Let C D Vectk be the standard dg-enhancement of the bounded derived category of k-vector spaces.“ p Thatq is, objects of C are bounded complexes E‚ of vector ‚ ‚ ‚ spaces and Hom E1 , E2 is the standard Hom-complex (with differential given by graded p q ‚ ‚ 0 commutation with the differentials in E1 and E2 . A morphism in H C is a homotopy class of morphisms of complexes. Thus, a complex over H0 C is a bigradedp q vector space C‚‚ i,j i,j`1 p q 2 together with an actual differential q0 : C C , of degree 0, 1 , satisfying q0 0 (so i i,‚ Ñ p q “ that each C C , q0 is a complex) and a system of homotopy classes of morphisms of complexes q : “Ci, p‚ Ci`q1,‚, squaring to 0 up to homotopy. Ñ ‚‚ An MC-lifting of this data is a differential q iě0 qi in C with qn of degree n, 1 n , satisfying q2 0. Since q is homogeneous of degree“ 1 with respect to the total grading,p ´ weq have a new complex“ T ‚, q , where T m řCij. p q “ i`rj“m In ther general case,r An MC-lifting q Àcan be seen as a structure of a twisted complex [15, 86] in C with the samer terms as C‚. We will denote this twisted complex by C‚. The pre-triangulated structure on C gives thenr the totalization Tot C‚ . We have the following general statement. p q r q q r Proposition 4.3.8. Let C‚ C´m C0 be situated in degrees m, 0 (this does not restrict the generality). Then,“ t weѨ¨¨ have aÑ bijectionu between the sets consistingr´ s of: (i) Gauge equivalence classes of MC-liftings of C‚.

(ii) Isomorphism classes of Postnikov systems in H0C

q C´mr´m ` 1s C´2r´1s ❴❴❴❴❴❴❴ / C´1 > ❉ r8 ▼▼▼ ✇; ●● ⑥⑥ ❉❉ `1 rr ▼▼hm `1 ✇✇ ●●h2 `1 ⑥ ❉ h1“q rr ▼▼ ✇✇ ●● ⑥⑥ ❉❉ rr ▼▼ ✇✇ ●● ⑥⑥ ❉ rr ▼& ✇ # ⑥ ❉! Dr´m,0s o Dr´m`1,0s o ¨ ¨ ¨ o Dr´2,0s o Dr´1,0s o D0 “ C0

124 with the property that each degree 1 map C´i i 1 C´i´1 i , i 2, , m, obtained as the composition (as indicated byr´ the` dotteds Ñ horizontalr´ s arrow)“ i¨s ¨ ¨equal to q : C´i C´ŒÕi`1 Ñ r´i,0s Proof (sketch): Suppose that we have a MC-lifting q qj. Let C be the truncation “ jě1 of C‚ obtained by retaining only the graded components Cl for l i, 0 . Then the compo- ř P r´ s nents of q that preserve Cr´i,0s, give a MC-lifting forr it, which we denote qr´i,0s. This gives rise to a twisted complex Cr´i,0s with the same terms as Cr´i,0s. Let Dr´i,0s Tot Cr´i,0s . “ p q Now, wer have a filtration of twisted complexes r r r C0 Cr´1,0s Cr´2,0s Cr´m,0s à à è¨¨Ă ´i with quotients being C i . Thisr filtrationr gives the desiredr Postnikov system for the total objects. r´ s 0 ´1 0 Conversely, suppose we have a Postnikov system as in (ii). Let q1 ´1,0 HomC C ,C ´1 p q0 P p q be a closed element representing the component q ´1,0 : C C of q. Then we can r´1,0s p q Ñ identify D with the canonical cone of q1 ´1,0 given by the pre-triangulated structure ‚ ´2 p r´q1,0s on C. Then, the complex HomC C 1 ,D can be identified with the cone of the morphism of complexes p r´ s q

Hom‚ C´2,C´1 Hom‚ C´2,C0 Cp q ÝÑ Cp q induced by q1 1,0. So the morphism h2 can be thought as the degree 0 cohomology class p q´ of this latter cone. A 0-cocycle h2 representing h2 consists of two components, which we 0 ´2 ´1 ´1 ´2 0 denote q1 ´2,´1 HomC C ,C and q2 ´2,0 HomC C ,C . Further, we can think r´2,p0s q P p q p q‚ ´P3 pr´2,0s q of D as the canonical cone ofrh2, so Hom C 2 ,D is the cone of the moprhism Cp r´ s q Hom‚ C´3 2 ,C´2 1 Hom‚ C´3 2 ,Dr´1,0s Cp r´ sr r´ sq ÝÑ Cp r´ s q 3 and h is a degree 0 cohomology class of this cone. Choosing a 0-cocycle h3 representing 0 ´3 ´2 h3, we decompose it into, first, the component denoted q1 ´3,´2 HomC C ,C and 0 ´3 r´1,0s p q P p qr´1,0s second, the component in HomC C 2 ,D which, using the representationr of D p r´ s q ´1 ´3 ´1 as a cone, is further split into two components q2 ´3,´1 HomC C ,C and q3 ´3,0 Hom´2 C´3,C0 . Continuing like this, we get componentsp q P p q p q P C p q 1´j i i`j qj i,i j Hom C ,C , j 1. p q ` P C p q ě 1´j ‚ ‚ Denote qj Hom C ,C the morphism with components qj i,i j and put q qj. P C p q p q ` “ jě1 h 0 The conditions that each j is a closed (of degree ) element of the correspondingř Hom- complex, imply together, in the standard way, that q is a Maurer-Cartan element.r r E. From the infrared Postnikov system tor an MC-lifing? We are interested in ‚ ‚ applying the above to C Rij, q being the i, j th infrared complex (4.3.4). We would like to construct an MC-lifting“ p of theq differentialp q.q

125 Note that we have the infrared Postnikov system P of Definition 4.2.5. But Proposition 4.3.8 does not apply directly since P is a more sophisticated diagram than the “linear” Postnikov systems appearing there. Only in the maximally nonconvex situation of Example 4.2.6 we get a linear diagram. In general, P does contain linear Postnikov systems (typically more than one) and we can apply Proposition 4.3.8 to those. However, the MC-liftings q ‚ thus obtained need not not compatible with the grading on Rij, i.e., may contain components m which act within the individual Rij . Nevertheless, we think that the following is true. r Conjecture 4.3.9. For each schober S Schob C, A , there is an MC-lifting q of the ‚ P p q differential q in Rij, canonical up to gauge equivalence, with the following property: The total ‚ object of R , q is quasi-isomorphic to the component Mij of the Fukaya-Seidel monad. p ij q r This should follow from a more systematic definition of schobers which will include various r coherence data from the very outset. We leave this question for future study.

5 Algebra of the Infrared associated to a schober on C, II

In this chapter we consider the more advanced aspects of the “infrared” approach to schobers, which use the notion of secondary polytopes. This corresponds to the approach via webs in [36, 37], see the discussion in [51] as well as above, in Introduction § F.

5.1 Background on secondary polytopes A. Basic definitions. The concept of secondary polytopes makes sense in the Euclidean space Rn of any number of dimensions [39]. In this paper we will be only interested in the case n 2 when R2 appears as the complex plane C. To fix the terminology, we give a brief reminder,“ under this and some other additional assumptions, see [39] for a systematic exposition.

Let A w1, ,wN C, N 3, be a set in linearly general position. We denote Q Conv“A t, a¨ convex ¨ ¨ u polygon. Ă Denoteě Vert Q the set of vertices of Q. We will refer to“ the pairp Q,q A as a marked polygon. Thus ap markingq of a given polygon Q consists in a choice of a setp A q Q in linearly general position which contains Vert Q . Ă p q A subpolygon of Q, A is a polygon Q1 such that Vert Q1 A. We write Q1 Q, A . A marked subpolygonp of qQ, A is a marked polygon Q1,p A1 qsuch Ă that A1 A.Ă We p writeq Q1, A1 Q, A . A markedp q subpolygon Q1, A1 p Q, Aq will be calledĂ geometric, if pA1 QqĂp1 A. A markedq triangle is a markedp polygonqĂpQ, A qsuch that Q is a triangle and A “VertXQ . p q “ p q A polygonal decomposition of a marked polygon Q, A is a finite collection P Qν , Aν p q “ tp qu of marked subpolygons in Q, A such that Q ν Qν and each intersection Qν Qµ, µ ν, is a common face of bothp (i.e.,q either empty,“ or a common vertex, or a commonX edge).‰ A Ť

126 triangulation of Q, A is a polygonal decomposition T Qν , Aν such that each Qν, Aν is a marked triangle.p Theq set of polygonal decompositions“ tp is partiallyqu ordered by thep relationq ĺ of refinement: we have P ĺ Q, if P induces a polygonal decomposition of any marked polygon from Q. In such a case Q gives a (necessarily) regular decomposition Qν of each Qν, Aν . p q Definition 5.1.1. Let P Qν , Aν be a polygonal decomposition of Q, A . A function ψ : A R is called: “ tp qu p q Ñ (a) P-piecewise affine, if there is a (necessarily unique) continuous function f : Q R Ñ which is affine-linear on any Qν and such that f w ψ w for any ν and any w Aν. p q“ p q P (b) P-normal, if (b1) ψ is P-piecewise-affine. (b2) The function f in (b) not piecewise affine on any subset of Q which strictly contains one of the Qν. In other words, f does break along any common edge of any two polygons in P. (b3) The function f is convex.

(b4) For w A not lying in any of the Aν, we have f w ψ w . P p qă p q We denote by RA the space of all functions ψ : A R. For a polygonal decomposition P we denote by D RA the linear space formedÑ by P-piecewise affine functions and P Ă CP DP the set of P-normal functions. It is clear that CP is an open convex cone in DP . A decompositionĂ P is called regular, if C . The following is by now standard. P ‰ H A Proposition 5.1.2. The space R is the disjoint union of the cones CP over all (regular) decompositions P of Q, A . p q Proof: Given ψ : A R, we construct, first of all, a piecewise-linear convex function f : Q R by looking atÑ the unbounded polyhedron Ñ 3 Gψ Conv w, t w A, t ψ w C R R “ p q P ě p q Ă ˆ “ and taking the finite (i.e., the bottom) ˇ part of its boundary( as the graph of f. The decom- ˇ position P Pψ is then given by the maximal regions Qν of piecewise affine behavior of f, “ and the marking of Qν is read off (b4). Cf. [51] Fig.2. Let V be an R-vector space. By a fan in V we will mean a decomposition of V into a disjoint union of locally closed convex cones. Examples 5.1.3. (a) Proposition 5.1.2 means that we have a fan in RA formed by the cones C . It is called the secondary fan of A and denoted Sec A . P p q (b) Let P V be a (bounded) convex polytope. It gives rise to the normal fan N P ˚ Ă p q in V whose cones NF P are labelled by the faces F P of all dimensions. By definition, p q Ă NF P , called the normal cone of P at F , consists of linear functionals l : V R such that Ñ l P achieves minimum precisely on the face F . | 127 The concept of secondary polytopes connects the examples of type (a) and (b) for V RA. In this case we identify V ˚ with RA as well, using the delta-functions at w A “as an orthonormal basis. P

Proposition-Definition 5.1.4. The secondary fan Sec A is realized as the normal fan of a convex polytope Σ A RA called the secondary polytopep q of A. More precisely: (a) Σ A is a convexp qĂ polytope of dimension A 3. (b) Facesp q F of Σ A are labelled by regular| subdivisions|´ P of Q, A . The inclusion of P p q p q faces FP FQ correspons to refinement P ĺ Q. (c) InĂ particular, vertices F of Σ A correspond to regular triangulations T of Q, A . T p q p q (d) The dimension of FP is equal to the codimension of the cone CP or, what is the same, A of the subspace DP , in R . The proof proceeds via an explicit construction of the vertices F RA, see [39]. T P B. Faces of small dimension and codimension.

Examples 5.1.5. Suppose that A is a circuit, i.e., A 4. Then Σ A has dimension 1, so it is an interval. There are precisely 2 triangulations| | “ of Q, A ,p bothq regular. These triangulations are coarse subdivisions of Q, A . There are twop possibilities:q p q (a) A is in convex position, so Q is a 4-gon. The two triangulations of Q, A are just the two says of splitting Q into two triangles. p q

(b) One point of A w1, ,w4 , say, w4, lies in the convex hull of the 3 others. In this case Q is a triangle“ t (but¨ ¨ ¨ Q, Au is not a marked triangle). One triangulation of Q, A splits Q into 3 smaller trianglesp q , and the other consists of a single marked trianglep q Q, w1,w2,w3 . p t uq

‚ ‚ ‚ ‚ ‚ ‚

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

Figure 37: Circuits and flips.

So for A a being a circuit, the interval Σ A represents one of the standard 2 2 and 3 1 flips of plane triangulations, see Fig.p 37.q Compare also with the diagramsØ in Figs. 26Ø and 29. For a general A it is known that the edges of Σ A correspond to flips of regular triangulations along such circuits. p q

Definition 5.1.6. A regular decomposition P of Q, A is called coarse, if FP Σ A is a face of codimension 1. p q Ă p q

128 Remark 5.1.7. P being coarse means that there is just one, up to rescaling and adding affine-linear functions, P-piecewise-affine continuous function f : Q R. So a simple way to construct a coarse subdivision would be to cut Q by a straight lineÑ into two subpolygons, so that f breaks only along this line. The triangulations in Example 5.1.5(a) give an instance of this construction. However, 5.1.5(b) gives a coarse subdivision into 3 triangles. This shows that coarse subdivisions can involve many polygons.

5.2 Variation of secondary polytopes and exceptional walls CN A. The phenomenon. Let A gen be in linearly general position. The partially ordered set of all polygonal decompositionsP of Q, A (regular on not) depends only on the convex geometry of the set A, i.e., on the orientedp matroidq O A , see Definition 2.2.1. In particular, it remains unchanged inside any configuration chamberp q (connected component) C CN . Ă gen However, the property of a given decomposition to be regular and therefore to correspond to a face of Σ A is more subtle and may not hold or fail uniformly on any configuration chamber C. Thisp q means that, as A varies in C, the structure of Σ A can change. This phenomenon, known in the theory of secondary polytopes, was rediscoveredp q in the dual language in [36] and called exceptional wall-crossing. This means that we have a further chamber structure on the chamber C given by a closed subset D C (union of “exceptional walls”) so that C D is dense and the structure of Σ A is unchangedĂ on each connected component of C D. Inz this section we quantify this picturep q in our case of two dimensions. z

B. The deformation complex of a subdivision. For any subset I C let Aff I RI be the subspace formed by the restrictions to I of R-affine-linear functionsĂ C R.p ThusqĂ Ñ dimR Aff I can be 1, 2 or 3. p q Let P be a polygonal subdivision of Q, A , regular or not. Denote by P0, P1, P2 the sets of all vertices, resp. edges, resp. polygonsp fromq P which do not lie on the boundary Q. For i B any i 0, 1, 2 and any σ Pi we denote or σ H σ, σ; R the 1-dimensional orientation R-vector“ space of σ. Thus,P for σ being ap polygonq“ p (i B 2),q the space or σ is canonically trivialized by the standard orientation of C. Also, for “i 0 (σ a point), thep q space or σ is trivialized as well. We now construct a 3-term cochain complex“ p q

(5.2.1) Def‚ P Aff σ or σ d Aff σ or σ d Aff σ or σ , p q “ p qb p q ÝÑ p qb p q ÝÑ p qb p q "σPP2 σPP1 σPP0 * à à à with grading normalized so that the leftmost term is in degree 0. More precisely, the dif- ferential d is defined to send Aff σ or σ to the sum of Aff τ or τ where τ runs over codimension 1 faces of σ (i.e., overp qb the edgesp q of σ, if σ is a polygonp qb andp overq the two vertices of σ, if σ is an edge). We define the matrix element

dσ,τ : Aff σ or σ Aff τ or τ p qb p q ÝÑ p qb p q 129 to be equal to rσ,τ εσ,τ , where: b • rσ,τ : Aff σ Aff τ is the restriction of functions from σ to τ, and p qÑ p q • εσ,τ : or σ or τ is the isomorphism given by canonical co-orientation of τ in σ. p qÑ p q Proposition 5.2.2. (a) We have d2 0, i.e., Def‚ P is indeed a cochain complex. “ p q 0 ‚ (b) The 0th cohomology space H Def P is identified with DP , the space of P-piecewise- affine functions. p q (c) We have H2 Def‚ P 0. p q“ Proof: (a) Let σ P2 be a polygon and let σ, f be an element of the σth summand of Def0 P , so that fP Aff σ . Then Then d2 σ,p f qis the sum over vertices w of σ, and the summandp q correspondingP p toq each such w is,p in itsq turn, the sum over the two edges of σ bounded by w, of the identical summands f w but with the opposite signs coming from the orientations. So d2 σ, f 0. p q p q“ 0 ‚ (b) By definition, H Def P consists of collections fσ of affine-linear functions on the polygons σ of P which matchp q over the internal edges,p i.e.,q precisely of P-affine-linear functions.

(c) For any i and any σ Pi we have the embedding R Aff σ of constant functions. This gives a subcomplex P Ă p q

(5.2.3) C‚ or σ or σ or σ Def‚ P . “ p q ÝÑ p q ÝÑ p q Ă p q "σPP2 σPP1 σPP0 * à à à ‚ This C is, up to shift, the relative cellular chain complex C‚ Q, Q; R for the cell decom- position given by P. So p B q R i ‚ , if i=0; H C H2´i Q, Q; R p q “ p B q “ #0, otherwise. Note that C2 Def2 P , so Def‚ P C‚ is situated in degrees 0 and 1, while H2 C‚ 0 by the above.“ Thereforep qH2Def‚ Pp q{ 0 as well. p q “ p q“ Let us denote by exc P and call the exceptionality of P, the dimension of H1 Def‚ P . We say that P is exceptionalp q , if exc P 0. p q p q‰ Proposition 5.2.4. Suppose P is regular and F be the corresponding face of Σ A . Then P p q codim F 3 P2 2 P1 P0 3 exc P . P “ | |´ | | ` | |´ ` p q Proof: If P is regular, then codim FP dim DP 3 is the dimension of the space of P- piecewise-affine functions modulo globally“ affine functions.´ To get our statement, it remains to equate the alternating sum of dimensions of the terms of Def‚ P with the alternating sum of the dimensions of the cohomology. p q So in the case exc P 0 we have an “unexpected jump” in the codimension of F . p q‰ P 130 C. The deformation complex and deformations of A. For a subset I C let Lin I : Aff I R be the quotient of Aff I by the subspace of constant functions.Ă Thus p q “ p q{ p q dimR Lin I can be 2, 1 or 0. Denote L Lin C HomR C, R . p q “ p q“ p q ‚ Let P be any polygonal subdivision of Q, A . Let Def Def P : Def‚ P C‚ be the quotient of Def P by the subcomplex C‚ pof constantq functions“ (5.2.3).p q “ Explicitly,p q{ p q Def Lin σ or σ Lin σ or σ , “ p qb p q ÝÑ p qb p q "σPP2 σPP1 * à à with the leftmost term in degree 0. Note that for σ P2 we have Lin σ L, while for P p q “ σ P1 we have a short exact sequence P ˚ 0 N C L Lin σ 0, Ñ σ{ ÝÑ ÝÑ p qÑ ˚ C where Nσ{C is the conormal space (i.e., the dual to the normal space Nσ{C) of σ inside . Therefore we have a short exact sequence (with maps written vertically) of complexes (with differentials written horizontally): (5.2.5) 0 0 0 0

    d K 0 / N ˚ or σ / L or σ “ σPP1 σ{C b p q σPP0 b p q " * À À     C‚ L L or σ / L or σ / L or σ b “ σPP2 b p q σPP1 b p q σPP0 b p q " * À À À     Def L or σ / Lin σ or σ / 0 “ σPP2 b p q σPP1 p qb p q " * À À     00 0 0 where in the middle we have the “constant” complex C‚ L, and K is defined as the kernel of the projection. Let us analyze this sequence. b First of all, C‚ L has H0 L and no other cohomology. So the long exact sequence of cohomology correspondingb to (5.2.5)“ gives a short exact sequence

(5.2.6) 0 L H0 Def δ H1 K 0 Ñ ÝÑ p q ÝÑ p qÑ and an isomorphism H1 Def » H2 K . p q ÝÑ p q 131 0 Now, H Def DP R is the space of P-piecewise affine functions modulo constants. So 0 p q “ { H Def L DP Aff C is the quotient of DP by the subspace of global affine-linear func- tions.p q{ “ { p q Further, K1 N ˚ or σ is the space of “normal derivative data” of linear “ σPP1 σ{C b p q functions with respect to various σ P1. So the coboundary À P δ : H0 Def H1 K K1 p q ÝÑ p qĂ is the “break data” map. It sends a piecewise affine function f fσ σPP2 (modulo global affine-linear functions) to the collection of breaks of f at all the intermediate“ p q edges.

Note that for σ P1 the orientation line or σ is the same as the tangent line to σ, and, since or σ b2 R,P it is also identified with thep cotangentq line, i.e., with Lin σ . Therefore ˚ p q » R p q Nσ{C or σ is identified with , in particular, we can speak about positive and negative elementsb ofp it.q

1 1 Proposition 5.2.7. Let P be a polygonal subdivision of Q, A . Let Ką0 K be the ˚ p q Ă octant formed by vectors gσ N or σ with all gσ positive. Then CP Aff C p q P σPP1 σ{C b p q { p q» H1 K K1 . In particular, P is regular if and only if H1 K K1 . p qX ` À p qX ą0 ‰ H Proof: For any P-piecewise affine function f D let f H0 Def be the image of f (the P P P p q class of f modulo global affine linear functions). The above analysis implies that CP consists of f DP such that δ f has all components positive. Our claim follows from this via the shortP exact sequence (5.2.6).p q Let us now consider the above picture from the point of view of deforming A. When we CN move A inside a given connected component C ‰ , we can uniquely move the subdivision P as well, as the convex geometry of A does notĂ change. The space of infinitesimal defor- N N mations of A is TAC C . Let us call a restricted deformation a deformation that moves “ only points of P0, leaving all the other elements of A (that is, the vertices of Q and the points inside Q not in P0) unchanged. The corresponding space of infinitesimal deformation is T res CP0 which we view as the leftmost (degree 2 ) term of the dual complex A “ p´ q

˚ ˚ B K L or w Nσ C or σ “ b p q ÝÑ { b p q "wPP0 σPP1 * à à Indeed, L˚ C and for a point w we have or w R. “ p q“ Proposition 5.2.8. H´2 K˚ Ker , viewed as a subspace of T resCN , consists of in- p q “ pBq A finitesimal deformations of P0 such that the corresponding infinitesimal deformation of P has all the edges moving in a parallel way. We call restricted infinitesimal deformations of A appearing in the proposition, parallel deformations.

Proof: Let ξ ξw wPP0 be an infinitesimal deformation of A, considered as an element of ˚ ´2 “ p q K . Let us define ξw 0 for w A P0. p q “ P z 132 1 1 Let σ w,w P1 so that either one or both vertices w,w lie in P0. Choose an “ r s P 1 ˚ orientation of σ, say from w to w . Denote pσ : L C Nσ C be the projection to “ Ñ { the normal space. With these preliminaries, σ-component of ξ is, by definition pσ ξw Bp q p q´ pσ ξw1 Nσ{C. The vanishing of this difference means that, under the infinitesimal move of p 1 q P 1 w,w given by moving w according to ξw and w according to ξw1 , the direction of the interval rdoes nots change. So vanishing of all the components of ξ means that the deformation is parallel. Bp q Corollary 5.2.9. A subdivision P of Q, A is exceptional, if and only if it admits parallel deformations. p q Proof: By definition, P is exceptional, if the space H1 Def‚ P H1 Def H2 K p p qq » p q » p q is nonzero, in other words if H´2 K˚ 0. p q‰ Example 5.2.10. Consider two “concentric” triangles in C, i.e., a triangle a, b, c contaning 0 inside and a contracted triangle a1 λa, b1 λb,c1 λc , λ 1. Let A r a, b,s c, a1, b1,c1 , so Q a, b, c and let P be the subdivisionr “ of“ Q, A “in Fig.s 38.ă Then P “is t exceptional. Itsu parallel“r restricteds deformation is obtained by varyingp q λ.

b ‚

1 b ‚

0 ‚ ‚ a1 c1 ‚ ‚ a c

Figure 38: An exceptional subdivision with an obvious parallel deformation.

CN D. Exceptionally general position. We would like to specify regions in gen where “the structure of the secondary polytope does not change”. A precise way of expressing this is as follows. For any convex polytope P let F P be the poset of nonempty faces of P (including P p q itself), ordered by inclusion. Two polytopes P1,P2 are called combinatorially equivalent, if there is an isomorphism of posets F P1 F P2 . For example, all plane n-gons with given n are combinatorially equivalent. p qÑ p q CN Definition 5.2.11. We say that A gen is in exceptionally general position, if, for any P CN CN regular subdivision P of Q, A , we have exc P 0. Denote by Xgen gen the set of A in exceptionally general positionp q and put D p CqN “ CN . Ă “ genz Xgen 133 CN R CN R Proposition 5.2.12. (a) Xgen is an -Zariski open subset in gen, i.e., D is an -Zariski closed subset (zero locus of a system of real polynomials). CN CN (b) Xgen is dense in gen with respect to the usual transcendental topology. In other words, codim D 1. p qě CN (c) For any connected component E Xgen, the combinatorial equivalence class of the polytopes Σ A , A E, is the same. Ă p q P Remark 5.2.13. The set D can be seen as the union of “exceptional walls” of [36] §8.3. However, we cannot exclude the possibility that some irreducible components of D have codimension 1, in which case they would not represent walls in the strict sense. ą CN Proof of Proposition 5.2.12: (a) Let A vary inside a component C of gen. As A moves, we can move P with A. This means that the deformation complex Def‚ P also moves, i.e., we get a complex of R-algebraic vector bundles on C, considered as anpRq-algebraic variety. Therefore the subset in C over which some particular cohomology group of this complex, in our case H1, is non-zero, is R-Zariski closed, given by vanishing of certain minors of the matrices of the differentials. So CN C is R-Zariski open in C. Xgen X CN CN (b) To show that Xgen is dense, it is enough to show that for any A gen and any excep- tional decomposition P of Q, A , there is a nearby deformation (actual,P not infinitesimal) A1 of A so that the correspondingp q decomposition P1 of Q1, A1 is no longer regular. So we suppose that P is exceptional, i.e., admitsp a parallelq infinitesimal deformation 1 1 ξ ξw wPP0 and exhibit an A as above. In searching for such an A , we are allowed to move the“p verticesq of Q as well. Let A1 be obtained by moving all the vertices of Q a little in the counterclockwise direction, while keeping all the other points, in particular, those from P0, unchanged, see Fig. 39.

‚ ‚

‚ ‚ ‚ ‚ ‚ ‚

‚ ‚ ‚ ‚

Figure 39: The deformation A1 of A obtained by moving the vertices of Q counterclockwise.

1 1 1 1 Let P be the subdivision of Q , A obtained by deforming P. Note that P0 P0. We claim that P1 is not regular. Indeed,p q the restricted deformation of P given by“ the same collection of vectors ξ, will now not be parallel, and instead we have the following property: 1 1 1 1 (*) If an edge w,w of P has both vertices in P0 P0, then the direction of w,w does r s 1 “ r s not change, as before. If, however, w,w has only one vertex in P0 and the other one on Q1, then the effect of ξ will rotater thes edge in the counterclockwise direction, see Fig.B 39.

134 We claim that (*) implies that P1 is not regular. Let us use Proposition 5.2.7 and denote K P1 the complex K for P1. Let d be the differential in K P1 , as in (5.2.5) and be thep dualq differential in K˚ P1 , as above. By 5.2.7, regularity ofp P1qmeans that the subspaceB 1 1 p q 1 1 Ker d H K P intersects the strictly positive octant K P 0, so that the corresponding p q“ p q p qą intersection is essentially the cone CP . In dual terms, this means that:

1 ˚ (**) Im intersects the non-strictlly positive octant K P ě0 only at 0 or along a single half-line.pBq p q

1 1 Indeed, the dual cone to Ker d K P ě0 is the image, denote it I, of the non-strict octant 1 1 ˚ 1˚ p qX1 p q 1 1 K P ě0 in Coker K P Im . So nonemptiness of Ker d K P ą0, means that I doesp q not containpBq a linear “ space,p q{ whichpBq is equivalent to (**). p qX p q But, as shown in the proof of Proposition 5.2.8, is given by the infinitesimal rotation of the edges, so (*) means that Im contains a vectorB which have some components 0 and other (at least 3) components strictlypBq positive. This means that P1 cannot be regular. This proves part (b) of Proposition 5.2.12. Let us show part (c). Let A move along a path in E. Then all subdivisions P of Q, A move as well, so we write P A for the subdivision for a given A. Now, since all regularp subdivisionsq P A remain non-exceptional,p q p q the dimension of the corresonding face FPpAq Σ A remains the same by Proposition 5.2.4. This means that the polytope Σ A cannot undergoĂ p q a combinatorial change. p q E. Factorization property of secondary polytopes. .

N Proposition 5.2.14. If A C , then for every regular subdivision P Qν , Aν , the face P Xgen “p q F of Σ A is combinatorially equivalent to the product Σ Aν . P p q ν p q This proposition corrects the treatment in [51], whereś the condition of exceptionally general position was overlooked.

Proof: The faces of FP are the FQ where Q ĺ P is a regular subdivision refining P. Such Q consists of a system of subdivisions Pν of Qν, Aν which are necessarily regular, since P is regular. This gives an embedding of posetsp q

a : F FP ã F Σ Aν F Σ Aν . p q Ñ ν p p qq “ ν p q ź ˆź ˙ We claim that a is a bijection, hence an isomorphism of posets. Since A CN , the P Xgen subdivision P is non-exceptional, and so Proposition 5.2.4 calculates dim FP in combinatorial terms. This calculation implies that dim F equal the sum of the dimensions of the Σ Aν , P p q i.e., the dimension of ν Σ Aν (we leave the elementary computation to the reader). So bijectivity of a is a particularp caseq of the following fact. ś Lemma 5.2.15. Let P1,P2 be two convex polytopes of the same dimension, and b : F P1 ã F P2 be an embedding of posets. Then b is bijective. p qÑ p q

135 Proof of the Lemma: Let δ dim P1 dim P2. The statement is clearly true for δ 0, 1, so we argue by induction, assuming“ the“ lemma true for the case when the common dimension“ of P1 and P2 is less than δ. For a poset S, we call the height of S and denote by h S the maximal d 0 such p ďq p q ě that there is a chain s0 ss of strictly increasing elements of S (it may be h S ). 㨨¨ă p q“8 For any convex polytope P we have dim P h F P . In particular, if F is a face of “ p p qq P1, then dim F is the height of the interval F F F P1 F . Since b is an embedding, p q p q “ p qď this means that dim b F dim F . Since dim P1 dim P2 , we conclude that dim b F p qě p q“ p q p q“ dim F for any face F of P1.

In particular, b takes the maximal face P1 F P1 to P2 F P2 , so P2 Im b . Let us P p q P p q P p q prove that all proper (non-maximal) faces of P2 lie in Im b as well. p q By the inductive assumption, for any proper face F F P1 the embedding b induces an isomorphism P p q bF : F P1 F F F F b F F P2 b F . p qď “ p q ÝÑ p p qq “ p qď p q Now, the geometric realization of the poset of proper faces of Pi, i 1, 2, is homeomorphic δ´1 “ to the sphere S . So the isomorphisms bF for proper F F P1 give, after passing to geometric realizations, a continuous embedding Sδ´1ã Sδ´1.P Byp topologicalq reasons, such an embedding must be a homeomorphism. ThereforeÑ b is surjective on proper faces as well.

5.3 The L8-algebra associated to a schober on C

The main feature of the Infrared Algebra formalism [36, 37] is a L8-algebra g associated to a “theory”. As shown in [51], g can be defined using secondary polytopes. In this section we show how such a datum can be constructed from any schober S Schob C, A . P p q We consider the situation of §4.1. Let A w1, ,wN C be in linearly general position and Q Conv A . “ t ¨ ¨ ¨ u Ă “ p q A. Framed subpolygons, their content and pasting. We start by adding some complements to the discussion of pasting diagrams in §3.2 E. By a subpolygon in Q we mean a set of the form P Conv B , where B A has cardinality at least 3. Let P Q be a subpolygon. By a framing“ of Pp weq mean a datumĂ f of two vertices α,ω of P (possiblyĂ equal) called the 0-dimensional source and target of P (with respect to f). These vertices subdivide the boundary P into two complementary polygonal arcs B

P, f α wi ,wi , ,wi ω , (5.3.1) B`p q“r “ 0 1 ¨ ¨ ¨ r “ s P, f α wj ,wj , ,wj ω , i0 j0, ir js, B´p q“r “ 0 1 ¨ ¨ ¨ s “ s “ “ with common beginning and end. More precisely, ` P, f is that of the two arcs which goes in the counterclockwise direction around P , and B pP, fq is the arc going in the clockwise B´p q

136 direction. These two arcs are referred to as the 1-dimensional source and target of P with respect to f. In this way a choice of a framing makes P into a pasting diagram with one 2-dimensional cell.

Example 5.3.2. As in Example 3.2.15(b), let ζ be a direction so that A is in linearly general position including ζ-infinity and pζ : C R be the projection to a line orthogonal to ζ. This Ñ datum defines a framing fζ on P , with α and ω being the vertices where pζ achieves the ζ ζ minimum and maximum, and ` P, fζ ` P resp. ´ P, fζ ´ P is the upper, resp. lower part of the boundary if lookedB p atq“B frompζRq, R 0B, seep Fig.q“B 28. p q " Definition 5.3.3. The content of a framed polygon P, f is defined as p q cont P, f A P P, f , p q “ | Xp zB`p qq| i.e., as the number of elements of A lying either in the interior of P or strictly inside P, f . B´p q

Let now ζ be generic as above and P Pi i I be a polygonal decomposition of P into “ p q P subpolygons, as in Example 3.2.15(b). By Example 5.3.2, we get framings (all denoted fζ ) on P and on each Pi. The following is straightforward.

Proposition 5.3.4. In the situation described we have cont P, fζ cont Pi, fζ . p q“ i p q ř B. The space of intertwiners associated to a polygon. Let now S Schob C, A be a schober with singularities in A. Let P Q be a subpolygon and f be aP framingp of Pq. Keeping the notation of (5.3.1), we define theĂ iterated transport functors

M P, f , M P, f : Φα S Φω S B`p q B´p q p q ÝÑ p q associated to the positive` and negative˘ ` boundary˘ paths given by f. To avoid confusion, let us emphasize the following:

• In the definition of M ˘ P, f we follow Convention 1.1.11, i.e., we identify the stalks of the local systems ΦBatp the intermediateq points by using clockwise monodromy from ` ˘ the incoming to the outgoing direction. This means that for M ` P, f we follow the arc going outside of the polygon P and for M P, f we followB p theq arc going ´ ` ˘ inside P , as shown in Fig. 40. B p q ` ˘

• To identify the sources of the two functors M ˘ P, f with each other (since a priori the sources are the stalks of the local system BΦp S qat the directions corresponding ` α ˘ to the two edges of P coming out of α), we use, similarlyp q to (1.1.10), the monodromy along the arc going inside P . In the similar way (inside arc) we identify the targets of these two functors. This is shown by two thick double arcs in Fig. 40.

137 Φjs´1

‚

Φω Φir Φjs ‚ ‚ “ “

‚ Φir´1 ‚ . . ‚ ð Φi1 ‚

Φα Φi Φj ‚ ‚ “ 0 “ 0 Φj1

Figure 40: The intertwiner dg-space and the identifications of the Φ-categories.

We define the (dg-)space of intertwiners associated to P, f to be p q (5.3.5) I P, f Hom‚ M P, f , M P, f cont P, f . p q “ dgFunpΦα,Φωq B`p q B´p q r p qs ˆ ˙ ` ˘ ` ˘ Proposition 5.3.6. The spaces I P, f for different framings of the same polygon P are canonically identified. p q We will therefore use the notation I P for I P, f with any framing f. p q p q Proof: Let us pass from a given framing f α,ω to a new framing f α,ω1 with the “ p q “ p q same α and ω wir´1 moved from ω wir wjs one step clockwise. This move shortens and enlarges“ by one, so cont P, f“1 cont“ P, f 1. Let B` B´ p q“ p q`

F Mi ,i , G M γ , H M P, f , “ r´1 r “ p q “ B´p q where γ is the part of P, f going from α to wi . Then` ˘ B`p q r´1 1 ˚ ˚ ´1 G M P, f , F M Mi ,i T 1 , “ pB`p qq “ ir´1,ir » r r´1 Φr r´ s the last identification following from Proposition 3.1.13. Recall that the monodromy of the local system Φi is T : TΦ 2 , see §3 A. Using this notation, r “ ir r s ˚ ´1 1 F H Mi ,r T M P, f 1 M P, f 1 . “ r r´1 pB´p qqr s “ pB´p qqr s Therefore the cyclic rotation isomorphism

Hom‚ FG,H Hom‚ G, F ˚H p q » p q given by (the dg-version of) Corollary A.9, gives an identification I P, f I P, f 1 . This shows the invariance of I P, f under our elementary move andp thus,q“ inductively,p q un- der any move that changes ω withoutp changingq α. In a similar way we prove an identification under any move that changes α without changing ω.

138 C. Pasting of the intertwiners. We now consider the situation of Example 3.2.15(b). That is, let P Q be a subpolygon and P Pν ν N be its polygonal subdivision into Ă “ t u P several other subpolygons Pν. Denote for short

(5.3.7) I P I Pν . p q “ νPN p q â In this situation we have the canonical pasting map

(5.3.8) Π : I P I P . P p q ÝÑ p q Its definition is as follows. As in Example 3.2.15(b), we choose a generic direction ζ and the corresponding linear projection pζ : C R. This defines a framing fζ on each Pν and on P , Ñ so P is made into a pasting scheme. Therefore, whenever we assign a category Ci to each vertex wi, a functor Fij : Ci Cj to each oriented edge wi,wj and a natural transformation Ñ r s Uν to each Pν , we can speak about the pasted natural transformation associated to P itself. We apply this to the situation where:

• Ci is an appropriate stalk of the local system of categories Φi;

• Fij is the transport functor Mij;

• Uν I Pν , ν N, is a given familly of elements of the intertwiner spaces. More P p q P precisely, each Uν is viewed, via Proposition 5.3.6, as a natural transformation acting from M Pν, fζ to M Pν, fζ . B`p q B´p q

We define Π ` Uν as˘ the result` of the˘ pasting. A different choice of the direction ζ P p ν q amounts to applying the tensor rotation (changing the source and target path) of each Uν and thus the result will differ by tensor rotation only, i.e., will be the same, according to our identifications.

D. Reminder on L8-algebras. Let h be a graded k-vector space. Recall that a L8-structure on h can be defined an algebra differential D in the completed symmetric algebra Sě1 h˚ 1 , continuous with respect to the natural topology3 and satisfying D2 0. Explicitly, Dp isr´ uniquelysq defined, by the Leibniz rule, by its value on the space of generators“ ˚ ˚ n ˚ h 1 , i.e.,p by the data of linear maps Dn : h 1 S h 1 given for n 1. The dual r´ s r´ sÑ p r´ sq ě map to Dn can be considered as linear map

bn ln : h h, deg ln 1 n ÝÑ p q“ ´ 2 The symmetry conditions on the Dn and the condition D 0 imply a series of well known “ relations on the ln, see [88, 43, 68], of which we recall the following:

3This is imposed to make sure that D is obtained from from an endomorphism ∆ of Sě1pgr1sq by dualization. We can of course work directly with ∆ but then we have to use the coalgebra, not algebra structure on Sě1pgr1sq, similarly to the use of comonads in §3.4.

139 2 (1) l1 : h h satisfies l 0 so it makes h into a cochain complex. Further, each ln is a Ñ 1 “ morphism of complexes with respect to the differentials given by l1.

b2 ‚ (2) l2 : h h is antisymmetric and makes the cohomology space Hl1 V into a graded Lie algebra.Ñ p q

b3 (3) l3 : h h provides an explicit chain homotopy for the Jacobi identity in (2), and so on. Ñ

A L -algebra with ln 0 for n 2 is the same as a dg-Lie algebra. 8 “ ą Sometimes it is convenient to start with a dg-vector space h,d . In this case by “a L -algebra structure on h,d ” one means a L -structure on h (consideredp q as just a graded 8 p q 8 vector space) such that l1 d. “ It is also convenient to write the value of ln on decomposable tensors as an n-ary multi- linear operation (bracket)

ln v1 vn v1, , vn . p b¨¨¨b q“r ¨ ¨ ¨ s By a bracket monomial in variables x1, , xm one means any operation obtained by iteration ¨ ¨ ¨ of brackets, for example x1, x2 , x3, x4, x5 . Such a monomial defines a map (called its m rr s r ss evaluation) h h for any L8-algebra h. We say that h is nilpotent [43], if there is m 2 (the degree of nilpotency)Ñ such that any bracket monomial in m1 m variables has evaluationě on h identically equal to 0. ě

E. The L8-algebra g. We now adapt the constructions of [51] §10 to our situation of a schober S Schob C, A . Assume that A is in exceptionally general position. For any P p q subset B A with B 3 we define P PB Conv B and I B I PB . Ă | |ě “ “ p q p q“ p q Recall that the secondary polytope Σ B has dimension B 3 and its faces F are p q | |´ P labelled by regular polygonal decompositions P Pν, Bν of P, B . In particular, we have “p q p q the dg-vector space I P defined in (5.3.7). Further, an inclusion of faces FP1 F P means that we have a refimenentp q relation P1 ă P and so the pasting maps (5.3.8) giveĂ rise to a map 1 Π 1, : I P I P . P P p q ÝÑ p q The “2-dimensional associativity” of pasting means that the maps ΠP1,P are transitive for 2 1 triple refinement P ă P ă P and so define a cellular complex of sheaves NB on Σ B . p q That is, the stalk of NB on F is I P and the generalization map from F 1 to F is Π 1, . P p q P P P P Proposition 5.3.9. The complex NB is factorizing, i.e., for any regular polygonal decom- position P Pν, Bν as above, the restriction of NB to F Σ Bν is identified with “ p q P “ p q bν NB . ν ś Proof: Follows directly from the definition of the the stalks and generalization maps. Let us now define

(5.3.10) EB I B or Σ B dim Σ B 1 , EP : EP A, P Q, A . “ p qb p p qqr´ p q´ s “ X Ăp q 140 Theorem 5.3.11. The differentials in the cellular cochain complexes of the NB unite to make ‚ the dg-vector space g pP.BqĂpQ,Aq EB into a L8-algebra. This L8-algebra is nilpotent. The subspace “ À g g S EP “ p q “ P ĂpQ,Aq ‚ à is a L8-subalgebra in g. Proof: The construction and argument are similar to those of [51]. Let us specialize the ‚ general discussion of L -structures above to h g and denote the product in the symmetric 8 “ algebra by . For any subdivision P Pν, Bν of a marked subpolytope P, B we have the subspaced “ p q p q

˚ ˚ I P or FP dim FP I Bν or Σ Bν dim Σ Bν p q b p qr s “ ν p q b p p qqr p qs » ˚ â ‚ ‚˚ EBν 1 S g 1 » ν r´ s Ă p r´ sq ä` ˘ Recall (Definition 5.1.6) the concept of a coarse subdivisiop n of a marked subpolygon P, B . Such subdivisions P (corresponding to taut webs of [36]) have the property that thep cor-q responding face F of Σ B has codimension 1. In this case we have the identification P p q (coorientation) κP : or FP or Σ B given by the outer normal to FP . Therefore we have a degree 1 map p q Ñ p p qq ˚ ˚ DP,P : EB 1 EBν 1 r´ s ÝÑ ν r´ s t ä` ˘ given by the tensor product of ΠP,P , the map dual to the pasting (5.3.8), and of the coori- entation κP . We now define D on each summand E˚ 1 to be the sum over all coarse subdivisions P Br´ s of P, B , of the maps DP,P . The Leibniz rule extends this uniquely to an algebra differential p ‚ q ‚ of S‚ g˚ 1 , which commutes with the differential induced from that in g by construction (all pastingp r´ sq maps commute with differentials). The property D2 0 follows from the “ similar,p obvious, property δ2 0 for the differentials δ in the cellular cochain complexes of “ the factorizing complexes of sheaves NB on the Σ B . p q 2 2 B1,¨¨¨ ,Bn More precisely, it is enough to prove that D 0 on the generators. Let D B be 2 “ ˚ p q the matrix element of D acting from the summand EB 1 of the space of generators to ‚ r´ s the summand E˚ 1 in S‚ g˚ 1 . In order for D2 B1,¨¨¨ ,Bn to not be 0 from the ν p Bν r´ sq p r´ sq p qB outset, we must have that P Pν , Bν is a regular polygonal subdivision of P, B and  “ tp qu p 2 B1q,¨¨¨ ,Bn the corresponding face FP Σ Bp has codimension 2. Now, if this is true, then D B is identical with the matrixĂ elementp q of δ2 corresponding to the maximal cell of Σp Bq and a codimension 2 cell F . So it is zero and therefore D2 0 identically. p q P “ ‚ This makes g into a L8-algebra. The claims that it is nilpotent and that g is a L8- subalgebra follow in the same way as in [51] by analyzing the brackets, see the next Remark 5.3.12.

141 ‚ Remark 5.3.12. It follows from the construction that the L8-operations in g are given by ‚ bn ‚ “pasting in coarse decompositions”. That is, the restriction of ln : g g to a summand n p q Ñ EB EB is zero unless P Pν, Bν , Pν : Conv Bν , is a coarse regular 1 b¨¨¨b n “ tp quν“1 “ p q polygonal decomposition of P, B where P Pν and B Bν (in particular, unless P p q “ “ is convex). In this is the case, then ln is the product of the pasting ΠP and the coorientation κ. This implies: Ť Ť

‚ (a) The maximal degree of a nonvanishing bracket monomial in g is bounded by the max- ‚ imal number of triangles in a triangulation of Q, A . Therefore g is nilpotent. p q ‚ (b) If each Bν Pν A represents a summand in g g, then the value of ln on the tensor “ X Ă product of the EBν also lies in a summand from g, as B P A in this case. Therefore ‚ “ X g g is a L -subalgebra. Ă 8 5.4 Deforming the infrared monad R A. The deformation complex of R. We fix a direction ζ such that A is in linearly general position including ζ-infinity. Let S Schob C, A and R Rζ be the corresponding infrared monad, see Proposition 4.3.2. P p q “ We define the deformation complex, or ordered Hochschild complex of the monad R as

‚ ‚ ‚ ‚ ÝÑC ÝÑC R, R Hom Rij, Rij Hom RjkRij, Rik “ p q “ p qÑ p qÑ " iăj iăjăk à à ‚ Hom RklRjkRij, Ril , Ñ p qѨ¨¨ iăjăkăl * à with the (horizontal) grading starting in degree 0. Compare with [51] §12. Elements of the component p ‚ ÝÑC Hom Rip,ip`1 Ri0,i1 , Ri0,ip`1 “ i0㨨¨ăip`1 ¨ ¨ ¨ à ` ˘ can be seen as p 1 -ary “operations” having as inputs, the p 1 functors Ri ,i and as p ` q ` ν n`1 output, the functor Ri0,ip`1 . Similarly to the case of the Hochschild complex of an associative (dg-)algebra, ÝÑC ‚ is a dg-Lie algebra with respect to the Gerstenhaber bracket [42]

p`1 q`1 degpfq degpgq p q f,g f ν g 1 g ν f, f ÝÑC , g ÝÑC . r s “ ν“1 ˝ ´ p´ q ν“1 ˝ P P ÿ ÿ Here f ν g is the “operadic composition” cf. [78], obtained by substituting the output of the operation˝ g into the νth input of the operation f (whenever this makes sense, and defined to be zero otherwise).

142 Example 5.4.1. As in the case with associative (dg-)algebras, a Maurer-Cartan element 1 α ÝÑC gives a unitriangular A8-deformation R α of R. More precisely, α, being of total degreeP 1, consists of components p q

p1q p1q 1 p2q p2q 0 α αij Hom Rij, Rij , α αijk Hom RjkRij, Rik , “p q P iăj p q “p q P iăjăk p q àp3q p3q ´1 à α αijkl Hom RklRjkRij, Ril , “p q P iăjăkăl p q ¨ ¨ ¨ à The deformed monad R α has: p q p1q (0) The same functors R α ij Rij. but with differential d α . p q “ ` ij p2q (1) The composition maps cijk α : RjkRij Rik defined by cijk α cijk α . p q Ñ p q“ ` ijk p3q (2) The cijk α may not be strictly associative but the α define the homotopy for the p q ijkl associativity and so on to give a full A8-structure.

h B. Reminder on L8-morphisms. Let h, k be two L8-algebras given by differentials D k ě1 ˚ ě1 ˚ and D in S h 1 and S k 1 respectively. Recall that a L8-moprhism ϕ : h k can be seen asp a morphismr´ sq of commutativep r´ sq dg-algebras Ñ p p ϕ7 : Sě1 k˚ 1 ,Dk Sě1 h˚ 1 ,Dh p r´ sq ÝÑ p r´ sq continuous with respect to` the natural topologies˘ ` on the source and˘ target. As above, we p p h bn can view the L8-structures on U and V in terms of the multilinear operations ln : h h and lk : kbn k, n 1. We also denote both differentials lh and lk by d and write the higherÑ n Ñ ě 1 1 ln as brackets. The morphism ϕ7, considered as just a morphism of free commutative algebras, is deter- mined by its values on the generators which dualize into a sequence of degree 0 maps

n (5.4.2) ϕn : S h 1 1 k p r sqr´ s ÝÑ or, equivalently, maps r bn ϕn : h k, n 1, deg ϕn n 1 ÝÑ ě p q“ ´ satisfying the symmetry/antisymmetry conditions derived from the above. The condition 7 that ϕ commutes with the differentials, gives a series of identities on the ϕn, on which the first two are:

(1) ϕ1 : h k commutes with the differentials. Ñ (2) For u1,u2 h we have P degpu1q ϕ1 u1,u2 ϕ1 u1 ,ϕ1 u2 d ϕ2 u1,u2 ϕ2 du1,u2 1 ϕ2 u1,du2 . pr sq´r p q p qs “ p p qq´ p q`p´ q p q ‚ ‚ In particular, the map Hd h Hd k defined` by ϕ1, is a morphism of graded Lie˘ algebras. p q Ñ p q

143 See [68] §3.2.6 or [72], Def. 5.2 for the full list of these identities, the second reference treating the case (which alone will be relevant for us) when k is a dg-Lie algebra, i.e., lk 0 for n 2. n “ ą A L -morphism ϕ is called of finite order, if ϕn 0 for n 0. 8 “ " C. From g to the deformation complex of R. Next, adapting the construction of ‚ [36, 37] as interpreted in [51] §10-11, we construct a L8-morphism η : g ÝÑC . Ñ bn ‚ In order to do this, for each n 1 we need to define a map ηn : g ÝÑC . Recall that ě Ñ the source and target of this desired ηn are direct sums of summands labelled by certain combinatorial data:

• We have g EP , see (5.3.10). Therefore “ P À bn g EP1 Epn , “ P1,¨¨¨ ,Pn b¨¨¨b à the summands being labelled by sequences of subpolygons P1, ,Pn Q. ¨ ¨ ¨ Ă

• We have Rij Rγ , the summands being labelled by ζ-convex polygonal paths “ γPΛpi,jq γ joining w and w , see (4.3.1). Therefore i À j ‚ (5.4.3) ÝÑC Hom Rγ1 Rγn , Rγ , “ γ1,¨¨¨ ,γm,γ p ˝¨¨¨ q à the summands being labelled by sequences of ζ-convex paths γ1, ,γm,γ such that ¨ ¨ ¨ γ1, ,γm form a composable sequence and γ has the same beginning and end as the ¨ ¨ ¨ composition γ1 γn. ˝¨¨¨˝

Therefore ηn should be given by matrix elements

n γ1,¨¨¨ ,γm|γ : Hom (5.4.4) ηP1,¨¨¨ ,Pn EPν Rγ1 Rγn , Rγ , ν“1 ÝÑ p ˝¨¨¨ q â where P1, Pn and γ1, ,γm,γ are as above. As in [51], we will define them by analyzing coarse subdivisions¨ ¨ ¨ of unbounded¨ ¨ ¨ polygons. More precisely, let us choose a Vladivostok point v rζ, r 0 far away in the direction ζ and let A A v and Q Conv A . Assume that A“is in exceptionally" general position. A subpolygon“ GYt Qu will be“ called boundedp q , if v G and unbounded, if v G. Ă R P A bounded subpolygon is the same as a subpolygon in A. An unbounded subpolygon G can be thought of as approximately ζ-convex, in particular, its unbounded edges (those containing v) can be depicted as half-lines going in direction ζ, see Fig. 41. Further, such G is uniquely determined by a ζ-convex path γ. Explicitly, γ is the bounded part of the ζ boundary G. Alternatively, γ ´ G is the “bottom part” of G if ζ is positioned B “ B p q ζ B 8 vertically. In what follows let us write for . B˘ B˘ 144 In the other direction, G Conv γ v , or G Convζ γ in the parallel representa- “ p Y t uq “ p q tion of Fig. 41. Therefore instead of convex paths γ1, ,γm,γ in (5.4.3) we can use the ¨ ¨ ¨ corresponding unbounded polygons G1, ,Gm,G. ¨ ¨ ¨ Let now G be an unbounded polygon and P be a polygonal decomposition of G. It consists of several bounded polygons which we denote P1, ,Pn (the precise numeration is ¨ ¨ ¨ not important) and several unbounded polygons which we denote G1, ,Gm and number ¨ ¨ ¨ consecutively clockwise around the boundary of G, see Fig. 41. Let γj Gj be the “ B´ ζ-convex path corresponding to Gj, oriented counterclockwise around the boundary of Gj. Then γ1, ,γm form a composable sequence. Let γ G. Then γ has the same beginning ¨ ¨ ¨ “B´ and end as γ1 γm. Note that γ may have nontrivial overlap with γ1 along the “left ˝¨¨¨˝ handle” λ and with γm along the “right handle” ρ, see Fig. 41.

Further, our choice of ζ gives a framing fζ of each Pν, ν 1, , n. In particular we “ ¨ ¨ ¨ have the positive and negative boundaries Pν and the pasting map B˘ n ‚ ‚ (5.4.5) ΠP : Hom M `Pν , M ´Pν Hom Mγ1 Mγm , Mγ . ν“1 pB q pB q ÝÑ p ˝¨¨¨˝ q â ` ˘

Now recal (4.3.1) that Rγ M γ or γ . Further, we have a canonical identification “ p qb p q max (5.4.6) or γ kGXA or Σ G A p q“ ÝÑ p p X qq coming from two steps: ľ (1) The point v A is distinguished, so wedge product with it on the left gives an identi- maxP max tification kGXA kGXA . p qÑ p q (2) By constructionŹ [39], the polytopeŹ Σ G A is embedded into kGXA and its affine span there is a vector subspace of codimensionp X 3qthe quotient by which is Aff A , the space of restriction to A of affine R-linear functions C R. This quotient subspacep q has a canonical orientation coming from the standard orientatioÑ n of C. So the orientation space of Σ G A , being the same as the orientation space of its affine span, is identified p X q with the orientation space of kGXA.

Definition 5.4.7. Let us call a sequence of bounded subpolygons P1, ,Pn and a sequence ¨ ¨ ¨ of ζ-convex paths γ1, ,γm,γ associated, if there exists a (necessarily unique) coarse subdi- ¨ ¨ ¨ vision P of an unbounded subpolygon G into bounded subpolygons Pν (in some order) and unbounded polyygons G1, ,Gm (in the counterclockwise order as above) so that γj ´Gj and γ G. ¨ ¨ ¨ “B “B´ Taking into account the identification (5.4.6) we see, just as in the definition of g, that in the situation of Definition 5.4.7 we have a canonical coorientation n m κ P : or Σ Pν A or γj or γ . ν“1 p p X qq b j“1 p q ÝÑ p q â â 145 It is just the coorientation of the codimension 1 face FP of the polytope Σ G A given by the outer normal to this face. Note also that p X q

‚ (5.4.8) EP Hom M Pν , M Pν or Σ A Pν 2 A P . ν “ pB` q pB´ q b p p X qqr ´ | XB` |s ` ˘ Right handle ρ G

γ1 G1

P1 . γ . 2 G2 ¨ ¨ ¨ Pn

γm Gm

Left handle λ ζ

Figure 41: A subdivision of an unbounded polygon G.

γ1,¨¨¨ ,γm|γ We now define the matrix element η in (5.4.4) to be 0 unless P1, ,Pn and P1,¨¨¨ ,Pn ¨ ¨ ¨ γ1, ,γm,γ are associated. If they are associated via a coarse subdivision P, we define the ¨ ¨ ¨ κ matrix element to be given by the pasting map (5.4.5), tensored with the coorientation P . Comparison of the shifts in (5.4.8) and with the horizontal grading in ÝÑC ‚ shows that in this way we get a degree 0 map of graded vector spaces.

γ1,¨¨¨ ,γm|γ Theorem 5.4.9. The matrix elements ηP1,¨¨¨ ,Pn thus defined form a finite order L8-morphism η : g ÝÑC ‚. Ñ

D. Proof of Theorem 5.4.9. We start with a general discussion. Let h be a L8- algebra, U be an associative dg-algebra and

‚ ‚ ‚ ‚ b2 C C U, U Homk U, U Homk U , U “ p q “ p qÑ p qѨ¨¨ be the standard Hochschild complex of U, with the horizontal grading starting( in degree 0. Similarly to the above (and more classically), C‚ is a dg-Lie algebra, so we can speak about ‚ L8-morphisms ϕ : h C . Let us recall a description of such morphisms in terms closer to those used earlier. Ñ The dg-algebra structure on U is encoded by a differential DU in the completed tensor algebra T ě1 U ˚ 1 , which has the form DU DU DU , where DU is the differential p r´ sq “ 1 ` 2 1 p 146 U induced by that in U while D2 is the graded derivation given on the space of generators by the map U ˚ U ˚ U ˚ dual to the multiplication in U. As before,Ñ let usb encode the L -structure on h by a differential Dh in Sě1 h˚ 1 . 8 p r´ sq Proposition 5.4.10. In the above notation, we have a bijection between: p (i) L -morphisms ϕ : h C‚ U, U . 8 Ñ p q (ii) Algebra differentials D in the completed tensor product Sě1 h˚ 1 T ě1 U ˚ 1 which are continuous in the natural topology, satisfy D2 0 andp r´ whosesqb valuesp onr´ thesq “ generators have the form: p p p

h U ϕ D h˚r´1sb1 D h˚r´1s 1,D 1bU ˚r´1s 1 D U ˚r´1s Dn,m, | “ | b | “ b | ` m,ně1 ÿ Dϕ :1 U ˚ 1 Sn h˚ 1 T m U ˚ 1 . n,m b r´ s ÝÑ p r´ sqb p r´ sq Proof: Given data in (ii), we dualize Dϕ and add a shift by 1 to get a degree 0 map n,m ´ n ‚ m m ϕn,m : S h 1 1 Hom T U 1 , U 1 C U, U m , p r sqr´ s ÝÑ p r sq pr sq “ p qr´ s the target being the mth term of the Hochschild` complex. These˘ maps unite into a collection r n ‚ of degree 0 maps ϕn : S h 1 1 C U, U . The condition that the ϕn form a L8- morphism, see (5.4.2), is thenp r sqr´ expresseds Ñ by pD2 q0. “ We now adapt thisr approach to our situation which, on the face of it, has twor differences:

(1) Instead of a dg-algebra U we have a dg-monad, i.e., a dg-endofunctor R : V V with ÑN a (closed, degree 0) associative natural transformation RR R. Here V Φi. Ñ “ i“1 (2) We consider not the full but the ordered Hochschild complex with respectÀ to the ordering on the summands Φi and the uni-triangular form R Rij i j of R. “p q ă To account for (1), we associate to the dg-monad R an associative dg-agebra U R as follows. Let F : C D be a dg-functor of dg-categories. Consider the dg-vector spacep q Ñ ‚ U F HomD d, F c . p q “ cPC,dPD p p qq à Given two composable dg-functors C F D G E, we have the morphism of dg-vector spaces U G U F U GF which consistsÑ ofÑ the composition maps p qb p q ÝÑ p q Hom‚ e, G d Hom‚ d, F c Hom‚ GF c , e , g f G f g, E p p qqb Dp p qq ÝÑ E p p q q b ÞÑ p q˝ and is zero one any other tensor products of summands. Further, a closed, degree 0 natural transformation t : F1 F2 between dg-functors F1, F2 : C D gives rise to a morphism of Ñ Ñ dg-vector spaces U t : U F1 U F2 . p q p qÑ p q 147 Therefore the (infrared) monad structure RR R on the endofunctor R : V V gives an associative dg-algebra structure U R U R Ñ U R . To account for theÑ difference p q b p q Ñ p‚ q (2) above, note that the ordered Hochschild complex ÝÑC of R is in fact a L8-subalgebra in ‚ bn ‚ C U R , U R . Therefore our morphisms of dg-vector spaces ηn : g ÝÑC give morphisms p p q p qq bn ‚ Ñ of dg-vector spaces U ηn : g C U R , U R and it is enough to show that the U ηn p q Ñ p p q p qq p q form a L8-morphism. For this, we can use Proposition 5.4.10 and argue similarly to the proof of Theorem 5.3.11. That is, we construct a differential D in Sě1 g˚ 1 T ě1 U R ˚ 1 (no comple- p r´ sqb p p q r´ sq tions needed in our particular case) incorporating the U ηn , by looking at the faces of the secondary polytopes. p q

More precisely, ηn is defined in terms of coarse (corresponding to codimension 1 faces of the secondary polytope) regular decompositions P of all possible unbounded polygons G, consisting of n bounded polygons and some number m of unbounded ones, see Fig. 41. The bounded polygons do not have any natural order, but the unbounded ones are ordered counterclockwise as in Fig. 41. Therefore the formula for ηn defines, for each m 1, a morphism ě DUpηq : U ˚ 1 Sn g˚ 1 T m U ˚ 1 , n,m r´ s ÝÑ p r´ sqb p r´ sq (we use the order on the unbounded polygons to lift the value into the tensor algebra). This, together with other components, defines a degree 1 graded derivation D of Sě1 g˚ 1 T ě1 U ˚ 1 , and all we need to show is D2 0. p r´ sq b p r´ sq “ This follows just like in the proof of Theorem 5.3.11, from comparison of matrix elements of D2 with those of the δ2 where δ runs over the differentials in the cellular chain complexes of appropriate cellular sheaves on the secondary polytopes Σ G A . p X q

5.5 Maurer-Cartan elements in the infrared L8-algebra

A. Maurer-Cartan elements in L8-algebras. Let h be a nilpotent L8-algebra. An element β h1 (of degree 1) is called a Maurer-Cartan element, if P 8 1 1 1 l β β dβ β, β β,β,β 0. n! n 2! 3! n“1 p b¨¨¨b q “ ` r s` r s`¨¨¨ “ ÿ This extends the more familiar concept for dg-Lie algebras, discussed in §4.3 D. We denote MC h the set of Maurer-Cartan elements in h. We also refer to [43] for the concept of gauge equivalencep q of Maurer-Cartan elements.

Let ϕ : h k be a finite order L8-morphism of nilpotent L8-algebras, with components bn Ñ ϕn : h k, n 1. Then for a Maurer-Cartan element β MC h the element Ñ ě P p q 8 1 ϕ β ϕ β β ˚ n! n p q “ n“1 p b¨¨¨b q ÿ is a Maurer-Cartan element in k, cf. [68] §3.2.6. This gives a map ϕ : MC h MC k . ˚ p qÑ p q 148 B. Maurer-Cartan elements in g explicitly. We specialize the above to the L8- morphism η : g ÝÑC ‚ R, R . Example 5.4.1 implies that any Maurer-Cartan element β of g Ñ p q gives an A8-deformation R β : R η˚ β of the infrared monad R. For this reason let us analyze the data encoded inp aq datum“ p ofp βqq MC g more explicitly. P p q We have g EP , where P runs over subpolygons P Q (with vertices in A). “ P Ă Therefore β βP is a system of elements βP EP , one for each subpolygon P . Let us write Σ P “Σ p AÀq P for the secondary polytopeP of P . Let us also choose a generic p q “ p X q direction ζ which gives a framing fζ on each P , in particular, it gives the positive and negative parts P of the boundary of P . B˘ Recalling (5.3.10) the definition of EP , taking into account the shifts as in (5.4.8) and setting aside the orientation factors (amounting to 1), we can say that βP is a natural transformation ˘

‚ βP Hom M P , M P , deg βP 3 A P . P pB` q pB´ q p q “ ´ | XB` | ` ˘ So the range of possible degrees of the components βP is 1, 0, 1, . It is convenient to analyze the Maurer-Cartan conditions on them recursively, starting´ ¨ ¨ ¨ from some “basic” components of degrees 1 and 0 and viewing the rest as a “system of higher coherence data” on these basic components. A subpolygon P Q, A will be called empty, if it contains no elements of A other than the vertices. Ăp q Basic level: multiplication and comultiplication. The “basic level” of β consists of the βP βijk where P Conv wi,wj,wk is an empty triangle. We number them so that “ “ t u wi ζ wj ζ wk, see Fig. 42. There are two possibilities: ă ă

(1) P wi,wk consists of a single edge, see the left of Fig. 42. In this case βijk : Mik B` “r s Ñ MjkMij is a natural transformation of degree 1, which we view as a comultiplication. The Maurer-Cartan condition on βijk amounts to d βijk 0. p q“

(0) P wi,wj wj,wk consists of two edges, see the right of Fig. 42. In this case B` “ r sY s βijk : MjkMij Mik is a natural transformation of degree 0, which we view as a Ñ multiplication. The Maurer-Cartan condition on βijk amounts to d βijk 0. p q“

wk wk ‚ ‚ ζ

β β wj ‚ ijk ijk ‚ wj ð ð ‚ ‚ wi wi

Figure 42: Empty triangles give comultiplications and multiplications.

149 Extremal empty polygons: A8-structures. We call an empty n-gon P extremal of the first kind, if P consists of a single edge. βP has degree 1. We say that P is extremal of B` the second kind, if P consists of a single edge. In this case βP has degree 3 n. B´ ´ The Maurer-Cartan conditions of the βP corresponding to extremal empty P of the first kind amount to saying that these βP form an A8 (homotopy co-associative) structure for the comultiplications βijk corresponding to empty triangles of the first kind.

ζ

wl wl wl ‚ ‚ ‚ w w w k‚ k‚ k‚ βjkl βikl βijkl

d βijl βijk ‚ “ ‚ ´ ‚ wj ð wj wj ‚ ‚ ‚ wi wi wi

Figure 43: An extremal empty 4-gon of the first kind gives a homotopy for coassociativity.

For example, let P be an empty 4-gon of the first kind. Labeling the vertices wi,wj,wk,wl as in Fig. 43, we get a degree 0 natural transformation βP βijkl : Mil MklMjkMij. “ Ñ The Maurer-Cartan condition on βijkl is that it ensures homotopy coassociativity of the elementary (triangular) comultiplications: d βijkl is the difference between two possible bracketing of four factors which correspond top twoq possible trianguilations of the 4-gon P , see Fig. 43. The Frobenius condition on the muiltiplication and comiltiplication. Consider now empty 4-gons P with P consisting of two segments, as in Fig. 44. The corresponding B` βP have degree 0 and can be seen as homotopies establishing certain relations between the multiplication and comultiplication arrows obtained by triangulating P into two empty triangles. There can be two different types of position depicted on the top and bottom of Fig. 44. The relations we obtain are analogous to the so-called Frobenius conditions in the theory of Frobenius algebras [65]. More precisely, suppose H is a vector space with a multiplication H H H, written a b ab and a comultiplication ∆ : H H H, written in b Ñ b ÞÑ p1q p2q Ñ b p1q p2q Sweedler’s notation as ∆ a a a , the latter being a shorthand for i ai ai . Then the top and bottomp partq “ of Fig.b 44, read right to left, can be written as b ř ∆ ab abp1q bp2q, ∆ ab ap1q ap2qb p q„ b p q„ b (with meaning being homotopic). Here the elements a and b are symbolically depicted on the „-arrows in Fig. 44. Compare with [65], Lemma 2.3.19. B` 150 ζ

‚ ‚ a a

mult. ‚ mult. ‚ „ ‚ ð b ‚ comult. ð b comult.ð ð ‚ ‚ ‚ ‚

comult. a a ‚ ‚ comult. ð „ ‚ ð mult. ‚ mult. ð ð b b ‚ ‚

Figure 44: Homotopy relations between multiplication and comultiplication.

A triangle with one point inside: 3 1 move. Let now P Conv wi,wj,wk be a Ñ “ t u triangle with one element wl of A inside. The Maurer-Cartan condition on βP has then the form that d βP equals to the pasting of the diagram of multiplications and comultiplications p q formed by the three triangles with vertex wl partitining P . In other words, this pasting is homotopically zero. In the case when P consists of two segments, βP has degree 0 and B` the pasting diagram we obtain is identical to that depicted in Fig. 29. So this βP gives an analog of the 3 1 move identity of Proposition 3.2.16. Ñ C. The infrared Maurer-Cartan element (conjectural). From the above analysys we see that the conditions on the basic components of a Maurer-Cartan element β g1 are completely analogous to those satisfied by the Picard-Lefschetz arrows in §3.2. The differenceP is that the Picard-Lefschetz arrows as we defined them, live in the homotopy category of dg-functors (are cohomology classes of closed natural transformations of degrees 1 or 0), while the components of β are natural transformations themselves. This makes natural the following conjecture. Conjecture 5.5.1. For any schober S Schob C, A there exists a Maurer-Cartan ele- 1 P p q ment β βS gA S , defined canonically up to gauge equivalence so that for any generic direction“ζ weP havep theq following properties:

(1a) For an empty triangle P Conv wi,wj,wk with `P consisting of one edge, as in “ t u B 1 ‚ the left of Fig. 42, the cohomology class of βP in H Hom Mik, MjkMij is equal to p q the Picard-Lefschetz arrow uijk u w ,w , w ,w , w ,w , see (3.2.3). “ r j ks r i j s r i ks 151 (1b) For an empty triangle P Conv wi,wj,wk with `P consisting of two edges, as in “ t u B 0 ‚ the right of Fig. 42, the cohomology class of βP in H Hom MjkMij, Mik is equal to p q the Picard-Lefschetz arrow vijk v w ,w , w ,w , w ,w , see (3.2.3). “ r j ks r i j s r i ks (2) The deformation Rζ β of the infrared monad Rζ corresponding to β, is quasi-isomorphic to the Fukaya-Seidelp monadq M v corresponding to the Vladivostok point v rζ, r 0 p q “´ " situated far away in the direction ζ and the v-spider Kv being the union of straight p´ q segments v,wi see Fig. 35. r s This conjecture should follow from a more systematic definition of perverse schobers where all the coherence data will be given at the outset as a part of the structure. Remark 5.5.2. Note that Conjecture 4.3.9 about an MC-lifting of the differential in the infrared complex is a particular case of Conjecture 5.5.1 obtained by focussing on the effect of β just on the differentials in the complexes Rij (level (0) of Example 5.4.1). Indeed, Fig. 34, describing the construction of the differential q in the infrared complex, can be seen as a particular case of Fig. 41 corresponding to m 1. “ Example 5.5.3. Consider the “maximally concave” situation of Example 4.3.3. That is, assume that A and ζ are positioned so that every element of A is a vertex of Conv´ζ A , see Fig. 36. Then every subpolygon P Q, A is, in the terminology of this section,p q Ă p q empty and extremal of the second kind. As we saw, in this case Rij Mij is the rectilinear ˚ “ transport functor. Moreover, it is identified with Mij a ai by deforming the interval wi,wj “ j r s to the Vladivostok path wi, v v,wj , as in Fig. 35. However, all the compositions R r sYr s M cijk : Rjk Rij Rik, i j k, are zero, while cijk : Mjk Mij Mik is given by the ˝ Ñ ă ă˚ ˝ Ñ counit for the adjoint pair aj, aj . Note that under our system of identifications Mij Mij, p q M » the cohomology class of the closed degree 0 natural transformation cijk corresponds to the M Picard-Lefschetz arrow vijk. Moreover, the cijk are associative by construction. Therefore, M putting βP cijk for P Conv wi,wj,wk being a triangle and βP 0 for all n-gons P with n 4, we get“ a Maurer-Cartan“ t element βuand R β M, thus establishing“ the conjecture in thisě case. p q» A natural strategy of proving Conjecture 5.5.1 would be to start from A A 0 in a “ p q maximally concave position above and then deform A via a path A t w1 t , ,wN t , t 0, 1 , to a new position A1 A 1 . We can then deform S accordingly,p q“p p q i.e.,¨ ¨ ¨ “isomon-p qq odromically”P r s to a new schober S“t p Schobq C, A t for each t. If the path A t is generic, the nature of g, R and M will bep unchangedq P p exceptp qq for finitely many “wall-crossing”p q points where we cross either a wall of collinearity/horizontality (§2.2) or an exceptional wall (§5.2), and the task is to describe explicitly the modification of the Maurer-Cartan element β t under such wall-crossing. We plan to return to this in a future work. p q

A Conventions on (enhanced) triangulated categories

1. DG-enhancements. Let k be a field of characteristic 0. We will work with k-linear triangulated categories (all Hom’s are k-vector spaces). The n-fold shift functor of a trian-

152 gulated category V will be denoted by x x n . An exact triangle in V will be written as ÞÑ r s x y z x 1 . ÝÑ ÝÑ ÝÑ r s In many cases we need an additional structure on a triangulated category called an enhance- ment. In this paper we are mostly interested in differential graded (dg-)enhancements [15] so we briefly recall the terminology and conventions. Recall that a (k-linear) dg-category is a category V in which each Hom-set is made into a ‚ k dg-vector space (=cochain complex) HomV x, y over , so that the differentials satisfy the Leibniz rules with respect to composition. Wep referq to [90, 91, 93] for a systematic backround on dg-categories. A dg-category V give a k-linear category H0 V by passing to the 0th cohomology of the Hom-complexes: p q

0 0 ‚ Ob H V Ob V , Hom 0 x, y H Hom x, y , p p qq “ p q H pVqp q “ V p q and a similar category H‚ V obtained by passing to the full cohomology spaces, not just H0. p q For example, all dg-vector spaces over k form a dg-category dgVectk. A dg-functor F : V W between dg-categories is a functor preserving the structures (vector space, differential,Ñ grading) on the Hom-complexes. All dg-functors V W form a new dg-categoru dgFun V, W . Any dg-category V has the Yoneda embedding Ñ p q op ‚ Υ: V dgFun V , dgVectk , x Hom , x . ÝÑ p q ÞÑ V p´ q A dg-functor F : V W is called a quasi-equivalence, if H‚ F : H‚ V H‚ W is an equivalence of categories.Ñ p q p q Ñ p q Any dg-category V has the pre-triangulated envelope Pre - Tr V , see [15, 90]. It is a new p q ‚ i dg-category whose objects are twisted complexes over V, i.e., data V V iPZ, qij iăj , i j´i i j “ p q p q where V are objects of V and qij Hom V ,V are such that P V p q ` ˘

dqij qjkqij 0. ` iăkăj “ ÿ 0 0 The category Tr V : H Pre - Tr V is always triangulated. A morphism f Hom - x, y p q “ p p qq P Pre TrpVqp q which is closed (i.e., df 0), has the canonical cone Cone f which fits into an exact triangle in Tr V “ p q r p q r x f y Cone f x r1 . ÝÑ ÝÑ p q ÝÑ r s Here f is the morphism in Tr V represented by the cohomology class of f. p q r A dg-category V is called pre-triangulated, if the canonical embedding κ : V Pre - Tr V is a quasi-equivalence. In this case H0 V is triangulated. Moreover, for ar pre-triangulatedÑ p q V we have a quasi-inverse dg-functorp toq κ which is called the total object functor Tot :

153 Pre - Tr V V. Note that a closed, degree 0 morphism f : x y is a particular type of a p qÑ ‚ ´1 0 Ñ twisted complex: in this case V has V x, V y and q 1,0 f. The corresponding “ “ ´ “ total object Tot V ‚ is simply the cone of f. This means thatr we can speak about canonical cones of closed,p degreeq 0 morphisms of V. r if V, W are dg-categories and W is pre-triangulated,r then so is dgFun V, W . In par- ticular, we can speak about cones of natural transformations of dg-functors.p q Note that op Pre - Tr V can be seen as the full dg-subcategory in dgFun V , dgVectk formed by taking iteratedp cones,q and shifts, starting from objects of Υ V . p q p q A dg-enhancement of a(k-linear) triangulated category V is an identification of triangu- lated categories V H0 V for a pre-triangulated category V. » p q Convention A.1. In the main body of the paper, we will often not distinguish notationally between V and V, saying that V is an “enhanced triangulated category” (meaning “V is an enhancement of V”) or speaking about “exact triangles in V” (which belong, strictly speaking, to V) etc.

Example A.2. Let R be a dg-algebra (=dg-category with one object). The category- modR op “ dgFun R , dgVectk of right dg-modules over R is pre-triangulated. A dg-module P is called perfectp, if it can be obtainedq from free modules (copies of R) by iterated application of shifts, taking of cones and of homotopy direct summands, see [91, 92]. Let PerfR - modR be the full dg-subcategory of perfect dg-modules. Ă Given two dg-algebras R1, R2 and an R1, R2 -dg-module M, perfect over R1 and R2, we have the dg-functor p q L FM : PerfR PerfR , P P M 1 ÝÑ 2 ÞÑ bR1 (the derived tensor product). See [91] for a more general treatment, where dg-algebras are replaced by dg-categories. In most of considerations of the present paper, one can assume the enhanced triangulated categories to be of the form PerfR and the functors to be of the form FM , thus simplifying the foundational issues.

2. -categorical enhancements. The theory of dg-categories can be embedded into the theory8 of -categories (or weak Kan complexes) [73] which provides a more general approach to “enhancements”.8 While we do not use this theory in the present paper, it will be used in the more systematic treatment of schobers in [32], so let us say a few words about the relation of the two approaches. An -category V gives an ordinary category ho V known as the homotopy category of V. From the8 point of view on -categories as simpliciallyp q enriched categories ( [73] §1.1.5), an 8 -categorical enhancement of a category V represents each HomV x, y as π0 of a simplicial 8set (instead of H0 of a cochain complex as is the case with dg-enhancements).p q To a dg-category V one can associate an -category NdgV known as the dg-nerve of V. 8 Viewed as a weak Kan complex, NdgV consists of Sugawara (weakly commutative) simplices 0 of closed, degree 0 morphisms of V, see [48, 74], and ho NdgV H V . p q“ p q 154 The role of pre-triangulated dg-categories is played, in the -categorical context, by 8 stable -categories [74]. If V is a pre-triangulated dg-category, then NdgV is stable [34], so the two8 approaches are compatible. The use of -categorical enhancements can clarify the treatment of several issues related to perverse schobers.8 For example, small dg-categories do not form a dg-category, but small -categories do form an -category. So the construction of NdgV leads to a definition of an 8-category dgCat formed8 by small dg-categories. This means that perverse schobers should, in8 a more systematic theory, form an -category. 8 3. Adjunctions for ordinary categories. Let V, W be categories. Recall [75] that a F pair F,G of functors in the opposite directions V o / W , is called an adjoint pair, if it p q G is equipped with natural isomorphisms „ (A.3) αx,y : Hom F x ,y Hom x, G y , x V, y W. W p p q q ÝÑ V p p qq P P One says that G is right adjoint to F and writes G F ˚, and also that F is left adjoint to ˚ “ G and writes F G. Equivalently, the data of αx,y are encoded in the unit and counit morphisms of functors“ p q

(A.4) e eF : Id GF, η ηF : FG Id , “ V ÝÑ “ ÝÑ W satisfying the standard axioms [75]. For example, for x V the morphism ex : x GF x P Ñ p q is αx,F pxq IdF pxq . Notep that theq adjoints, if they exist, are defined canonically (i.e., uniquely up to a unique isomorphism). For example if F is given, then (A.3) means that G y represents the functor x Hom F x ,y , and the representing object of a functor, if it exists,p q is defined canonically. ÞÑNote furtherp p q thatq passing to the adjoints reverses the order of compositions:

˚ ˚ ˚ ˚ ˚ ˚ (A.5) F1 F2 F F , G1 G2 G2 G1. p ˝ q “ 2 ˝ 1 p ˝ q “ ˝ We can also speak about iterated adjoints F ˚˚ F ˚ ˚, ˚˚G ˚ ˚G etc. provided they exist. “ p q “ p q Remark A.6. The concept of adjoint functors between categories is a particular case of the concept of adjoint 1-morphisms between objects of a 2-category C (corresponding to C Cat being the 2-category of categories). Most of the constructions related to adjoints generalize“ to that 2-categorical setting. On the other hand, a 2-category C with one object pt is the same as a monoidal category V, whose objects are 1-morphisms pt pt and is the composition of 1-morphisms. Thep bq2-categorical concept of left and rightÑ adjoints specializeb s then to the concept of left and right duals ˚V,V ˚ of an object V V, see, e.g., [33] §2.10. The features below are perhaps more familiar in such “tensor context”.P

Given two functors F1, F2 : V W and a natural transformation k : F1 F2, we have the right and left transposes Ñ Ñ

t ˚ ˚ t ˚ ˚ (A.7) k : F F , k : F 2 F1. 2 ÝÑ 1 ÝÑ 155 For example, kt is the composition

˚ ˚ ˚ ˚ eF ˝F F ˝k˝F F ˝ηF ˚ 1 2 ˚ ˚ 1 2 ˚ ˚ 1 2 ˚ F2 / F1 F1F2 / F1 F2F2 / F1 , and tk is defined similarly. Proposition A.8. (a) Passing the the left or right transposes gives natural identifications ˚ ˚ ˚ ˚ Hom F2, F1 Hom F1, F2 Hom F , F . p q » p q » p 2 1 q (b) The identifications in (a) are compatible with composition. That is, for natural trans- k1 k2 formations F1 F2 F3 we have Ñ Ñ t t t t t t k2k1 k k , k2k1 k2 k1 p q “ p 1qp 2q p q“p qp q (provided the adjoints exist). (c) Let C, D, E be categories and C G D F E and C H E be functors. Then we have natural idenfitications (partial transposes),Ñ providedÑ the adjointsÑ exist: Hom ˚HF G, Id Hom FG,H Hom G, F ˚H . p Dq » p q » p q Proof: Similar to the tensor case, see [33] (Ex. 2.10.7 and Prop. 2.10.8). For example, the last identification in (c) is the composition

F ˚˝p´q ue Hom FG,H Hom F ˚FG,F ˚H F Hom G, F ˚H , p q ÝÑ p q ÝÑ p q where the first arrow is the application of F and the second is the substitution of eF : Id F ˚F . Ñ Corollary A.9. Assuming that the relevant adjoints exist, in the situation of Proposition A.8 we have natural identifications (“tensor rotation”) Hom FG,H p q“ Hom G, F ˚H Hom GH˚, F ˚ Hom H˚,G˚F ˚ Hom F ˚˚G˚˚,H˚˚ “ p q“ p q“ p q“¨¨¨“ p q“¨¨¨ Hom F,H ˚G Hom ˚HF, ˚G Hom ˚H,˚ G ˚F Hom ˚˚F ˚˚G , ˚˚H “ p p qq “ p p qq “ p p qq“¨¨¨“ p p q p qq“¨¨¨

4. Adjunctions at the dg-level. We will need the formalism of adjoints in the setting of dg-categories and dg-functors. In this case there are extra subtleties: the αx,y need to be not isomorphisms but only quasi-isomorphisms of Hom-complexes, the naturality should hold not “on the nose” but only up to a coherent system of homotopies and so on. A practical way of handling these issues was proposed in [3] for dg-functors coming from bimodules, cf. Example A.2. In this paper we will mostly ignore these coherence issues, assuming implicitly the ansatz of [3]. A more satisfactory general treatment of adjunctions can be provided in the framework of -categories [73] §5.2 and will be used in [32]. 8 We will use the following statement which is clear in the bimodule setting of [3].

156 Proposition A.10. Suppose V, W are pre-triangulated dg-categories and F1, F2 : V W ˚ ˚ Ñ are dg-functors with right adjoints F1 , F2 . Let k : F1 F2 be a closed natural transformation of degree 0, so that we have a triangle of dg-functorsÑ

k Cone k 1 F1 F2 Cone k . p˚q p qr´ s ÝÑ ÝÑ ÝÑ p q Then Cone k ˚, the right adjoint to Cone k , exists and is identified with Cone kt 1 , so that the trianglep q p q p qr s t Cone k ˚ F ˚ k F ˚ Cone k ˚ 1 p q ÝÑ 2 ÝÑ 1 ÝÑ p q r s dual to (*), is identified with the triangle

t Cone kt 1 F ˚ k F ˚ Cone kt . p qr´ s ÝÑ 2 ÝÑ 1 ÝÑ p q

5. Serre functors. Calabi-Yau categories. Let V be a k-linear triangulated category with all HomV x, y finite-dimensional. Recall [14] that a Serre functor for V is a self- equivalence S :pV q V equipped with natural isomorphisms (the Serre structure) Ñ ˚ (A.11) ρx,y : Hom x, y Hom y,S x . V p q ÝÑ V p p qq The basic example is V DbCoh X , the coherent derived category of a smooth projective k “ p q n variety X of dimension n. Then S F F ΩX n os a Serre functor, with (A.11) being the classical{ Serre duality. p q“ b r s Note that S, if it exists, is defined canonically (uniquely up to a unique isomorphism of functors). This is because (A.11) means that S x is the representing object for the functor y Hom x, y ˚. However, when we say thatp q a given functor F : V V “is” a Serre functor,ÞÑ wep meanq that it is made into a Serre functor by equipping it withÑ an appropriate Serre structure. This Serre structure does not have to be unique, it is unique only up to an automorphism of F as a functor. For example, we say that V is a Calabi-Yau category of dimension n Z, if the shift func- tor x x n is (made into) a Serre functor, i.e., if V is equipped with naturalP isomorphisms (Calabi-YauÞÑ r s structure)

˚ (A.12) κx,y : Hom x, y Hom y, x n . V p q ÝÑ V p r sq For instance, in the example V DbCoh X , the functor x x n , n dim X , can be made into a Serre functor, if X admits“ a globalp q nowhere vanishingÞÑ volumer s “ form ωp qΓ X, Ωn (so it is a Calabi-Yau manifold in the usual sense). A choice of such a form (uniqueP onlyp upq to a scalar) gives a Calabi-Yau structure. We will also use the dg-enhanced versions of these concepts. That is, let V be a pre- ‚ triangulated dg-category such that each complex HomV x, y has the total cohomology space finite-dimensional. Then by a Serre functor for V we meanp q a dg-functor S : V V which is Ñ 157 a quasi-equivalence and is equipped with natural quasi-isomorphisms of Hom-compleces ρx,y, like in (A.11) but with Hom‚ instead of Hom. As this amounts to (homotopy) representing dg-functors of the form y Hom‚ x, y ˚, the object S x is defined homotopy canonically (uniquely up to a contractibleÞÑ spacep of choices)q and so onep caq n use the -categorical approach to address various issues of coherence etc. 8 Similarly, we will speak about enhanced versions of Calabi-Yau structures.

6. Local systems of (pre-triangulated) dg-categories. Let X be a topological space. The natural categorification of the concept of a sheaf (of sets, vector spaces etc.) on X is the concept of a stack of categories. Let us explain it in a convenient language analogous to that of sheaves, see, e.g., [18] for a more systematic treatment.

Definition A.13. A pre-stack (or a pseudo-functor on open sets) on X is a datum F of:

(0) For any open set U X, a category F U . Ă p q

(1) For any any inclusion U1 U, a functor (called the restriction) rU,U : F U F U1 , Ă 1 p qÑ p q so that rUU Id. “

(2) For any triple inclusion U2 U1 U, an isomorphism of functors (transitivity of Ă Ă restrictions) ρU,U ,U : rU ,U rU,U rU,U , so that: 1 2 1 2 ˝ 1 Ñ 2

(3) For any quadruple inclusion U3 U2 U1 U0 U we have a compatibility (cocycle) Ă Ă Ă “ condition on the 4 isomorphisms ρU ,U ,U , 0 i j k 3. i j k ď ă ă ď

A stack is a pre-stack satisfying the descent conditions for any open covering U a Ua which say that objects and morphisms in F U can be glued out of families of such“ on the p q Ť Ua equipped with compatibility data on the intersections. A standard example of a stack is the correspondence U Sh U , the category of all sheaves on U. The descent conditions in this case mean that aÞÑ sheafp onq U can be glued out of sheaves on the Ua with compatible identifications on the intersections. Let X be an n-dimensional C8-manifold, possible with boundary. A local system of categories i(or a locally constant stack) on X is a stack F with the following property: whenever U1 U are two balls in X, the functor rU,U1 is an equivalence of categories. Here by a ball in XĂ we mean an open set homeomorphic either to the n-dimensional open ball n Bn x 1 R , or to the “ball at the boundary” Bn x1 0 . “ t} }ă uĂ X t ě u Given a local system F on X, for each x X we have the category Fx, the stalk of F at x, defined as F U for sufficiently small balls containingP U. Given a path γ joining two points p q x, y X, we have the monodromy (or categorical parallel transport) functor T γ : Fx Fy by coveringP γ by a sequence of small balls in a standard way. These objectsp q are definedÑ canonically in the following sense. The category Fx is defined uniquely up to an equivalence of categories which, in its turn, is defined uniquely up to a unique isomorphism of functors, and the functor T γ is defined uniquely up to a unique isomorphism. p q 158 Alternatively, a local system can be defined as a pseudo-functor U F U as above but ÞÑ p q defined only on disks and such that any rU,U1 is an equivalence. It is straightforward to generalize to the case of dg-categories. As mentioned above, all dg-categories are united into an -category dgCat, see §A2. A (homotopy) stack of dg-categories is, formally, a sheaf F on8X with values in dgCat in the sense of [73]. Explicitly, this means that F is, similarly to Definition A.13 a datum consistinng of dg- categories F U at level (0), dg-functors rU,U at level (1), but at level (2) we have closed p q 1 degree 0 natural transformations ρU,U1,U2 , and level (3) is replaced by an infinite chain of higher coherence conditions. Similarly, the descent conditions all involve infinite systems of higher coherences.

A local system of dg-categories on a manifold X is a stack F such that rU,U : F U 1 p q Ñ F U1 is a quasi-equivalence of dg-categories whenever U, U1 are disks. As before, we p q can speak about stalks Fx and monodromies T γ which are dg-categories and dg-functors defined homotopy canonically, i.e., up to a contractiblep q space of possible choices. If X is a complex manifold, a local system of pre-triangulated dg-categories should be certainly considered as an example of a perverse schober. However, it does not seem possible to realize more general perverse schobers as stacks of dg-categories in the sense we described.

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M.K.: Kavli IPMU, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583 Japan. Email: [email protected] Y.S.: Department of Mathematics, Kansas State University, Manhattan KS 66508 USA. Email: [email protected] L.S.: Department of Mathematics, Higher School of Economics, 109028 Moscow, Russia. Email: [email protected]

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