Copyright by Rustam Darius Riedel 2019 the Dissertation Committee for Rustam Darius Riedel Certiﬁes That This Is the Approved Version of the Following Dissertation

Copyright by Rustam Darius Riedel 2019 The Dissertation Committee for Rustam Darius Riedel certiﬁes that this is the approved version of the following dissertation:

Free loop spaces, Koszul duality, and shifted Poisson geometry

Committee:

David Ben-Zvi, Supervisor

Andrew Neitzke

Andrew Blumberg

David Nadler Free loop spaces, Koszul duality, and shifted Poisson geometry

Rustam Darius Riedel

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulﬁllment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN December 2019 To my grandfather Dr. Pirozshaw Dara Antia, for whom I was able to ﬁnish this. “Monks, the All is burning. Which All is burning? The eye is burning. [...] The ear [...] nose [...] tongue [...] body is burning. [...]

The mind is burning. Ideas are burning. Mind-consciousness is burning. Mind-contact is burning. And whatever there is that arises in dependence on mind- contact – experienced as pleasure, pain or neither-pleasure-nor-pain – that too is burning. Burning with what? Burning with the fire of passion, the fire of aver- sion, the fire of delusion. Burning, I say, with birth, aging & death, with sorrows, lamentations, pains, distresses, & despairs.

Seeing thus, the instructed disciple of the noble ones [...] grows disenchanted with the mind, disenchanted with ideas, disenchanted with mind-consciousness, disenchanted with mind-contact. And whatever there is that arises in dependence on mind-contact, experienced as pleasure, pain or neither-pleasure-nor-pain: With that, too, he grows disenchanted. [...]”

Aditta-pariy¯aya¯ Sutta (Samyutta Nik¯aya 35:28) Acknowledgments

I was fortunate in David Ben-Zvi accepting me as a student after my long odyssey of trying to find an advisor. Without his patience, kindness and continued support through difficult times, I would have never been able to finish my PhD. Mathematically, besides the huge debt I owe my advisor, I am particularly grateful to Sam Raskin, Rok Gregoriˇc,and Pavel Safronov for their help. Andy Neitzke’s kindness and the sympathetic ear he lent me on various occasions meant a lot to me. I am also immensely grateful to my undergraduate advisor Annette Werner for her continued caring support and mentorship.

On a more purely personal level I would like to in addition thank my par- ents, my sister, my extended family, and all of the following people. Whether they are more casual or close friends or are or were part of my life in other ways, their presence (or just my interactions with them) has (have) been meaningful and valu- able to me in diﬀerent ways, and in some cases very important even though this might not have been evident to them: The Venerable Bhante Ananda,¯ Nihal Arju, Behnam Arzaghi, Tobias Baumann, Santiago Benavides, Gaurav Chaudhary, Adrian Clough, Abhranil Das, Arun Debray, the Derryberrys: Richard, Dakota and Za- havi, Stefan Eccles, Diney Ether, Iordan Ganev, Tom Gannon, Rok Gregoriˇcand Neˇza Zagerˇ Korenjak, Anne Lise Haenni, Elham Heidari, Andy Hutchison, Annika Jansson, Dan Kaplan, Pat Koch, Max Kurandt, Jonathan Lai, Manasvi Lingam,

vi Tim Magee, Tom Mainiero, George Miloshevich, Vaibhav Murali, the Otto/Otto- Schödels: Matthias Siddhartha, Anja, and Jonna, Sumati Panicker and Wolfgang Kaim, Rebecca P. (please find happiness!), Travis Schedler and Marie-Amelie Lawn, Ellie Schmidt, Isabelle Scott, Akarsh Simha, the Stratis/Serwe clan: Yorgos, Sarah, and Lola, Sivaramakrishnanan Swaminathan, the Venerable Ajaan Geoff (T. h¯anissaro Bhikkhu) for his writings and dhamma talks, Megan Deepti Thomas, Haoran Wang, Michael Wong, Yan Zhou, and Yaoguang Zhu, and anyone I may have inadvertently omitted writing this, as is my wont, at the last minute and while sleep deprived.

vii Free loop spaces, Koszul duality, and shifted Poisson geometry

Rustam Darius Riedel, Ph.D. The University of Texas at Austin, 2019

Supervisor: David Ben-Zvi

Let X be a smooth aﬃne scheme, and let LX be its (derived) free loop space. We have a Koszul duality equivalence between the ∞-category of ind-coherent sheaves on LX, and the ∞-category of sheaves on (the total space of) the shifted cotangent bundle T ∗X[2] given by modules for the commutative (diﬀerential graded)

• algebra π (O ∗ ) ∼ Sym (T [−2]) (π is the projection of the 2-shifted cotangent ∗ T X[2] = OX X space T ∗X[2] to the base X). The action by loop rotation of S1 considered as a homotopy type on the category of ind-coherent sheaves on LX induces an action on the Koszul dual ∞-category. This thesis describes this action on the Koszul dual ∞-category in terms of the shifted Poisson structure on T ∗X[2]: The main result is that the action is the exponential of the Poisson bracket.

viii Table of Contents

Acknowledgments vi

Abstract viii

Chapter 1. Introduction 1 1.1 Introduction ...... 1 1.2 Notation and conventions...... 14

Chapter 2. Preliminaries 15 2.1 A few words on our use of ∞−Categories ...... 15 2.2 Derived Algebraic Geometry ...... 20 2.3 Quasi-coherent and Ind-coherent Sheaves ...... 24 2.4 Shifted Symplectic and Poisson Geometry ...... 29 2.5 (Algebraic) Families of ∞−Categories ...... 34 2.6 Group actions on ∞−Categories ...... 39 2.7 Hochschild Homology and Cohomology ...... 48 2.8 Formal moduli problems ...... 54 2.9 Koszul duality ...... 59 2.10 Free derived loop spaces ...... 63

Chapter 3. The main theorem 67

Bibliography 69

Vita 85

ix Chapter 1

Introduction

1.1 Introduction

Let X be a smooth scheme, and let LX be its (derived) free loop space (see below for an explanation of what this is). Using formality of cochains on the circle S1, we can identify LX with (the total space of) the (−1)-shifted tangent bundle TX[−1]. We have a Koszul duality equivalence between certain homotopical categories of sheaves on TX[−1] and on the Koszul dual space given by (the total space of) the shifted cotangent bundle T ∗X[2]. The S1-action by loop rotation on the category of sheaves on LX induces an action on the Koszul dual category of sheaves on T ∗X[2]. This thesis describes this action on the Koszul dual category in terms of the shifted Poisson structure on T ∗X[2].

1.1.1 Free loop spaces

The free loop space LX of a topological space X is the space of continuous maps from a circle S1 into X. Free loop spaces are of fundamental interest in topology and geometry. Let us mention some examples: In topology its rich structure is studied in string topology (see [35, 37, 38, 26]). In geometry one considers for X a smooth manifold only suitably regular maps from S1. The topology of LX is then of fundamental importance when studying the existence of closed geodesics on

1 Riemannian manifolds ([95]). Free loop spaces LG of compact Lie groups X = G are known as loop groups ([99]), and are the simplest example of infinite dimensional Lie groups. Loop groups (and their central extensions) have been studied both in geometry and representation theory, for example in the algebraic guise as mapping spaces out of formal disks through their connections to the affine Grassmannian and moduli of G-bundles on a curve (see e.g. [138] for an expository account), or through the theory of affine Kac-Moody algebras/groups ([64]). In symplectic geometry Viterbo identifies symplectic cohomology (a version of Hamiltonian Floer theory) of the cotangent space T ∗X with the homology of the free loop space ([130, 131, 2, 109, 3]). The circle acts on LX by loop rotation in such a way that the fixed point locus is X. In [132, 133] (see also [8] for an expository account) Witten showed that for a Spin manifold X equivariant localization to the fixed point locus in equivariant cohomology and K-theory yields the index of the Dirac operator and the elliptic genus (which is related to elliptic cohomology) of X, thereby connecting the study of LX to physics, geometry, and homotopy theory. A version of the free loop space in algebraic geometry was constructed in [124, 126] as a recipient of a Chern character for categorical sheaves [125]. This was further studied in [20] and applied to geometric representation theory in [19]. It is this algebro-geometrical version of the loop space, called the derived free loop space and its Koszul dual that we will consider in this thesis.

2 1.1.2 Derived free loop space

Let us try to make sense of the free loop space LX of a smooth scheme X. We will consider S1 as a homotopy type, and think of it as two points connected by two arcs (i.e. we are given two essentially diﬀerent identiﬁcations of the two points). Mapping the two points into X we have are free to choose any pair of two points x, y in X, that is the mapping space from the two points is X × X. The arcs tell us that for any pair (x, y) we want to identify x with y twice (once for each arc). Thus, the free loop space is given by imposing the equation x = y in X × X twice, i.e. it is the ∼ self intersection of the diagonal LX = X ×X×X X.

This intersection is highly degenerate, and in the usual classical algebraic geometry setting just gives us back X. To make better sense of it one needs to embed the world of algebraic geometry into the more homotopical world of derived algebraic geometry. One of the major motivations for the development of derived algebra is to make sense of such highly degenerate intersections. The elementary building blocks of classical algebraic geometry are algebras, i.e. vector spaces with a multiplication. In derived algebraic geometry the elementary building blocks are an appropriate notion of derived algebras, i.e. (connective) cochain complexes with a multiplication (a.k.a (connective) commutative diﬀerential graded algebras). Working with derived algebras allows us to remember that we imposed the equation x = y twice, see 2.2 for an explanation of this. On the level of algebras - say X = Spec(A) is aﬃne - intersections correspond to tensor products, and passing to the derived world means

L that we have to derive these tensor products. Thus, LSpec(A) = Spec(A ⊗A⊗A A).

3 1.1.3 Hochschild-Kostant-Rosenberg

To compute the derived tensor product we need to replace A by an equivalent but homotopicaly better behaved A ⊗ A-algebra. This is in analogy with classical

h topology, where for a topological space X the homotopy ﬁber product X ×X×X X may be computed by replacing one of the factors X by the weakly equivalent space of paths PX = Maps([0, 1],X) in X, where the maps PX → X × X is given by ∼ evaluating a path at the endpoints of [0, 1]. It is easy to see that PX ×X×X X = Maps(S1,X) = LX.

In the world of algebra one may replace A by a Koszul resolution of the

L diagonal. One then computes that A ⊗A⊗A A (which is the Hochschild homology V 1 1 of A) may be identified with with the exterior algebra A ΩA on the module ΩA of Kählerdifferentials (the analogue in algebraic geometry of differential 1-forms in differential geometry), i.e. with algebraic forms. This is the Hochschild-Kostant-

V 1 ∼ 1 Rosenberg isomorphism. We can write A ΩA = SymA(ΩA[1]) and interpret this algebra as a diﬀerential graded algebra with vanishing diﬀerential. Derived algebraic

1 1 geometry then allows us to interpret LX as Spec(SymA(ΩA[1])). Since ΩA[1] is linear functions on the fibers of (the total space of) the −1-shifted tangent bundle TX[−1], we can interpret Hochschild-Kostant-Rosenberg as a linearization LX ∼= TX[−1], and this holds in the non-affine setting for general derived schemes as well ([20]). Under this linearization constant loops are identified with the zero section, and loop rotation corresponds to the Connes B-operator on Hochschild homology/the de- Rham differential on algebraic forms.

4 1.1.4 Koszul duality

Koszul duality starts with the following observation ([113]): Given an arbitrary k-linear category C and a distinguished object c, the functor HomC (c, −)

op deﬁnes a functor C → A − mod where A is the endomorphism algebra HomC (c, c) of c. Under suitable conditions on c (one needs c to be a compact generator) this is an equivalence. The simplest and most classical version of Koszul duality is that there is an equivalence

IndCoh(Λ) ∼= S − mod between (suitable versions of) module categories for the ”Koszul dual pair“ of algebras given by the exterior (commutative differential graded) algebra Λ = Sym•(V ∗[1]) and the symmetric algebra S = Sym•(V [−2]), where V is a finite-dimensional vector space. The module categories for these algebras contain a distinguished object given by the augmentation module k, and the duality of module categories for Λ and S comes from computing the endomorphism algebras of the augmentation modules: ∼ ∼ One finds that HomΛ(k, k) = S and EndS(k, k) = Λ. Here IndCoh(Λ) is a slight modification of Λ − mod that guarantees that k is a compact generator. We can interpret this geometrically: There is a linearization identifying the (derived) based ∼ loop space Ω0V = 0 × 0 of V at the origin 0 with V [−1] = Spec(Λ). For x ,→ X V a pointed smooth scheme, one has that the (derived) based loop space x × x only X ∼ depends on the formal neighbourhood of x in X, so that we can identify ΩxX = Ω0V for V = TX,x. Thus, Koszul duality can be interpreted as an equivalence

∼ • IndCoh(ΩxX) = Sym (TX,x[−2]) − mod.

5 Koszul duality generalizes to certain more general pairs of associative algebras given by quadratic relations, and to (suitable quadratic) operads, and plays an important role in algebraic geometry, rational homotopy theory, representation theory, and derived deformation theory.

1.1.5 The Koszul dual of LX

We are now in a position to explain what the Koszul dual space of LX for X a smooth scheme is. By Hochschild-Kostant-Rosenberg we can linearize and consider ∼= TX[−1] in place of LX. Instead of an algebra, we have a (derived) scheme, so instead of modules we need to consider (a suitable algebro-geometric notion of)

1 sheaves IndCoh(TX[−1]) on TX[−1]. The structure sheaf OX on the zero section X,→ TX[−1] is the analogue of the augmentation ideal. It turns out to be a compact generator of IndCoh(TX[−1]), so that one can identify this category with modules for the (sheafy) endomorphism algebra of OX . This algebra may be computed using

• a Koszul resolution to be S := Sym (T [−2]), that is (the pushforward π O ∗ X OX X ∗ T X[2] to X of) functions on the 2-shifted cotangent bundle T ∗X[2]. One thus has

∼ IndCoh(LX) = SX − mod,

where an SX -module is to be understood in the diﬀerential graded sense, i.e. is a sheaf of chain complexes of quasi-coherent sheaves with an action of the sheaf of diﬀerential graded algebras SX . We thus think of the derived free loop space LX and the 2-shifted cotangent bundle T ∗X[2] as Koszul duals, giving rise to an equivalence

1Technically, ind-coherent sheaves are not sheaves, but they are close enough that at ﬁrst pass the casual reader may wish to ignore this distinction.

6 on suitable categories of sheaves. This is a version in families of the Koszul duality between appropriate module categories for an exterior algebra (functions on TX[−1]) and for a symmetric algebra. If we restrict to a point we recover the pointwise Koszul

∼ • duality IndCoh(ΩxX) = Sym (TX,x[−2]) − mod. The at ﬁrst sight somewhat exotic choice of IndCoh as our theory of sheaves instead of, say, quasi-coherent sheaves is again explained by the fact that we want the augmentation OX to be a compact generator.

1.1.6 Shifted Symplectic and Poisson geometry

In classical differential geometry or algebraic geometry (if one considers smooth varieties) one has a notion of symplectic and Poisson structures. In derived algebraic geometry one has an analogous derived notion of n-shifted symplectic and Poisson structures (see 2.4, the foundational papers [96, 32, 101], the review [97], and the introductions [29, 108]). This is a rich and powerful formalism that is currently an active area of investigation. Shifted symplectic and Poisson structures are central in the study of higher deformation quantizations of ∞-categories of perfect complexes on derived Artin stacks. In physics, they are used in the approach of Costello-Gwillam to perturbative quantization of field theories [40, 41], and allow for a version of the AKSZ and BV constructions [96]. They also have applications to the categorification of Donaldson-Thomas invariants [17], geometric representation theory, non-abelian Hodge theory, quantum groups, string topology, and algebraic L-theory. The relevance to us is that just as in classical differential geometry the cotangent bundle of a smooth manifold is canonically symplectic and thus Poisson,

7 the 2-shifted cotangent bundle T ∗X[2] of a derived scheme (or even derived Artin stack [30]) X is naturally 2-shifted symplectic and thus 2-shifted Poisson.

1.1.7 The E2-structure and circle action on the Koszul dual category

1.1.7.1. The circle action. Since IndCoh(LX) has a circle action by loop rotation, one gets a circle action on the Koszul dual category SX − mod, which a priori did not look like it came with a circle action. We can now state our main result:

1 Theorem 1.1.1. Let X be an aﬃne smooth scheme. The S -action on SX − mod is the exponential (in a way that we will make precise) of the (2-)shifted Poisson structure on T ∗X[2].

We are certain that the proof extends to the case of a non-aﬃne scheme, but need a result from the theory of formal moduli problems in families we do not have.

1.1.7.2. The E2 structure. Besides its intrinsic interest one motivation for under- standing this circle action is that it provides insight on an E2(= little 2-disk)-algebra structure on SX − mod. An En-algebra structure on a (∞ or ordinary) category or on a cochain complex is roughly a family of monoidal structures parametrized in a homotopically locally constant way by the space of conﬁgurations of embeddings of two n-disks2 in a ﬁxed n-disk. These disks are to be thought of as rigid, i.e. we are not allowed to rotate them. In homotopical algebra En-algebras interpolate between

(homotopy) associative (= E1 = A∞) and commutative (= E∞) algebras or monoidal

2The n-disk is the space {x ∈ Rn : |x| ≤ 1}.

8 and symmetric monoidal ∞-categories, becoming more and more commutative as n grows. For more details on En-structures, see [45, 82]. The E2-structure on SX −mod arises Koszul dually on IndCoh(LX) from a pair of pants bordism (i.e. two little disks cut out of a big disk)3. The bordism can be thought of as a correspondence from the ingoing pair of circles to the outgoing circle, which then deﬁnes a correspondence of mapping spaces to X. Since we can push and pull (ind-coherent) sheaves, pushing and pulling along the legs of this correspondence we get a family of products on IndCoh(LX) parametrized by the space of conﬁgurations of a pair of disks embedded in a big disk (cp. [18, Corollary 6.7]).

1.1.7.3. The framed E2-structure. How does the circle action help us understand the

1 E2-algebra structure? The S -action combines with the E2-algebra structure to an fE2(= framed little 2-disk)-algebra structure: having the circle action means we can rotate the disks, so that we have monoidal structures parametrized by the space of embeddings of two 2-disks in a 2-disks where are now allowed to rotate the disks.4 We also get unitary operations parametrized by the space of embeddings (allowing for rotations) of one 2-disk into a 2-disk. This space of embeddings is homotopy

∼ 1 equivalent to SO(2) = S . Now, an E2-algebra structure on an ordinary (non- homotopical) category C is the structure of a braided monoidal category on C (for any two objects A, B ∈ C we have operations RA,B : A ⊗ B → B ⊗ A satisfying

3 Of course there is also the less interesting obvious E2-structure gotten by forgetting down from E∞ (i.e. ordinary symmetric monoidal) structure on SX − mod 4More generally for framed little n-disk algebras we are allowed to rotate the n-disks by the action of SO(n).

9 the braid relations), and a fE2-algebra structure is that of a ribbon braided category

5 structure on C (we have in addition to the braiding a twist operator θA : A → A for every object A of C such that θA⊗B = (θA ⊗ θB)RB,ARA,B). The braiding comes from interchanging the two disks, and the twist comes from rotating the circle by

2π. In the homotopical setting of fE2-actions on homotopical categories we should in some sense be able to recover the braiding from the twist and the monoidal structure by rotating the two inner disks in the same direction by π, and the outer disk in the opposite direction by π. This would allow us to understand the E2-action on the Koszul dual side in terms of the circle action, and thus in terms of the Poisson structure.

Remark For historical reasons the terminology is a little confusing, since for framed

E2-algebras one is allowed to rotate, i.e. change the framing. In some parts of the literature on factorization homology the less confusing terminology of Diskn,fr and

Diskn,or-algebras in place of En and fEn-algebras is now used.

1.1.8 Rozansky-Witten theory

Our interest in the E2 action on SX − mod comes from Rozansky-Witten theory. Rozansky-Witten theory is a partially defined oriented 3-dimensional topological quantum field theory ZRW (−) defined as a sigma model with target a holomorphic symplectic variety, which for us will be T ∗X ([107, 69, 68, 65, 66, 67]). For X =

BunG(Σ) the moduli stack of G-bundles on a Riemann surface Σ (i.e. with target

5a.k.a ”balanced braided category“

10 ∼ ∗ (roughly) the Hitchin moduli space of G-Higgs bundles HiggsG(Σ) = T BunG(Σ)) Rozansky-Witten theory arises as a reduction along S1 × Σ of ’Theory X’, the mys- terious and beautiful 6-dimensional (2, 0)-superconformal field theory discovered in [134, 117] with connections to a myriad of topics in topology, geometry, and geometric representation theory (see [92] for an overview of various reductions, and for connections to Geometric Langlands in particular see [135]). For example, Rozansky- Witten theories define finite type (Vassiliev) knot invariants ([105, 106]).

The characteristic feature of Rozansky-Witten theory is that it gives the B-

∗ 1 1 model TQFT for T X after reduction with S : ZRW (S × −) = ZB−model(−), so

1 ∗ that we get on a circle that ZRW (S ) = QCoh(T X). In fact, we actually only get a Z/(2)-graded version QCoh(T ∗X) ⊗ C[, −1], deg() = 2. Rozansky-Witten theory was originally defined in the language of physical field theories, and it is of interest to fit it in the axiomatized picture of fully extended quantum field theories worked out by Baez-Dolan and Lurie ([12, 86, 44]). In this mathematical picture of topological quantum field theories the locality of the path integral translates into the field theory being determined by what it assigns to a point, i.e. the boundary conditions - this is the cobordism hypothesis - and so we should ask what Rozansky- Witten theory assigns to a point. The answer should be something like the (∞, 2)- category QCoh(X) − mod of modules for the monoidal category of quasicoherent sheaves on X. Smooth schemes are affine in some categorical sense, so that we can identify this with the (∞-)category ShvCat(X) of quasicoherent sheaves of ∞- categories over X (they are ”1-affine“, see 2.5, or [48]). Thus we should have that

ZRW (−) on a circle is the Hochschild homology QCoh(X)⊗QCoh(X)⊗QCoh(X)QCoh(X)

11 of the associative algebra object (QCoh(X), ⊗), which by the integral transforms formalism of [18] is given by the (∞-)category QCoh(LX). The two answers for a circle are related by Koszul duality, since the Koszul dual space of LX is T ∗X[2]. We see that we are oﬀ by a shift by [2],6 and this is somehow the reason that we only get Z/(2)-graded categories. What goes wrong is that instead of considering the ordinary scheme T ∗X we really should have considered the shifted cotangent

∗ 7 bundle T X[2], and as an E3 algebro-geometrical object In this case local operators

0 ∗ H (T X[2]; OT ∗X[2]) do naturally form an E3-algebra as they should, and then the category of line operators (QCoh(T ∗X[2]), ⊗) = (Sym• (T [−2])−mod, ⊗)8 aquires OX X 9 an E2-monoidal structure that from the point of view of the ﬁeld theory comes from colliding line operators, and is not a priori evident on the category. The circle

6We are also off by the issue of IndCoh vs. QCoh, but we will ignore this here. This is the reason that what we wanted to assign to a point is a little off. 7 ∗ E.g. as a functor of points on E3-algebras. However, T X[2] is naturally only 2-shifted Poisson = P3, not E3. What is going on here? It turns out that P3 is the same as the homology H∗(E3). By ∼ Kontsevich formality En is formal [118, 73, 76], but the equivalence Pn = En is highly nontrivial and depends on a non-canonical choice of a Drinfeld associator, of which there are many, so that the ∗ E3-structure on T [2] is not natural. The issue is that to find the Koszul dual we passed through Hochschild-Kostant-Rosenberg, which morally comes from the formality of cochains on the circle C∗(S1) =∼ H∗(S1) (see [20]), so that we were taking the Koszul dual of the ’wrong’ space. The correct Koszul dual is related to En D-modules studied in forthcoming work by David Ben-Zvi and Pavel Safronov ([22]). 8Actually, QCoh(T ∗X[2] and Sym• (T [−2]) − mod are not quite equivalent, because T ∗X[2] OX X is a relative affine stack (Sym• (T [−2]) is not connective), however the latter ∞-category maps OX X to the former by a functor that is fully faithful on coconnective objects for the natural t-structures on both sides, see [80][Prop 4.5.2(7)]. For the purposes of our story we can ignore this here. 9 We can equivalently think of an En-structure as a monoidal structure parametrized by the space of configurations of two n-cubes embedded in a fixed n-cube, i.e. n compatible associative monoidal structures (corresponding to the axes of n-space). Passing to modules for an En-algebra eats up one monoidal structure so that we get an En−1-monoidal ∞-category, whereas taking endomorphisms of the unit in a En-monoidal ∞-category adds a multiplication, so that we get an En+1-algebra. Thus if remembered the E2 structure when Koszul dualizing, we would have found that the dual is naturally E3.

12 action comes because Rozansky-Witten theory is oriented so that we can also rotate conﬁgurations of disks/spheres. It is this (f)E2-structure that we hope our result might shed some light on.

13 1.2 Notation and conventions.

We work over a ﬁxed base ﬁeld k of characteristic zero.

We lay out notation for commonly used ∞-categories in 2.1.

We prefer the term family of (∞-)categories to quasi-coherent sheaf of (∞- )categories.

We denote (co)tangent sheaves (respectively complexes) on a space X by sub-

∗ ∗ ∗ ∗ scripts, i.e. TX ,TX (TX , TX ) and reserve TX,T X (TX, T X) for the total spaces.

∗ ∗ For smooth schemes X the (co)tangent sheaves TX ,TX and complexes TX , TX agree and thus also the corresponding total spaces agree. We have chosen to use the underived notation in the introduction for the sake of familiarity for the casual reader, and the derived notation in the main body of the thesis.

Remark We mention a few background results that are relevant to an alternative proof (which would have also worked in the non-aﬃne case) of the main theorem that we were not quite able to complete.

14 Chapter 2

Preliminaries

2.1 A few words on our use of ∞−Categories

The idea of an ∞-category is that we have a sort of category where we have a topological space of morphisms between any two objects, and composition and associativity are deﬁned only up to coherent homotopy. We will not review the theory of ∞-categories here, or even attempt to give an introduction, but rather just explain our philosophy in how we use them. The foundational reference is [83], and introductions may be found in [49][Chapter I.1][55, 33].

2.1.0.1. Quasi-categories. There are many different models formalizing the idea of an ∞-category, the most extensively developed being the theory of quasi-categories (see [16] for a comparison of the different notions). A quasi-category is a simplicial set satisfying a certain inner horn filling condition. One thinks of the 0-simplices as the objects of the ∞-category, the 1-simplices as morphisms, the 2-simplices as 2-morphisms between 1-morphisms, and so on. The inner horn filling condition translates to the existence of compositions of sequences of composable 1-morphisms. Quasi-categories were developed by Joyal in [63], and extensive foundations for the end-user laid out by Lurie in [83].

15 2.1.0.2. A more radical approach through univalent foundations. Another in a way more natural way to think of ∞-categories one might take is by changing the foundations on which mathematics is built. Formalizing this is difficult (this is the project of Voevodsky’s univalent foundations [127]), but the basic idea is simple: we replace the fundamental building block of mathematical thinking given by the notion of equality or identity by the notion of identification, i.e. for any particular identity we are as- serting we always ask for the data of a “witness” of the identification we are making. The fundamental structure is now not that of a set, but that of an ∞-groupoid, which consists of objects or points, identifications (alias paths, homotopies or 1-morphisms) between objects, identifications between identifications (alias homotopies of homotopies, 2-morphisms between 1-morphisms) etc, such that all n-morphisms for n ≥ 1 are (weakly) invertible. This is essentially the same notion as a topological space (by which we always mean a homotopy type). The notion of category we get is thus one where morphisms form an ∞-groupoid and associativity (and invertibility) holds up to coherent identifications, i.e. homotopy. From this perspective ∞-categories are completely natural, and not of a topological or combinatorial nature, but rather more algebraic.

2.1.0.3. Our use of the language of ∞-categories. We will not need to access any particular construction of ∞-categories. Our approach to the use of the language of ∞- categories is that we assume the existence of a notion of ∞-category as well as of suitable ∞-categorical extensions of the categorical notions and constructions we need such as functors, equivalences, (co)limits, adjunctions, Kan-extensions, (co)monads,

16 the Barr-Beck theorem, Ind-completions, (symmetric) monoidal structures, (commutative) algebra objects in (symmetric) monoidal ∞-categories etc. and that everything works mostly as expected. All constructions we mention can be made rigorous in the setting of quasi-categories, but we have not attempted to explain this.

2.1.0.4. The linear setting of dg-categories. The classical deﬁnition of a (pretrian- gulated) dg-category is that of a category enriched chain complexes such that the homotopy category (the category enriched in vector spaces whose vector space of morphisms are given by H0 of the mapping chain complexes) is triangulated. Ref- erences for dg-categories are [70, 71]. We will use the equivalent approach of [49] deﬁning dg-categories as k-linear stable ∞-categories. See [39] for a precise comparison.

2.1.0.5. Model categories. For computations it is often convenient to work in the more rigid setting of model categories. A model category is an ordinary category with three distinguished classes of morphisms (W, F, C) - weak equivalences, fibrations, and cofibration - satisfying properties similar to properties of the category of topological spaces equipped with the class of weak equivalences given by maps inducing isomorphisms on homotopy groups π∗, fibrations given by Serre fibrations, and cofibrations given by (retracts of) relative cell complexes. References for model categories are [102, 42, 53, 58, 57]. Any model category induces a ∞-category by a process of weakly inverting the weak equivalences. The fibrations and cofibrations allow one to compute nonabelian derived functors (i.e. functors on the associated

17 ∞-category) by replacement/resolution by providing for an analogue of the injective and projective objects of homological algebra. (Co)limits in associated ∞-categories correspond to homotopy (co)limits in the model categories.

2.1.0.6. Common ∞-categories. Let us set some notation here and record the following ∞-categories:

We denote by Sp the ∞-category of (compactly generated, weakly Hausdorﬀ) topological spaces/Kan complexes/∞-groupoids. This is the analogue of the category of sets in classical category theory. This ∞-category is presented by the model structure mentioned above.

We denote by V ect the dg-category of cochain complexes. We will always use cohomological grading, i.e. the diﬀerential increases the degree by +1, so that complexes that are more naturally homologically graded (e.g. singular chains on a space) will be placed negative degrees. We denote by V ect≤0 the full subcategory of connective (i.e. concentrated in negative degrees) complexes.

The tensor product ⊗k of cochain complexes induces symmetric monoidal ∞- structures on these ∞-categories. This ∞-category is presented by a model category whose weak equivalences are the quasi-isomorphisms, i.e. morphisms of chain complexes inducing isomorphisms of cohomology.

We denote by dga the ∞-category of diﬀerential graded algebras, that is associative algebra objects in V ect, i.e. cochain complexes A equipped with a

multiplication A ⊗k A → A that is unital and associative. This ∞-category

18 is presented by a model category whose weak equivalences are the quasi- isomorphisms, i.e. morphisms of chain complexes inducing isomorphisms of cohomology.

We denote by cdga the ∞-category of commutative diﬀerential graded algebras, that is commutative algebra objects in V ect, i.e. cochain complexes

A equipped with a multiplication A ⊗k A → A that is unital, associative, and commutative (in the graded sense: for homogenous elements a, b we have a·b = (−1)deg(a)deg(b)b·a). We denote by cdga≤0 the full subcategory of connective objects. This ∞-category is presented by a model category whose weak equivalences are the quasi-isomorphisms, i.e. morphisms of chain complexes inducing isomorphisms of cohomology. The cofibrant objects in cdga≤0 are (retracts of) the so-called quasi-free cdga’s, which are those cdga’s whose underlying graded algebra is free. In particular Koszul complexes are cofibrant, and thus give cofibrant replacements for regular sequences.

We denote by dg − Lie the ∞-category of diﬀerential graded Lie algebras.

We denote by P rL the ∞-category of presentable ∞-categories with morphism spaces given by the space of continuous (i.e. colimit preserving) functors. We

L denote by P rA for A ∈ dga the ∞-category of A-linear ∞-categories with morphism spaces given by the space of continuous A-linear functors.

We denote by DGCatcont the ∞-category whose objects are k-linear stable cocomplete ∞-categories and whose morphisms are k-linear continuous functors.

19 2.2 Derived Algebraic Geometry

Here we motivate and explain some basic ideas in derived algebraic geometry, the language in which this thesis is written. The foundational references are [122, 123, 79, 85]. For an excellent very brief introduction see [29], for more leisurely expositions see [6, 119], the review article [120], or the introductions to [85, 79].

2.2.1 Derived Schemes

The local models in algebraic geometry are affine schemes which are in (con- travariant) correspondence with k-algebras. The local models in derived algebraic geometry - affine derived schemes - are in correspondence with a notion of derived k-algebras, which for us will mean connective cdga’s. This solves the problem that in degenerate cases intersections, or more general fiber products or limits in schemes, do not give the geometrically meaningful “right” answer. Consider for example the intersection of two lines in the plane, which we would like to always be point- like. When the lines are parallel, one is forced to deal with the projective completion to see the intersection point at infinity, but when the lines coincide, say {x = 0} ∩ {x = 0} ⊂ Spec(k[x, y]), the intersection in the world of schemes is not even 0-dimensional. Intersecting on an affine scheme Spec(A) corresponds on the algebra side to imposing equations on A, that is taking quotients of A. We want to remember the identifications made, and see that the problem lies with the quo- tient construction. One way to solve this is to pass to (commutative algebra objects in) ∞-groupoids, or, linearizing, connective cochain complexes of k-vector spaces. Thus one is lead to consider as local models derived affine schemes dAff, that is

20 (by definition) the opposite of the ∞-category of connective commutative differential graded algebras cdga≤0. These glue together to an ∞-category of derived schemes, and just as in classical algebraic geometry one may import the (homotopical) commutative algebra of the local model to do geometry. One may think of a derived scheme as a classical scheme with a sheaf of connective cdga’s which in degree ≤ 0 correspond to “higher nilpotents”. In the example above the derived intersection {x = 0} ∩ {x = 0} is now Spec(k[y, ξ]), where ξ is a generator in degree −1 with dξ = 0. One can compute this noting that from another point of view intersecting affines corresponds on the algebra side to taking the tensor product, which in the derived world has to be taken in the derived sense. In the model category present- ing cdga’s we cofibrantly resolve one of the factors, here conveniently by a Koszul resolution k[x, y]/(x) ∼= k[x, y, ξ], dξ = x, deg(ξ) = −1. There is a notion of virtual dimension, which for Spec(k[y, ξ]) is 0 as desired. We think of ξ ∈ k[y, ξ] as a 1-cell witnessing the identification of 0 with itself, i.e. the second identification of x with 0. The underlying classical scheme is Spec(H0(k[y, ξ])) = Spec(k[y]), the ordinary intersection.

2.2.2 Derived Stacks

2.2.2.1. Derived prestacks. Just as passing from schemes to derived schemes corrects limits, passing to derived stacks corrects quotients, gluings and more general colimits. One takes the functor of points view and considers as a ﬁrst step the ∞-category of derived prestacks, that is (∞-)functors dAff → Sp. This is a free cocompletion into which dAff embedds via Yoneda.

21 2.2.2.2. Derived stacks. Wanting to keep the meaningful colimits in dAff given by Zariski or étaleatlases we single out derived “sheaves”, that is (the ∞-category of) derived stacks dSt. By definition this is the full subcategoryof derived prestacks X satisfying the descent condition that for any étalecover f : U = Spec(B) → Y = Spec(A) of derived affines the induced morphism

X(Y ) → lim X(U •/Y ) is an equivalence, where U •/Y is the Cech-nerveˇ of f 1.

2.2.2.3. Derived Artin stacks. One has a “geometric” class of derived stacks given by derived Artin stacks. One deﬁnes n-Artin derived stacks by induction as repeated quotients by smooth groupoids of (n − 1)-Artin derived stacks, starting with derived schemes as 0-Artin derived stacks (see [123] for details). A stack is then Artin if it is n-Artin for some n. These are geometric in that one can do geometry on a smooth cover.

2.2.2.4. (Relative) affine stacks. For us these will not be important, but we will need affine derived stacks. These are the derived stacks give by the functor of points Spec(A) of a coconnective cdga A.A relative affine stack is a morphism of derived stacks X → Y such that the fiber over any derived affine is an affine derived stack.

1Really, one should consider hypercoverings.

22 2.2.2.5. Affinizations, in particular of the circle. The affinization of a derived prestack X is the affine stack Spec(Γ(X; OX )). This is the universal affine stack through which X factors. The affinization of the constant derived group prestack S1 is given

1 by the morphism of derived group prestacks S → BGa = Spec(k[η]) where η has degree 1, and in particular satisﬁes η2 = 0. Here we used the formality (in characteristic

1 ∼ ∗ 1 ∼ ∗ 1 ∼ zero) of cochains on S : OS1 = C (S ; k) = H (S ; k) = k[η].

2.2.2.6. Cartesian closed structure. The ∞-category of derived (pre)stacks is symmetric monoidal and Cartesian closed. In particular given derived (pre)stacks X,Y there is an internal Hom-object: the mapping derived (pre)stack whose S points for S a derived aﬃne are Maps (X,Y )(S) = Maps (X × S,Y ). d(P )St d(P )St

2.2.2.7. Completions. Let X → Y be a morphis of derived prestacks. Then the (de-

Rham) completion YbX of Y along X is the derived prestack whose space of R points

≤0 0 red for R ∈ cdga is given by Y (R) ×Y ((H0(R))red) X((H (R)) ). We will only need this for X ∼= pt. There is an alternative notion of completion given by a functor of derived Artin algebras (see 2.8.0.3). The two notions correspond via restriction and Kan extension, see [50].

2.2.2.8. Real life examples. Naturally occuring examples of genuinely derived objects besides the derived free loop spaces are for example the derived moduli stack of local system on a curve, or moduli stack of vector bundles on a surface in geometric representation theory.

23 2.3 Quasi-coherent and Ind-coherent Sheaves 2.3.1 Quasi-coherent sheaves

2.3.1.1. The aﬃne case. We want to deﬁne the dg-category QCoh(X) for X a de-

≤0 rived prestack. Consider first the case where X = Spec(A) is affine, A ∈ cdgak = dRingconn. In this case we define QCoh(X) to be the k-linear stable ∞-category A − Mod of (possibly unbounded) dg-A-modules. This ∞-category is presented by a model structure for which the weak equivalences are the quasi-isomorphisms, i.e. those morphisms of differential graded A-modules which induce isomorphisms on cohomologies. The quasi-isomorphisms thus are invertible in the ∞-category A−Mod. See [70, 71, 56] for further details on this model structure. For our purposes it is suffi- cient to know that the Koszul resolution produces good (i.e. cofibrant) replacements not just as cdgas, but also as modules. If A is classical, then QCoh(Spec(A)) is an ∞-categorical enhancement of the triangulated category D(A) of unbounded cochain complexes of A-modules localized at quasi-equivalences. The general definition is:

2.3.1.2. The case of a general derived prestack. Let X be a derived prestack. Then the ∞-category of quasi-coherent sheaves QCoh(X) is deﬁned to be

QCoh(X) = limSpec(A)∈dAff/X A − Mod

where the limit is computed in DGCatcont.

That is, a quasi-coherent sheaf F on X is given by the data of an A-module ∼ Ff for each morphism f : Spec(A) → X, an identiﬁcation Ff ⊗A B = Fh for each identiﬁcation of h and f ◦g where g is a morphism g : Spec(B) → Spec(A) of derived

24 aﬃnes, and higher coherences.

QCoh(X) is k-linear, stable, and presentable.

2.3.1.3. Structure sheaf. The structure sheaf OX of a derived prestack X is given by the assignment Ff = A to f : Spec(A) → X.

2.3.1.4. Functoriality and symmetric monoidal structure. The ∞-category QCoh(X) can be equipped with a symmetric monoidal structure, which on derived aﬃnes just comes from the (derived) tensor of modules. Given a morphism of derived prestacks f : X → Y , there is an associated symmetric monoidal continuous ∗-pullback functor

∗ f : QCoh(Y ) → QCoh(X) that to a morphism g : Spec(A) → X assigns Ff◦g. This has a right adjoint f∗ : QCoh(X) → QCoh(Y ) by the adjoint functor theorem. If f is a morphism of derived aﬃnes f : X = Spec(A) → Y = Spec(B), then the adjunction is just the usual tensor/forgetful adjunction between module categories

∗ f (−) = (−) ⊗B A : B − Mod A − Mod : f∗ where f∗ forgets the A-module structure along f. In general the functor f∗ is not well behaved, but under nice

∗ conditions the adjoint pair (f , f∗) satisﬁes base change.

2.3.1.5. Descent. The derived prestack QCoh(−) satisfies faithfully flat descent and is insensitive to stackification. Given a faithfully flat surjection Y → X, we can ˇ compute QCoh(X) as the totalization of the Cech-nerve QCoh(Y•/X).

25 2.3.1.6. Cotangent complex. The natural notion of cotangent sheaf in derived al-

∗ gebraic geometry is the already derived cotangent complex TX , which represents appropriately deﬁned derived derivations. For classical schemes (considered as derived schemes) this is the cotangent complex of Andre-Quillen ([5, 103, 104, 61]) and Illusie ([60]).

2.3.1.7. Perfect sheaves. Sitting inside QCoh(X) we have the small subcategory of perfect complexes. The ∞-category of perfect (quasi-coherent) sheaves P erf(X) is the full subcategoryof QCoh(X) given by dualizable objects. Alternatively the ∞- category P erf(X) may be described as the smallest subcategorycontaining OX that is closed under shifts, finite colimits, and direct summands. For classical schemes this agrees with the classical definition of complexes locally quasiisomorphic to bounded complexes of finitely generated projectives. When the cotangent complex is perfect, for example for a derived Artin stack locally of finite presentation, its dual TX is the tangent complex.

2.3.1.8. Coherent sheaves. Under some conditions on X we have another small subcategory of QCoh(X). Recall the following definitions: A derived affine scheme Spec(A) is Noetherian if H0(A) is Noetherian, and each Hi(A) is finitely generated over H0(A). A derived scheme is locally Noetherian, if every (equivalently just one) Zariski cover by derived affines is a cover by Noetherian derived affines. A derived scheme is Noetherian if it is quasi-compact and locally Noetherian.

Assume that X is a Noetherian derived scheme. The ∞-category of coherent

26 sheaves Coh(X) is deﬁned to be the full subcategorygiven by objects with bounded cohomological amplitude, and coherent cohomologies.2

2.3.1.9. Perfect vs. coherent sheaves. When both are deﬁned, we have that P erf(X) is a full-∞-subcategory of Coh(X), equivalent to all of Coh(X) if and only if X is a smooth classical scheme. Neither P erf nor Coh are cocomplete. In many cases (“perfect stacks”, [18]), one has that Ind(P erf(X)) ∼= QCoh(X). The ∞-category Ind(Coh(X)) on the other hand is the subject of the next subsection.

2.3.2 Ind-coherent sheaves

References for ind-coherent sheaves are [49, 47]. The notion of Ind-coherent sheaves arose in geometric representation theory where many ∞-categories of sheaves appear there most naturally as IndCoh, not QCoh. A particularly noteworthy example is the candidate ∞-category on the spectral side of the Geometric Langlands program. A useful analogy with functional analysis to keep in mind is that quasi- coherent sheaves are to ind-coherent sheaves what nice functions on a space are to distributions.

We will only need the theory for schemes, where the theory is a sort of “renor- malization” of the theory of quasi-coherent sheaves. In this subsection all derived schemes are assumed to be Noetherian.

2In the world of classical algebraic geometry for X a classical scheme, Coh(X) is (an ∞- categorical enhancement of) the bounded derived category of coherent sheaves usually denoted by Db(Coh(X)).

27 2.3.2.1. Basics on ind-coherent sheaves. Let X be a (Noetherian) derived scheme. The ∞-category of ind-coherent sheaves on X is the ind-completion of Coh(X).

For X classical IndCoh(X) is equivalent to (an ∞-categorical/dg enhancement of) Krause’s stable unbounded derived category ([75]) of injective complexes modulo homotopies (not modulo quasi-isomorphisms). The ∞-category of compact objects in IndCoh(X) is Coh(X). The action of P erf(X) on QCoh(X) preserves the subcategory Coh(X), and thus deﬁnes an action of QCoh(X) on IndCoh(X).

2.3.2.2. Ind-coherent vs quasi-coherent sheaves. One way of measuring the difference between IndCoh and QCoh is via t-structures. There is a natural t-structure on QCoh(X), hence on Coh(X), and there is unique t-structure compatible with filtered colimits on IndCoh(X) extending it. The natural functor ΨX : IndCoh(X) → QCoh(X) which sends a formal filtered colimit in Coh(X) to its realized colimit in QCoh(X) induces an equivalence on eventually coconnective objects and realizes

QCoh(X) as the left completion of IndCoh(X). The functor ΨX is not conservative - for X classical this is reﬂected in Krause’s model by the existence of complexes of injectives that are weakly equivalent to the zero complex but not nullhomotopic - and admits a right adjoint ΞX if and only X is eventually coconnective. In this case the unit of the adjunction is an equivalence and realizes QCoh(X) as a colocalization of IndCoh(X).

2.3.2.3. Functoriality and symmetric monoidal structure. The functoriality of ind- coherent sheaves is complicated. Given a morphism of derived schemes f : X → Y ,

28 IndCoh there is a ∗-pushforward f∗ : IndCoh(X) → IndCoh(Y ) that agrees with the ordinary pushforward on Coh(X) ⊂ IndCoh(X), and a !-pullback f ! : IndCoh(Y ) →

! IndCoh IndCoh(X). When f = j is an open immersion, j is left adjoint to j∗ , and when

! IndCoh f is proper f is (by deﬁnition) right adjoint to f∗ . Given a cartesian diagram of derived schemes, one gets a base change natural isomorphism. Note that since

IndCoh ! f∗ , f are in general not adjoint, this is additional data, and does not come by adjunction. IndCoh(X) can be equipped with a symmetric monoidal structure via !-

! ! pullback along the diagonal ∆, i.e. F ⊗ G = ∆ (F G). One can deﬁne IndCoh more generally not just on schemes, but on a much more general class of prestacks, and one can package functoriality into a lax symmetric monoidal functor from a certain (∞, 2)-category of correspondences of prestacks to a (symmetric monoidal) (∞, 2)- categorical reﬁnement of DGCatcont encoding the six-functor formalism and Serre- duality. One may think of this functor as giving some sort of bi-variant functoriality.

2.4 Shifted Symplectic and Poisson Geometry

2.4.0.1. Classical symplectic and Poisson geometry. The geometry of classical symplectic and Poisson structures on smooth manifolds has origins and applications in classical mechanics and ﬁeld theory and is a vast and rich subject interlinked with many other areas of mathematics. For example it is connected to algebraic geometry through Fukaya categories in homological mirror symmetry [9, 116], to low-dimensional topology e.g. through the many variants of Floer theory (see e.g. [88] for a survey for the case of three-manifolds), to mathematical physics through (e.g. deformation or geometric) quantization [72], to representation theory through

29 viewing symplectic singularities ([27, 46]) as the “Lie algebras of the 21st century” (Andrei Okounkov), etc. See [34, 90, 129] for introductions to symplectic and Poisson geometry.

2.4.0.2. Recollection of classical deﬁnitions. Let us recall the basic deﬁnitions. A

2 symplectic structure on a smooth manifold X is a two form ω ∈ ΩX that is de Rham closed, i.e. satisﬁes ddRω = 0 and non-degenerate, i.e. such that the associated mor-

] ∗ phism of vector bundles given by contraction w : TX → TX is an isomorphism. A

Poisson structure is given by the data of a Poisson bracket on smooth functions OX on X, that is a biderivation {−, −} : OX ⊗ OX → OX that is also a Lie bracket, or V2 equivalently a Poisson bivector Π ∈ , i.e. such that [Π, Π]Sch = 0, where [−, −]Sch is the unique extension of the Lie bracket on vectorﬁelds to a Gerstenhaber algebra V• structure on polyvectorﬁelds TX called the Schouten bracket. A symplectic structure induces a Poisson structure, and a Poisson structure is induced by a symplectic

∗ structure if and only if it is non-degenerate, i.e. if the induced morphism TX → TX is non-degenerate.

2.4.0.3. Analogues in the derived world, examples and relevance. Shifted symplectic and Poisson structures are the analogues in derived algebraic geometry of ordinary symplectic and Poisson structures in diﬀerential geometry. This is currently a very active area of research for which references are the foundational papers [96, 32], the review article [97], and the excellent expositions in [29, 108]. Let us mention one of the interesting results that have been proven. Shifted symplectic structures allow

30 via transgression for a version of the AKSZ construction of symplectic structures on derived mapping stacks from derived stacks equiped with a sort of orientation into shifted symplectic derived stacks. Since the classifying (derived) stacks of affine smooth group schemes G and the derived stack of perfect complexes are (2−)shifted symplectic one finds that as a consequence many derived moduli stacks are shifted symplectic and thus Poisson. This holds for example for the derived moduli stacks of perfect quasi-coherent sheaves on a Calabi-Yau surface, of principal G-bundles or Higgs G-bundles on a smooth proper curve, or of flat G-bundles on the underlying topological space of such a curve. An n-shifted Poisson structure then allows one, for n 6= 0, to canonically deformation quantize the symmetric monoidal ∞-categories of perfect quasi-coherent sheaves on such stacks into E|n|-monoidal ∞-categories. There are other applications or connections to other areas of mathematics, for example to the categorification of Donaldson-Thomas invariants ([17]), to Costello-Gwilliam’s to perturbative quantization of field theories [40, 41], to geometric representation theory, non-abelian Hodge theory, quantum groups, string topology, or algebraic L-theory.

2.4.0.4. Shifted symplectic structures. Having explained how shifted symplectic and Poisson structures are interesting, let us explain what they are. For our application

∗ we need the case of n-shifted cotangent bundles T X[n] = SpecX SymOX (TX [−n]) (for n positive), i.e. of relative affine derived stacks Y , over a smooth classical base X. Let us consider shifted symplectic structures first. The differences with

∗ the classical case arise because we have to replace the (co)tangent bundles TY , TY

31 ∗ by the (co)tangent complexes TY , TY which are of homotopical nature and have internal degree and diﬀerential. Thus there will be a space of 2-forms, and de Rham closedness3 of a 2-form ω is not a condition, but the extra data of a homotopy of the de Rham diﬀerential of ω to the zero form. Further a closed 2-form has an internal degree n, which can be non-zero, so that we have a shift by n in the contraction

] ∗ morphism: ω : TY → TY [n].

Deﬁnition (Somewhat vague) Let Y be a relative aﬃne stack over a smooth classical base X.A n-shifted symplectic form on Y is a closed 2-form ω of internal degree n

] ∗ such that ω : TY → TY is an equivalence.

This definition works for the case of derived Artin stacks locally of finite presentation as well. In general the contraction morphism of an n-shifted symplectic structure on a derived stack trades off stacky and derived directions in the (co)tangent complex.

2.4.0.5. Shifted Poisson structures. For shifted Poisson structures the situation is more involved. The general deﬁnition for derived Artin stacks is complicated since e.g. polyvectors don’t have nice functorial properties with respect to smooth morphisms. Recall that a Pn-algebra structure (“n-shifted Poisson”) on a cdga A is an operation (“bracket”) {−, −} : A⊗k A → A[1−n] of degree 1−n that endows A[n−1]

3Note a possible point of confusion: There are two differentials on differential forms: One coming from the internal differential of the cotangent complex and one from the de Rham differential. The structure one gets on differential forms is called a graded mixed cdga. See e.g. [96] for details.

32 with a Lie algebra structure, and is a biderivation with respect to the commutative product on A.

π Deﬁnition Let Y = SpecX (π∗OY ) → X be a relative aﬃne stack over a smooth classical base X. A n-shifted Poisson structure on Y is a lift of π∗OY to a sheaf of

Pn+1-algebras.

2.4.0.6. Polyvectorﬁeld approach. The commutative dg-algebra of n-shifted polyvec-

0 torﬁelds is P oly(X, n) = H (Y, SymOY (TY [−n − 1])). This is bigraded by internal degree and weight (TY has weight 1), and admits a Schouten style bracket of degree −n − 1 and weight −1. There is an equivalent description of n-shifted Poisson structures in terms of n-shifted bivectors Π satisfying [Π, Π]Sch ∼ 0 up to a prescribed homotopy [91].

2.4.0.7. Comparisons between shifted symplectic and Poisson, and with the classical notions. For X smooth classical one ﬁnds that there are no n-shifted symplectic or Poisson structures for n 6= 0, and that one recovers the classical notions for n = 0. It is a theorem that (n-shifted) symplectic is the same as non-degenerate Poisson. This holds in great generality for derived Artin stacks once one makes sense of shifted Poisson structures on them, but is quite diﬃcult [32].

2.4.0.8. Shifted cotangent bundles. Similarly to the cotangent space in diﬀerential geometry being canonically symplectic and thus Poisson, the n-shifted cotangent bundle Y = T∗X[n] over a smooth base X (and even over a derived Artin stack locally

33 of finite presentation [30]) is canonically n-shifted symplectic and thus n-shifted Poisson. Locally on a derived affine Spec(A) ⊂ X we resolve A by a quasifree cdga k[xi], in which case the n-shifted cotangent complex over Spec(k[xi]) is Spec(k[xi, pi]), where the pi are variables of degree deg(xi) − n corresponding to fiber directions

∂ pi = . The symplectic form is the de Rham diﬀerential of the canonical Liouville ∂xi P 1-form locally given by ω = ddRxi ∧ ddRpi, and the Poisson bivector is given by i P ∂ ∂ ∧ . The data of the homotopies expressing closedness ddRω ∼ 0 respectively ∂xi ∂pi i [Π, Π]Sch ∼ 0 are trivial since these identiﬁcations hold on the nose.

2.5 (Algebraic) Families of ∞−Categories

We will need a notion of a “nice” (algebraic) family of k-linear stable presentable ∞-categories, which will be formalized by quasi-coherent sheaves of ∞- categories. These are a categoriﬁcation of the notion of “nice” algebraic families of modules, i.e. of quasi-coherent sheaves of modules. This is not just an instance of idle abstract generalization that happens to be convenient to us. Families of ∞- categories are important in formulating the Geometric Langlands program through their connection to (strong)4 representations of loop groups G((t)), i.e. to the ∞- category of families of ∞-categories on B(G((t))dR) (see next section). The main reference for this section is [48].

4The group actions on ∞-categories we deﬁned are usually known as weak group actions. From the point of view of the sheafy/categorical group algebra formulation of an action, a strong group action on an ∞-category C is a module structure on C for the ∞-category DG − mod of D-modules on G equipped with the convolution monoidal structure (instead of QCoh(G)). This corresponds to considering the classifying stack of the de-Rham space G((t))dR of G((t)).

34 2.5.0.1. Definition. We will need a functor of points approach to defining families of ∞-categories. For S = Spec(A) a derived affine scheme, the right thing to consider is the ∞-category QCoh(S) − mod(DGCatcont) of modules in DGCatcont for QCoh(S) equipped with the symmetric monoidal structure given by the tensor product. We will usually shorten this to QCoh(S) − mod. Given a morphism of derived affine schemes S → T , we can use the tensor/induction functor (−) ⊗QCoh(T ) QCoh(S) to

op deﬁne a functor dAff → DGCatcont. Taking the right Kan extension we get a functor ShvCat : P reStk → DGCatcont.

Deﬁnition Let X be a derived prestack. The ∞-category of families of ∞-categories or quasi-coherent sheaves of ∞-categories on X is 5

ShvCat(X) = limS∈dAff/X QCoh(S) − mod.

Thus a family of ∞-categories F on a derived prestack X is the data of an A- linear stable presentable ∞-category F(A) = Γ(Spec(A), F) for each Spec(A) → X, ∼ together with an identiﬁcation F(A) ⊗A B = F(B) for each morphism of derived aﬃnes Spec(B) → Spec(A), and higher coherences.

The functor ShvCat tautologically sends colimits to limits.

2.5.0.2. Alternative description: maps to derived prestack of ∞-linear ∞-categories Cat. Let Cat denote the derived prestack that to a connective cdga A associates

5We prefer to call these families of ∞-categories, but the standard terminology is quasi-coherent sheaves of ∞-categories.

35 L the ∞-category of A-linear stable presentable ∞-categories P rA, and to a morphism

A → B of such cdga’s the tensor functor (−) ⊗A B. We can recover the maximal

∞-subgroupoid of ShvCat(X) as the mapping space MapsdSt(X, Cat).

2.5.0.3. Structure sheaf. Give a derived prestack X we have the family of ∞-categories

QCoh/X which is to ShvCat(X) as the structure shead OX is to QCoh(X). Con- cretely, this is the assignment of QCoh(S) as a QCoh(S)-module to an derived aﬃne scheme S over X.

2.5.0.4. Global sections. Given a derived prestack X, we have a global sections func-

tor Γ(X, −): ShvCat(X) → DGCatcont given by Γ(X, F) = limS∈dAff/X Γ(S, F).

Thus for example, we have that Γ(X, QCoh/X ) = QCoh(X). We have Γ(X, F) =

MapsShvCat(X)(QCoh/X , F). The functor Γ(X, −) is lax symmetric monoidal and upgrades to an enhanced functor ShvCat(X) → QCoh(X) − mod, which by an abuse of notation we will continue denoting by Γ(X, −).

2.5.0.5. Enhanced global sections. The enhanced global sections functor Γ(X, −) admits a left adjoint LocX : QCoh(X)−mod → ShvCat(X). For C ∈ QCoh(X)−mod, the sections over S ∈ dAff/X are Γ(S, LocX (C)) = C ⊗QCoh(X) QCoh(S). Evidently

LocX is symmetric monoidal.

2.5.0.6. Descent. ShvCat satisfies fppf descent. It is insensitive to sheafification, and for an fppf surjection of derived prestacks f : Y → X, we can compute ShvCat(X)

36 as the totalization T ot(ShvCat(Y •/X)) of ShvCat on the Cech-nerveˇ Y •/X of f.

2.5.0.7. 1-Aﬃneness. We say that a derived prestack X is 1-aﬃne if Γ(X, −) and

LocX are inverse equivalences. One of the main results of [48] is that certain prestacks are 1-affine. This holds for example for quasi-compact quasi-separated derived schemes, classifying spaces of finite type derived affine group schemes and their formal completions, de-Rham spaces of finite type derived schemes, and algebraic stacks (with some conditions). For finite type derived group schemes G, the statement for BG can be read as categorical representations being determined by their invariants (i.e. global sections), something that is very far from true in the less categorified setting of representations on vector spaces.

2.5.0.8. Functoriality. Families of ∞-categories admit functoriality. Let f : X → Y

6 be a morphism of derived prestacks. Then we have a pullback functor coresf for families of ∞-categories given by restriction: If F is a family over Y , then given a derived aﬃne scheme S and an S-point S → X of X we get an S-point of Y , and then have Γ(S, coresf (F)) = Γ(S, F). For example, we have coresf (QCoh/Y ) = QCoh/X .

The corestriction functor coresf admits (by the adjoint functor theorem) a right

7 adjoint pushforward coindf , and coresf and coindf satisfy various natural properties with respect to composition and with respect to compatibility with the global sections

Γ(X, −) and localization functors LocX .

6“corestriction” 7“coinduction”

37 2.5.0.9. Definition of a deformation of an ∞-category. A deformation of a presentable k-linear ∞-category C over a pointed derived prestack (X, pt) is a family of ∞-categories C over X together with an identification of the fiber over pt wth C. A first order deformation of a presentable k-linear ∞-category C of degree d (pos-

2 sibly positive) is a deformation of A over (D, pt). Here D denotes Spec(k[]/( )), where is a degree d inﬁnitesimal. This is a derived scheme if d is negative and a coaﬃne stack when d is positive, and is naturally pointed via the augmentation map k[]/(2) → k.

2.5.0.10. Kodaira-Spencer first order deformation map. Let C be a presentable stable k-linear ∞-category, and C a deformation of C over a pointed prestack (X, pt). We wish to define a corresponding Kodaira-Spencer type first order deformation map. Considering the family as a pointed map of prestacks (X, pt) → (Cat, C), we see that this should just be the derivative at pt of this pointed map. For this to make sense, we clearly need to, and will, assume that X admits a tangent complex at pt. We have an issue in that Cat does not in general admit a tangent complex at C. We can see this by considering the associated completion Catd =: DefCatC of Cat at C. By [81, 85] this is not a formal moduli problem (see 2.8), so does not admit a tangent complex, but it is very close to being one in that it is a 2-proximate moduli

∧ problem. It thus has a best approximation by a formal moduli problem DefCatC ,

∧ which comes equipped with a universal morphism DefCatC → DefCatC . Thus

∧ we can consider the composition of Xbpt → DefCatC → DefCatC . This is now a morphism of which we can take the derivative, i.e. the corresponding map on tangent

38 complexes. By [100] the derived Lie algebra TDefCatC [−1] controlling DefCatC is the 1-shifted Hochschild cohomology HH•(C)[1] (see 2.7) with bracket the Gerstenhaber

• bracket induced from the E2 structure on HH (C). Thus the derivative corresponds

• to a morphism of cochain complexes TX,pt → HH (C)[2]. We now see that the following deﬁnition is reasonable:

Definition Let C be a family of ∞-categories over a pointed prestack (X, pt) with fiber over pt given by a category C, and assume X admits a tangent complex at pt. The first order deformation map of C or derivative of C is the associated morphism

• of cochain complexes TX,pt → HH (C)[2] we explained above. If TX,pt is naturally identiﬁable with the one-dimensional k-vectorspace k[n] concentrated in a single degree n (in particular for X = BBGa, n = 2), the ﬁrst order deformation Hochschild cohomology class of C is the image in HH•(C)[2] of 1 ∈ k[n].

2.6 Group actions on ∞−Categories

Useful references for this section are [36, 94, 48]. We deﬁne inertial group actions and unipotent circle actions, and prove some minor results that we couldn’t

ﬁnd in the literature, in particular a comparison of group actions of BGa and BGb a.

2.6.1 Actions of homotopical groups in spaces

2.6.1.1. Homotopical/∞-groups. The ∞-category of homotopical groups or ∞-groups is the ∞-category of ∞−group objects in the monoidal ∞-category of spaces. Thus, a homotopical group is a space G equipped with a multiplication that is invertible and associative up to (higher) coherences.

39 2.6.1.2. Categorical representations/group actions. A homotopical action or categorical representation of a homotopical group G on a presentable k-linear category C is a morphism of ∞-groups G → Aut(C). Here we consider C as an object of the ∞-

L category P rk of presentable k-linear ∞-categories, so that by Aut(C) we mean based L loops ΩCP rk at C, i.e. continuous invertible k-linear endofunctors of C. Letting BG denote the ∞-category with one object and G as its ∞-group of automorphism, we can equivalently describe a categorical G-representation on C as a functor of pointed

=∼ ∞-categories (BG, EG = pt) → (P rk , C). Categorical G-representations organize L G into a ∞-category (P rk ) .

Remark A functor BG → P rk may be though of as a family of (∞-)categories. This is in complete analogy to the case one category level downwards of, say, group actions of an ordinary group G on a vector space V , which can also be described as (classical) local systems on BG, i.e. (locally constant) families of vector spaces on BG.

2.6.1.3. Inertial group actions. An inertial group action of a homotopical group G on a presentable k-linear category category C is a morphism of homotopical groups

G → AutAutk (IdC) = ΩIdC Autk(C). We call these actions inertial because they do not move around objects.

2.6.1.4. Circle actions and automorphisms of the identity. We will really only need the case of categorical representations of the topological circle group S1. Since S1 ∼=

BZ is 1-connective, a categorical representation of S1 is equivalent to the inertial

40 1 action Z → Autk(IdC) we get by taking based loops of S → Autk(C). Since Z is the free homotopical group on one generator (e.g. by the Jones construction the free

∼ 1 ∼ 1 ∞-group is given by ΩΣ+pt = ΩS = Z), we see that the S action is determined by the (k-linear) automorphism of the identity given by the image of 1 ∈ Z.

L G L 2.6.1.5. (Co)invariants. The G-invariants functor (P rk ) → P rk is the functor

G G (−) : C 7→ C = limBGC. The invariants functor has a right adjoint given by equip- ping a category with the trivial G-action. We also have a (homotopy) coinvariants functor (−)G : C 7→ CG = colimBGC.

2.6.2 Actions of (homotopical) group prestacks

We will also need to deal with the case of a derived group prestack acting on a category.

2.6.2.1. Derived group prestacks. A (derived) group prestack is a ∞-group object in the ∞-category of derived prestacks.

2.6.2.2. The group prestack of automorphisms. Let Cat denote the derived stack of linear presentable ∞-categories, and let C be a k-linear presentable category. The derived group prestack of automorphisms Aut(C) of C is the derived group prestack

ΩCCat of based loops in Cat at C. Its space of R-points for a connective cdga

R is AutR(C ⊗k R), and it associates in a homotopy coherent way to a morphism

R → S of connective cdga’s a morphism of spaces AutR(C ⊗k R) → AutS(C ⊗k R ⊗R

41 ∼ S) = AutS(C ⊗k S) that on underlying points is given by φ → φ ⊗R IdS. Here we

L consider C ⊗k R as an object in P rR, so that we are considering continuous R-linear endofunctors.

2.6.2.3. Categorical representations/group actions. Let G be a derived group prestack. An action or categorical representation of G on a presentable k-linear category C is a morphism of derived group prestacks G → Aut(C).

2.6.2.4. Description as family of ∞-categories over BG. Since by the classifying space/based loops adjunction the morphism of derived group prestacks G → Aut(C) is equivalent to a morphism of pointed derived stacks (BG,EG = pt) → (Cat, C) we see that a categorical representation of G on C is equivalent to a family C of ∞-categories on BG with an equivalence of the ﬁber over EG = pt with C. This is the family of invariants C ∼= CG.

2.6.2.5. Description as modules for categorical group algebra (QCoh(G),?). Under mild conditions satisfied in our applications, a family of ∞-categories over BG with an identification of the fiber over EG = pt with G is the same data as the category C on the cover EG = pt with G together with descent data. We have that ShvCat(BG) is equivalent to the totalization of the cosimplical ∞-category (QCoh(G•), ⊗) − mod ˇ where G• is the Cech nerve of EG → BG, and one can check the Beck-Chevalley conditions to conclude that ShvCat(BG) is equivalent to comodules for QCoh(G) equipped with the comonoidal structure m∗ induced by the multiplication m : G ×

42 G → G. Equivalently, since QCoh(G) is self-dual, ShvCat(BG) is equivalent to modules for the categorical convolution algebra, i.e. again QCoh(G), but now by passing to adjoints considering it to be equipped with the convolution monoidal structure ? = m∗.

2.6.2.6. Analogy with classical notion. Note the analogy with the classical notion of a linear representation V of an algebraic group G which we can describe as either

A group morphism G → Autk(V )

A quasicoherent sheaf on BG, or

A comodule for functions OG, with the coalgebra structure coming from pulling back along G × G → G. If G is ﬁnite, then we can dualize to consider modules

for the group algebra instead, i.e. functions OG equipped with the convolution product.

2.6.2.7. Remark on strong group actions. What we deﬁned as a group action is what is more commonly known as a weak group action. There is also a notion of strong group action, which replaces the ∞-category of quasicoherent sheaves G by the ∞- category of D-modules on G (or, from another point of view BG by B(GdR)). This encodes in particular that the action is in some sense “inﬁnitesimaly trivial”. We have no use for strong group actions in this thesis.

43 2.6.2.8. Relation between the two notions of action for homotopical groups. For a homotopical group G we now have two notions of categorical representations, since we can consider actions of the constant derived group prestack G. These two notions agree since one has a geometric morphism p of ∞-topoi from the ∞-topos of derived stacks to the terminal ∞-topos of spaces, such that p∗(G) ∼= G, and ∼ ∼ ∼ p∗(Aut(C)) = Γ(Aut(C)) = Aut(C)(k) = Autk(C). We thus have an adjunction ∼ HomGrp(G, Aut(C)) = HomGrp(G, Autk(C)). Here the ﬁrst Hom is happening in the ∞-category of derived group prestacks, and the second in the ∞-category of homotopical groups. We will not usually distinguish notationally between G and G.

2.6.2.9. Derivative of an action. Let now C be a categorical G-representation, where G is a group derived prestack that admits a tangent complex at the identity, and let us try to make sense of the derivative of the action G → Aut(C). Since we only care about tangent directions, we can consider the morphism of formal completions G → Aut\(C) . We want to take the tangent map of this, but Aut(C) ∼= Ω Cat bId IdC C does not in general admit a tangent complex at the identity, since the completion of Cat at C is only a 2-proximal formal moduli problem. However, it admits a best approximation by a formal moduli problem Aut(C) → Aut(C)∧ ∼ Ω DefCat∧ d IdC d IdC = C C controlled by the (derived, i.e. dg) Lie algebra given by the 1-shifted Hochschild cohomology HH∗(C)[1] with the Gerstenhaber bracket.

We can now deﬁne the derivative of a categorical G-representation C as the

∗ ∧ derivative TG,Id → HH (C)[1] of the composition GbId → Autd(C)IdC → ΩCDefCatC .

If TG,Id is naturally identiﬁable with the one-dimensional k-vectorspace k[n] concen-

44 trated in a single degree n (in particular for G = BGa, n = 1), we identify the derivative with the image of 1 ∈ k[n] in HH•(C)[1].

2.6.2.10. The two notions of derivatives agree. Let us restrict to the case that G = on an ∞-category C we now have two notions of derivative: We can take the derivative

∗ TG,Id → HH (C)[1] of the action G → Aut(C) as above, and we can take the deriva- ∗ ∼ G tive TBG,EG → HH (C)[2] of the family C = C over the pointed derived prestack (BG,EG ∼= pt). The two notions of derivative agree in that the second is a shift by [1] of the ﬁrst, because tangent complexes commute with ﬁnite limits. In particular

∼ 0 for G = BGa the two deformation classes in HH (C) agree.

2.6.2.11. Inertial group actions. In complete analogy to the case of homotopical groups, an inertial group action of a derived group prestack G on a category C is a morphism of derived group prestacks G → AutAut(C)(IdC). There is an evident notion of derivative of an inertial group action. Note that AutAut(C)(IdC) has an hon- est tangent complex (without having to pass to an associated formal moduli problem at the level of completions).

2.6.2.12. Unipotent circle actions. The homotopical circle group S1 has aﬃnization

1 ∼ ∗ 1 given by the derived group prestack Aff(S ) = Spec(C (S ; OS1 )). By formality of ∗ 1 ∼ ∗ 1 ∼ cochains on the circle C (S ; OS1 ) = H (S ; OS1 )) = k[η], where η is a generator of

∗ 1 2 1 H (S ; k) of degree 1 (so η = 0), so that Aff(S ) = BGa ([20]). We say that an action of S1 on a presentable k-linear ∞-category C is unipotent if it factors through

45 the aﬃnization BGa.

2.6.2.13. A BGa action can be recovered from its restriction to BdGa. Let BdGa denote the de-Rham completion at the origin 0 ∈ BGa. In this subsection we will prove that the restriction morphism (P rL)BGa → (P rL)BdGa coming from the inclusion i : BdGa ,→ BGa is fully faithful.

This restriction morphism is equivalent to the corestriction coresi :

ShvCat(BGa) → ShvCat(B[Ga), or equivalently the morphism (QCoh(BGa),?) − ∗ mod → (QCoh(BdGa),?) − mod induced by i . By Cartier duality, this is equivalent to the morphism (QCoh(Gb a), ⊗) − mod → (QCoh(Ga), ⊗) − mod induced by j∗, where j is the inclusion j : Gb ,→ Ga. This morphism admits a left adjoint (i.e. we have an induction functor for categorical representations) induced by j∗, so that we have an adjoint pair

(QCoh(Ga), ⊗) − mod (QCoh(Gb a), ⊗) − mod

∗ Note that j is symmetric monoidal, and j∗ is fully faithful. Furthermore, by the projection formula j∗ is QCoh(Ga)-linear.

We are thus in a situation where we have an adjunction of modules ∞- categories res : A − mod B − mod : les = (−) ⊗A B coming from an adjunction between symmetric monoidal ∞-categories

L : A B : R where the left adjoint L is symmetric monoidal, and the right adjoint is A-linear and fully faithful. We denote the symmetric monoidal products on A and B by ⊗A and

46 ⊗B and the units by 1A and 1B. If we can show that in this situation the restriction morphism res : B − mod → A − mod is fully faithful, we are done. Note that we get a natural (A, ⊗A) module structure on B via A · B := L(A) ⊗B B for A ∈ A,B ∈ B.

Let us ﬁrst show that the monoidal structure µ = (−) ⊗A (−): B ⊗A B → B is an isomorphism. We start by showing that the following diagram commutes: µ B B ⊗A B

=∼ R⊗AidB A ⊗A B, where the vertical morphism is 1A ⊗A (−). Indeed, let B1,B2 ∈ B. Then

∼ (R ⊗A idB)(B1 ⊗A B2) = (R(B1) ⊗A B2) ∼ = 1A ⊗A (R(B1) · B2) ∼ = 1A ⊗A (LR(B1) ⊗B B2) ∼ = 1A ⊗A (B1 ⊗B B2) ∼ = 1A ⊗A (µ(B1 ⊗A B2)).

∼ For any B ∈ B we have µ(1B ⊗AB) = B and so µ is essentially surjective, and because

R is fully faithful R ⊗A idB is fully faithful as well. Hence µ is an isomorphism.

We can now prove that res : B − mod → A − mod is fully faithful by showing that the counit η : les ◦ res → idB is an isomorphism. Let M ∈ B − mod. Then the following diagram commutes:

=∼ =∼ (les ◦ res)(M) B ⊗A M B ⊗A B ⊗B M

η α ∼ M = B ⊗B M, where α = µ ⊗B idM . Since α is an isomorphism so is the counit η.

47 2.6.2.14. Unipotent S1 actions are determined as the exponential of the derivative of

1 the associated inertial Ga-action. A unipotent S action on a category C is determined by its BGa-action BGa → Aut(C), or equivalently, since BGa is 1-connective, by an analogue of May’s recognition principe determined by the inertial Ga-action act : Ga → Aut(IdAut(C)) given by taking based loops of the BGa-action. The

BGa-action in turn, as we have shown above, is determined by its restriction to the morphism of formal derived groups BdGa → Autd(C), or equivalently by the based looping l : Gb a → Autd(IdAut(C)). The (derived, i.e. dg) Lie algebra of Autd(IdAut(C)) is ∗ ∼ the Hochschild cohomology HH (C) = End(IdC) of C (2.8), and Lie(Gca) is equivalent to the free derived Lie algebra k on one generator in degree 0. Thus, since by the formalism of [50] the morphism l is equivalent to the morphism of derived Lie

∗ 1 algebras Lie(Gca) → HH (C), we see that a unipotent S -action is determined by 0 ∼ 0 the element of HH (C) = End (IdC) given by the image of 1 ∈ k under the tangent morphism of cochain complexes underlying the morphism of derived Lie algebras,

d which we suggestively denote by dx (act). In conclusion, we ﬁnd that the automorphism of the identity act(1) determining the S1 action is given by the automorphism of the identity given by the exponential of the endomorphism of the identity a:

d {unipotent S1-action on C} exp( (act : → Aut(Id (C)))) ∈ Aut(Id ). ! dx Ga Aut C

2.7 Hochschild Homology and Cohomology

There is much one could write about Hochschild homology and cohomology that we will will not touch upon - see e.g. [93]. We will summarize the basic

48 deﬁnitions and results necessary for us, and mention some interesting tidbits along the way for the ediﬁcation of the reader.

2.7.1 Hochschild homology

2.7.1.1. Deﬁnition of Hochschild homology. The Hochschild homology of a dga A is deﬁned via the the tensoring of the ∞-category of spaces or Kan complexes with

1 the ∞-category of cdga’s as tensoring with a circle: HH∗(A) = S ⊗ A. Note that despite the name we consider the chain complex, and do not pass to cohomology.

1 This definition makes evident the S action on HH∗(A). Choosing a presentation of the circle S1 as the the simplicial set given by the standard 1-simplex with the two endpoint 0 simplices identified, one can show that this is equivalent to the more traditional definition as A ⊗A⊗Aop A.

0 2.7.1.2. Interpretations. We have for A classical that H (HH∗(A)) is the cocenter

0 A/[A, A], i.e. H (HH∗(A)) is universal with respect to maps out of A that vanish on commutators, so that we may think of A → HH∗(A) as a sort of derived trace ([21]). Let us also mention the following interpretation: The oriented fully extended 2-dimensional topological quantum ﬁeld theory valued in the symmetric monoidal Morita (∞, 2)-category of dga’s, bimodules, and intertwiners, deﬁned by assigning to a point the dga A evaluates to HH∗(A) on the circle. Another way of saying R this is that Hochschild homology HH∗(A) is the factorization homology S1 A of the 1 8 E1-algebra A over the circle S . See [11, 4] for expositions of this circle of ideas.

8The reader may ask the following pointed question: Is this a pun?

49 Remark Versions for ring spectra of Hochschild and (negative) cyclic homology are important as the recipients of trace maps in algebraic K-theory (”Dennis/cyclotomic trace“).

2.7.1.3. Relation to free loop spaces. Hochschild homology allows one to access the free loop space: By Jones’ theorem ([62, 78]) given a topological space X we can compute cochains (i.e. derived functions ) C∗(LX) on the free loop space LX =

∗ X ×X×X X as HH∗(C (X)). We also get a chain level equivalence of (homotopy) equivariant cochains and the (homotopy) S1 invariants of Hochschild homology, called negative cyclic homology. Applying equivariant localization to the ﬁxed point locus X ⊂ LX of the S1-action we can, after inverting a generator

∗ 1 ∼ ∗ −1 u ∈ H (BS ) = k[u], identify equivariant cochains on LX with C (X) ⊗k k[u, u ], or, if X is a smooth manifold, equivalently with de Rham chains made periodic:

• −1 ΩX,dR ⊗k k[u, u ]. This has analogues in algebraic geometry: For A commutative, we have that Spec(HH∗(A)) is equivalent to the derived free loop space LX = X×X×X X of X = Spec(A)s.

2.7.1.4. Categoriﬁed equivariant localization in derived algebraic geometry. Categori- fying the above equivariant localization statement [20] show that for X a smooth scheme the ∞-category IndCoh(LX)S1 of S1-equivariant ind-coherent sheaves on

≤i i LX is equivalent to modules in QCoh(X) for the Rees algebra DX,~ = ⊕iDX ~ of DX -modules (see 2.10.0.6, 2.10.0.7) with respect to the ﬁltration by degree of

50 diﬀerential operator and with Rees parameter (now called ~) of degree 2:

S1 ∼ IndCoh(LX) = DX,~ − mod.

2.7.1.5. Hochschild-Kostant-Rosenberg as linearization. By the Hochschild-Kostant- Rosenberg theorem (see [20]) we have that for X a derived scheme we have a lineariza-

∼ ∼ ∗ tion LX = X[−1] (recall X[−1] = Spec (SymOX ( X [1]))). This uses formality T T OX T 0 1 ∼ ∗ 1 ∼ ∗ 1 ∼ of the algebra of cochains on the circle H (S ; OS1 ) = C (S ; k) = H (S ; k) = k[η], where deg(η) = +1 and η2 = 0. The idea is that one can pass through the aﬃnization

1 0 1 ∼ 1 Aff(S ) = Spec(H (S ; OS1 )) = Spec(k[η]) = BGa of S . The last identification is by definition. For X = Spec(A) affine, we have

∼ 1 LX = MapsdP St(S ,X)

∼ 1 = MapsdP St(Aff(S ),X) ∼ = MapsdP St(BGa,X) ∼ = TX[−1].

The last equivalence states that maps out of Spec(k[η]) pick out shifted tangent vectors, which can be shown to follow from the universal property of the cotangent complex. See [20] for more details. Under this linearization the circle action corresponds to the vector field of degree −1 on TX[−1] given by the de Rham differential. Another nice heuristic geometric argument for Hochschild-Kostant-Rosenberg is as follows: Hochschild homology is the self intersection of the diagonal, and we should ∼ ∼ ∼ have X ×X×X X = X ×N∆ X = X ×TX X = TX[−1]. Here N∆ is the normal bundle (complex, really) of the diagonal, which can be identified with the tangent complex

51 TX , and the ﬁrst equivalence heuristically holds because the self intersection should only see the formal neighbourhood of the diagonal in the product. The last equivalence states that (even in families) looping9 linear objects corresponds to shifting by [−1].

2.7.2 Hochschild cohomology

Useful references for Hochschild cohomology are [7, Appendices E, F], [1].

2.7.2.1. Deﬁnition of Hochschild cohomology. The Hochschild cohomology HH∗(C) of a dg-category C ∈ DGCatcont is End(IdC), where End is taken in the category of (k-linear) continuous endofunctors of C. Since this category is monoidal, HH∗(C) is naturally E2 (Deligne conjecture). Note that by Hochschild cohomology, we usually mean the derived object, i.e. the cochain complex, not its cohomology. For Z a derived prestack we deﬁne the Hochschild cohomology of Z to be HH∗(Z) =

∗ ∼ HH (QCoh(Z)) = EndOZ×Z (∆∗OZ ). If Z is a quasi-compact derived scheme, then HH∗(Z) ∼= HH∗(IndCoh(Z)) ([7, 100]).

2.7.2.2. Explicit cochain model. Hochschild cohomology is Morita invariant, in par-

∗ ticular if C = A − mod for a dg-algebra A, then HH (C) = EndA⊗Aop (A, A) can be computed as the direct product totalization of the bigraded cochain complex

Q ⊗n n≥0 Homk(B ,B)[−n] coming from the Bar resolution, where the bigrading is by ⊗n internal degree and Hochschild degree (i.e. Homk(B ,B)[−n] sits in Hochschild

9based

52 degree n) and the diﬀerentials are the internal and the usual Hochschild cochain diﬀerential.

2.7.2.3. Interpretation as derived center. We have for A classical that H0(HH∗(A)) is equivalent to the center Z(A) = {a ∈ A : ab = ba for all b ∈ A} of A, so that we may interpret HH∗(A) → A as a sort of derived center ([21]).

2.7.2.4. Relation to deformations. It is classical that for A an underived associative algebra the (ﬂat) ﬁrst order deformations of A as an associative algebra are controlled

2 by HH (A): The deformed multiplication a ·def b = ab + c(a, b) on the vector space A ⊕ · A ∼= A ⊗ (k[]/(2)) is determined by a Hochschild 2-cochain c : A ⊗ A → A which defines an associative product if and only if it is a Hochschild cocycle. Equivalent deformations define equivalent cocycles, and obstructions to deforming to higher order lie in HH3(A). See e.g. [51] for details. In the derived setting we see that a derived deformation of a dga A as an A∞-algebra similarly defines Hochschild cochains, now of arbitrary Hochschild degree since we can deform higher multiplications as well, which suggests a similar picture holds. This is in fact the case: as explained in the section on formal moduli problems the formal moduli problem of derived associative deformations of A is controlled by the dg-Lie algebra of truncated

∗ Hochschild cochains HH+(A)[1]. The dg-Lie algebra structure on HH (C)[1] induced from the E2-structure is the Gerstenhaber bracket.

53 2.7.2.5. The Atiyah algebra and cohomological Hochschild-Kostant-Rosenberg. There is a dual version of Hochschild-Kostant-Rosenberg identifying Hochschild cohomology of a scheme as a chain complex with polyvectorfields. At least in the smooth classical case it has been shown that this can be twisted to respect the rich algebraic structure on both sides (see e.g. [28] and related papers). (A sheafified version of) Hochschild cohomology of a derived scheme X can be identified with the universal enveloping algebra of the Atiyah Lie algebra TX [−1], and under this identification Hochschild-Kostant-Rosenberg becomes a version of the PBW theorem.

2.8 Formal moduli problems

Nice expository accounts of derived deformation theory can be found in [84, 87]. For review articles see [121, 31], and for thorough foundational references see [81, 85]. For the derived deformation problem associated to a category see [25].

2.8.0.1. Small/Artinian cdga’s. To study derived deformation theory, we need a notion of ”small“ cdga’s that are in some sense close to k. The correct ∞-category is given by derived Artinian commutative algebras or Artinian cdga’s. By deﬁnition

≤0 this is the full subcategoryof the ∞-category of augmented connective cdga’s cdga/k consisting of augmented cdga’s A → k such that Hn(A) ∼= 0 for n negative enough, H0(A) is local Artinian, and each Hn(A) is ﬁnite dimensional as a vector space over k.

54 2.8.0.2. Deﬁnition of a formal moduli problem. The main deﬁnition is functor-of- points style: The ∞-category of derived deformation problems or (more commonly) pre-formal moduli problems is given by the ∞-category of functors X : dArt → Sp such that X(k) ∼= pt (i.e. we are ”deforming a single object“). The ∞-category FMP of formal moduli problems is given by the ∞-category of derived deformation problems that satisfy the following gluing condition: Given a pullback square determined by morphisms B → A and C → A in dArt that are surjective on H0, the induced morphism of spaces X(B ×A C) → X(B) ×X(A) X(C) is an equivalence.

2.8.0.3. Completion. Given a derived prestack X pointed by a morphism x : Spec(k) →

X we can study X near x by taking the completion Xbx of X at x, which is the associated derived deformation problem Xb(A) = X(A) ×X(k) {x} of moduli of points in a formal neighbourhood of x. If X is Artin this is a formal moduli problem. See also 2.2.2.7.

2.8.0.4. Examples. Typical examples of derived deformation problems are given by functors describing deformations of objects of geometric or algebraic nature, such as schemes, vector bundles, G-bundles, local systems, (homotopical) algebras over various operads ([137]), etc. One case important to us is the derived deformation problem DefAlgB associated to an associative dg-algebra B. The space of A-points

DefAlgB(A) classiﬁes associative A-deformation of B, i.e. associative A-dg-algebras ∼ BA together with an identiﬁcation BA ⊗A k = B. Given a morphism of Artinian

0 0 0 cdga’s A → A we get an A -deformation by considering BA0 = BA ⊗A A .

55 2.8.0.5. Associated formal moduli problem. The inclusion of derived deformation problems into formal moduli problems admits a right adjoint (−)∧, which may be thought of as a sort of sheafification. Sometimes a derived deformation problem X is not quite a formal moduli problem, but close enough in the following sense. We say that X is n-proximate if the unit of the adjunction above X → X∧ is (n − 2)- truncated i.e. X(A) → X∧(A) has (n − 2)-connective fiber for every Artinian cdga A. This is equivalent to the pointwise defined n-fold based loops ΩnX being a formal moduli problem.

2.8.0.6. Tangent complex. The sequence {k ⊕ k[n]}, n ≥ 0 of square zero extensions ∼ ∼ of k forms a spectrum object in dArt, i.e. k⊕k[n−1] = Ω(k⊕k[n]) = k×k⊕k[n] k, and the sequence X(k ⊕ k[n]) then forms k-module spectrum, equivalently a complex, which is called the tangent complex of X. This notion agrees with the usual notion of tangent complex for completions of derived prestacks.

2.8.0.7. The main theorem on derived deformation theory. The main deep and beautiful theorem of derived deformation theory due to Lurie [81], but with a long and distinguished history preceding it, is:

Theorem 2.8.1. There is an equivalence between the ∞-categories of formal moduli problems and dg-Lie (or L∞) algebras.

This theorem reﬂects the Koszul duality between the commutative and Lie operads. In one direction, the equivalence roughly associates to a formal moduli

56 ∼ problem the derived Lie algebra TΩX = TX [−1] associated to the group object ΩX in formal moduli problems. If X = Yby arises as the completion of a pointed derived

Artin stack {y} ,→ Y , this is the Atiyah bracket structure on TY,y[−1]. In the other directions in good cases the inverse equivalence can be described by associating to a derived Lie algebra L the formal moduli problem XL whose A-points for an Artinian

0 cdga A with maximal ideal mA ∈ H (A) are an enhancement to a space of the usual set of Maurer-Cartan solutions of the dg-Lie algebra L ⊗k mA. One can make sense of the ∞-category of ind-coherent sheaves on a formal moduli problem X. By the main theorem above there must be a description of this ∞-category in terms of the corresponding dg-Lie algebra L, and it is (dg-linear) representations of L: IndCoh(X) ∼= Rep(L).

2.8.0.8. Deforming associative algebras. Let B be a associative dg-algebra B. The derived deformation problem DefAlgB of associative dg-algebra deformations of B is 1-proximate. There is thus a best approximation by a formal moduli problem

∧ DefAlgB which is controlled by the dg-Lie algebra of derived derivations Der(B). Notice that deforming B is equivalent to deforming the pointed ∞-category (B,B − mod) (i.e. deforming both the pointing and the category).

2.8.0.9. Deforming ∞-categories. k-linear ∞-category C defines a derived deformation problem DefCatC whose space of A-points for connective cdga A classifies deformations over A, i.e. A-linear ∞-categories CA together with an identification ∼ CA ⊗A k = C. The derived deformation problem DefCatC is 2-proximate. There is

57 ∧ thus again a best approximation by a formal moduli problem DefCatC which is controlled by the dg-Lie algebra of shifted Hochschild cochains with the Gerstenhaber bracket HH∗(C)[1] [100]. We can consider this for C = B − mod, and this gives us another derived deformation problem we can associate to B.

2.8.0.10. Relating deformations of algebras and of module categories. Taking ∞- categories of modules deﬁnes a morphism of derived deformation problems DefAlgB

→ DefCatB−mod, thus of the associated formal moduli problems. One may equivalently think of DefAlgB as the derived deformation problem of the pointed category (B,B − mod), and we see then that the fiber is the derived deformation problem describing deformations of the pointing, i.e. of B as an object of B − mod. The fiber is controlled by the dg-Lie algebra B (with Lie bracket given by the commutator). This fiber sequence corresponds to the fiber sequence of dg-Lie algebras B → Der(B) → HH∗(B)[1] ([43]). Noting that HH∗(B)[1] is (the direct product totalization of the

Q ⊗n bigraded by internal and Hochschild degrees object ) n≥0 Homk(B ,B)[−n + 1], we can identify Der(B) with Hochschild cochains truncated into Hochschild degrees

∼ Q ⊗n ∗ greater than or equal to 1: Der(B) = n≥1 Homk(B ,B)[−n + 1] := HH+(B)[1] . This reﬂects that the homotopy theory of associated dg-algebras is equivalent to the homotopy theory of A∞-algebras, where we can deform k-ary multiplications by Hochschild cochains in Hochschild degree k. The morphism Der(B) → HH∗(B)[1]

∧ ∧ of dg-Lie algebras, and thus the morphism DefAlgB → DefCatB−mod of formal moduli problems, then corresponds to the inclusion of the truncated into the full

58 shifted Hochschild cohomology

∗ ∗ HH+(B)[1] → HH (B)[1].

2.9 Koszul duality

Koszul duality in its simplest form is a Morita/Fourier style equivalence (”duality“) of module categories for symmetric and exterior algebras (see [24] relating to representation theory, and [23] relating to algebraic geometry), and generalizes to certain quadratic10 algebras and ”dual“ (co)algebras. There are many different variants, for example depending on different weight or cohomological gradings (see for example [136]), or for En-algebras [81]). Usually one does not quite get an equivalence of module categories, and one needs to modify the categories slightly, e.g. by considering completions or finiteness conditions. There is also a version for quadratic operads [52], for example the commutative and Lie operads are Koszul dual. The associative operad is self-dual. Koszul duality plays an important role in representation theory [24], in rational homotopy theory relating the Quillen (Lie) and Sullivan (commutative) models for simply connected pointed rational topological spaces, and in derived deformation theory (2.8 and [81, 85, 84, 121, 31]). There is also a generalization of both Koszul duality relating to topological field theories through factorization algebras [10]

10A graded algebra is quadratic if it is generated by its degree 1 vector subspace with relations in degree 2, i.e. if it is of the form ⊕V ⊗n/(R) where R ⊂ V ⊗ V .

59 2.9.0.1. Fiberwise version of Koszul duality. Let us first explain a version of the classical Koszul duality we need. This will become the fiberwise version of the Koszul duality we will be doing later in families. Geometrically it arises by comonadic descent from including the origin into a finite dimensional vector space V ∗[2] shifted

∼ • into degree 2. Consider the symmetric cdga SV := OV ∗[2] = Symk(V [−2]) of func- ∗ • tions on V [2] with the natural augmentation Symk(V [−2]) → k corresponding to the inclusion of the origin. We have an adjoint pair of pull-push/tensor-forget func- tors1112

(−) ⊗SV k : SV − mod k − mod : forget, where we think of the left adjoint as restriction to the origin, and we can identify the

comodules over the corresponding comonad as comodules for the coalgebra k ⊗SV k. ∼ • Computing the coalgebra k ⊗SV k using the Koszul resolution k = Symk(V [−2] ⊕ ∼ • V [−1]) = SV ⊗Symk(V [−1]), where the diﬀerential maps the generating vector space V [−1] to V [−2] by the identity, we ﬁnd that

∼ • k ⊗SV k = (SV ⊗ Symk(V [−1])) ⊗SV k

∼ • = Symk(V [−1])), where the diﬀerential is now zero. This is an equivalence in V ect, and one checks easily that the coalgebra structures correspond. Dualizing this coalgebra we get the

11 Recall that by our conventions e.g. SV − mod means the (k-linear stable) ∞-category (or dg category) of unbounded dg-modules with quasi-isomorphisms inverted. 12 Note that since SV is coconnective, Spec(SV ) is a coaﬃne stack, and so SV − mod is not equivalent to QCoh(Spec(SV )). There is an obvious functor SV − mod → QCoh(Spec(SV )) which induces an equivalence on coconnective objects. For a precise comparison of the two ∞-categories, see [80][4.5]. We only care about the ∞-category of modules.

60 • ∗ 13 exterior algebra Symk(V [1]), which we will denote by ΛV . We can then dualize

• the k-linear stable ∞-category of Symk(V [−1])-comodules to the k-linear stable ∞- ∼ category of ΛV -modules. Note that we also have HomΛV (k, k) = SV (see below for this calculation in the relative case). We now have the adjunction

(−) ⊗SV k : SV − mod ΛV − mod : HomΛV (k, −).

This is not quite monadic, since the augmentation ΛV -module k is a generator, but not compact/perfect. We remedy this by noting that k is however coherent, so that it generates IndCoh(ΛV ). We thus have the Koszul duality equivalence

∼ IndCoh(ΛV ) = SV − mod.

2.9.0.2. Koszul duality in families. We can do the same thing in familes:

Theorem 2.9.1 (Koszul duality in families over a smooth scheme). Let X be a smooth Noetherian classical scheme, and E be a ﬁnite rank vector bundle over X. Consider the (total space of) the (−1)-shifted bundle E[−1] of) as a derived scheme Spec (Sym• (E∗[1])). Then we have an equivalence X OX

IndCoh(E[−1]) ∼ Sym• (E[−2]) − mod. = OX

This is essentially the same argument as above. Let us ﬁrst record that by i we denote the inclusion of the zero sesction i : X → E[−1].

13Note that we can ignore the completions for symmetric algebras on (dg-)vector spaces concentrated in odd degree.

61 IndCoh(X) Proof. We have that i∗ OX is compact since it is coherent, and a generator for IndCoh(E[−1]) since any nonzero Ind-coherent sheaf has to be supported on X (which is the underlying classical scheme of E[−1]). By the compatibility of

IndCoh(X) pushforwards for IndCoh and Coh we have i∗ OX = i∗OX . Resolving i∗OX by the Koszul resolution i O ∼ Sym• (E∗[1] ⊕ E∗[2]), where the diﬀerential on ∗ X = OX the E∗[2] part kills the E∗[1] part (i.e is the identity), we can compute the internal/ sheafy Hom-algebra:

∼ • ∗ ∗ HomSym• (E∗[1])(i∗OX , i∗OX ) = HomSym• (E∗[1])(SymO (E [1] ⊕ E [2]), i∗OX ) OX OX X ∼ • ∗ = HomSym• (E∗[1])(SymO (E [1])⊗ OX X Sym• (E∗[2]), i O ) OX ∗ X ∼ Hom (Sym• (E∗[2]), i O ) = OX OX ∗ X ∼ = SymOX (E[−2]).

One easily checks that the algebra structures agree.

2.9.0.3. Koszul duality from topology. One can get Koszul duality from the topology of the circle. One story goes as follows: For G a connected homotopical group we get an equivalence of the ∞-categories of spaces with homotopical G-action and spaces over BG. Passing to cochains, we get an equivalence of C∗(G; k) − comod ∼= C∗(BG; k)−mod, and under this equivalence ordinary cohomology with its C∗(G; k)- comodule structure corresponds to equivariant cohomology ([54]). We can dualize the comodule category, but need to be careful about ﬁniteness issues. Goresky- Kottwitz-MacPherson ([54]) prove that we have an equivalence given by an adjoint

62 pair

∗ ∗ perf f.g. (−) ⊗ C (EG): C (BG) − mod C∗(G) − mod : HomC∗(G)(C∗(EG), −) C∗(BG)

Taking G to be S1, and passing to Ind-completions, we get Koszul duality. Note that we can interpret C∗(G; k) as a sort of homotopical group algebra, so that we can

fin.gen. interpret C∗(G; k) − mod as (ﬁnitely generated) homotopical representations. On the other hand, C∗(BG; k) is (homotopical) functions on BG, so if BG is aﬃne in some sense we can think of C∗(BG; k) − modperf as families of (homotopical/dg)- modules on BG, i.e. local systems.

2.10 Free derived loop spaces

Here we collect some results on free derived loop spaces ([19, 20, 126], see also 2.7).

2.10.0.1. Deﬁnition and basic properties. Let X be a derived prestack. The free (derived) loop space LX of X is the derived mapping prestack Maps (S1,X). dSt Since we can present the circle S1 as a pushout of two points along two arcs connecting them, we ﬁnd that the free loop space is the self-intersection of the diagonal

a LX ∼ Maps (pt pt, X) = dSt pt ` pt ∼ = X ×X×X X

If X is a derived stack (or derived Artin stack, or derived scheme), then so is

63 LX.

2.10.0.2. Hochschild-Kostant-Rosenberg as linearization. By a result of [20] we have that the following generalization of the classical Hochschild-Kostant-Rosenberg theorem holds (see 2.7):

Theorem 2.10.1. Let X be a quasi-compact derived scheme with aﬃne diagonal. Then the free loop space LX is isomorphic to the (total space of) the (−1)-shifted tangent complex TX [−1].

If X is stacky, this breaks down, because the free derived loop space does not see ”big“ loops. Nonetheless we have ([20]):

Theorem 2.10.2. Let X be a derived Artin stack with aﬃne diagonal. Then the completion LdXX of LX along constant loops X free loop space LX is isomorphic to the completion of TX [−1] along the zero section T\X [−1]X .

2.10.0.3. The circle action. The free loop space LX = Maps (S1,X) of a derived dSt stack X aquires a natural S1-action by precomposition with the natural action of S1 on itself. Thus we get an induced S1-action on IndCoh(LX). Explicitly, g ∈ S1 acts on an Ind-coherent sheaf F via g ·F = g!(F). Note that g! is continuous, so this really deﬁnes an action. For derived schemes the S1-action factors through the

1 aﬃnization Aff(S ) = BGa.

We see that this is the right notion by analogy with the case of functions. In any reasonable geometric setting, given a space with an action by a group G we get

64 an action on functions by pulling back. Thus when we go one categorical level up to the ∞-category IndCoh we should act via (!-)pullbacks as well.

2.10.0.4. The framed E2-structure on IndCoh(LX). As explained in the introduction by the (∞, 2)-categorical functoriality of IndCoh under spans IndCoh(LX) actually inherits a framed-E2 algebra structure subsuming both the circle action and an E2- structure.

2.10.0.5. Koszul duality for free derived loop spaces. Let X be a smooth classical scheme, that is furthermore Noetherian with aﬃne diagonal. Let LX be its free derived loop space, and let S denote the relative symmetric algebra Sym• ( [−2]) X OX TX ∼ over X. Taking E = TX in 2.9.1, and use that LX = TX [−1], we ﬁnd that we have a Koszul duality equivalence of (presentable k-linear stable) ∞-categories

∼ IndCoh(LX) = SX − mod

2.10.0.6. Brief reminder on the algebra of differental operators DX . Let X be a smooth classical scheme. Then the sheaf of differential operators DX on X is the k subalgebra of Homk(OX , OX ) generated by OX and TX thought of as the sheaf of derivations TX = Derk(OX ). If X = Spec(A) is affine, then Γ(X; DX ) is isomorphic

to the algebra hA, TAi/(∂1 · ∂2 − ∂2 · ∂1 = [∂1, ∂2]TX , ∂ · a − a · ∂ = ∂(a)), where

∂1, ∂2, ∂ ∈ TX , a ∈ A, the bracket [−, −]TX is the commutator of derivations, and by hA, TAi we denote the noncommutative algebra generated by symbols in A and TA

≤n subject to the relations in A. This algebra has a ﬁltration DX given by the order

65 of the diﬀerential operator (i.e. in the above local description for X = Spec(A) we count the number of occurences of symbols in TA). The associated graded algebra is

• Sym (TX ) = π∗(OT ∗X ).

2.10.0.7. Circle Invariants of IndCoh(LX). Taking S1-invariants of IndCoh(LX),

∗ 1 ∼ we get a family of ∞-categories over the coaffine stack Spec(H (BS ; k)) = Spec(k[~]) ∼ 1 ∗ 1 = A [2], where ~ is of degre 2. Note that Spec(H (BS ; k)) can also be thought of S1 as BBGa. If X is a scheme then the ∞-category of invariants IndCoh(LX) is equivalent to the category of (quasicoherent sheaves of) modules for the DX,~ − mod, where DX,~ − mod is the sheaf of cdga’s on X given by the graded Rees algebra ≤n n DX,~ = ⊕nDX · ~ of differential operators on X with Rees parameter ~ of degree 2 equipped with the zero differential. We can also for X = Spec(A) affine describe

DX,~ explicitly as hA, TA, ~i/(∂1 · ∂2 − ∂2 · ∂1 = ~ · [∂1, ∂2]TX , ∂ · a − a · ∂ = ~ · ∂(a)), where we think of ~ as central, i.e. commuting with everything. This equivalence may be viewed as an instance of the nonhomogenous Koszul duality of [98]. Turning on S1-equivariance, i.e. the de-Rham diﬀerential, corresponds on the Koszul dual side to considering the ﬁltered noncommutative deformation of SX − mod given by D-modules.

66 Chapter 3

The main theorem

Theorem 3.0.1. Let X = Spec(A) be a smooth aﬃne scheme. Then S1-action on the Koszul dual category SX − mod of IndCoh(LX) is the given by the exponential

∗ of the 2-shifted Poisson structure on TX [2].

Proof. Since X is aﬃne, the S1 action LX is unipotent (2.6.2.12, 2.10.0.3), thus

1 so is the S -action on SX − mod. We are thus reduced to considering the BGa- action. The derivative of the action BGa-action is the derivative of the family of

BGa ∼ ∞-categories SX − mod over BBGa = Spec(k[~]), with ~ of degree 2 (2.6.2.10). By the Koszul duality equivalence (2.10.0.5), this family is equivalent to the family

IndCoh(LX)BGa , and by 2.10.0.7 this is equivalent to the family of ∞-categories

∼ ≤i i of modules over the dg-algebra that is the Rees algebra DX,~ = ⊕i≥0D (X) · ~ of diﬀerential operators on X with Rees parameter ~ of degree 2. Since the family of ∞-categories is a family of module ∞-categories arising from a family of (derived) associative algebras, the derivative/Kodaira-Spencer deformation class of this family is given by the Hochschild deformation class of the family of (derived) associative algebras (2.8.0.10) DX,~. This class {−, −} : A ⊗ A → A is given on generators

67 a ∈ A, ∂1, ∂2, ∂ ∈ TA by

{a, b} = 0

{∂1, ∂2} = [∂1, ∂2]TA

{∂, a} = ∂(a).

∗ We see that this is just the canonical 2-shifted (i.e. P3) Poisson bracket on TX [2]. Note that the bracket has degree −2 so as a Hochschild class A ⊗ A → A it has total degree 0 as desired. Thus the derivative of the action is the Poisson bracket {−, −} so that by 2.6.2.14 the S1-action is given by the exponential of {−, −}.

68 Bibliography

[1] Hossein Abbaspour, On algebraic structures of the Hochschild complex, 2013, arxiv1302.6534 [math.AT].

[2] Alberto Abbondandolo and Matthias Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006), no. 2, 254316. arxiv:0408280 [math.SG].

[3] Mohammed Abouzaid, Symplectic cohomology and Viterbo’s theorem, 2013, arxiv:1312.3354 [math.SG].

[4] Araminta Amabel, Artem Kalmykov, Lukas M¨uller,and Hiro Lee Tanaka, Lectures on Factorization Homology, Inﬁnity-Categories, and Topological Field Theories, 2019, arxiv:1907.00066 [math.AT].

[5] Michel Andr´e, Homologie des Alg`ebres Commutatives, Grundlehren der mathema- tischen Wissenschaften, 206, Springer-Verlag (1974).

[6] Mathieu Anel, The Geometry of Ambiguity: An introduction to the ideas of De- rived Geometry, 2016, available at http://mathieu.anel.free.fr/mat/doc/Anel- DerivedGeometry.pdf.

[7] Dima Arinkin and Dennis Gaitsgory, Singular support of coherent sheaves, and the geometric Langlands conjecture, 2012, arxiv:1201.6343 [math.AG].

69 [8] Michael Atiyah, Circular symmetry and stationary-phase approximation, Colloque en l’honneur de Laurent Schwartz - Volume 1, Ast´erisque,no. 131 (1985), p. 43-59.

[9] Denis Aurox, A beginner’s introduction to Fukaya categories, 2013, arxiv:1301.7056 [math.SG].

[10] David Ayala and John Francis, Poincar´e/Koszul duality, 2014, arxiv:1409.2478 [math.AT].

[11] David Ayala and John Francis, A factorization homology primer, 2019, arxiv:1903.10961 [math.AT].

[12] John Baez and James Dolan, Higher dimensional algebra and Topological Quantum Field Theory, J.Math.Phys. 36 (1995), p. 6073-6105, arXiv:9503002 [math.QA].

[13] Dario Beraldo, The center of H(Y), 2017, arxiv:1709.07867v3 [math.AG].

[14] Dario Beraldo, Sheaves of categories with local actions of Hochschild cochains, 2018, arxiv:1801.03752v3 [math.AG].

[15] Dario Beraldo, The topological chiral homology of the spherical category, 2018, arxiv:1802.08118v3 [math.AG].

[16] Julia Bergner, A survey of (∞, 1)-categories, 2006, arxiv:0610239 [math.AT].

[17] Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, and Dominic Joyce, A ’Dar- boux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications, 2013, arxiv:1312.0090 [math.AG].

70 [18] David Ben-Zvi, John Francis, and David Nadler, Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, 2008, arxiv:0805.0157 [math.AG].

[19] David Ben-Zvi and David Nadler, Loop Spaces and Langlands Parameters, 2007, arxiv:0706.0322 [math.RT].

[20] David Ben-Zvi and David Nadler, Loop Spaces and Connections, 2010, arxiv:1002.3636 [math.AG].

[21] David Ben-Zvi and David Nadler, Nonlinear Traces, 2013, arxiv:1305.7175 [math.AG].

[22] David Ben-Zvi and Pavel Safronov, Higher diﬀerential operators and Algebroids, in preparation.

[23] Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, Algebraic vector bundles on

Pn and problems of linear algebra, Funct. Analysis, 12 (No. 3)(1978), pp.66-67.

[24] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul Duality Pat- terns in Representation Theory, J. Amer. Math. Soc. 9 (1996), 473-527.

[25] Anthony Blanc, Ludmil Katzarkov, Pranav Pandit, Generators in formal deformations of categories, 2017, arxiv:1705.00655 [math.AG].

[26] Andrew Blumberg, Ralph Cohen, Constantin Teleman, Open-closed ﬁeld theories, string topology, and Hochschild homology, 2009, arxiv:0906.5198 [math.AT].

71 [27] Tom Braden, Anthony Licata, Nicholas Proudfoot, Ben Webster, Quantizations of conical symplectic resolutions, Societé Mathématique de France, 2016, Astérisque384.

[28] Damien Calaque and Michel Van den Bergh, Hochschild cohomology and Atiyah classes, 2007, arxiv:[math.AG].

[29] Damien Calaque, Three lectures on derived symplectic geometry and topological ﬁeld theories, Indagationes Mathematicae Volume 25, Issue 5, October 2014, Pages 926-947.

[30] Damien Calaque, Shifted cotangent stacks are shifted symplectic, 2016, arxiv:1612.08101 [math.AG].

[31] Damien Calaque and Julien Grivaux, Formal Moduli Problems and Formal Derived Stacks, 2018, arxiv:1802.09556v2 [math.AG].

[32] Damien Calaque, Tony Pantev, Bertrand To¨en,Michel Vaqui´e,and Gabrielle Vez- zosi, Shifted Poisson Structures and Deformation Quantization, J. of Topology, Volume 10, Issue 2 June 2017 Pages 483 584, arxiv:1506.03699 [math.AG].

[33] Omar Antol`ın Camarena, A Whirlwind Tour of (∞, 1)-categories, 2013, arxiv:1303.4669v3 [math.CT].

[34] Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, Berlin (2001).

[35] Moira Chas, Dennis Sullivan, String Topology, 1999, arxiv:9911159v1 [math.AT].

72 [36] Harrison I-Yuan Chen, A Localization Theorem for Derived Loop Spaces and Periodic Cyclic Homology, PhD thesis, UC Berkeley, 2018, available at https://escholarship.org/uc/item/05m960s2.

[37] Ralph Cohen, John Jones, A homotopy theoretic realization of string topology, 2002, arxiv:0107187 [math.AT].

[38] Ralph Cohen, Alexander Voronov, Notes on string topology, 2005, arxiv:0503625 [math.AT].

[39] Lee Cohn, Diﬀerential Graded Categories are k-linear Stable Inﬁnity Categories, 2016, arxiv:1308.2587 [math.AT].

[40] Kevin Costello and Owen Gwilliam, Factorization algebras in quantum ﬁeld theory, vol. 1, Cambridge University Press, 2016.

[41] Kevin Costello and Owen Gwilliam, Factorization algebras in quantum ﬁeld theory, vol. 2, to appear, http://people.math.umass.edu/ gwilliam/vol2may8.pdf.

[42] William Dwyer, Jan Spali´nski, Homotopy theories and model categories, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology (1995).

[43] John Francis, The tangent complex and Hochschild cohomology of En-rings, 2011, arxiv:1104.0181 [math.AT].

[44] Daniel Freed, The cobordism hypothesis, 2012, arxiv:1210.5100v2 [math.AT].

[45] Benoit Fresse, Little discs operads, graph complexes and Grothendieck-Teichm¨uller groups, 2018, arxiv:1811.12536 [math.AT].

73 [46] Baohua Fu, A survey on symplectic singularities and resolutions, Annales math´ematiques Blaise Pascal, Volume 13 (2006) no. 2, p. 209-236, arxiv:0510346v1 [math.AG].

[47] Dennis Gaitsgory Ind-coherent sheaves, 2011, arxiv:1105.4857 [math.AG].

[48] Dennis Gaitsgory, Sheaves of categories and the notion of 1-aﬃneness, 2013, arxiv:1306.4304v4 [math.AG].

[49] Dima Gaitsgory and Nick Rozenblyum, A study in derived algebraic geometry. Vol. I. Correspondences and duality, volume 221 of Mathematical Surveys and Mono- graphs. American Mathematical Society, Providence, RI, 2017.

[50] Dima Gaitsgory and Nick Rozenblyum, A study in derived algebraic geometry. Vol. II. Deformations, Lie theory and formal geometry, volume 221 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2017.

[51] Victor Ginzburg, Lectures on Noncommutative Geometry, 2005, arxiv:0506603v1 [math.AG].

[52] Victor Ginzburg, Mikhail Kapranov, Koszul duality for operads, Duke Math. J., Vol- ume 76, Number 1 (1994), pp 203-272. Available at arxiv:0709.1228 [math.AG].

[53] Paul Goerss, Kristen Schemmerhorn, Model Categories and Simplicial Methods, 2006, arxiv0609537 [math.AT].

74 [54] Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. math. 131, 25-83 (1998).

[55] Moritz Groth, A short course on ∞-categories, 2010, arxiv:1007.2925 [math.AT].

[56] Vladimir Hinich, Homological algebra of homotopy algebras, 1997, arxiv:9702015 [math.QA].

[57] Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002).

[58] Mark Hovey, Model Categories Mathematical Surveys and Monographs, Volume 63, AMS (1999).

[59] Mark Hovey, The Eilenberg-Watts theorem in homotopical algebra, 2009, arxiv:0910.3842 [math.AT].

[60] Luc Illusie, Complexe cotangent et d´eformationsI, Lecture Notes Math. 239, Springer (1971).

[61] Srikanth Iyengar, Andre-Quillen homology of commutative algebras, 2006, arxiv:0609151 [math.AC].

[62] John Jones, Cyclic homology and equivariant homology, Invent. Math. 87, pp. 403- 423 (1987).

[63] Andr´eJoyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra, 175 (2002), 207-222.

75 [64] Victor Kac, Inﬁnite dimensional Lie algebras, Cambridge University Press, (1990).

[65] Anton Kapustin, Topological Field Theory, Higher Categories, and Their Applica- tions, 2010, arxiv:1004.2307 [math.QA].

[66] Anton Kapustin, Lev Rozansky, Natalia Saulina, Three-dimensional topological ﬁeld theory and symplectic algebraic geometry I, 2008, arxiv:0810.5415 [hep-th].

[67] Anton Kapustin, Lev Rozansky, Three-dimensional topological ﬁeld theory and symplectic algebraic geometry II, 2009, arxiv:0909.3643 [math.AG].

[68] Anton Kapustin, Natalia Saulina, Chern-Simons-Rozansky-Witten topological ﬁeld theory, 2009, arxiv:0904.1447 [hep-th].

[69] Mikhail Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71-113. arxiv:9704009 [math.AG].

[70] Bernhard Keller, Deriving DG-categories, Annales scientiﬁques de l’Ecole´ Normale Sup´erieure,Serie 4, Volume 27 (1994) no. 1, pp. 63-102.

[71] Bernhard Keller, On diﬀerential graded categories, 2006, arxiv:0601185 [math.KT].

[72] Maxim Kontsevich, Deformation quantization of Poisson manifolds, I, Letters in Mathematical Physics 66, p. 157216, 2003, available at arxiv:9709040 [math.QA].

[73] Maxim Kontsevich, Operads and Motives in Deformation Quantization, 1999, arxiv:9904055 [math.QA].

76 [74] Maxim Kontsevich and Yan Soibelman, Deformation Theory, I, 1991, https://www.math.ksu.edu/ soibel/Book-vol1.ps.

[75] Henning Krause, The stable derived category of a Noetherian scheme, 2004, arxiv:0403526 [math.AG].

[76] Pascal Lambrechts, Ismar Voli´c, Formality of the little N-disks operad, 2008, arxiv:0808.0457 [math.AT].

[77] Janko Latschev and Alexandru Oancea (eds.), Free Loop Spaces in Geometry and Topology, 2015, IRMA Lectures in Math. and Th. Physics, Vol. 24, EMS.

[78] Jean-Luis Loday, Free loop space and homology, 2011, arxiv:1110.0405 [math.AT].

[79] Jacob Lurie, Derived Algebraic Geometry, PhD thesis, MIT, 2004, available here: http://www.math.harvard.edu/ lurie/papers/DAG.pdf.

[80] Jacob Lurie, Derived Algebraic Geometry XVII: Quasi- Coherent Sheaves and Tannaka Duality Theorems, 2011, http://www.math.harvard.edu/ lurie/papers/DAG-VIII.pdf.

[81] Jacob Lurie, Derived Algebraic Geometry X: Formal Moduli Problems, 2011, available at http://www.math.harvard.edu/ lurie/papers/DAG-X.pdf.

[82] Jacob Lurie, Higher Algebra, 2017, available online at http://www.math.harvard.edu/ lurie/papers/HA.pdf.

77 [83] Jacob Lurie, Higher Topos Theory, Annals of Mathematics Stud- ies 170, Princeton University Press (2009), available at http://www.math.harvard.edu/ lurie/papers/HTT.pdf.

[84] Jacob Lurie, Moduli Problems for Ring Spectra, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), pp. 1099-1125 (2011), available at http://www.math.harvard.edu/ lurie/papers/moduli.pdf.

[85] Jacob Lurie, Spectral Algebraic Geometry, preprint available at http://math.harvard.edu/ lurie/papers/SAG-rootﬁle.pdf, Feb. 2018.

[86] Jacob Lurie, On the Classiﬁcation of Topological Field Theories, Current Devel- opments in Mathematics, Volume 2008 (2009), p. 129-280, arxiv:0905.0465 [math.AT].

[87] Marco Manetti, Deformation theory via diﬀerential graded Lie algebras, 2005, arxiv:0507284 [math.AG].

[88] Ciprian Manolescu, Floer theory and its topological applications, 2015, arxiv:1508.00495 [math.GT].

[89] Aaron Mazel-Gee, The Zen of ∞-categories, introductory chapter of PhD thesis, https://etale.site/writing/zen-of-infty-cats.pdf.

[90] Dusa McDuﬀ and Dietmar Salamon, Introduction to Symplectic Topology, Oxford University Press (2017).

78 [91] Valerio Melani, Poisson bivectors and Poisson brackets on aﬃne derived stacks, Adv.Math. 288 (2016), pp. 10971120, available at arXiv:1409.1863 [math.AG].

[92] Gregory Moore, Lecture Notes for Felix Klein Lectures, 2012, https://www.physics.rutgers.edu/ gmoore/FelixKleinLectureNotes.pdf.

[93] Answers to https://mathoverﬂow.net/questions/28472/book-on-hochschild- cohomology.

[94] Thomas Nikolaus, Urs Schreiber, and Danny Stevenson Principal ∞-bundles - theory, presentations and applications, Journal of Homotopy and Related Struc- tures, June 2014 (arXiv:1207.0248 [math.AT]).

[95] Alexandru Oancea, Morse theory, closed geodesics, and the homology of the free loop spaces, 2014, arxiv:1406.3107 [math.DG].

[96] Tony Pantev, Bertrand To¨en,Michel Vaqui´e,and Gabrielle Vezzosi, Shifted Symplec- tic Structures, Publ. Math. IHES, June 2013, Volume 117, Issue 1, pp 271-328, available at arxiv:1111.3209 [math.AG].

[97] Tony Pantev, Gabrielle Vezzosi, Symplectic and Poisson derived geometry and deformation quantization, 2016, arxiv:1603.02753 [math.AG].

[98] Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series 37, American Mathematical Society, Providence, RI, 2005.

[99] Andrew Pressley, Graeme Segal, Loop groups, Oxford University Press (1988).

79 [100] Anatoly Preygel, Thom-Sebastiani and Duality for Matrix Factorizations, and Re- sults on the Higher Structures of the Hochschild Invariants, PhD thesis, MIT, 2012, available at https://dspace.mit.edu/handle/1721.1/73373.

[101] Jon Pridham, Shifted Poisson and symplectic structures on derived N-stacks, 2015, arxiv:1504.01940 [math.AG].

[102] Daniel Quillen, Axiomatic homotopy theory in Homotopical algebra, Lecture Notes in Mathematics, No. 43, Berlin (1967).

[103] Daniel Quillen, On the (co-)homology of commutative rings , Proc. Symp. on Categorical Algebra , (1970), 6587, American Math. Soc.

[104] Daniel Quillen, Homology of commutative rings, unpublished notes.

[105] Justin Roberts, Rozansky-Witten theory, 2001, arxiv:0112209 [math.QA].

[106] Justin Roberts, Simon Willerton, On the Rozansky-Witten weight systems, 2006, arxiv:0602653 [math.DG].

[107] Lev Rozansky, Edward Witten, Hyper-Khler geometry and invariants of 3- manifolds, Selecta Math., New Ser. 3 (1997), 401-458. arxiv:9612216 [hep-th].

[108] Pavel Safronov, Lectures on shifted Poisson geometry, 2017, arxiv:1709.07698 [math.AG].

[109] Dietmar Salamon and Joa Weber, Floer homology and the heat ﬂow, Geom. Funct. Anal. 16 (2006), no. 5, 1050-1138. arxiv:0304383 [math.SG].

80 [110] Paolo Salvatore and Nathalie Wahl, Framed discs operads and Batalin-Vilkovisky algebras, Q. J. Math., 2003, 54, p. 213-231.

[111] Travis Schedler, Deformations of Algebras in Non-commutative Algebraic Ge- ometry, book chapter in “Noncommutative Algebraic Geometry” (2016) by Gwyn Bellamy et al, MSRI Publications 64), CUP, available online at http://math.mit.edu/ trasched/deformations.pdf.

[112] Stefan Schwede, Morita theory in abelian, derived and stable model categories, 2003, arxiv:0310146 [math.AT].

[113] Stefan Schwede and Brooke Shipley, Classiﬁcation of stable model categories, 2001, arxiv:0108143 [math.AT].

[114] Brooke Shipley, Morita theory in stable homotopy theory, Handbook of Tilting Theory (eds. Angeleri Huegel, Happel, Krause) London Math. Soc. Lecture Notes Series. 332 (2007) 393-409, available at http://www.math.uic.edu/ bship- ley/tilting7.pdf.

[115] Dev Sinha, The homology of the little disks operad, 2006, arxiv:0610236 [math.AT].

[116] Ivan Smith, A symplectic prolegomenon, 2014, arxiv:1401.0269 [math.SG].

[117] Andrew Strominger, Open p-branes, Phys.Lett. B383 (1996), pp. 44-47, arXiv:9512059 [hep-th].

[118] Dmitry Tamarkin, Another proof of M. Kontsevich formality theorem, 1998, arxiv:9803025 [math.QA].

81 [119] Bertrand To¨en, Simplicial presheaves and derived algebraic geometry, in Simplicial Methods for Operads and Algebraic Geometry, Springer, Basel, pp.119-185, 2010.

[120] Bertrand To¨en, Derived Algebraic Geometry, EMS Surv. Math. Sci. 1 (2014), no. 2, 153-245, available at arxiv:1401.1044 [math.AG].

[121] Bertrand To¨en, Probl`emesde modules formels. Expos˜eBourbaki 2016.

[122] Bertrand To¨enand Gabrielle Vezzosi, Homotopical Algebraic Geometry I: Topos theory, 2002, arxiv:0207028 [math.AG].

[123] Bertrand To¨enand Gabrielle Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications, 2004, arxiv:0404373v7 [math.AG].

[124] Bertrand To¨enand Gabrielle Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry, 2008, https://arxiv.org/abs/0804.1274 [math.AG].

[125] Bertrand Toënand Gabrielle Vezzosi, Caractères de Chern, traces équivarianteset géométriealgébriquedérivée, 2009, arxiv:0903.3292 [math.AG].

[126] Bertrand To¨enand Gabrielle Vezzosi, S1-Equivariant simplicial algebras and de Rham theory, 2009, https://arxiv.org/abs/0904.3256 [math.AG].

[127] The Univalent Foundations Program, Homotopy Type Theory: Univalent Founda- tions of Mathetics, 2013, https://homotopytypetheory.org/book.

[128] Bruno Vallette, Algebra + Homotopy = Operad, 2012, arxiv:1202.3245 [math.AT].

82 [129] Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathe- matics, vol. 118, Birkh¨auser,Basel and Boston, 1994.

[130] Claude Viterbo, Generating functions, symplectic geometry, and applications, 1994 ICM address.

[131] Claude Viterbo Functors and computations in Floer homology with applications. I, Geom. Funct. Anal. 9 (1999), no. 5, 9851033.

[132] Edward Witten, The Index of the Dirac Operator in Loop Space, Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, N.J., Sep 1986.

[133] Edward Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109-525, 1987.

[134] Edward Witten, Some comments on string dynamics, Strings ’95 (Los Angeles, CA, 1995), World Sci. Publ., River Edge, NJ, 1996, pp. 501-523, arxiv:950712 [hep-th].

[135] Edward Witten, Geometric Langlands from six dimensions, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, 2010, pp. 281-310. arXiv:0905.2720 [hep-th].

[136] Michael Wong, A Geometric Approach to Hochschild Cohomology of the Exterior Algebra, 2016, arxiv:1607.02661 [math.KT].

83 [137] Sinan Yalin, Moduli spaces of (bi)algebra structures in topology and geometry, 2016, arxiv:1611.03662v2 [math.AT].

[138] Xinwen Zhu, An introduction to aﬃne Grassmannians and the geometric Satake equivalence

84 Vita

Rustam Darius Riedel, more commonly known as Rustam Darius Antia, re- ceived a Diplom degree in mathematics with a minor in theoretical physics from the Goethe University of Frankfurt am Main, Germany, and is/was a graduate student in mathematics at the University of Texas at Austin.

This dissertation was typeset with LATEX by the author.

LATEX is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEX Program.