Cluster-Like Coordinates in Supersymmetric Quantum Field Theory

Cluster-like coordinates in supersymmetric quantum SPECIAL FEATURE field theory

Andrew Neitzke1

Department of Mathematics, University of Texas at Austin, Austin, TX 78712

Edited by Lauren K. Williams, University of California, Berkeley, CA, and accepted by the Editorial Board April 30, 2014 (received for review November 20, 2013)

Recently it has become apparent that N = 2 supersymmetric quan- enlargement of the usual cluster story, and we are hopeful that it tum field theory has something to do with cluster algebras. I re- may be of some interest for the cluster community. view one aspect of the connection: supersymmetric quantum field The connection between N = 2 theories and the structures theories have associated hyperkähler moduli spaces, and these mod- studied in the cluster community goes well beyond what I can uli spaces carry a structure that looks like an extension of the notion describe here, and presumably there is much more still to be of cluster variety. In particular, one encounters the usual variables and discovered. For some work in the physics literature, see refs. 5–9. mutations of the cluster story, along with more exotic extra variables There is also another way in which cluster algebras have been and generalized mutations. I focusonaclassofexampleswherethe connected to supersymmetric quantum field theory, namely in underlying cluster varieties are moduli spaces of flat connections on perturbative scattering amplitudes in N = 4 theories (10, 11). I surfaces, as considered by Fock and Goncharov [Fock V, Goncharov A do not know any relation between that story and the one re- (2006) Publ Math Inst Hautes Études Sci 103:1–211]. The work viewed here, beyond the fact that cluster algebras show up in reviewed here is largely joint with Davide Gaiotto and Greg Moore. both places.

character varieties | Hitchin systems | supersymmetry 2. Expectations We begin with a brief rundown of the expectations that come from quantum field theory: given an N = 2 theory (whatever that 1. Introduction is), we expect to get all of the following objects and conditions. MATHEMATICS Supersymmetric quantum field theory has been a rich source of The relation to cluster algebras is most apparent in X5 and mathematical ideas and insights over the last few decades. This X6 below. paper is a review of one new facet of that story, which concerns a As we will see below, we expect in particular that we have a relation between N = 2 supersymmetric quantum field theories in hyperkähler space M, which, considered as a complex space, four dimensions, or N = 2 theories for short, and cluster algebras. looks very different depending on which complex structure we × Quantum field theory is far from a mathematically rigorous use. In any complex structure Jζ for ζ ∈ C , M has a structure subject. Nevertheless, the study of quantum field theory some- much like that of a cluster variety in the sense of ref. 4. Con- times leads to precise predictions about concrete mathematical versely, in complex structure J0, M looks very different: it is a objects. In particular, every N = 2 theory is expected to give rise complex integrable system, fibered by compact complex tori over to a family of moduli spaces M½R, parameterized by R > 0. a complex base B. From now on, we write M for any M½R. For the rest of this paper, we will mainly focus on the complex × M is expected to carry a number of interesting structures. For structures Jζ for ζ ∈ C , because these are the ones that make example, M is a complex integrable system, carrying a compati- direct contact with the cluster story. I emphasize, however, that – ble hyperkähler metric. M is a Calabi Yau space, which arises some of the most interesting applications arise from the interplay × naturally as a special Lagrangian fibration. This setup is precisely between the cluster-like spaces at ζ ∈ C and the complex in- the one introduced by Strominger et al. (1) for studying mirror ζ = 2.8 – tegrable system at 0, expressed by Eq. . For example, this symmetry of Calabi Yau manifolds. Moreover M is essentially interplay is the key to a proposed scheme for describing the self-mirror. hyperkähler metric on M (2, 12). In joint work of the author with Davide Gaiotto and Greg Moore The reader who knows a bit of quantum field theory should (2, 3), it was appreciated that quantum field theory suggests that read this section in parallel with section 5 below, where we give an M has another important structure, a collection of canonical indication of where these expectations come from. local Darboux coordinate systems fX γg. These coordinate systems are related to one another by transformations that gener- alize the behavior of cluster coefficients under mutation (see Eq. Significance 2.4). The main aim of this paper is to describe the expected properties of the functions X γ′ and some examples of their construction. The subject of cluster algebras was born out of the study of Most of the paper is written with a mathematical reader in mind; in concrete mathematical questions such as “how can we detect section 5, I describe a bit more of the physical underpinnings of when a matrix will have all eigenvalues positive?” Recently it the story. has turned out that cluster algebras show up in all kinds of un- As we will describe, in some cases, the X γ are essentially the expected places, even in the physicists’ playground of quantum “cluster X coordinates” on moduli spaces of local systems in- field theory. This paper is a review of one way in which quantum troduced by Fock and Goncharov in ref. 4. Even in these cases, field theory and cluster algebras interact. In particular, the paper the field theory perspective leads to a new geometric way of argues that geometric ideas coming from quantum field theory thinking about the cluster X coordinates. It appears, however, lead to a natural extension of the theory of cluster algebras. that there are other cases in which the X γ are not cluster coordinates, but rather a generalization thereof. Moreover, even in Author contributions: A.N. wrote the paper. the cases where X γ are cluster coordinates, our perspective leads The author declares no conflict of interest. to considering coordinate transformations that are not cluster This article is a PNAS Direct Submission. L.K.W. is a guest editor invited by the Editorial transformations (see Eq. 2.4 for the most general possibility and Board. Eq. 4.4 for a concrete instance). Thus, we seem to be meeting an 1Email: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1313073111 PNAS | July 8, 2014 | vol. 111 | no. 27 | 9717–9724 Downloaded by guest on September 28, 2021 C1. There is a complex manifold B. (In the best-studied exam- + − hγ;γ i = + γ h ; [2.5] ples, B is noncompact and indeed an affine-linear space.) X γ X γ 1 X h C2. There is a local system of lattices Γ over the complement of some divisor D ⊂ B, obtained as an extension for some fixed γh ∈ Γ. This rule will look familiar to cluster enthusiasts: it is very close to the mutation law for coefficients 0 → Γ → Γ → Γ → 0; [2.1] f g in the sense of ref. 15 or, equivalently, the coordinate change under mutation in a cluster X variety in the sense of ref. 4. Γ Γ where f is a fixed lattice (trivial local system). carries an 2.5 X integral antisymmetric pairing〈,〉, which vanishes on Γ and thus Eq. is not quite identical to the mutation in the vari- f ety, but this difference arises for an easy reason: the cluster induces a pairing on Γg; this induced pairing on Γg is unimod- coordinates xi are labeled by the elements i of some seed, ular (therefore, in particular, Γg has even rank). whereas our functions X γ are labeled by elements γ ∈ Γ; M1. For any R > 0, there is a hyperkähler space M½R. thus, to relate the two, we need to assign a basis element γ ∈ Γ For background on hyperkähler spaces, see refs. 13 and 14. i to each i; after this is done, the two transformation Here we quickly review the basics. As with any hyperkähler laws become literally identical. space, M has a family of complex structures Jζ parameter- The changes of coordinates (Eq. 2.4) are a generalization of ized by ζ ∈ CP1 and a family of holomorphic symplectic the mutation in the X variety; therefore, in the rest of this forms ϖζ. The complex structures Jζ are organized into a paper, we will call them generalized mutations. single twistor space Z. Z is diffeomorphic to M × CP1 and × ζ As an aside, the integers Ω(γ) in Eq. 2.4 are expected to has a complex structure that restricts on each M f g to Jζ, – such that the projection Z → CP1 is holomorphic. We will be generalized Donaldson Thomas invariants, in the sense × 1 of refs. 16 and 17. In the case K = 2, this expectation has mainly be interested in the Jζ with ζ ∈ C ⊂ CP ; thus, let × Z′ be the subset of Z corresponding to M × C ⊂ M × CP1. been made precise in ref. 18. The celebrated wall-crossing × – We denote points of Z′ by ðx; ζÞ ∈ M × C . formula for generalized Donaldson Thomas invariants, in the form written in ref. 16, has a simple interpretation in our M2. Viewed as a complex space using complex structure J0, M setting: it is the fact that if ðu; ϑÞ travel around a contractible π → = 1 u;ϑ has a holomorphic projection : M B, where dimB loop in B × S , the X γ undergo a sequence of jumps of the − ð1=2ÞdimM. Each generic fiber M = π 1ðuÞ is a compact form in Eq. 2.4, whose product must be the identity. complex torus. X1. For a generic point ðu; ϑÞ ∈ B × S1 there is a canonical col- X6. There exists a distinguished collection {La} of regular func- × ;ϑ ;ϑ ′ ; ϑ lection of C -valued functions u . Each X u is de- tions on Z , such that for each ðu Þ and each a,onecan fX γ gγ∈Γu γ fined on some dense open set in Z′. expand La in the form X ; ϑ = Ωu;ϑ ; γ u;ϑ; [2.6] We sometimes suppress the labels ðu Þ; whenever they are La ða ÞX γ suppressed, they should be considered to be fixed. γ

X2. The X γ are multiplicative, in the sense that u;ϑ u;ϑ where Ω ða; γÞ ∈ Z. The coefficients Ω ða; γÞ jump when hγ;γ′i ðu; ϑÞ cross a wall. [The precise jump can be computed from X γX γ′ = −1 X γ+γ′: [2.2] u;ϑ the fact that X γ jumps by Eq. 2.4 when ðu; ϑÞ cross a wall, whereas La is independent of ðu; ϑÞ.] γ n Γ In particular, for any basis f igi=1 of , all of the X γ are n In some examples, the La resemble the “dual canonical ba- determined by fX γ g . i i=1 sis,” which was part of the motivation for the definition of X3. For fixed x, X γ is holomorphic in ζ. For fixed ζ, X γ is hol- cluster algebra, e.g., compare the last section of ref. 19 with omorphic in x, with respect to complex structure Jζ on M.In ref. 20. (I thank Hugh Thomas for explaining this to me.) other words, X γ is holomorphic on Z′ ⊂ Z. The La are also close to the functions appearing in the × X4. For any fixed ζ × C , the X γ form a Darboux coordinate duality conjectures of Fock and Goncharov (4). ; ; ϖ system on the holomorphic symplectic manifold ðM Jζ ζÞ, X7. If ζ → 0 while remaining in the half-space centered on ϑ,andif in the following sense. The holomorphic symplectic structure induces a holomorphic Poisson bracket, and this bracket is very u = πðxÞ; [2.7] simple in terms of the X γ u;ϑ then each X γ ðx; ζÞ has asymptotics of the form X γ; X γ′ = γ; γ′ X γ+γ′: [2.3] u;ϑ X γ ðx; ζÞ ∼ cγðxÞexp ZγðuÞ ζ ; [2.8] u;ϑ X5. The coordinate system X γ depends on ðu; ϑÞ in a piecewise constant fashion, but jumps when ðu; ϑÞ reach some real- where Zγ(u) depends holomorphically on u. codimension-1 walls in B × S1. Each wall is transverse to the S1 factor and therefore has a + 3. Theories of Class S u;ϑ side and a − side. The relation between the X γ on the two We now specialize to the theories of class S, for which all of the sides is computable in terms of data attached to the wall, namely story of section 2 has been developed. More specifically, we will an element γh ∈ Γ,integersfΩðγÞgγ∈Γ,andsignsσn =±1 consider theories associated to the Lie algebra u(K), which is the only case for which the functions X γ have been thoroughly in- ∞ + − nΩðnγ Þhγ;γ i vestigated t far. = ∏ − σ γ h h : [2.4] X γ X γ 1 nX n h Thus, let g = uðKÞ and let h be the Cartan subalgebra con- n=1 sisting of diagonal matrices. Let C be a compact Riemann surface of genus g with n > 0 marked points z1,...,zn. Also fix Ω γ = Ω γ = C ∈ R ∈ An important special case arises when ( h) 1, (n h) parameters m hC and mi h for each marked point, with the > σ = − C i Z 0forn 1, and 1 1. In this case mi chosen linearly independent over . We think of the marked

9718 | www.pnas.org/cgi/doi/10.1073/pnas.1313073111 Neitzke Downloaded by guest on September 28, 2021 points and parameters as included in the definition of the surface corresponds to a different lens through which to view the C. Given these data, there is an N = 2 theory S½g; C (3, 21, 22), harmonic bundles, by “forgetting” part of their structure. SPECIAL FEATURE × which we call a theory of class S. We will consider ζ = 0 in M2 below, and ζ ∈ C in section 4. C M Wenowdescribethedata and in these theories; this M2. Given a harmonic bundle (E, D, φ), let ∂ be the (0, 1) part amounts to a lightning review of the basic facts about Hitchin’s D of D and let ϕ be the (1, 0) part of φ. The pair ðE; ∂ DÞ define integrable system. a holomorphic vector bundle over C; call this holomorphic 3.3b ϕ C1. B is the space of all tuples bundle Eh. Then, Eq. says that is a meromorphic section of End(Eh) ⊗ KC. Thus, the pair (Eh, ϕ) are a G- = φ ; ...; φ ; [3.1] u ð 1 K Þ Higgs bundle over C, with first-order poles at zi, whose C residues are determined by mi . φ where k is a meromorphic k-differential, i.e., a section of Let MH denote the moduli space of G-Higgs bundles with φ the kth power of T*C, such that k has poles only at the these singularities. This space carries a natural complex struc- points zi, of order k, and the residue of φk at zi is the co- ; ; φ ↦ ; ϕ − ture. Remarkably, the forgetful map ðE D Þ ðEh Þ indu- λK k C − λ H efficient of in the characteristic polynomial detðmi Þ. ces a diffeomorphism M’M and thus induces a complex B is an affine-linear space: the difference between any two structure on M. This induced complex structure is J0. tuples in B is an element of the vector space of tuples Now suppose a given point of MH , represented by a Higgs φ~ ; ...; φ~ φ~ ð 1 K Þ where k is a meromorphic k-differential, with bundle (E , ϕ). The characteristic polynomial of ϕ is of ≤ − h poles only at the zi,oforder k 1. the form C2. Given a point u = ðφ ; ...; φ Þ ∈ B, we define the corre- 1 K XK sponding spectral curve K−i Pϕ λ = φ λ ; [3.5] ( ) ð Þ i = XK i 1 Σ = λK−kφ = ⊂ p ; [3.2] φ ⊗i u k 0 T C where each i is a meromorphic section of KC , with poles at = k 0 the marked points zi. Passing from (Eh, ϕ) to the tuple H (φ1,...,φk) gives the holomorphic projection π: M → B. where we set φ = 1. Σ is a K-fold branched cover of C∖{z ,..., 0 u 1 The discussion above admits various possible generalizations: MATHEMATICS zn}.ItcanbecompactifiedtoabranchedcoverΣ u of C, un- • φ branched over the zi. One could consider harmonic bundles where has poles of order > 1 (irregular punctures). The mathematical theory of The singular divisor D ⊂ is the locus for which Σ is a B u such bundles has been worked out in ref. 26. In fact, some singular curve. For u ∉ D, we define Γ = H Σ ; Z . The u 1ð u Þ of the technically simplest examples of the whole story are of quotient (Γ ) is H Σ ; Z , with the quotient map Γ → g u 1ð u Þ u this sort. We suppress this generalization here only to (Γ ) induced by the inclusion Σ ⊂ Σ . g u u u lighten the notation. M1. M½R is a moduli space of harmonic G bundles on C, de- • One could consider surfaces C with no punctures at all, or fined as follows. with punctures where the residue of φ is nilpotent. These cases are technically more difficult to treat, although all of A harmonic G bundle on C is a triple (E, D, φ), where E is the formal expectations from quantum field theory (QFT) a rank K Hermitian vector bundle on C, D is a unitary con- are the same. nection in E,anφ is a section of End(E) ⊗ KC obeying the Hitchin equations (23) Finally, although in this paper we are focusing on the theories of class S, we should note that there is another class of N = 2 − 2 φ; φ† = ; [3.3a] FD R 0 theories that has been studied extensively, namely the class of quiver gauge theories. For these theories, the data in C1, C2, M3, and M4 have been described comprehensively in a recent work ∂ φ = 0; [3.3b] D (27) (and many special cases were known earlier). However, in these cases, the functions X γ of X1–X7 have not yet been de- such that D and φ are smooth on C except at the marked scribed. Thus, it seems that the cluster structure in quiver gauge points zi and in a neighborhood of zi have first-order singu- theories is waiting to be discovered. It should be very interesting larities of the form (in some gauge) to do so. 4. Spectral Coordinates = + R dz − dz ; [3.4a] D Dreg mi z − zi z − zi In the next few sections, following refs. 3 and 28, we will describe the data in X1–X7 in the theories of class S, first in the special = = dz case of G U(2) and then more generally for G U(K). φ = φ + mC ; [3.4b] As we have mentioned, these data are most directly related to reg i z − z × i the complex structures Jζ on M for ζ ∈ C . Thus, we first describe what those complex structures look like. ζ ∈ C × φ where Dreg and φreg are regular at zi. Fix some . Given a harmonic bundle (E, D, ), we may then build the combination By an application of the gauge-theoretic tools developed in ref. 24, it has been shown that solutions to Eq. 3.3 indeed − † ∇ = ζ 1φ + D + ζφ : [4.1] exist (23, 25) and that one can define a generically smooth, finite-dimensional moduli space M parameterizing harmonic The resulting ∇ is a flat GC connection (as follows from Eq. 3.3). Its G bundles on C modulo gauge equivalence. Moreover, as ex- − C R C monodromy around z is given by μ = exp 2πiðζ 1m + m + ζm Þ. plained in ref. 23, M is naturally hyperkähler. ♭ i i i i ♭ i Let M denote the moduli space of such connections. M carries As expected for a hyperkähler space, M admits a canon- a natural complex structure (coming from the complex structure 1 ical family of complex structures Jζ, ζ ∈ CP . Each fixed ζ in GC).

Neitzke PNAS | July 8, 2014 | vol. 111 | no. 27 | 9719 Downloaded by guest on September 28, 2021 Fig. 1. Behavior of the foliation F(u, ϑ) around a second-order pole of φ2.

The forgetful map ðE; D; φÞ ↦ ∇ induces a diffeomorphism ♭ M’M. This diffeomorphism induces a complex structure Jζ Fig. 3. Cells in the complement of the critical graph CG(u, ϑ). × on M (depending on the parameter ζ ∈ C , which entered ♭ through the Eq. 4.1). Moreover, M also carries a holomorphic symplectic form [the complexification of the one described by endomorphisms Mi, given by monodromy of ∇ around the four Atiyah and Bott for G connections (29)], so we obtain a family of marked points zi on the boundary of Qc. Because ∇ is generic, ϖ holomorphic symplectic forms ζ on M. each Mi is semisimple and hence decomposes V into two eigenlines. Moreover, because in Eq. 3.4 the singular parts of D 4.1. Case G = U(2). We now specialize to G = U(2). In this case, the and φ commute, these two eigenlines are also the eigenlines C construction of the X γ was given in ref. 3. of the residue m of φ. Let ℓi be the eigenline of M corre- i i − ϑ Suppose a generic ðx; ζÞ ∈ Z′.Asjustexplained,(x, ζ)determine sponding to the eigenvalue β of mC with the largest Re(e i β). ; C ∇ 4.1 i aflatGLð2 Þ connection up to equivalence, via Eq. . We thus have four distinguished lines ℓi ⊂ V, i.e., four points u;ϑ ; ζ “ × Roughly, the desired X γ ðx Þ will be complexified shear coor- of the projective line PðVÞ.Letxc ∈ C be the cross-ratio of these dinates” of the connection ∇. These coordinates have been studied four points. u;ϑ by many authors; for us, the most relevant reference is ref. 4, where Now we are ready to build the desired functions X γ . Given c, they were linked to cluster structure on moduli spaces of flat con- there is a canonical γc ∈ Γu, pictured in Fig. 6. We define nections. In our setup, for each generic ðu; ϑÞ, we will consider u;ϑ ;ϑ a preferred shear coordinate system, from which we build the X γ . u X γ = xc: [4.2] First we should understand what ðu; ϑÞ mean in this case. As c φ φ we have reviewed above, B is a space of pairs ( 1, 2). In what γ Γ follows, φ plays no role, so we focus on φ : it is a meromorphic The c are a basis for the sublattice of u, which is odd under the 1 2 automorphism σ of Σ exchanging the two sheets. Thus, Eqs. 4.2 quadratic differential on C, with a second-order pole at each zi, u;ϑ 2 φ and 2.2 together determine X γ ðx; ζÞ for all σ-odd γ. of residue mi . Fixing u means fixing one such 2. Fixing u generic σ Γ φ The -even part of u plays a minor role and could be elimi- means that 2 should have only simple zeroes. = = We consider a singular foliation Fðu; ϑÞ of C whose leaves are nated by considering G SU(2) instead of G U(2). We will not 2iϑφ ; ϑ discuss it explicitly here (it is automatically incorporated in the the trajectories along which e 2 is real. Fðu Þ has singularities φ φ more general formalism of section 4.2). only at the zeroes and poles of 2. Around a pole zi of 2, the u;ϑ picture depends on e−iϑmC, as indicated in Fig. 1. Around a zero The functions X γ so constructed have all of the properties of i X1–X7. In particular of φ2, Fðu; ϑÞ looks like Fig. 2; there are three distinguished “critical trajectories” emerging from the zero. Let CGðu; ϑÞ de- • The generalized mutations (X5) emerge when we consider the u;ϑ note the union of the critical trajectories (critical graph). Using dependence of X γ on ðu; ϑÞ. The data ðu; ϑÞ are entered in results from ref. 30, we can describe the global topology of the definition of CGðu; ϑÞ and in the choice of the eigenlines ; ϑ ; ϑ CGðu Þ for generic ðu Þ: it divides C up into cells of two types, ℓ . We consider these two dependences in turn. illustrated in Fig. 3. From now on, for simplicity, we suppose all i ; ϑ ; ϑ ∈ × 1 of the cells are like the one on the left of Fig. 3. In this case, the First, consider how CGðu Þ depends on ðu Þ B S . Be- critical graph CGðu; ϑÞ induces a corresponding ideal triangulation ginning at a generic ðu; ϑÞ, small variations of ðu; ϑÞ lead to ; ϑ u;ϑ of C as shown in Fig. 4. CGðu Þ varying by an isotopy. The construction of X γ evi- u;ϑ Each cell c in C is contained in a quadrilateral Q ⊂ C,as dently depends on CGðu; ϑÞ only up to isotopy. Hence X γ is c ; ϑ shown in Fig. 5. We will build one coordinate function xc from not changed by sufficiently small variations of ðu Þ. However, × 1 ; ϑ each c. Because Qc is simply connected, the space of flat sections there are codimension-1 walls in B S where CGðu Þ jumps of ∇ over Qc is a 2D vector space V. V has four natural discontinuously. These walls are the ones that appear in X5.

One such jump is illustrated in Fig. 7: at ϑ = ϑc, the critical graph CGðu; ϑÞ degenerates, because of the presence of a saddle connection. The corresponding triangulations are related ± ;ϑ by a flip. Let X denote the value of X u for ðu; ϑÞ on the ±

Fig. 4. Relation between the critical graph CG(u, ϑ) and the corresponding

Fig. 2. Behavior of the foliation F(u, ϑ) around a simple zero of φ2. induced ideal triangulation of C.

9720 | www.pnas.org/cgi/doi/10.1073/pnas.1313073111 Neitzke Downloaded by guest on September 28, 2021 SPECIAL FEATURE

Fig. 5. Quadrilateral Qc containing a cell c. Fig. 7. A jump of the critical graph CG(u, ϑ), induced by the appearance of a saddle connection. + − side; then, a straightforward calculation shows X and X are related by quantities that appear in Eq. 4.4. [They also have an independent γ;γ interpretation as local coordinate systems that include complex- + − − h hi – X γ = X γ 1 + X γ : [4.3] ified Fenchel Nielsen coordinates on M, as well as complexified h shear coordinates (32).] This limit was analyzed directly in ref. 3 and more indirectly (but more elegantly) in refs. 28 and 32. This equation matches the mutation law (Eq. 2.5). A closely related construction recently appeared in ref. 33: • ± There is a second jumping phenomenon, associated with roughly our X should be understood as monomials attached ; ϑ γ a different way in which CGðu Þ can degenerate: a closed to their limit triangulations. trajectory can appear. The associated transformation is of 1 the form • There are also some walls in B × S where the eigenvalues β of C −ϑβ mi have equal Re(e ). Upon crossing one of these walls, −2hγ;γ i + − − h the eigenline ℓi changes, and the critical graph CGðu; ϑÞ jumps. X γ = X γ 1 − X γ : [4.4] h When both of these phenomena are taken into account, the + − functions X γ are invariant, i.e., X γ = X γ (3). This transformation differs from Eq. 4.3 by the factor −2 in the • The functions Lc of X6 correspond to multicurves c on C. If c

exponent, and hence, Eq. 4.4 cannot be interpreted as a muta- is a single curve on C that is not isotopic to a loop around any MATHEMATICS tion. It is a generalized mutation (Eq. 2.4). zi, carrying weight k, Lc(x, ζ) is the trace of the monodromy of + − ∇ around c, taken in the kth symmetric power of the defining We must explain precisely what is meant by the X and X in 4.4. ; ϑ representation of U(2). If c consists of several curves, then Lc Eq. Care is needed here, because if ðu cÞ lies on the locus is a product of such traces for each curve. where a closed trajectory appears, the behavior of Fðu; ϑÞ in • – ; ϑ ϑ+ ϑ− The asymptotic property in X7 comes from the Wentzel a neighborhood of ðu cÞ is wild: there are phases n and n , – ϑ− < ϑ < ϑ+ ϑ ± = ϑ Kramers Brillouin approximation applied to the family of with n c n ,suchthatlimn→∞ n c, and at each ∇ ζ ζ → ; ϑ ± connections ( )as 0. The function Zγ appearing in the ðu n Þ, there is a saddle connection. Each of these two infinite asymptotic is a period sequences of saddle connections consists of trajectories wind- þ ing more and more tightly around a cylinder, which becomes γ = λ; [4.5] foliated by closed trajectories at ðu; ϑcÞ (see ref. 3 for pictures Z or ref. 31 for a movie illustrating this phenomenon). γ × 1 Thus, the walls in B S associated with closed trajectories are λ never isolated: rather they are loci where walls coming from where is the Liouville 1-form on T*C. saddle connections accumulate. Then we may ask: what happens This whole story is closely related to the cluster X variety u;ϑ ϑ → ϑ ± to the functions X γ in the limit as c ? They infinitely structure on M (or more precisely a covering space of M) in ref. undergo many mutations (Eq. 4.3) as we approach this locus, “ ” ± u;ϑ ± 4 and the cluster algebras from surfaces in ref. 34. Each ge- but nevertheless, X γ = limϑ→ϑ ± X γ exists. These X γ are the c neric ðu; ϑÞ induces a seed in the corresponding cluster X variety, u;ϑ and the functions X γ are the coordinates on the cluster torus. The mutations described above, which occur when a saddle connection appears in CGðu; ϑÞ, and the corresponding triangulation flips are the same ones that appear in refs. 4 and 34.

4.2. Case G = U(K). Now we consider the case G = U(K)whereK ≥ 2. Fix a generic ðx; ζÞ ∈ Z′.(x, ζ) determine a flat GLðK; CÞ connection ∇ up to isomorphism, via Eq. 4.1. The desired u;ϑ X γ ðx; ζÞ are a new kind of coordinates associated with ∇. Their construction has not been completely worked out in general, but some special cases are understood. The construction involves an auxiliary gadget Wðu; ϑÞ, which we call a spectral network, generalizing the critical graphs CGðu; ϑÞ that appeared in the K = 2 case. We now describe the construction of Wðu; ϑÞ. Over any contractible U ⊂ C containing no branch points of Σ → C, we may trivialize the covering Σ, labeling the sheets by i = 1,...,K. The 1-form λ on Σ then induces holomorphic 1-forms λ1,...,λK on U. We want to consider paths r(t)onC that locally obey an equation of the form ½λiðrÞ − λjðrÞr_ = 1. More precisely,

Fig. 6. Cycle γc ∈ Γu corresponding to a cell c. let a trajectory be an oriented ray r: [0, ∞) → C with a closed

Neitzke PNAS | July 8, 2014 | vol. 111 | no. 27 | 9721 Downloaded by guest on September 28, 2021 Fig. 8. Three distinguished trajectories emerging from a branch point.

image, such that for any t ∈ (0, ∞), there exists a neighborhood N Fig. 10. Example of a spectral network. of t and a pair (i, j) of sheets of Σ over r(N), with of Σ → C over U. Let L denotethelinebundleoverΣ underlying r_ðtÞ = 1 λi − λj ½rðtÞ for t ∈ N: [4.6] ab ab ∇ .OnU, the vector bundle underlying πp∇ is canonically decomposed as Given initial data, consisting of a point z ∈ C that is not a branch point and a pair (i, j) of sheets of Σ over a neighborhood of z, K π = ⊕ : [4.7] there is a unique trajectory r determined by these initial data. L Lk * = To draw concrete pictures, it is convenient to trivialize Σ away k 1 from a collection of “branch cuts” on C. On the complement of the cuts, we can label each trajectory by a pair (i, j) according to We require which equation it obeys. When a trajectory of type (i, j) crosses σ σ ι+ = ðId + SÞ○ ι−; [4.8] a branch cut, its label changes to (i , j ), where σ is the permu- tation of sheets associated to the cut. π → π → Consider a simple branch point b of Σ → C at which two sheets where S: pL pL maps Li Lj and maps all of the other collide. Monodromy around b exchanges those two sheets. A summands to zero, and (i, j) is the labeling of the trajectory short computation shows that there are three distinguished tra- where z sits. [Thus, in a basis respecting Eq. 4.7, S is a matrix jectories emerging from b (Fig. 8). whose only nonzero entry is in the (i, j) position.] Now we can define the spectral network Wðu; ϑÞ: it is the Refs. 28 and 32 provide more details on abelianization. We smallest collection of trajectories on C such that the three dis- expect (but have not proven!) that a generic connection ∇ can be tinguished trajectories emerging from each branch point belong ; ϑ to Wðu; ϑÞ and if Wðu; ϑÞ contains two trajectories that intersect, Wðu Þ abelianized only in finitely many ways and that fixing (u, ϑ) also gives a way of fixing this discrete choice to get a preferred and near the intersection point these trajectories carry labels ; ϑ = (i, j) and (j, k), then Wðu; ϑÞ contains another trajectory, which Wðu Þ abelianization. (If K 2, this discrete choice is the choice of lines ℓi we used in section 4.1.) originates from the intersection point and carries the label (i, k), ; ϑ as shown in Fig. 9. One class of examples of Wðu Þ where everything works as In the case K = 2, trajectories never cross, and in that case, expected was considered in ref. 35. One can construct them by ; ϑ ; ϑ beginning with the critical graph for a quadratic differential and Wðu Þ is just the critical graph CGðu Þ that we considered in − K > u; ϑ then replacing each branch point by a cluster of K(K 1)/2 section 4.1. For 2, the structure of Wð Þ can be much “ ” more intricate. There are examples of C for which Wðu; ϑÞ is branch points, in one of two standard structures called Yin and “ ” always well behaved—in particular, it is locally finite on C. Fig. Yang ; the Yin case is illustrated in Fig. 11. Each of the tra- = 10 shows an example. However, for general C, even with punc- jectories that appear in the K 2 case then gets replaced by a “ ” − tures, Wðu; ϑÞ may be dense in parts of C. cable of K(K 1)/2 trajectories. The global structure is gov- Having defined Wðu; ϑÞ, we return to our original goal: the erned by an ideal triangulation of C just as in the K = 2 case (Fig. ;ϑ construction of the functions X u . The construction works by 12). When all triangles are of the Yin type, it is shown in ref. 35 γ γ ∈ Γ associating to the GL(K) connection ∇ over C a corresponding that for particular choices of cycles i , the spectral coor- ab u;ϑ ∇ Σ dinates X γ are the coordinate functions in the X cluster variety GL(1) connection over , in a way depending on the net- i u;ϑ × work Wðu; ϑÞ. The desired X γ ðx; ζÞ ∈ C will simply be the introduced by Fock and Goncharov (4). Thus, we get a new way holonomies of ∇ab around cycles γ. of thinking about these cluster coordinates: they can be explained The connections ∇ab and ∇ are supposed to be related by Wðu; ϑÞ abelianization, which means an additional datum ι as follows: ab • Consider the pushforward connection πp∇ , defined on the complement of the branch locus of Σ → C. ι is an isomorphism ι : ∇ ’ π ∇ab on C∖W(u, ϑ). * • The limits ι± of ι as we approach Wðu; ϑÞ need not be equal, but their difference is constrained, as follows. Fix a point z on Wðu; ϑÞ, a contractible neighborhood U of z, and a trivialization

Fig. 11. The local structure of a spectral network around a cluster of branch Fig. 9. The rule by which colliding trajectories in W(u, ϑ) give birth to new ones. points, in the Yin case, where we took K = 4.

9722 | www.pnas.org/cgi/doi/10.1073/pnas.1313073111 Neitzke Downloaded by guest on September 28, 2021 5. Some QFT Finally, for readers with an interest in quantum field theory, we SPECIAL FEATURE briefly revisit the list of key properties from section 2 and describe where they come from. C1. B is the Coulomb branch of the N = 2 theory. This is one branch of the moduli space of Poincare invariant quan- Fig. 12. (Left) Relation between a spectral network with K = 2 and corre- tum vacua of the theory. Its general structure was de- sponding ideal triangulation. (Right) Corresponding picture with K = 3, scribed in ref. 37, where one particularly important example where each branch point has been replaced by a cluster of three branch was worked out, namely pure SU(2) super Yang–Mills the- points like that of Fig. 11, and each trajectory has been fattened into a cable ory. [That theory is in fact also an example of a theory of of three trajectories. class S with G = SU(2), as explained in ref. 21 and later revisited in a context close to that of this paper in ref. 3.] C2. Γ is the lattice of charges carried by states in the low-energy by the abelianization construction. We believe that this is likely to approximation to the full N = 2 theory. The quotient Γg con- be a useful point of view. sists of electromagnetic charges in the usual sense. The sub- u;ϑ – It is argued in ref. 28 that the X γ have the properties of X1 lattice Γ consists of pure flavor charges, related by Noether’s X7, generalizing what we said in section 4.1 in the case G = U(2). f ; ϑ theorem to a global symmetry rather than a gauge symmetry. In particular, for sufficiently small variations of ðu Þ, the net- M1. M½R is the target of a 3D sigma model obtained by compac- work Wðu; ϑÞ varies by a generalization of isotopy, called = “ ” tifying our N 2 theory on a circle of length R and considering equivalence in ref. 28. As before, there are codimension-1 the physics at energies E 1/R. Such a target space is neces- walls in B × S1 where Wðu; ϑÞ undergoes a topology change. The u;ϑ sarily hyperkähler as explained in refs. 38 and 39. The various jump of the X γ at such a wall can always be written as a gen- complex structures Jζ correspond to various subalgebras eralized mutation (Eq. 2.4). In the case G = U(2), we had just ; ϑ of the supersymmetry algebra of this sigma model. two possible kinds of topology change for CGðu Þ and two M2. The existence and holomorphy of the projection π follow from 4.3 4.4 G = U K corresponding jump formulas (Eqs. and ). For ( ), the fact that supersymmetric local operators of the original K > 2, the story seems to be much wilder: there is a recipe in ref. N = 2 theory also give supersymmetric local operators in the 28 for determining the jump associated to any particular topol- 3 1 theory on R × S (invariant under a subalgebra of the super- ogy change of Wðu; ϑÞ, but there is no classification of all pos- MATHEMATICS symmetry algebra, corresponding to Jζ = 0). sible topology changes. ;ϑ X1. The u are the vacuum expectation values of certain su- The simplest topology change that can occur for U(K) but not X γ for U(2) is shown in Fig. 13. This particular change leads to persymmetric line defects in the low energy approximation to the N = 2 theory, when these defects are placed at the a mutation. At the moment when the mutation occurs, a new 1 3 1 ; ϑ locus {point} × S ⊂ R × S . finite subnetwork appears in Wðu Þ; this three-pronged sub- u;ϑ X2. The product law for X γ follows from the operator product network is a generalization of the saddle connections from the = K = 2 case. A few other simple examples are discussed in ref. 28. algebra of line defects in the low energy N 2 theory with A more elaborate topology change and its corresponding gen- the abelian gauge group. eralized mutation are considered in ref. 36. X3. The holomorphy of X γ for fixed x is a consequence of the We emphasize that, in contrast to the G = U(2) case, for G = supersymmetry of the corresponding line defect. U(K), a generic Wðu; ϑÞ is not governed by an ideal triangulation. X4. The simplicity of the Poisson brackets is shown in refs. 2 u;ϑ and 40 but in a somewhat indirect way. It should follow Correspondingly, the spectral coordinates X γ are more general than the ones assigned to triangulations. We do not know whether from the fact that the Poisson bracket can be expressed – they exhaust the set of coordinate systems in the cluster atlas of ref. 4. in terms of topological field theory (Rozansky Witten theory), In addition, it seems possible that some of the coordinate although I believe this has not been fully worked out. u;ϑ Ω γ – – systems X γ are not part of the cluster atlas at all. The reason is X5. ( ) is an index counting Bogomolny Prasad Sommerfield 1 γ = that there can be domains in B × S where the walls are actually (BPS) states of charge in the N 2 theory. The general- dense (36). To escape from such a domain, one would have to ized mutations (Eq. 2.4) are most easily understood in terms cross infinitely many walls; thus, if (u, ϑ) sit in such a domain, of framed wall-crossing (see below). u;ϑ then there seems to be no reason why the coordinate system X X6. The functions La are vacuum expectation values of line defects γ = Ω ; γ should be connected to a cluster coordinate system by finitely of the original N 2 theory. The coefficients ða Þ govern many mutations. The role of these coordinate systems should the decomposition of the line defects of the original theory be similar to that of the limiting coordinate systems we de- into line defects of the low energy theory. They can also be scribed in the U(2) case above: those limiting coordinate systems understood as dimensions of certain vector spaces: the spaces Ω ; γ also did not belong to the cluster atlas, because they were sep- of framed BPS states. The jumps of ða Þ arise from framed arated from the cluster coordinate systems by infinite chains wall-crossing: as the parameters (x, ζ) of the field theory and of mutations. line defect vary, framed BPS states may enter or leave the Hilbert space from infinity. The picture is an infinitely heavy particle sitting at the origin of R3, which decays by emitting a bulk particle or forms a bound state by attracting a bulk particle. X7. This asymptotic property was deduced indirectly in refs. 2 and 40, but (as far as I know) has not yet been explained in a physically satisfying way.

ACKNOWLEDGMENTS. Most of the information reviewed here is joint work with Davide Gaiotto and Greg Moore; I thank them for a very enjoyable collaboration. I also thank Clay Cordova, Dmitry Galakhov, Alexander Goncharov, Lotte Hollands, Pietro Longhi, Tom Mainiero, and Hugh Thomas for related collaborations and discussions. This work is supported in part by National Fig. 13. A jump in the topology of a spectral network with K = 3. Science Foundation Grant DMS-1151693.

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