K-decompositions and 3d gauge theories

Journal of High Energy Physics, Nov 2016

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, ℂ)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space ℒ K (M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of ℒ K (M) as a Lagrangian subvariety in the symplectic moduli space \( {\mathcal{X}}_K^{\mathrm{un}}\left(\partial M\right) \) of framed flat connections on the boundary — and more so, as a “K2-Lagrangian,” meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of ℒ K (M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, ℂ)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of ℒ K (M) is K2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that ℒ K (M) is K2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d \( \mathcal{N}=2 \) superconformal field theories T K [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories T K [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N f = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T K [M] grow cubically in K.

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K-decompositions and 3d gauge theories

Received: February K-decompositions and 3d gauge theories Tudor Dimofte 0 2 4 Maxime Gabella 0 3 4 Alexander B. Goncharov 0 1 Davis 0 CA 0 U.S.A. 0 0 Open Access , c The Authors 1 Yale University Mathematics Dept. , New Haven, CT 06520 , U.S.A 2 U.C. Davis, Dept. of Mathematics and Center for Quantum Mathematics and Physics 3 Institut de Physique Theorique, CEA/Saclay , 91191 Gif-sur-Yvette , France 4 Institute for Advanced Study , Einstein Dr., Princeton, NJ 08540 , U.S.A manifolds M with boundary. We introduce a moduli space LK (M ) of framed nections on the boundary @M that extend to M . Our goal is to understand an open part of LK (M ) as a Lagrangian subvariety in the symplectic moduli space XKun(@M ) of framed at connections on the boundary | and more so, as a \K2-Lagrangian," meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of LK (M ) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M , in a way that generalizes (and combines) both Thurston's gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed at PGL(K; C)connections on surfaces. By using a canonical map from the complex of con gurations of decorated ags to the Bloch complex, we prove that any generic component of LK (M ) is K2-isotropic as long as @M satis es certain topological constraints (theorem 4.2). In some cases this easily implies that LK (M ) is K2-Lagrangian. For general M , we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement. Di erential and Algebraic Geometry; Supersymmetric gauge theory; Super- - symmetry and Duality 1 Introduction The 3d-3d correspondence A mathematical perspective Organization Basic tools and de nitions Gluing admissible 3-manifolds from truncated tetrahedra Con gurations of ags The basic moduli spaces From framed at bundles to con gurations of ags Hypersimplicial K-decomposition Con guration of ags and hypersimplices Localization of framed Boundary phase spaces Coordinates from the K-triangulation Symplectic structure The Lagrangian pair assigned to a 3d triangulation of M The Lagrangian pair for the PGL(2)-tetrahedron The Lagrangian pair for the PGL(K)-tetrahedron The coordinate phase space for a tetrahedron via symplectic reduction 41 Gluing constraints for 3-manifolds Holonomy representation Small torus boundaries 3.5 2{3 moves via 4d cobordism 3d snakes in a tetrahedron 3d snakes in a 3-manifold K2-Lagrangians and the Bloch complex From decorated ags to the Bloch complex Decorated ags BiGrassmannian complex Conclusion: from decorated ags complex to the Bloch complex LK (M ) is a K2-isotropic subvariety Di erential of the volume of a framed at PGL(K; C)-connection Poisson brackets and the quotient Remarks on quantization 5.5 2{3 moves and path coordinates Paths on slices Big boundary coordinates Small torus coordinates Unipotent conditions Composing elementary moves Tetrahedron and polyhedron theories Review: the octahedron theory Warmup: elementary 2{3 move theories Tetrahedra at higher K The K = 3 bipyramid K = 3 K = 4 K = 5 Combinatorics of K-decompositions Octahedra and slices Coordinates on the product phase space Knot complement theories Flavor symmetry and marginal operators of TK [M ] Real masses and boundary holonomy K3 scaling in TK [M ] Trefoil knot invariants Figure-eight knot gluing A Symplectic data and class R A.1 The symplectic data A.2 Constraints on the data A.3 The gauge theory PGL(K) volume from octahedra From free energy to volumes The SL(3) character variety Symplectic gluing data A.4 Equivalences A.5 The existence of operators Ok A.6 Associated invariants B The Poisson bracket for eigenvalues B.1 fC; U g = fX ; U g = 0 B.2 fU ; U g = B.2.2 Slices in di erent families This paper presents a combination of mathematical and physical results. Its main goal is a eld theories TK [M ] labeled by an oriented topological 3-manifold M and an integer K The theories TK [M ] are meant to coincide with the compacti cation of the six-dimensional implies that the theories TK [M ] should possess several important properties, relating their observables to the topology and geometry of M | in particular, to the geometry of the moduli space of at SL(K; C)-connections on M and its quantization. These properties are summarized in table 1 below (page 6). The combinatorial de nition we give of TK [M ] makes many of its expected properties manifest. were de ned in [1]. The key idea of [1] was to decompose M into topological ideal tetrahedra (i.e. to choose a triangulation of M ). Then, after assigning a canonical \tetrahedron theory" T2[ ] to each tetrahedron, T2[M ] was constructed by \gluing" together the tetra M = SiN=1 i T2[M ] = T2[ 1 T2[ N ] = Physically, each T2[ i] contained a 3d chiral multiplet, and the gluing operation ` ' added superpotential interactions and gauged some global symmetries, producing an abelian Chern-Simons-matter theory with an explicit Lagrangian. The nal step in the gluing ow to the infrared, de ning T2[M ] as the infrared limit of the Chern-Simons The gluing of (1.1) was done in such a way that, generically, the de nition of T2[M ] would be independent of the choice of triangulation. Topologically, any two ideal triangulations can be related by local 2{3 Pachner moves as depicted in gure 1 [2, 3]. The basic theory T2[bipyramid] associated with a triangular bipyramid and constructed from with two chiral multiplets. The theory constructed from the triangulation on the r.h.s. was the \XYZ model," consisting of three chiral multiplets and a cubic interaction. These two 3d theories are equivalent (dual) in the infrared [4], ensuring a local triangulationindependence. Lifting the local independence to a global independence of any theory T2[M ] turned out to be subtle for two reasons: 1) it required interchanging infrared limits for di erent pieces of a Chern-Simons-matter Lagrangian; and 2) not all triangulations could be used to produce sensible Chern-Simons-matter Lagrangians, and the \good" triangulations that work are not all known to be related by sequences of 2{3 moves. Thus, strictly speaking, the triangulation-independence of T2[M ] was only conjectural. The observables of theories T2[M ] were related to classical and quantum hyperbolic geometry, because at SL(2; C)-connections on M are (roughly speaking) hyperbolic metrics.1 The properties that allowed T2[M ] to be de ned as in (1.1) were a direct generalization of the symplectic properties that Neumann and Zagier [8] observed in Thurston's gluing equations [5] for ideal hyperbolic tetrahedra. These same symplectic properties allowed Thurston's gluing equations to be quantized in [9] (following [10, 11]). The lift of Thurston's gluing equations to theories T2[M ] amounts to a categori cation of hyperbolic geometry | though many details of this categori cation remain to be worked out. In this paper, we extend the triangulation methods of [1] to general K 2. This requires developing some new mathematics. At the classical level, we need to describe at PGL(K; C)-connections on triangulated 3-manifolds in a manner analogous to Thurston's description of hyperbolic metrics. To achieve this, we enhance moduli spaces of at connections on a 3-manifold M with additional framing data, much as in the 2d constructions of [12]. Namely, we consider at connections together with a choice of invariant ags along certain loci on the boundary @M . This requires the introduction of extra topological data on @M .2 Then, for admissible 3-manifolds M , we construct two algebraic varieties LeK (M ) ' fframed at PGL(K; C)-connections on M g ; XKun(@M ) ' fframed unipotent at PGL(K; C)-connections on @M g : Although we do not give precise de nitions here, let us present a basic example. 1Precisely, hyperbolic metrics on M are in 1-1 correspondence with at PGL(2; C)-connections whose holonomy representation 1(M ) ! PGL(2; C) is discrete, faithful, and torsion-free, cf. [5]. The is the holonomy representation of the hyperbolic metric. The relation between hyperbolic geometry and PGL(2; C)-connections was heavily exploited in [6] and subsequent works to study quantum gravity in three dimensions, and in [7] and subsequent works to understand aspects of quantum Chern-Simons theory. 2Examples of this \extra structure" include the laminations discussed in early versions of [13], and expanded on in [14]. The slightly more general notion of framing data that we present in section 2 arose in collaboration with the authors of [15]. be a polyhedron. Then the moduli space XKun(@M ) parametrizes PGL(K; C)-connections on the sphere @M punctured at the vertices of the polyhedron, with unipotent monodromies around the vertices, plus a choice of an invariant ag near each of the vertices. Turning to LeK (M ), it makes little sense to talk about a moduli space of just at connections on a polyhedron, since a polyhedron is simply connected, and so any at connection is trivial. However, the framed moduli space of a polyhedron parameterizes con gurations of ags in CK , labelled by the vertices of the polyhedron. For example, in the most fundamental example when M is a tetrahedron we get a con guration space of The space XKun(@M ) is a singular complex symplectic space, which carries a canonical symplectic form on its non-singular part, and more so, a K2-avatar of the symplectic form [12]. We consider the image of the natural projection LK (M ) := im(LeK (M ) ! XKun(@M )) : K2-isotropic.3 by the A-polynomial of the knot [16]. The spaces LK (M ) and LeK (M ) may have several irreducible components. We introduce a notion of a generic component, and conjecture that any generic component of LK (M ) is a K2-Lagrangian subvariety of XKun(@M ). This implies that it is Lagrangian for the symplectic form on XKun(@M ). We prove that, assuming that M has no toric components on the boundary, any generic component of LK (M ) is The moduli space XKun(@M ), as well as a larger moduli space XK (@M ) where the unipotence condition is dropped, were introduced and studied in [12], generalizing the classical Teichmuller theory. The moduli space XKun(@M ) is birationally isomorphic to Hitchin's moduli space MH for @M in any xed complex structure. Hitchin's moduli space, as a hyperkahler manifold, has played a major role in the study of 4d supersymmetric gauge theories, cf. [21, 22]. In applications it is often important to consider a hyperkahler resolution of singularities in MH . The framing data of XKun(@M ) in (1.2) partially resolves the space of at PGL(K; C)-connections on @M and can be thought of as an algebraic counterpart for the hyperkahler resolution of MH . LeK (M ) was studied in [24], following [25{27]. We will comment on the relation between these works and the current paper at the end of section 1.2. In [12] it was shown that an open part of XK (@M ) is covered by a ne charts XK (@M; t) of speci c type, the cluster Poisson coordinate charts, labeled by 2d ideal triangulations t of @M . Here we similarly glue together LeK (M ) from a ne varieties LeK (M; t3d), labeled by 3d ideal triangulations t3d. To de ne LeK (M; t3d), we use an important tool: the 3The graph of any cluster transformation is K2-isotropic, and hence K2-Lagrangian: this follows immediately from the basic relation of cluster transformations to the Bloch complex [17, Sec 6], and provided rst general examples of K2-Lagrangians. In particular, when M is a cobordism of triangulated 2d surfaces generated by ips of the triangulation, this implies that LK(M ) is a K2-Lagrangian. The K2-Lagrangian property of the cluster transformations is crucial for their quantization. The K2-Lagrangian property of A-polynomials was discussed in [18, 19], and argued to be necessary for quantization in [20]. 1) octahedra to form an AK 1 tetrahedron (for K = 4). hypersimplicial K-decomposition of an ideal triangulation, or K-decomposition for short. j ( gure 2). To each octahedron we assign a triple of \crossratio parameters" z; z0; z00 with zz0z00 = 1, and the variety LeK [M; t3d] is cut out from z00 + z 1 = 1 : 1 N K(K2 j , we propose that These gluing equations are analogues of Thurston's gluing equations from hyperbolic geometry. Symplectic properties of the gluing equations (which we prove in some cases) provide one of the ways to prove that LK (M ) XKun(@M ) is K2-isotropic. Some features of the cluster coordinate charts for XK (@M ) were recently generalized by the spectral networks of [28, 29]. The generalization uses the geometry of K-fold spectral covers of the surface @M . It would be very interesting to describe the variety LeK (M ) in a similar way, using K-fold covers of the 3-manifold M , perhaps along the lines of [30, 31]. Coming back to 3d gauge theories: we use the structure of the space LeK (M ), and in particular the symplectic properties of gluing equations, to formulate a de nition of TK [M ] = in direct analogy to (1.1). Here T [ j ] is the canonical theory of a single chiral multiplet (identical to T2[ ]), and gluing is again implemented by superpotentials and gauging. Thus TK [M ] acquires a description as the infrared limit of an abelian Chern-Simons-matter theory. The invariance of TK [M ] under 2{3 Pachner moves follows (again, conjecturally) from basic 2{3 moves for the octahedra. The theory TK [ ] associated with a single tetrahedron at higher K is no longer so 1) chiral multiplets. For the rst few K, we nd in section 6: 10 chirals + degree-six superpotential 20 chirals, U(1)2 gauge group, W = P(four monopole ops) In general, for all K 5, the gluing rules force TK [ ] to have both a nontrivial superpotential and a nontrivial gauge group. Constructing theories TK [M ] for all K 2 allows us to study some features of the K ! 1 limit. In particular, by relating degrees of freedom of theories TK [M; ] to the volume of PGL(K; C)-connections on M , we will nd evidence of the large-K scaling # degrees of freedom of TK [M ] This agrees beautifully with predictions from M-theory and holography [32, 33]. We note, however, that K does not appear as a continuously tunable parameter in TK [M ], such as the rank of a gauge group. (This is obvious, for example, in (1.6).) Recently a number of 3d theories that do allow analytic continuation in K were studied in [34]; it would be very interesting to relate these to the present constructions. As we review momentarily, part of the 3d-3d correspondence relates partition functions of TK [M ] on spheres (and more generally lens spaces) to partition functions of ChernSimons theory on M itself, with complex gauge group PGL(K; C). Thus, the combinatorial construction of UV Lagrangians for theories TK [M ] implies a combinatorial construction of partition functions for PGL(K; C) Chern-Simons theory on M . Alternatively, one may simply say that the symplectic properties of PGL(K; C) gluing equations allow a systematic quantization of the pair of spaces LK (M ) XKun(@M ), generalizing [9]. The result is a construction of a subsector of PGL(K; C) Chern-Simons theory that generalizes a circle of ideas initiated in [7, 11] for K = 2.4 In the remainder of this introduction, we review some basic features of the 3d-3d correspondence between observables of TK [M ] and the geometry of at PGL(K; C)-connections on M ; then we provide a more detailed summary of our main mathematical results. The 3d-3d correspondence The 3d-3d correspondence was rst conjectured in [38], and has since been studied in a multitude of papers. Some of its fundamental ideas were developed in [1, 30, 39], and the rst physical proofs of the correspondence appeared in [40, 41]. (The recent review [42] contains further discussion and references.) New and interesting aspects of the correspondence are still being developed, cf. [35{37, 43], which appeared after the rst version of The basic idea behind the 3d-3d correspondence is that the compacti cation of the 6d (2; 0) theory of type AK 1 on R M , with a topological twist along M that preserves may be de ned as the e ective eld theory on a stack of K M5-branes wrapping M in the M-theory background R T M . More generally, compacti cation of the 6d (2; 0) theory (or of 5-branes) on a d-manifold Xd should produce an e ective theory TK [Xd] in 4The PGL(K; C) Chern-Simons partitions constructed with the current methods have limited TQFT-like properties under cutting-and-gluing operations that preserve a triangulation and some additional structures. It is nevertheless an open problem to de ne PGL(K; C) Chern-Simons theory (for any K TQFT, with complete cutting-and-gluing rules. This has recently been emphasized in [35{37], along with new proposals for resolving the problem. coupling to TK [@M ] rank of avor symmetry group twisted superpotential on C SUSY parameter space on C avor line operators identities for line operators partition function on lens space L(k; 1) holomorphic blocks on C q S1 classical GC Chern-Simons functional quantized algebra of functions on XKun(@M ) quantization of LK (M ) GC Chern-Simons theory on M at level k GC Chern-Simons theory on M at level k = 0 analytically cont'd CS wavefunctions on M 6 d. Notable examples include the 2d-4d correspondence developed in [22, 44{46] and many other works, and the 4d-2d correspondence introduced in [47]. It is often expected, but so far not strictly proven, that the theory TK [M ] only depends on the topology of M , along with a bit of boundary data discussed below. From a eldtheory perspective, this statement crucially depends on de ning TK [M ] as the infrared limit of the compacti cation of the 6d (2; 0) theory on M . Away from the infrared, all bets are o . Indeed, one may view a particular triangulation (or K-decomposition) of M as a choice of discretized metric, and it is clear even in the simplest examples (such as SQED ' XYZ associated to the Pachner move of gure 1) that two triangulations only de ne the same theory after owing to the infrared. In the infrared, the expectation5 is that TK [M ] only depends on a choice of uniformized, constant-curvature metric on M . If M happens to be hyperbolic, then this choice is unique, whence metric invariants are actually topological invariants. However, if M is not hyperbolic and (more importantly) the constant-curvature metric is not rigid, then TK [M ] should admit additional parameters corresponding to metric deformations. We will not encounter this latter scenario in this paper. Various properties of the parent (2; 0) theory imply relations between observables in the theory TK [M ] and the geometry of at GC connections on M , where GC is a complex group with Lie algebra AK 1 .6 Some of these relations are summarized in table 1. An important aspect to mention is that when M has a boundary (which will be true throughout this paper), the theory TK [M ] is not truly an isolated 3d theory, but rather a boundary condition for a 4d theory TK [@M ]. By the 2d-4d correspondence, the rank of the gauge group of TK [@M ] is half the dimension of the space of at connections on the boundary XKun(@M ). To obtain an isolated 3d theory, we must choose a weak-coupling 5See some relevant discussion in the Introduction of [42]. The statement about uniformization comes in part from holography. It was analyzed carefully by [48] for the analogous 2d-4d correspondence, though it has yet to be fully developed for the 3d-3d correspondence. 6Some subtle discrete choices can modify the theory TK[M ] and the appropriate form of the group GC. See, for example, [49]. limit7 for the theory TK [@M ] [15]. This choice amounts to a polarization denoted TK [M; ]. It acquires a avor symmetry of rank 12 dim XKun(@M ). a ne charts of the complex symplectic space XK (@M ). The resulting 3d theory should be By the 2d-4d correspondence, putting the 4d theory TK [@M ] in a background with angular momentum quantizes its algebra of line operators (see [50] and similar ideas in [51]). An example of such a background is (C R+, where (C q S1) := f(w; t) 2 C L^K (M ) is a quantization of the Lagrangian LK (M ). [0; 1]g=(w; 1) (qw; 0). The resulting quantum algebra X^Kun(@M ) is a quantization of for K > 2. When acting on a 3d boundary theory TK [M; ], line operators of TK [@M ] satisfy additional relations. These relations de ne an X^Kun(@M )-module L^K (M ) whose characteristic variety (or \classical limit") should be LK (M ). In other words, the module conjectured in [1, 54], and (after the rst version of this paper) in [55] that these partition functions are equivalent to partition functions of GC Chern-Simons theory at level k on (in fact, two commuting copies of it) acts on the L(k; 1)b partition function of TK [M; ] in such a way that the relations of the module L^K (M ) are satis ed. Correspondingly, a quantization of the Lagrangian variety LK (M ) should annihilate a GC Chern-Simons partition function on M [7]. A somewhat simpler observable of TK [M; ] is the e ective twisted superpotential of the theory on C parameter space that also appears on the l.h.s. of table 1. As a concrete example, let's U(1) avor symmetry (see section 6.1). Its e ective twisted superpotential on C Wf(z) = Li2(z 1) : The dilogarithm here is a standard 1-loop contribution to the twisted superpotential coming from Kaluza-Klein modes of the 3d chiral [56]. The complex parameter z is the twisted mass of , associated with its U(1) symmetry. The supersymmetric parameter space is then de ned by L2[ ] := = z00 = z00 + z 1 = 1 : By the 3d-3d correspondence, this should correspond to the Lagrangian L2[ ]. Indeed, (1.9) is identical to (1.4), and we recognize it as the canonical K2-Lagrangian subvariety of X2[@ ] ' (C )2, with symplectic form d log z00 ^ d log z and K2 form z00 ^ z. The superpotential Wf(z) itself is interpreted as the complexi ed hyperbolic volume of an ideal hyperbolic tetrahedron. 7More precisely, we mean here a weak-coupling limit for the abelian Seiberg-Witten theory that describes TK[@M ] on its Coulomb branch. Such a weak-coupling limit always exists. Now consider a bipyramid as in gure 1. Triangulating by two tetrahedra (with a 2 SQED, i.e. a U(1) gauge theory with two chiral multiplets. Its twisted superpotential is 1 2 Wf(2)(x; y) = Li2(s 1) + Li2(s=x) + log(s)2 + log s(log y Each dilogarithm comes from a chiral multiplet of SQED (geometrically: from a tetrahedron), and the e ect of the U(1) gauge group is to extremize the superpotential with respect to a dynamical variable s. In contrast, triangulation by three tetrahedra gives the XYZ model, whose twisted superpotential is Wf(3)(x; y) = Li2(x 1) + Li2(y 1) + Li2(xy) + This time there is no minimization (since there is no dynamical gauge group), but the arguments of the dilogarithms are constrained by the superpotential of the XYZ model, so that their product is one. Geometrically, the constraint is associated with the internal edge of the triangulation. ow. (Supersymmetry protects them from quantum corrections beyond one-loop.) Then the infrared equivalence of SQED and the XYZ model implies that we should have Wf(2)(x; y) = Wf(3)(x; y) ; with appropriate choices of branch cuts. Mathematically (1.12) is the 5-term identity for the dilogarithm. The superpotentials Wf(2) and Wf(3) simply compute the complexi ed hyperbolic volume of the bipyramid in two di erent ways. The SUSY parameter space of algebraic variety that may equivalently be computed from either Wf(2) or Wf(3). Heuristically, the theory TK [M ] should be thought of as a categorical analogue of the motivic PSL(K; C)-volume of M , or (equivalently) the PSL(K; C)-class of M in the Bloch to [1]. In brief, the Bloch group B2(F ) of a eld F is the abelian group generated by elements fzg2 with z 2 F f1g = F f0; 1g, modulo ve-term relations = 0 ; regular component of) the moduli space of framed at PGL(K; C)-connections on M . B2(F ), obtained by summing the parameters zi associated to each octahedron in a Kdecomposition of M , viewing these parameters as functions on LeK (@M ). The relations ensure that the element fM g2 is invariant under (generic) 2{3 moves. The real PGL(K; C)-volume of M is obtained by applying the Bloch-Wigner dilogarithm map (4.57) to the motivic volume. We may compare this to the theory TK [M; ]. It is de ned by taking a tensor product of octahedron theories and adding some interactions to enforce the gluing | similar to the formal sum of octahedron parameters de ning fM g2. Just like fM g2, the theory TK [M; ] is invariant under (generic) 2{3 moves. The real PGL(K; C)-volume of M may be recovered (roughly) by evaluating the real part of the twisted superpotential of the theory on C S1; but by instead evaluating partition functions of TK [M; ] one also recovers many quantum invariants, as in table 1. Finally, we note that while in this paper we construct theories TK [M ] as abelian Chern-Simons-matter theories, one might generally expect them to have dual non-abelian descriptions as well, related to the su(K) symmetry of the underlying 6d (2,0) theory. In principle, a non-abelian construction of TK [M ] would arise from decomposing M into pieces (e.g. handlebodies) glued along smooth surfaces, as apposed to the triangulations and K-decompositions of this paper. Some isolated examples of non-abelian TK [M ] have recently appeared (after the initial version of this paper) in [36, 37, 43], but a systematic A mathematical perspective Given a 3-manifold M with boundary, consider the moduli space LocK (@M ) of at PGL(K; C)-connections on the boundary @M . This space is symplectic, with the symplectic form at a generic point given by the Atiyah-Bott/Goldman construction. The moduli space of at connections on the boundary @M that can be extended to M is expected to be a Lagrangian subvariety LK (M ) LocK (@M ). One can quantize the symplectic space LocK (@M ) by de ning a non-commutative q-deformation of the -algebra of regular functions on LocK (@M ), and constructing its -representation in an in nite-dimensional Hilbert space HK (@M ). This has nothing to do with a 3-manifold: the problem makes sense for any oriented 2d surface C. Assuming that the surface is hyperbolic, that is ( ) < 0, and has at least one hole, the quantization was done in [12, 13, 17]. It generalizes the quantization of Teichmuller spaces [52, 53], related to the SL(2) case. The next goal is to quantize the Lagrangian subvariety LK (M ). By this we mean de ning a line in HK (@M ) that must be annihilated by the q-deformations of the equations de ning the subvariety LK (M ). In the SL(2) case, this was discussed in a series of physics papers [7, 9{11, 20, 54, 57, 58], and in the closely related mathematical works [59{62]. (An alternative mathematical approach to the quantization of LK (M ) has been proposed using the theory of skein modules, cf. [63, 64].) Subsequent to the initial version of this paper, further constraints on a consistent quantization were uncovered in [55, 65]. The following problem motivated this project. We would like to have a local procedure for the quantization of the moduli space of at PGL(K; C)-connections on 3-manifolds, by decomposing the manifold into tetrahedra, quantizing the tetrahedra, and then gluing the quantized tetrahedra. This approach for SL(2) was implemented in [9] by introducing the phase space and the Lagrangian subvariety related to a hyperbolic tetrahedron. However, already for SL(2), an attempt to understand even the Lagrangian subvariety itself as a moduli space of at connections immediately faces a serious problem: Any at connection on a tetrahedron is trivial, so the corresponding moduli space is just a point, and thus cannot produce a Lagrangian subvariety in the phase space assigned to the boundary, which in this case is a four The problem is that the tetrahedron has trivial topology, while the moduli space of at connections is a topological invariant, and hence also becomes trivial. We suggest a solution to this problem based on the following idea. We consider moduli spaces of at connections on 3-manifolds with framings. A framing amounts to introducing invariant ags on each of the so-called small boundary components, which we de ne below, invariant under the holonomy around the component. This, remarkably, allows one to produce the missing Lagrangian subvariety for the tetrahedron. The corresponding moduli spaces are de ned for arbitrary admissible manifolds, and can be \symplectically" glued from the ones assigned to the tetrahedra. So we use the invariant ags to localize connections to tetrahedra. This notion of framing generalizes the key idea used in [12] to introduce cluster coordinates on the moduli space XK (C) of framed at PGL(K)-connections on a surface C with punctures: invariant ags were invoked to localize at connections on ideal triangles. In the three-dimensional case, a related notion of \decorations" was used by [25{27] to study representations of 1(M ). In section 2.1 we start with a careful discussion of a class of 3-manifolds with boundary, which we call admissible 3-manifolds, which are glued from truncated tetrahedra. Since the boundary faces of truncated tetrahedra are of two di erent types, triangles and hexagons, the boundary of the manifold obtained by gluing them along the hexagonal faces also has two kinds of boundary components, big (formed by the unglued hexagons) and small (formed by the triangles). We say that such a manifold is admissible if 1) the fundamental groups of the small boundary components are abelian; 2) the small boundary components are not spheres; and 3) the big boundary components have negative Euler character. The second and third conditions are technical, and simply allow us to avoid stacks in our constructions. The rst condition, however, is dictated by the notion of framing: every vector bundle with at connection on an admissible 3-manifold admits at least one choice of framing. Indeed, if the fundamental group of the small boundary is abelian, then the holonomies of a at connection on the small boundary all commute with each other, and a family of commuting operators in a vector space VK always has an invariant ag. (Typically there are just K! invariant ags; in a basis in which all operators are diagonal with di erent eigenvalues, choosing a ag amounts to ordering the basis.) So by adding a framing to a at vector bundle we enlarge the moduli space by taking its cover and partially resolving its singularities, rather than cutting it down. In section 2.3 we discuss the moduli spaces presented in (1.2): the space LeK (M ) at PGL(K; C)-connections on M , the space XKun(@M ) of framed unipotent im LeK (M ) ! XKun(@M ) . The simplest example of an admissible 3-manifold is the tetrahedron , whose boundary is understood as a sphere with four punctures. We associate to the boundary the moduli at PGL(2; C)-connections on the four-punctured sphere with unipotent monodromy around the punctures. It is a two-dimensional symplectic space. Its Lagrangian subspace L2( ) consists of the connections that can be extended to the bulk with framings at the four vertices at the boundary. Since any connection on the ball is trivial, the only data left is the four ags, which in this case amounts to a con guration of four lines in a two-dimensional space V2. The resulting Lagrangian pair is our main building block. It has a natural Zariski open part, which deserves a special P@ , described in coordinates as follows: P@ := fz; z0; z00 2 C j zz0z00 = L := fz; z0; z00 2 P@ j z00 + z 1 1 = 0g: (1.15) The natural compacti cation of the symplectic space P@ space (i.e. a stack) X2un(@ ) should help to deal with non-generic framings. = C C given as a moduli at connections from octahedra. Our rst major goal is to build the moduli space LeK (M ) of framed at PGL(K; C)-connections on an admissible 3-manifold M out of these building blocks. To achieve this, we choose an ideal triangulation t3d of M , and a further hypersimplicial K-decomposition of each tetrahedron in t3d ( gure 2). We show in section 3 that this K-decomposition has precisely the combinatorial and geometric data that we need to describe LeK (M ) and its projection LK (M ) to the boundary moduli space. We use the framing data on a at connection to assign to each tetrahedron in the triangulation t3d a con guration of four ags at its vertices. One can think of these as four ags in the K-dimensional space VK , considered modulo the action of PGL(K; C).8 The K-decomposition of tetrahedra is used to construct various generalized cross-ratios that determine the con guration of four ags. These generalized cross-ratios correspond precisely to the parameters zi; zi0; zi00 assigned to the vertices of each octahedron the left of gure 2). Once the parameters are identi ed with cross-ratios, they naturally hedron parameters sitting at any vertex v of the K-decomposition is trivial. These gluing constraints generalize Thurston's gluing equations for an ideal hyperbolic triangulation. These projective geometry constructions can be restated as follows. Let t3d be an ideal triangulation of M , inducing an ideal triangulation t of the big boundary. We construct a Zariski-open subset 2d = dimCXKun(@M ) of the space of framed unipotent at connections on @M , generalizing the cluster coordinate charts de ned by [12]. We write the C -coordinates of XKun(@M; t) as monomials of the 8It is important to notice that the sets of con gurations of objects of any kind associated with a vector space depend only on the dimension of the space and not on the choice of vector space itself; thus con gurations assigned to isomorphic vector spaces are canonically isomorphic. octahedron parameters zi; zi0 (where i ranges over the octahedra). This just means that we get a projection : QiN=1 P@ i ' (C )2N is not quite canonical in the presence of small-torus boundary components, but canonical if they are absent. We de ne an open subset of the space Le(M ) by intersecting the product of octahedron Lagrangians L i with the gluing constraints LeK (M; t3d) = QiN=1 L i \ fcv = 1gvertices v : By showing that a set of octahedron parameters that satis es the gluing equations can be used to uniquely reconstruct a framed at connection on M (section 3.3), we prove that Theorem 1.1 (Theorem 3.1 (page 44)). The intersection (1.18) parameterizes a Zariskiopen subset of the space of framed at connections on M . P@ i \ fcv = 1g ! XKun(@M; t): It is an open subset of the space LK (M ). Theorem 3.1 implies that changing the bulk (t3d) and boundary (t) triangulations amounts to birational transformations of the spaces LeK (M; t3d), LK (M; t3d) Changes of bulk triangulation are generated by 2{3 moves, which can be decomposed into elementary 2{3 moves acting on octahedra in a K-decomposition, as described in section 3.5. Changes of boundary triangulation correspond to cluster transformations on XKun(@M; t) [12]. Taking the union of the spaces assigned to a given bulk triangulation t3d over the set of the bulk triangulations compatible with a xed boundary triangulation t, we obtain a triple LeK (M; t) ! LK (M; t) It depends only on t. One may then vary t to eliminate the dependence on boundary triangulation. We emphasize that even then we do not cover the whole moduli spaces LeK (M ) and XKun(@M ). In particular, only components of LeK (M ) corresponding to irreducible at connections on M will be detected. Symplectic gluing. Our next major goal is to understand the symplectic properties of LK (M ) and XKun(@M ). They are summarized in the Symplectic Gluing Conjecture (Conj. 4.1, page 59): for any admissible 3-manifold M with bulk triangulation t3d and corresponding big-boundary triangulation t, we expect that The moduli space XKun(@M; t), equipped with the canonical complex symplectic form, is isomorphic to a holomorphic symplectic quotient of the product of octahedron d log zi0. Precisely, it is the symplectic quotient for the Hamiltonian (C ) whose Hamiltonians are given by the gluing monomials cv. Thus QiN=1 P@ i \ fcv = 1g the product Lagrangian QiN=1 L i QiN=1 P@ i under the symplectic quotient (1.21). XKun(@M; t) is a Lagrangian subvariety; it coincides with the image of : QiN=1 P@ i ! XKun(@M; t) from (1.17). We prove that the gluing monomials cv Poisson commute, and that they Poisson commute with the pullbacks of the cluster coordinates on XKun(@M; t), given by the monomials in the octahedron parameters. Thus the rst claim (1.21) of the Symplectic Gluing Conjecture reduces to the claim that exactly N d of the equations cv = 1 are independent. An easy Euler-characteristic count shows that in the presence of t small-torus boundaries, the total number of gluing monomials cv equals N 1)t; therefore, it remains to show that there are exactly (K 1)t relations among the gluing monomials. Also recall the map from (1.19). The claim that (XKun(@M; t); ), as a holomorphic symplectic space, is the reduction of the product of octahedron spaces means that = PiN=1 d log zi ^ d log zi0 cv=1 : Since the number of monomials cv is N 1)t, the dimension of Le(M ; t3d) is at least d (K 1)t, see (1.18). The second claim of the Symplectic Gluing Conjecture is that the image of Le(M ; t3d) under the projection by the Hamiltonian ows of the Hamiltonians cv is Lagrangian. When the only small-boundary components of @M are discs, theorem 4.2 guarantees that the image of Le(M ; t3d) is isotropic; thus it would su ce to show that its We refer to the relationship between the pair LK (M; t3d) ementary octahedron pairs L i P@ i as symplectic gluing. Since our parameters are assigned to the octahedra of the K-decomposition of M , the Symplectic Gluing Conjecture The Lagrangian pair LK (M ) XKun(@M ) is obtained by symplectic gluing of the elementary Lagrangian pairs (1.14) parametrized by the octahedra of the K-decomposition of M corresponding to an ideal triangulation The symplectic gluing is studied in sections 4 and 5 from two di erent perspectives. In both settings, it is very natural to promote the notion of a Lagrangian subvariety to the much stronger notion of a K2-Lagrangian. Let us explain what this means. K2-Lagrangians. Let F := F f0g be the multiplicative group of a eld F . Recall that with a ^ b = 2F is the abelian group generated by elements of the form a ^ b, a; b 2 F , F ^ F by the subgroup generated by Steinberg relations (1 K2(F ) = F ^ F =f(1 Next, let X be a complex algebraic variety. Denote by C(X) the eld of rational functions on X. Then there is a homomorphism to the space logarithmic singularities on X: l2og(X) of holomorphic 2-forms with d log : C(X) ^ C(X) The map kills elements (1 depends only on its class in K2(C(X)). f ) ^ f , and so the image of an element W 2 C(X) ^ C(X) It was proved in [12] that the symplectic form on the moduli space XKun(C) of unipotent at connections on a 2d surface C can be upgraded to its motivic avatar, a class W in K2 of XKun(C). The symplectic form is recovered as d log(W). From our current perspective, the construction of [12] applies directly to the big boundaries of admissible 3-manifolds, and we explain the simple generalization to small boundaries in sections 4.3. (An even wider class of examples is provided by cluster A-varieties in [17].) This motivates the following de nition. De nition 1.1. Let X be a complex variety with a class W in K2(C(X)) such that d log(W) is a symplectic form at the generic part of X. A subvariety L X is called a K2-Lagrangian Examples: 1. The space P@ has the symplectic form d log z ^d log z0. It lifts to a symbol z ^ z0, which is invariant, up to 2-torsion, under the cyclic shift z; z0; z00 7 ! z0; z00; z. The curve L is a K2-Lagrangian, since z ^ z0 restricts to (1 2. The graph of any cluster transformation A ! A of a cluster A-variety A is a K2-Lagrangian subvariety of the product A A, see section 6 of [17]. Theorem 1.2 (Theorem 4.2). (i) If @M has only big boundary and small discs, any generic component of LK (M ) is a K2-isotropic subspace of XKun(@M ), i.e. the restriction of the K2-class W is zero. (ii) If M is a convex polyhedron, then LK (M ) is a K2-Lagrangian subvariety of XKun(@M ). Part (ii) of theorem 4.2 follows from part (i) and an easy dimension count. Indeed, LK (M ) is just the con guration space of ags parametrized by vertices of the polyhe LK (M ) = GnBfvertices of M g; dim LK (M ) = v(M ) dim B dim XKun(M ) = dim(G)(v(M ) rk(G)v(M ) = 2dim LK (M ): Since the cluster coordinates on XKun(@M; t) and the gluing constraints cv are monomials of the octahedron parameters zi; zi0, a proof of Conjecture 4.1 implies a K2-analog W = PiN=1 zi ^ zi0 cv=1 (mod torsion) : It would immediately imply that LK (M; t3d) XKun(@M; t) is K2-isotropic. In section 4 we prove theorem 4.2 by using the canonical map of complexes [66]: : Complex of generic con gurations of decorated ags in VK ! Bloch complex. To de ne it, we de ne rst a closely related homomorphism of complexes, where the notation will be explained momentarily: Here the abelian group A(nK) is given by formal integral linear combinations of the con gurations of n + 1 generic decorated ags in VK . The di erential d is the standard simplicial di erential. This way we get the complex of generic con gurations of decorated Let us de ne the bottom complex. We denote by Z[F with a basis fzg, where z runs through the elements of F f1g] the free abelian group 1 . The homomorphism in (1.27) is de ned by setting fzg := (1 z) ^ z. So its cockerel is the group K2(F ) f1g] be the subgroup generated by the ve-term relations So the map of complexes (1.27) induces a map of complexes (1.26). where (z0; : : : ; z4) run through generic con gurations of 5 points on P1(F ), and r( ) is the cross-ratio. One shows that the restriction of the map to the subgroup R2(F ) is zero. So we get the bottom complex, where i is the natural embedding. The Bloch group B2(F ) is the quotient of the group Z[F f1g] by the ve-term Since (R2(F )) = 0, the map descends to the Bloch group, and we get the Bloch complex : B2(F ) := Z[F Li2(z) := Im Li2(z) + log(1 It is a single-valued function, well de ned for all z 2 C. So it de nes a group homomorphism It satis es the Abel ve-term relation, i.e. its restriction to the subgroup R2(C) is zero. Therefore it gives rise to a group homomorphism The Bloch group is famously related to 3d hyperbolic geometry, in the following way. Consider the scissor congruence group P2 of ideal hyperbolic polyhedra. It is an abelian group with the generators assigned to ideal oriented hyperbolic tetrahedra I(z1; z2; z3; z4) with the vertices z1; : : : ; z4. The generators [I(z1; z2; z3; z4)] satisfy two kinds of relations. First, the cutting and gluing relation: cutting an ideal hyperbolic bipyramid in two di erent ways into 2 or 3 ideal tetrahedra amounts to the same element of P2. Second, changing [I(0; 1; 1; z)]. Denote by B2(C) the aniinvariants of the action of complex conjugation on the group B2(C). Then there is a canonical group isomorphism fzg2 ! [I(0; 1; 1; z)]: Finally, the map shows up in the formula for the di erential of the dilogarithm: dLi2(z) = log j1 log jzjdarg(1 Let us return nally to the homomorphism of complexes (1.27). To de ne it, one uses a key construction of [66] relating con gurations of ags to con gurations of vectors: Complex of generic con gurations of decorated ags in VK ! biGrassmannian complex : Given a single generic con guration of (n + 1) decorated ags in VK , one assigns to it a of simplices is crucial here: viewing the (n + 1) ags as assigned to the vertices of an n, each hypersimplex in the K-decomposition of gives rise to a single point in a Grassmannian that matches the type of the hypersimplex. Combining (1.33) with the homomorphism of complexes, de ned in [67], which we review in section 4.2.3: biGrassmannian complex we arrive at the homomorphism of complexes (1.27), and hence (1.26). The homomorphism of complexes (1.26) controls a number of features of the geometry at connections on an admissible three-manifold M and its boundary. Let us elaborate on this. First, choosing compatible triangulations t3d and t of M and its big boundary, we assign to points in the moduli spaces XKun(@M; t) and LeK (M; t3d) elements in the complex of generic con gurations of decorated ags in VK . For example, we may start from a generic point of LeK (M; t3d), representing a framed at connection on a triangulated manifold M . We pick a decorated ag representative for each ag of the framing: we can do this thanks to the unipotence condition. Then we assign to each tetrahedron of the triangulation the con guration of four decorated ags obtained by the restriction of the framed connection to the tetrahedron. The formal sum of these con gurations over all tetrahedra of the triangulation is an element of the group A(K) which we assign to the framed at connection. Similarly, we assign to a generic point of XKun(M; t) an element of A(2K). Then we nd: 1) The component 3 of the map (1.27) was used in [12, section 15] to de ne the K2class W on the space XKun(C) for a 2d surface C. It provides a K2-class on the space 2) Theorem 4.2 tells that the restriction of the K2-class W on XKun(@M ) to LK (M ) is zero. The component 4 of map (1.27) tells how exactly it becomes zero. Precisely, given an ideal triangulation t of @M , the class W has a natural lift to section 15]. The map 4 presents its restriction to LK (M ) as a sum of Steinberg zi) ^ zi. These zi's are just our octahedron parameters. They are precisely the terms of the map 4 in (1.27). The very existence of the map of complexes (1.27) implies that the image Pfzig2 of the map 4 in the Bloch group B2(C) is independent of the choice of the balk triangulation t3d. We call it the motivic volume. Applying the dilogarithm homomorphism (1.31) to it we de ne in section 4.4 the volume of a generic framed PGL(K; C)-connection on M . We stress that although the motivic volume of a framed at connection does not depend on the balk triangulation t3d, it does depend on the boundary triangulation t. Using formula (1.32) and the commutativity of the last square in (1.27), we get a formula for the variation of the volume of generic framed at connections on M . It generalizes the Neumann-Zagier formula for variation of volumes of hyperbolic 3-manifolds with toric boundary [8], and the work of Bonahon [68] on hyperbolic 3-manifolds with geodesic boundary. The PGL(3; C)-analog of the variation formula was beautifully established by Bergeron-Falbel-Guilloux in [23]. If the big boundary is absent, the motivic volume lies in the kernel of the Bloch complex map (1.29), and thus de nes an element of Ki3nd(C) Q due to a theorem of Suslin [69]. This follows immediately from the construction and the fact that the map (1.26) is a map of complexes. The value of the regulator on it is the ChernSimons invariant of the connection. 3) What exactly happens under the 2{3 moves changing the triangulation t3d? In section 3.5 we prove that the PGL(K) pentagon relation can be reduced to a sequence tions. The elementary pentagon relations match the terms of the component 5 of the map (1.26). Simons couplings for the background gauge elds associated with U(1) avor symmetries and with the R-symmetry.31 The gauge theory T(g; ) is built in the following steps. +1 under a background U(1) avor symmetry and R-charge zero, and an additional The Chern-Simons level matrix can be represented as 1=2 : coordinates Zi on P . generators. Then 1. Form the product theory T = T 1 T N . It is a theory of N chirals i, and has maximal abelian avor symmetry U(1)N . The real masses of the U(1)'s (scalars in the background gauge multiplets) can be associated with the real parts of the 2. Apply the Sp(2N; Z) symplectic transformation g to the theory T to obtain a theory avor symmetry. The Sp(2N; Z) action on 3d N = 2 gauge theories was de ned in [92], and can be implemented (e.g.) by factoring g into a. \U-type" generators of the form ing the basis of U(1)N avor currents. If V = (V1; : : : ; VN )T is a vector of U0 U 01T , where U 2 GL(N; Z), act by changbackground gauge multiplets corresponding to the U(1)N symmetry, we send b. \T-type" generators of the form N identity and k is N symmetric, act by adding supersymmetric Chern-Simons terms for the background U(1)N avor symmetry with (mixed) level matrix k. c. \S-type" generators of the form with entry 1 at the i-th spot on the diagonal and zeroes everywhere else, act by gauging the i-th U(1)i symmetry and replacing it with a new topological avor symmetry. A background gauge multiplet for U(1)J is coupled to the gauge multiplet of the gauged U(1)i via a supersymemtric FI term, which is the same thing as a mixed Chern-Simons term with level matrix ( 01 10 ). This Sp(2N; Z) action can be extended to rational matrices Sp(2N; Q) if one allows rescaling of the charge lattice of T . 31More precisely, (g; ) encodes avor- avor and mixed avor-R Chern-Simons couplings. It does not uniquely specify R-R couplings. Such background couplings and their e ect on partition functions and dualities were recently discussed in [86, 87]. 3. Now the theory Te from Step 2 has a U(1)N avor symmetry with real masses associated with the top N rows of g, i.e. to the coordinates Xj and Ck. Change the R-charge assignment of the theory Te by adding i (1 of each U(1)i avor charge to the R-charge. For each component N+i in the bottom half of the shift vector, 1 N+i units of mixed Chern-Simons coupling between the background gauge for the U(1)i avor symmetry and the background gauge eld for the R-symmetry. This interpretation of a ne shifts de nes an action of the full a ne symplectic group ISp(2N; Z) on 3d SCFT's. In particular, the full relations of ISp(2N; Z) are satis ed (after owing to the IR), up to an integer ambiguity in the background R-R Chern 4. Finally, break N d avor symmetries U(1)k associated with the undecorated coordinates Ck. This is done by adding N d operators Ok to the superpotential, charged under each respective U(1)k. We'll review the construction of the operators below. The breaking can be thought of as setting the respective mass parameters Ck ! 0. In the end, this produces a theory T(g; ) with manifest U(1)d avor symmetry. The mass parameters of the d unbroken U(1)'s are associated with the coordinates Xj (i.e. with the decorated rows of (g; )). The rank of the gauge group of T(g; ) depends on exactly how the symplectic transformation g is applied in Step 2. Di erent decompositions into generators lead to slightly di erent UV descriptions that ow to the same thing in the IR. (In the IR the relations of the symplectic group hold.) It is fairly easy to see that a lower bound on the rank of the UV gauge group is given by rank(B). Often the bound can be realized | i.e. there exists a decomposition of g that involves exactly rank(B) S-type generators. Alternatively, if B has maximal rank and is unimodular, there exists a way to apply the a ne-symplectic transformations of Steps 0{3 all at once, cf. [110]. We de ne a theory Te by starting with N chirals i, each charged under an independent dynamical gauge symmetry U(1)i (and having zero R-charge). Thus the gauge group is U(1)N . The avor group is also U(1)N , since each dynamical U(1)i has an associated withpological U(1)J;i. We then specify the full Chern-Simons coupling matrix to be (Here `G' stands for the N dynamical gauge elds, `J ' for the N background avor gauge elds, and `R' for the background U(1)R eld. For instance, the G{J blocks encode FI terms. The G{G blocks encodes dynamical CS terms that couple the otherwise independent gauge elds.) The N avor symmetries of Te correspond directly to the coordinates (Xj; Ck). To obtain T(g; ) from Te we \simply" have to apply Step 4, adding appropriate operators to the superpotential to break the Ck symmetries. Two elements (g; ); (g0; 0) 2 IcR can sometimes de ne theories that are equivalent after owing to the IR, T(g; ) ' T(g0; 0). There are three basic ways in which this can happen. (In the following, we will assume that both T(g; ) and T(g0; 0) possess the chiral operators needed to break avor symmetries in Step 4 of the gauge theory construction; we return to this in section A.5.) First, there are some trivial equivalences. We'll describe them in terms of initial and nal coordinates on a phase space P , as in (A.1). We can permute pairs of initial coordinates (Zi; Zi00) $ (Zi0 ; Zi000) (which corresponds to permuting the N chirals of T(g; )). Similarly, we can permute the nal coordinates Ck. Moreover, we can rede ne the Xj and Pj coordinates by any integer linear combination of the Ck, as long as we simultaneously rede ne the k so that g remains symplectic. We can also rede ne the k by integer linear combinations of the Ck and any multiples of i . In gauge theory, these latter rede nitions of coordinates all correspond to rede ning the avor currents (or background CS levels) for avor symmetries U(1)k that will ultimately be broken. We might also allow ourselves rede ne the Pj by integer linear combinations of the Xj and by multiples of i , while keeping g symplectic. This changes background ChernSimons levels for the avor and R-symmetries of T(g; ), which one is sometimes interested in keeping track of and sometimes not. The second type of equivalence is an \octahedron rotation." In terms of phase-space coordinates, it corresponds to a cyclic permutation Zi ! Zi0 ! Zi00 ! Zi for any xed i ; in other words, should be an IR duality.32 This is an operation on two columns of g, and on . In terms of gauge theory, the cyclic rotation corresponds to taking a chiral i of T(g; ) and replacing it by a U(1) gauge theory coupled to another chiral 0i to obtain T(g0; 0), as discussed in detail in section 6.1. This The third type of equivalence is the algebraic version of an elementary \2{3 move" on octahedra. It replaces (g; ) of rank N and d decorated rows with (g0; ) of rank pairs of initial coordinates, say (Z1; Z100) and (Z2; Z200), and replace them with new initial coordinates (Wi; Wi00)i3=1, where Z1 = W200 + W30 ; Z100 = W100 + W20 ; Z2 = W20 + W300 ; Z200 = W30 + W100 ; 32As usual, this statement must be taken with a grain of salt. We are assuming that the mirror symmetry between a free chiral T ; Z and a free \vortex" T ; Z0 , as in (6.2){(6.3), implies IR duality between any gauge theories T(g; ); T(g0; 0) in which the free chiral and the free vortex (respectively) are embedded. Arguing this carefully may require a subtle interchange of limits of IR ow. The same comments apply to the 2{3 moves below. rows (C0; 0) = (W1 + W2 + W3 Wi00. We also add a new undecorated pair of 2 i; W100). The new (g0; 0) can be worked out in a straightforward manner. In the opposite direction, a 3 ! 2 move can only be applied to an element (g0; 0) that is already the image of a 2 ! 3 move. In particular, an undecorated The interpretation of the 2{3 move in gauge theory was reviewed in section 6.2. To apply a 2 ! 3 move, we take any pair of chirals 1 ; 2, think of them as an embedded SQED (6.9). In the opposite direction, we isolate any three chirals coupled by a cubic superpotential, treat them as a copy of the XYZ model (6.13), and replace them with a move do not necessarily preserve all of the operators Ok associated with additional broken symmetries. In order to hope that T(g; ) and its putative image T(g0; 0) are truly IR dual, both of them must independently contain all the necessary Ok's. Algebraically, we note that the Angle Constraint (A.4) is preserved by all three types equivalences we have just described. In contrast, the Superpotential Constraint is preserved by the rst and second types, but not by all 2{3 moves. A preliminary guess for a set that from abelian theories in the UV) is (g; ) 2 IcR (g; ) satisfy the Angle and Superpotential Constraints modulo equivalences IR := R := 3d N = 2 abelian Chern-Simons-matter theories with any U(1)R{preserving superpotentials modulo IR duality : In de ning IR, we only quotient out by equivalences (including 2{3 moves) that do preserve the Superpotential Constraint. The potential correspondence IR ' R deserves much The existence of operators Ok For completeness, let us brie y describe the construction of the operators Ok. We assume that we have followed Steps 0{3 of the gauge theory construction to build a theory Te, associated with symplectic data (g; ), and we want to apply Step 4. If (g; ) satis es the Angle and Superpotential Constraints, then for every symmetry U(1)k that must be broken there is another pair (g; )k, related to (g; ) by \octahedron rotations," such that the corresponding coordinate Ck takes the form (A.6). The pair (g; )k de nes a theory Tek after Step 3 that is mirror symmetric to Te. In Tek one de nes O^k = a simple product of elementary chiral elds of the theory. (Recall that the nonnegative integers!) Because of the a ne term 2 i in (A.6), this operator is guaranteed to have UV R-charge 2. By using the mirror symmetry between Te and Tek, one can then \pull back" the operator O^k to an equivalent chiral operator Ok in Te, charged precisely under the desired U(1)k symmetry. This can be repeated for each of the N that must be broken. Then the nal superpotential of Tg; takes the form PN k=1 Ok. Associated invariants Finally, let us touch upon some mathematical objects associated with an element (g; ) 2 IcR, and their physical signi cance. All of these objects are invariant under the rst two types of equivalences presented in section A.4. However, further restrictions are sometimes necessary to ensure invariance under 2{3 moves, such as the non-degeneracy discussed in section 5.5.1. Phase space and K2-Lagrangian. We explained how to build phase spaces and classical Lagrangian submanifolds (and their geometric signi cance) in section 5. The phase space is a symplectic reduction P(g; ) = P and has the holomorphic symplectic structure whereas the putative Lagrangian submanifold L(g; ) P(g; ) is the image of the product = fzi00 +zi 1 this means inverting the transformation (A.1) to re-write zi; zi00 as Laurent monomials in exponentiated phase space (with C coordinates xj and pj ), so long as L the moment maps, and to slices of P(g; ) at generic constant xj . The Lagrangian submanifold is the supersymmetric parameter space for the theory on S1 at nite radius is governed by a twisted superpotential Wf(X1; : : : ; Xd; ). It is the function of complex twisted masses that correspond to the decorated position coordinates of P(g; ), as well as dynamical twisted-chiral elds . The Lagrangian equations are is transverse to @Zi @W~ =@ =0 = pi : We saw in section 5.5.1 that L(g; ) is invariant under elementary 2{3 moves so long as the transversality condition is preserved. (In particular, it was necessary that at generic In terms of gauge theory, transversality means that T(g; ) has isolated massive vacua on S1. The vacua correspond to the points p( )(x) on L(g; ) at xed mass parameters x. and to slices at constant x, as above. Then by using the equations zi00 + zi 1 Volume. Suppose that the product Lagrangian L is transverse to the gluing constraints 1 = 0 and the exponentiated form of (A.1) we can express zi and zi00 as rational functions on L(g; ) | i.e. rational functions of x; p. (Geometrically, we would be solving for octahedron parameters given xed boundary coordinates on @M .) The real volume function associated with (g; ) can be de ned as Vol(g; )(x; p) = volume is a function on L(g; ), and is invariant under algebraic 2{3 moves (that preserve transversality) due to the 5-term relation for Li2(z). de ned modulo 6 Z using methods of [70] (or as in [110, Sec 5.2]). The data (g; 2) 2 ISp(2N; Z) actually allows us to promote (A.16) to a complex volume For the theory T(g; ), the volume Vol(g; )(x; p( )(x)) is the free energy density (the value of Wf) in the vacuum p( )(x) on C S1. As we already observed in section 7.2.2, the volumes of particular solutions p( )(x) dominate the asymptotics of various partition functions of T(g; ). Quantum Lagrangian. We reviewed in section 5.4 how to use the data (g; ) to promote a classical Lagrangian L(g; ) to a left ideal of quantum operators L^(g; ). The basic idea promote (A.1) to linear relations among quantum operators with a canonical correction ! (i + ~=2) on the r.h.s. . This then allows the left ideal z^i00 + z^ 1 i Ward identities for line operators satis ed by partition functions of T(g; ). 1 iN=1 to be Quantum partition functions. The element (g; ) gives rise to several closely related wavefunctions. Geometrically, they are versions of Chern-Simons partition functions on a 3-manifold M ; they are also partition functions of T(g; ) on simple curved backgrounds. The supersymmetric index of T(g; ) on S2 S1 was described in [54] following [111{ 113]. As a wavefunction, it corresponds to quantizing the exponentiated phase space P(g; ) ' (C )2d with a real symplectic form ! polarization, it depends on d integers mj (magnetic ~ 1Im . In a particular real are identi ed with the position coordinates xj . Geometrically, the index matches the M . The index is well-de ned and invariant under 2{3 moves so long as the data (g; ) admits a so-called semi-strict angle structure [61]. 33An alternative method of quantizing one-dimensional Lagrangians involves the topological recursion of [57], adapted to the current setting in [20, 58]. It is expected that the topological recursion is equivalent to the algebraic approach outlined here. The ellipsoid (Sb3) partition function of T(g; ) can be computed by methods of [1, of the current paper appeared), this turns out to be identical to the quantization of the real slice of the phase space P(g; ) with Xj ; Pj 2 R. Thus, geometrically, one may identify the ellipsoid partition function with either a PGL(K; C) Chernfunction. It depends analytically on d continuous variables Xj (masses). Geometric partition functions of this type (\state integral models") rst appeared in [10], with later developments including [9, 11, 59, 60]. pected to agree with a PGL(K; C) Chern-Simons partition function at level k [55, 65]. By expanding the formal integrals for ellipsoid partition functions around complex critical points, it is possible to derive a series of perturbative invariants associated with classical solutions p( )(x) on L(g; ) (i.e., geometrically, to at PGL(K; C) xed boundary conditions x) [110]. After the volume (A.16), the rst subleading invariant is a Reidemeister-Ray-Singer torsion. In the geometric setting, these perturbative invariants are expected to match the asymptotic expansion of colored HOMFLY polynomials on M . The partition functions B (x; q) of T(g; ) on a twisted (spinning) geometry C were recently discussed in [117]. These \holomorphic blocks" depend locally on a solution p( )(x) to the classical Lagrangian equations, but are fully nonperturbative quantum objects. It was conjectured that both the index and the ellipsoid partition functions can be written as sums of products of the same holomorphic blocks. Geometrically, the holomorphic blocks seem to come from nonperturbative completions of analytically continued Chern-Simons theory on a manifold M , as in [94, 109]. The Poisson bracket for eigenvalues Here we prove that the eigenvalues of holonomies around noncontractible cycles on small boundaries (small tori or annuli) have the expected Atiyah-Bott-Goldman Poisson brackets discussed in sections 3.4 and 5.3. We consider an admissible 3-manifold M with an ideal triangulation and a subsequent K-decomposition. Let P = P@ i be the linear product phase space of all the octahedra. We work with logarithmic coordinates, as in section 5. For every internal and external point in the K-decomposition we de ne a ne-linear functions Ck, Xj (respectively) on P | the sums of octahedron parameters at vertices that touch those points. For any closed on a small boundary, we choose K 1 representative paths a on the slices parallel must analytically continue the answer. to that small boundary; then we use the rules of section 5.2.1 to de ne K a ne-linear path-coordinates Ua . We already argued in section 3.2.3 (using cancellation of arrows on octahedra) that fCk; Ck0 g = fCk; Xj g = 0 ; which is the expected Poisson structure on the boundary. We now want to show that fCk; Ua g = fXj ; Ua g = 0 ; fUa ; Ub g = ab h ; i ab = < a = b bj = 1 where h ; i is the signed intersection number of the paths ; on the small boundary, and is the Cartan matrix of SL(K). This justi es all of the brackets summarized in section 5.3, and generalizes a central result of [8] for K = 2. Our proof basically extends that of [8], using the framework of slices and path coordinates. There are two basic steps. First we show that the Ua commute with all other coordinates. The fact that Ua commutes with the Ck immediately implies that the Poisson bracket of Ua only depends on the homotopy class of the chosen path a on the a-th slice. In particular, it indicates that fUa ; Ub g should only depend on the intersection number h ; i and some universal function of a; b. The second part of the proof xes the normalization of fUa ; Ub g, through what is unfortunately a brute-force, case-by-case computation. fC; U g = fX ; U g = 0 Let p be an internal or external point in the K-decomposition of M . Let X generically denote the sum of octahedron parameters at p. Let U be the coordinate for any closed path (we drop the subscript a in this subsection) on any global slice S in the K-decomposition. We want to show that fX ; U g = 0. The approach will be to look at how the octahedron parameters contributing to X can appear on the slice S. There are two basic cases: either X contains a sum of parameters in a crown-shaped or wheel-shaped arrangement of small triangles on S. These two arrangements are shown in gure 66, with the parameters contributing to X marked by red dots. Any path disjoint from these arrangements automatically commutes with X . Any path and exits these arrangements (without stopping) also has a coordinate U that commutes with the sum of contributions to X , due to pairwise cancellations in fX ; U g. For example, p internal to ∆ p ∈ internal face p ∈ internal edge pearing in X marked by red dots. Z2 + Z300 + : : :, and the coordinate for the path drawn through Z400 + : : :. In each case there is a pair of 1 contributions to fX ; U g that cancel, coming from the entry and exit points. Now let us go through the various options for locations of p more carefully, and verify that the crown and wheel arrangements arise. First suppose that p is a point internal to the K-decomposition of a tetrahedron. Recall that there are four families of K 1 slices through the octahedron (one family is centered around each tetrahedron vertex). In each family there are three slices that contain parameters contributing to p: the slice containing p and the slices directly above and below it. On the slice containing p, the parameters appear in a six-sided wheel; whereas on the slices above and below the parameters appear in three-spiked crowns ( gure 67, left). If p is on a glued face shared by two tetrahedra, then there are ve relevant families of slices | one centered around each vertex of the big bipyramid that contains p. In the three families centered around the vertices of the big glued face, the slices containing p and those above and below p have X parameters, in wheels or crowns ( gure 67, center). In the two families of slices parallel to the glued face, only the (K 1)-st slice has X parameters, in a three-spiked crown. If p is on an internal edge E of the triangulation shared by N tetrahedra (not necessarily distinct) in an N -gonal bipyramid, there are two types of families of slices to consider. The two families centered around the ends of E each have two slices with X parameters, in an N -sided wheel and an N -spiked crown ( gure 67, right). The N families centered around the \equatorial" vertices of the bipyramid each have the (K parameters, in a 2-spiked crown. 1)-st slice containing X Finally, suppose that p is on a big boundary of M . Then the slices containing X parameters are basically as described above, except some families of slices may get cut in half. For example, if p is on a face of a tetrahedron on the big boundary, the slices centered around the three vertices of that face look like the left side of gure 68. If p is on an external edge, then the slices centered around the endpoints of the edge look like the right side of gure 68. This does not a ect the argument for cancellations above. Any closed in one of these slices must both enter and exit the \half-crown" or \half-wheel" regions without stopping, and contributions to fX ; U g will continue to cancel in pairs. fU ; U g = In the second part of the proof, we consider any two closed paths ; on slices in the K-decomposition of M . Again we suppress the subscripts (a; b) denoting the slice number. We want to show that the corresponding path-coordinates U ; U have Poisson bracket proportional to the signed intersection number of the projection of the paths to the small boundary; and that the proportionality constant equals 2 if the paths are on the same slice, 1 if the paths are on immediately neighboring slices, and 0 otherwise. From section B.1 we know that fU ; U g only depends on the homotopy class of (since closed paths commute with all gluing constraints Ck). Therefore, we may \uniformize" the paths, so that 1) they are smooth (have no bounces); 2) whenever they turn counterclockwise (left) inside a big tetrahedron they stay close to the edge they wind around; and 3) whenever they turn clockwise (right) they stay as far away from the edge as possible ( gure 69). (It may be useful to recall that slices are always viewed from above, from the perspective of a small boundary, in the 3-manifold M . Thus clockwise/right and counterclockwise/left orientations are well de ned.) The goal now is to show that fU ; U g gets contributions from points of intersection of the paths and nowhere else. We x and assume it lies on the a-th global slice (parallel to some small boundary component). We will look at how octahedron parameters that contribute to U appear in other slices of M , and how they interact with other putative on some b-th slice. We simply go through an exhaustive case-by-case analysis. Any putative contribution to fU ; U g comes from segments of that run along a common octahedron i | so that potentially non-commuting parameters Zi; Zi0; Zi00 from that octahedron occur in both U and U . Moreover, in order for segments of share an octahedron, the segments must run along slices in a common big tetrahedron There are two basic cases to consider: these slices are in the same family within they are parallel to each other); or they are in di erent families. Suppose that segments of run on parallel slices within a single tetrahedron they may continue running on parallel slices in a second tetrahedron and into a third, etc. As long as the paths stay together, they simultaneously turn left or right within each successive tetrahedron. We will see in a second that the contributions to fU ; U g from such synchronous segments (where they turn together) vanish identically. which the paths converge, starting their synchronous run, and a tetrahedron the paths diverge, ending the synchronous run. We will show that if the run then the contribution to fU ; U g = 0 equals ab, and otherwise vanishes. Thus, let us rst consider slices of a tetrahedron in which the paths turn the same way, modulo orientation (which may be equal or opposite). The path , on the a-th slice, either turns left or right. If it turns left (making a small turn) then the octahedron parameter (call it Zi) contributing to this segment of also appears on the (a + 1)-st slice. This is shown on the left of gure 70 with a red `+' sign, indicating that the parameter contributes positively to U . We also draw possible locations for , in green. Depending on the precise value of K and whether is oriented equal or opposite to , the path can appear in several di erent positions. It can also appear on any slice. However, in any position to pass through small triangles containing Zi (red +'s), the coordinate U −1 −1 ··a ····−1 −1 a − 1 −1 and the octahedron parameters contributing to it are drawn in red, while possible options for only picks up Zi | never Zi0 or Zi00. Thus, in these parallel slices of , the contribution turns right inside , it picks up a series of (distinct) octahedron parameters with opposite signs, which also appear on the (a + 1)-st and (a on the left of gure 70. In any possible con guration, the path for 1)-st slices, shown at worst picks up these same octahedron parameters. So again the contribution to fU ; U g is zero. Knowing that the intermediate stages of a synchronous run contribute nothing to the Poisson bracket, we can focus on the beginning and end of the run. We might as well assume that the tetrahedra 1 at the beginning and end are immediately adjacent (i.e. there are no intermediate segments). a − 1 −1 · λ · −1 1 on the left (right). If the projections of to the common small boundary do not have a net crossing during the run, there are two possible cases to analyze, depicted in gure 71. On the right, turns left (making a small turn), and its octahedron parameters appear on two slices. Call the octahedron parameters Z0; Z1. If runs along these slices, it can be in several di erent positions, depending on K and its orientation. In each possible position, any parameter picked up by U in 0 that does not commute with Z0 is paired with a parameter in that does not commute with Z1, so that the total contribution to fZ0 + Z1; U g cancels. For example, one of the paths on the a slice picks up Z100g = 0. turns to the left, its octahedron parameters appear in slices a, a + 1, 1. Thus paths on these slices might have segments that don't commute with . But all non-commuting contributions from 0 cancel with those from Altogether, with no net crossing, the total contribution to fU ; U g from a full synchronous run of the paths vanishes. Finally, we consider the most interesting case: a nontrivial crossing of ; going from . There are two possible arrangements. This time we keep careful track of orientations of both and . The path must have segments turning both right and left, so in both cases its octahedron parameters appear on the (a 1)-st slices as well as the a-th. On the left (resp., right) of gure 72 the intersection number of the projections h ; i equals minus one (resp., plus one). Correspondingly, on the left of gure 72, we see that if is in the a-th slice then each of the two small triangles around the intersection point of is on the (a + 1)-st or (a 1)-st slices then U picks up a single parameter that doesn't commute with part of U , contributing +1 to fU ; U g . On the right of gure 72, the same observations hold, with opposite signs. Altogether, we nd that the K − a + 1 K − a K − 1 contribution to the bracket from these crossing segments of path is fU ; U g = ab h ; i lies on the b-th slice. Slices in di erent families To complete the proof, we need to show that whenever run through the same tetrahedron (or tetrahedra) but along two di erent families of slices (centered locally around two di erent tetrahedron vertices) the contribution to the Poisson bracket vanishes. As for parallel slices, is useful to begin with a single tetrahedron in which the two paths turn the same way | meaning that they enter and exit in the same two tetrahedron faces. This identi es an edge E of that both paths turn around. We depict this situation gure 73, choosing to place on the family of slices centered around the \top" vertex around the \bottom" vertex. There are two possible cases. If (on the a-th slice with respect to the top) turns to the left, then it picks up a single octahedron parameter Zi. This parameter appears on a + 1 with respect to the bottom. Any possible paths pick up Zi itself, and so commute with this segment of . Alternatively, if turns to the right, then its octahedron parameters appear on the (K 1)-st (i.e. the last) slice with respect to the bottom vertex. They appear in two horizontal rows | the position of these rows depending on a. Again, all possible paths at worst pick up the same octahedron parameters as in , so that the contribution to fU ; U g vanishes. Now we consider the general situation. The paths may run together through a collection of tetrahedra, moving on di erent families of slices the entire way. In some 0 they must enter on di erent faces and exit on the same face to start the run. In some other tetrahedron 1 they enter together and exit on di erent faces to end We may skip intermediate tetrahedra where they move together (because we know the segments in these tetrahedra commute), and simply assume that immediately adjacent. There are four basic cases to analyze. First, it is possible for 1 to coincide | i.e. both enter and exit a single tetrahedron separately. This is depicted in gure 74. If turns to the left, then its K − a + 1 K − 1 K − 1 K − a K − a + 2 K − a +1−1 −1 K − a + 1 −1 +1 −1 single octahedron parameter appears on the last slice with respect to the bottom vertex (at a distance a away from the edge E). The only possible path crossing the small triangle with this parameter just picks up the parameter itself. If turns to the right, its 2a octahedron parameters appear on the rst through the (K a)-th slices with respect to the bottom vertex. On the rst through the (K a + 2)-th slice the octahedron parameters are arranged in two 2-spiked crowns (as in section B.1, so every closed path commutes with them. On the (K a)-th and (K a + 1)-st slices the possible problematic pick up the parameters themselves. Otherwise, we may assume that 1 are distinct and adjacent. Then there are three remaining arrangements in which may occur. They are depicted in gures 75{77. Since run along two di erent families of slices, they distinguish two di erent vertices common to 1, which in turn distinguishes a common edge E. We choose to position on slices centered around the \top" vertex of E and centered around the bottom. In each of these three cases, there are two options for orienting . It is then shown how its octahedron parameters appear on various slices with respect to the bottom vertex of E (the arrangements of +'s and 's follow by combining the arrangements in gures 73 and 74). Now the potential paths do pick up parameters that don't commute with those in U . However, the non-commuting contributions always cancel in pairs | either within 1 independently, or between . We let the gures speak for themselves. This nally exhausts all possible ways in which the two paths may come close enough to share an octahedron | and thus potentially have nontrivial contributions to their Poisson bracket. The only contributions that don't vanish identically or cancel in pairs are those that come from paths that cross on adjacent (or identical) parallel slices, K − 1 K − a + 2 K − a + 2 K − 1 −1 K − a + 1 −1 −1 −1 −1 −1 K − a −1 −1+1 −1 −1 −1 −1 −1 −1 as in gure 72. The contribution (B.6) found there is therefore the full Poisson bracket fU ; U g. This nishes the proof. Open Access. 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Tudor Dimofte, Maxime Gabella, Alexander B. Goncharov. K-decompositions and 3d gauge theories, Journal of High Energy Physics, 2016, 151, DOI: 10.1007/JHEP11(2016)151