Gauge Theories Labelled by Three-Manifolds
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Gauge Theories Labelled by Three-Manifolds
Tudor Dimofte and Davide Gaiotto and Sergei Gukov , Institute for Advanced Study, Einstein Dr., Princeton, NJ 08540, USA California Institute of Technology, Pasadena, CA 91125, USA Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
Abstract:
We propose a dictionary between geometry of triangulated 3-manifolds andphysics of three-dimensional N = 2 gauge theories. Under this duality, standard operationson triangulated 3-manifolds and various invariants thereof (classical as well as quantum)find a natural interpretation in field theory. For example, independence of the SL (2)Chern-Simons partition function on the choice of triangulation translates to a statement that S b partition functions of two mirror 3d N = 2 gauge theories are equal. Three-dimensional N = 2 field theories associated to 3-manifolds can be thought of as theories that describeboundary conditions and duality walls in four-dimensional N = 2 SCFTs, thus making thewhole construction functorial with respect to cobordisms and gluing. CALT-68-2847 a r X i v : . [ h e p - t h ] A ug ontents
1. Introduction 22. Geometry of 3-manifolds 7
3. Operations on 3d abelian theories 19 Sp (2 N, Z ) action on 3d CFTs with U (1) N flavor symmetry 203.1.1 Generalities 203.1.2 Adding supersymmetry 223.2 A Z action on 3d N = 2 theories with a chiral operator 233.3 Useful N = 2 mirror symmetries 24
4. Construction of T M T M as a boundary condition 364.5 The octahedron 364.6 Figure-eight knot 38
5. Moduli space on R × S S b partition functions 44 R × S
7. Line operators and q –difference equations 51 – 1 – . Introduction One of the predictions of String Theory/M-Theory is the existence of a discrete family ofmaximally symmetric six-dimensional conformal field theories, labeled by a simply-laced Liealgebra g . These theories lack a Lagrangian definition, but some of their properties areknown. The existence of such six-dimensional SCFT’s has a simple, but perhaps surprising,consequence: it allows a geometric description of many lower-dimensional supersymmetricfield theories. Indeed, one can define large families of 6 − d dimensional theories T [ M d , g ]via compactification of the six-dimensional theory labeled by g on a d -dimensional manifold M d . If the compactification is accompanied by an appropriate twist, it will lead to theorieswith 6 − d dimensional supersymmetry. In order to fully exploit this type of construction,one should ideally give an alternative explicit definition of these “effective” theories directlyin 6 − d dimensions. If that can be accomplished, the result is a large family of theoriesdefined in 6 − d dimensions, whose properties are controlled by the geometry of d -dimensionalmanifolds.This program was pursued successfully for d = 2 [1, 2, 3]. The compactification ofthe six-dimensional theories on a Riemann surface C leads to N = 2 supersymmetric gaugetheories T [ C , g ] in four dimensions. The geometry of the Riemann surface controls a varietyof protected quantities in the four-dimensional gauge theories: the space of exactly marginaldeformations, the space of vacua in flat space and upon compactification on a circle, thepartition function of the Ω-deformed theory, the S partition function, the superconformalindex, etc. It is natural to wonder if there is a similar d = 3 dictionary. A twisted compactificationof a 6d theory on a three-manifold M will give an N = 2 field theory T [ M , g ] in threedimensions. Some properties of these theories follow from the definition. For example, one ofthe basic properties of the 6d theories is that they reduce to 5d SYM upon compactificationon a circle. If we consider a 6d SCFT on S × M , we find that the moduli space of vacua of T [ M , g ] is the same as the space of flat complex g -connections on M [4].One way to find other properties of this d = 3 correspondence is to draw lessons from its d = 2 version. Indeed, consider a three-dimensional cobordism, i.e. a 3-manifold M whichinterpolates between two (or, more generally, several) Riemann surfaces, as in Figure 1. Thecompactification of the six-dimensional theory on the cobordism should give a domain wallbetween the 4d theories associated to the Riemann surfaces. Note, in particular, that a half-BPS domain wall ( cf. Figure 1) or a boundary condition ( cf.
Figure 2) in a 4d N = 2field theory preserve the same amount of supersymmetry as a three-dimensional N = 2 fieldtheory.Therefore, one possible strategy for understanding T [ M , g ] is to directly leverage the d = 2 correspondence to construct the three-dimensional field theories: take a closed manifold M , and stretch it to a configuration of long tubes with a Riemann surface as cross sections,joined by appropriate plumbing fixtures, i.e. cobordisms. One could then reduce the six-dimensional theory on the tubes of section C to give known four-dimensional gauge theories– 2 – (cid:38)(cid:10)(cid:48) (cid:55)(cid:11)(cid:48)(cid:12) (cid:55)(cid:11)(cid:38)(cid:10)(cid:12)(cid:55)(cid:11)(cid:38)(cid:12)(cid:68)(cid:12) (cid:69)(cid:12) Figure 1: ( a ) A cobordism M between C and C (cid:48) gives rise to a domain wall ( b ) between 4d N = 2theories T ( C ) and T ( C (cid:48) ). T [ C , g ] on segments, cf. Figure 2. These theories would be coupled through the domain wallsassociated to the plumbing fixtures, and the whole setup taken to define a three-dimensionalgauge theory in the IR.This strategy is hampered by the rapid proliferation of possible “elementary” plumbingfixtures: one would need to find a way to construct the corresponding domain walls by hand,and demonstrate a large set of mirror symmetries which relate different ways to glue togetherthe same manifold. This should be contrasted with a similar approach in d = 2, where thetubes are all cylinders with S cross-section, and the only plumbing fixture is the pair ofpants.We will follow an alternative, simpler strategy. Namely, we will abandon the restrictionto cut the manifold along tubes only, and instead propose a candidate N = 2 SCFT T M for the theory T [ M, su (2)] based on a decomposition (triangulation) of a 3-manifold M intotetrahedra, glued together along the triangular faces. Note, here and in the rest of the paperwe focus (mainly for simplicity) on g = su (2). Moreover, since we are interested only in thecase d = 3, so here and in what follows we denote M simply by M .We do not derive our construction of the N = 2 theory T M directly from properties ofthe six-dimensional theory. Instead, we wish to associate a simple “building block” theory T ∆ to each tetrahedron ∆, and to define the field theory analogue of the geometric gluingwith a simple constraint in mind: different triangulations of the same manifold must giveequivalent definitions of the corresponding theory, in the sense that they flow to the sameSCFT in the IR. In d = 2 different decompositions of the same Riemann surface were relatedby known S-dualities. In d = 3 we aim to relate different triangulations of M through knownmirror symmetries, so that every 3-manifold M is associated to a well-defined, triangulation-independent 3d N = 2 SCFT.We describe the theory T M as the IR fixed point of an abelian Chern-Simons-mattertheory whose Lagrangian depends on the choice of triangulation of M (plus some extradecoration Π that one encounters in SL (2) Chern-Simons theory on M ). Intuitively, given a– 3 – (cid:55)(cid:11)(cid:38)(cid:12) (cid:55)(cid:11)(cid:48)(cid:3)(cid:3)(cid:3)(cid:12)(cid:55)(cid:11)(cid:48)(cid:3)(cid:3)(cid:3)(cid:12)(cid:68)(cid:12) (cid:69)(cid:12) (cid:14) (cid:16) (cid:48)(cid:48) (cid:14) (cid:16) Figure 2: ( a ) A 3-manifold M stretched along a ‘neck’ R × C becomes a 4d N = 2 superconformaltheory ( b ) on R × I coupled to 3-dimensional theories T ( M + ) and T ( M − ) at the boundary. The 4d N = 2 gauge theory in the bulk is determines by the cross-section C of the 3-manifold M . triangulation M = (cid:83) Ni =1 ∆ i , we construct a theory for each tetrahedron ∆ i ∆ i (cid:32) T ∆ i , (1.1)and glue the tetrahedra together to build M (cid:32) T M ∼ (cid:79) i T ∆ i . (1.2)The gluing of theories T ∆ i involves a bit more than just taking a tensor product, and one ofthe main technical aims of this paper is to develop a proper understanding of the sign ‘ ∼ ’in (1.2). Loosely speaking, the gluing involves two steps, which require a careful explanationand depend on a choice of the extra data Π (defined below): gauging some flavor symme-tries, with carefully chosen Chern-Simons couplings, and adding a superpotential couplingfor each internal edge of the triangulation. The choice of the operators which enter the su-perpotential couplings is the most subtle part of the construction. In general, they cannot besimultaneously realized as products of elementary fields, but are defined as ’t Hooft monopoleoperators.Regardless of the compactification from six dimensions, the family of 3d N = 2 SCFTs T M associated to 3-manifolds M is an interesting object, and we hope it will lead to interestingconnections between three-dimensional SCFTs and three-dimensional geometry and topology.For example, quantities like the superconformal index of T M or the partition function on S should map to interesting three-manifold invariants, as summarized in Table 1. In this paperwe specialize to a very simple building block theory for the tetrahedron, which is essentially thetheory of a single chiral multiplet. We believe our approach is much more general though, andwith an appropriate choice of tetrahedron building block one can produce natural candidatesfor T [ M, g ]. – 4 – -manifold M N = 2 theory T M ideal tetrahedron theory T ∆ change of triangulation mirror symmetrychange of polarization Π Sp (2 N, Z ) duality actionboundary flip F transformationgluing along superpotentialan internal edge couplingWilson lines line operatorsboundary C = ∂M coupling to 4d N = 2 theoryflat SL (2 , C ) connections SUSY moduli on R × S Vol( M ) + i CS( M ) twisted superpotential (cid:102) W eff SL (2) Chern-Simons partition function on S b partition functionSeiberg-Witten invariants superconformal index Table 1:
The dictionary between geometry and physics.
We will be able to motivate our proposal for T M = T [ M, su (2)] in a wide variety of ways,and to check that it has expected properties. In particular, we take inspiration from tworelated facts: • The moduli space of vacua of the 3d theory must coincide (with some caveats) with thespace of flat SL (2) connections on M . • The partition function of T [ M, g ] on an ellipsoid S b , as in [5], should coincide with the(analytically continued) g Chern-Simons partition function on M .We engineer T M = T [ M, su (2)] in such a way that these two properties are automaticallytrue.One may wonder why the IR dynamics of the non-abelian six-dimensional theory on a3-manifold M should admit a dual 3d description based on abelian gauge fields. A likelyanswer is that in a generic vacuum of the 3d theory, the 6d theory is deep in its Coulombbranch on most of M . Far on the Coulomb branch, the 6d theory reduces to an abeliantheory of self-dual forms. It is conceivable that the abelian gauge fields in our descriptionarise from these 6d abelian fields, and the matter fields arise from excitations localized inthe regions of M where the 6d theory is close to the origin of the Coulomb branch. Similarideas are useful for d = 2, but they give rise to IR-free, non-UV complete four-dimensionalabelian gauge theories. On the other hand, a three-dimensional abelian gauge theory is a UVcomplete description of an IR fixed point. – 5 –inally, we should describe in more detail the class of 3-manifolds M to which our con-struction applies. In the d = 2 case, it is useful to introduce codimension two defects of thesix-dimensional (2 ,
0) theory, which sit at points of the Riemann surface and fill the entire 4dspace-time. These defects do not break any further supersymmetry, and greatly extend thespace of four-dimensional N = 2 theories which are amenable of a geometric construction.The presence of even a single puncture allows one to use some interesting tools based on“ideal” triangulations of Riemann surfaces, which have vertices at the defects only. Similarly,in d = 3 one can add the very same kind of defects, which fill the entire 3d space-time and aresupported on a line (or, better, on a knot/link) inside M . Again, our construction employs an“ideal” triangulation: the tetrahedra have vertices at the defects. In particular, the manifoldshould have at least one defect. In d = 2 a defect can represent a semi-infinite tubular regionof the surface, and the same is true in d = 3.Our construction does not actually force us to glue all the faces of the tetrahedra pairwisetogether, to get a closed manifold with defects. We can also do a partial gluing, and obtaintheories associated to manifolds with boundaries made by faces of the tetrahedra. The defectsand boundaries both have an interpretation in terms of coupling to four-dimensional N = 2theories. The difference is that defects represented by semi-infinite tubular region with a crosssection C correspond to couplings of theories T M to N = 2 theories in the UV. In particular,for our theories T M = T [ M, su (2)] that come from compactification of the (2 ,
0) theory oftype g = su (2), the corresponding N = 2 theories associated to C in the UV typically have SU (2) gauge groups. For example, closed cusps in M represented by semi-infinite tubularregion with a 2-torus C = T as a cross section correspond to coupling to four-dimensional N = 4 super-Yang-Mills with gauge group SU (2).On the other hand, a big, “geodesic” boundary of M (formed from unglued tetrahe-dron faces) of topology C represents coupling of theory T M to the IR limit of the N = 2four-dimensional theory T [ C ] ( cf. Figure 2). In contrast to its UV version, this IR theoryis usually abelian. Therefore, to summarize, each boundary of M corresponds to a possiblecoupling of the 3d N = 2 theory T M to either IR or UV limit of the 4d N = 2 gauge theory T [ C ], depending on whether the boundary C is big and “geodesic” or small and “defect-like.”This is very natural because a typical example of a boundary condition for a weakly coupledfour-dimensional N = 2 field theory consists of a three-dimensional N = 2 field theory livingat the boundary and coupled to the bulk degrees of freedom. Looking at the same boundarycondition or domain wall in different weakly coupled regions of the bulk parameter spaceleads to different descriptions involving different three-dimensional degrees of freedom.The paper is organized as follows. In section 2 we will review the geometric properties oftriangulated three-manifolds that will inspire the construction of T M . In fact, we will needto generalize the standard constructions a little bit in order to describe triangulations of 3-manifolds that support irreducible flat SL (2 , C ) connections. Section 3 reviews the physicaltools needed for the construction of T M , whereas the definition of the 3d N = 2 theory T M is presented in section 4. Section 5 describes the match between the moduli space of flat– 6 –onnections on M and the moduli space of vacua of the theory T M on a circle. Section 6describes a similar match between the SL (2) Chern-Simons partition function of M and theellipsoid partition function of T M . Finally, section 7 extends the dictionary between geometryof M and physics of T M to line operators.
2. Geometry of 3-manifolds
In this section, we discuss the geometric construction of oriented 3-manifolds M from basicbuilding blocks: ideal tetrahedra. Such “ideal triangulations” in three dimensions were initi-ated by Thurston [6]. More precisely, we wish to build 3-manifolds that support irreducibleflat SL (2 , C ) connections A . For this purpose, it is often convenient to replace flat SL (2 , C )connections with hyperbolic metrics on M — that is, metrics of constant curvature − SL (2 , C ) structures become geometric, and can be manipulated in a much moreintuitive manner. The 3-manifolds we consider have two geodesic boundaryannular cuspstorus cusp Figure 3:
Types of boundaries for M different types of boundary, geodesic bound-aries and generalized cusps . Geometrically,the geodesic boundaries are (possibly punc-tured) geodesic surfaces of any genus, andcome with an induced 2-dimensional hyper-bolic metric. Any triangulation of M willdetermine a triangulation of the geodesic bound-ary, which will be part of the data in even-tually defining a 3d gauge theory.In contrast, “cusp” boundaries do nothave a triangulation that is relevant in defin-ing 3d gauge theories. Geometrically, cuspsare knotted loci where the hyperbolic metricon M develops a cone angle, or the SL (2 , C )connection has a specified monodromy de-fect. Such loci can be resolved to boundarieswith the topology of either tori T or annuli S × I . In either case, the induced metricon cusp boundaries is Euclidean. Well-studied examples of 3-manifolds with torus cusps areknot complements in S . More generally, a cusp might begin and end at punctures on thegeodesic boundary of M (Figure 3). Then, the resolved cusp has the topology of an annulus.The total boundary of M , with potential components of both types, determines a bound-ary moduli space of flat connections, P ∂M = { flat SL (2 , C ) connections on ∂M } (cid:14) (gauge equivalence) . (2.1) The equivalence between flat connections and hyperbolic geometry results from the fact that the isometrygroup of hyperbolic three-space is ( P ) SL (2 , C ), cf. [6, 7, 8, 9]. Almost all flat connections can be realized as(possibly degenerate) hyperbolic metrics; for further remarks on this in the context of ideal triangulations, see[10, 11], and Section 4 of [12]. – 7 –his is a symplectic phase space, with a natural holomorphic symplectic form ω ∂M = 1 (cid:126) (cid:90) ∂M Tr (cid:0) δ A ∧ δ A (cid:1) . (2.2)The semiclassical parameter (cid:126) here governs the normalization of the symplectic form. Ge-ometrically, ω ∂M is an analytic continuation of the Weil-Petersson form in 2-dimensionalhyperbolic geometry. In addition to the phase space P ∂M , we can also define a Lagrangiansubmanifold [8] L M = { flat SL (2 , C ) connections on M } (cid:14) (gauge) ⊂ P ∂M , (2.3)which is the set of flat connections on ∂M that can be extended as flat connections inside the3-dimensional bulk of M . Mathematically, L M is described as the image of the “charactervariety” of M inside the character variety of ∂M .Our goal now is to construct a manifold M together with the pair ( P ∂M , L M ) from idealhyperbolic tetrahedra. This will give us an extremely explicit realization of boundary phasespaces, Lagrangians, and the symplectic structure (2.2), which in turn will enable us in Section4 to explicitly build the 3d gauge theory associated to M . As previewed in the introduction,this 3d theory will depend on M , a triangulation of its geodesic boundary, and a polarization Π of its phase space P ∂M — with additional ingredients such as L M playing roles like modulispaces of vacua. ∞ z zz z z z ∂ H z Figure 4:
An ideal hyperbolictetrahedron in H , with verticeson ∂ H The fundamental building block used in building our 3-manifolds M is an ideal hyperbolic tetrahedron (Figure 4).Geometrically, an ideal tetrahedron ∆ has faces that aregeodesic surfaces and vertices that lie right on the boundaryof hyperbolic 3-space H . As shown in Figure 4, hyperbolic3-space can be viewed as the interior of a 3-ball, with theRiemann sphere as its boundary.The full hyperbolic structure of ∆ is determined by asingle complex cross-ratio of the positions of its vertices on ∂ H . There are three different ways to write this one cross-ratio, encoded in three different edge parameters ( z, z (cid:48) , z (cid:48)(cid:48) ).Geometrically, the edge parameters are dihedral angles onpairs of opposite edges of the tetrahedron [6]. Explicitly, z ≡ exp( Z ) with Z = (torsion) + i (angle) , (2.4)and similarly for z (cid:48) = exp( Z (cid:48) ) and z (cid:48)(cid:48) = exp( Z (cid:48)(cid:48) ), where “torsion” measures the twisting ofthe hyperbolic metric as one moves around an edge. As discussed in [12], the edge parameters We note that topologically, one might engineer (resolved) cusp boundaries that look identical to geodesicboundaries. In particular, networks of annular cusps can assume the topology of nontrivial punctured Riemannsurfaces [13]. Formally, the phase spaces P ∂M associated to the two types of boundary would then be equivalent.However, the natural coordinate systems — and in particular the polarizations — for phase spaces on cuspand geodesic boundaries are very different. In turn, the 3d gauge theories associated to 3-manifolds with thetwo different types of boundary will be quite different. – 8 –atisfy zz (cid:48) z (cid:48)(cid:48) = −
1, which leads to the definition of the boundary phase space P ∂ ∆ = (cid:8) ( z, z (cid:48) , z (cid:48)(cid:48) ) ∈ ( C ∗ \{ } ) (cid:12)(cid:12) zz (cid:48) z (cid:48)(cid:48) = − (cid:9) (cid:39) ( C ∗ \{ } ) , (2.5)or in a lifted, logarithmic form, P ∂ ∆ = (cid:8) ( Z, Z (cid:48) , Z (cid:48)(cid:48) ) ∈ ( C \ πi Z ) (cid:12)(cid:12) Z + Z (cid:48) + Z (cid:48)(cid:48) = iπ (cid:9) . (2.6)This is an affine linear space, with symplectic form ω ∂ ∆ = (cid:126) dZ ∧ dZ (cid:48) or, equivalently, aPoisson structure such that { Z, Z (cid:48) } = { Z (cid:48) , Z (cid:48)(cid:48) } = { Z (cid:48)(cid:48) , Z } = (cid:126) . (2.7)The edge parameters also obey a second relation z + z (cid:48)− − L ∆ = { z + z (cid:48)− − } = { e Z + e − Z (cid:48) − } ⊂ P ∂ ∆ . (2.8)Any cyclic permutation of the Lagrangian equation (with z → z (cid:48) → z (cid:48)(cid:48) → z ) could also beused. Topologically, it is convenient to truncate or regularize the z zz z z z Figure 5:
A truncated idealtetrahedron four vertices of an ideal tetrahedron, as in Figure 5. The tetra-hedron then has four large, geodesic boundaries, whose inducedmetric is hyperbolic; and four small boundaries at the truncatedvertices, whose induced metric is Euclidean. In fact, the condi-tion Z + Z (cid:48) + Z (cid:48)(cid:48) = iπ that defines the phase space in (2.6) simplysays that the sum of angles in the small Euclidean triangles atthe vertices is always π .While the Lagrangian equation z + z (cid:48)− − z, z (cid:48) , z (cid:48)(cid:48) ) as equivalent cross-ratios, it also has anintrinsic description in terms of SL (2 , C ) connections. If we view the boundary ∂ ∆ of a tetra-hedron as a four-punctured sphere, the phase space P ∂ ∆ is the set of flat SL (2 , C ) connectionswith unipotent monodromy around the four punctures. The Lagrangian L ∆ is then the sub-space of flat connections with trivial monodromy — in other words, the flat connections thatcan be extended from the boundary into the bulk of the tetrahedron. Understanding thisdescription explicitly in coordinates ( z, z (cid:48) , z (cid:48)(cid:48) ) requires a bit of further background, which wedefer to Section 2.3.In order to define the gauge theory associated to a tetrahedron, we will need to choosea polarization Π for its boundary phase space. This means choosing affine linear coordinateson P ∂ ∆ that are canonically conjugate to each other with respect to the Poisson structureabove, with one coordinate thought of as “position” and the other as “momentum.” There– 9 –re three natural possibilities, which we call Π Z , Π Z (cid:48) , and Π Z (cid:48)(cid:48) ,position X conjugate momentum P Π Z : Z Z (cid:48)(cid:48) Π Z (cid:48) : Z (cid:48) Z Π Z (cid:48)(cid:48) : Z (cid:48)(cid:48) Z (cid:48) (2.9)Each of these polarizations can be encoded in a choice of opposite edges on the tetrahedron,such that the edge parameters of the distinguished edges act as “positions” (Figure 6). z zz z z z z zz z z z z zz z z z Π Z Π Z Π Z Figure 6:
Natural polarizations for a tetrahedron, with the thickened pairs of opposite edges corre-sponding to the “position” coordinate.
We can define a larger class of polarizations by starting with any of those in (2.9), andacting with an affine symplectic transformation Sp (2 , Z ) (cid:110) ( iπ Z ) . By this we mean takingthe vector (cid:0) position , momentum (cid:1) , multiplying by Sp (2 , Z ) (cid:39) SL (2 , Z ) matrices, and shiftingboth position and momentum by integer multiples of iπ . For example, instead of Π Z , we couldhave considered polarization Π − Z in which X − = Z is position and P − = − Z (cid:48) is momentum;then the transformation from Π Z to Π − Z isΠ Z → Π − Z : (cid:32) X − P − (cid:33) = (cid:32) (cid:33) (cid:32) XP (cid:33) + (cid:32) − iπ (cid:33) . (2.10)Similarly, to go from Π Z to Π Z (cid:48) , we transform (cid:32) Z (cid:48) Z (cid:33) = (cid:32) − −
11 0 (cid:33) (cid:32) ZZ (cid:48)(cid:48) (cid:33) + (cid:32) iπ (cid:33) , (2.11)where the matrix involved is ST = (cid:0) −
11 0 (cid:1) ( ) ∈ Sp (2 , Z ). The identity ( ST ) = I corre-sponds to the fact that three cyclic permutations of shape parameters brings us back wherewe started. Any 3-manifold M with a combination of geodesic and cusp boundaries can be constructedfrom a collection of ideal tetrahedra { ∆ i } Ni =1 , by gluing together their faces one pair at atime. Topologically, the geodesic boundary of M comes from faces of tetrahedra that remain– 10 –nglued. The torus or annular cusps of M , however, arise from assembling collections of smalltruncated-vertex triangles, as in Figure 7. Geometrically, it is clear that the geodesic boundaryof M will be endowed with a hyperbolic metric, since all the faces of ideal tetrahedra aregeodesic, hyperbolic surfaces. Similarly, the cusp boundaries become resolved into Euclideantori or annuli, triangulated by the Euclidean truncated vertices. a ) b ) Figure 7:
Triangulations by Euclidean vertex triangles of (a) an annular cusp attached to a geodesicboundary, and (b) a torus cusp.
In order for the hyperbolic metric on M resulting from such a gluing to be smooth, onemust impose that the total dihedral angle around every internal edge of the triangulation is2 π , and that the hyperbolic torsion vanishes. In other words, for every internal edge I j , thesum of complex edge parameters Z i , Z (cid:48) i , Z (cid:48)(cid:48) i meeting this edge must equal exactly 2 πi . Thiscould be written formally as C I ≡ N (cid:88) i =1 (cid:104) n ( I, i ) Z i + n (cid:48) ( I, i ) Z (cid:48) i + n (cid:48)(cid:48) ( I, i ) Z (cid:48)(cid:48) i (cid:105) = 2 πi ( ∀ internal edges I ) , (2.12)where n ( I, i ) ∈ { , , } is the number of times the edge I in M coincides with an edgeparameter Z i of tetrahedron ∆ i in the triangulation M = (cid:83) Ni =1 ∆ i .Given individual phase spaces P ∂ ∆ i for each tetrahedron ∆ i , I z z z Figure 8:
Illustrationof gluing at an internaledge, with C I = Z + Z (cid:48) + Z . one can construct a product phase space P { ∂ ∆ i } = (cid:81) Ni =1 P ∂ ∆ i witha product symplectic structure. The edge coordinates in this spaceobey a Poisson algebra { Z i , Z (cid:48) j } = { Z (cid:48) i , Z (cid:48)(cid:48) j } = { Z (cid:48)(cid:48) i , Z j } = (cid:126) δ ij , (2.13)with all other brackets vanishing. It is a wonderful fact that in theproduct phase space all the “gluing constraints” C I defined in (2.12)commute with each other [14]. It turns out that the remaining linearcombinations of edge coordinates in P { ∂ ∆ i } that commute with (butare independent of) the gluing constraints C I precisely parametrizethe remaining boundary phase space of the glued 3-manifold M . Thisincludes both geodesic and cusp-like boundary components, and wewill momentarily give explicit examples of both.– 11 –ormally, the fact that all gluing constraints C I commute with each other and with thecoordinates of flat connections on ∂M means that P ∂M can be obtained as the symplecticquotient of the product phase space P { ∂ ∆ i } by the flows of the C I viewed as moment maps [12], P ∂M = (cid:18) N (cid:89) i =1 P ∂ ∆ i (cid:19)(cid:46)(cid:46)(cid:0) C I = 2 πi (cid:1) , (2.14)where I runs over all internal edges. The individual Lagrangian submanifolds L ∆ i can alsobe carried through this symplectic reduction. One forms a product Lagrangian L { ∆ i } = (cid:81) Ni =1 L ∆ i ⊂ P { ∂ ∆ i } cut out by N polynomial equations z i + z (cid:48) i − − C I (projecting L { ∆ i } along the flows of the C I ); and sets C I = 2 πi in the equations that remain (intersectingthe projection with the moment map conditions). This leads to a Lagrangian submanifold L M ⊂ P ∂M . Subject to several technical caveats discussed in [12], it is precisely the desiredset of flat connections on M . We proceed to provide some details of the phase spaces P ∂M associated to the various typesof boundary for M , and to give explicit examples of their construction. A more complete,mathematical analysis of boundaries and phase spaces will appear in [13].It is perhaps simplest to begin with geodesic boundaries. As discussed above, these arisewhen tetrahedra ∆ i are impartially glued; then some tetrahedron faces are left over to formone or more disjoint boundaries C ⊂ ∂M , each a triangulated, punctured Riemann surface.The punctures are places where vertices of the tetrahedra ∆ i are located, and can ultimatelybe regularized into cusps that end on C — we will say a bit more about this later. The induced2d triangulation of C is “ideal” in the sense that all edges begin and end on punctures.The phase space P C , a factor in P ∂M , is the moduli space of flat SL (2 , C ) connections on C , with specified (fixed) holonomy eigenvalues at every puncture. These eigenvalues becomecentral elements in the algebra of functions on P C . Geometrically, we can also describe P C as the complexified Teichm¨uller space of C , a complexification of the moduli space of 2-dimensional hyperbolic metrics. From this perspective, the puncture eigenvalues reflect thegeometric size of holes in C .We can construct coordinates on P C by associating to every edge E in the triangulationof C the total complexified dihedral angle around it. In other words,edge E (cid:32) coordinate X E = N (cid:88) i =1 (cid:104) n ( E, i ) Z i + n (cid:48) ( E, i ) Z (cid:48) i + n (cid:48)(cid:48) ( E, i ) Z (cid:48)(cid:48) i (cid:105) , (2.15)where n ( E, i ) ∈ { , , } is the number of times an edge of tetrahedron ∆ i with parameter Z i coincides with the glued edge E , and similarly for n (cid:48) ( E, i ) and n (cid:48)(cid:48) ( E, i ). This definition isanalogous to (2.12), except that now E is an external edge of M . In fact, these are coordinates on algebraically open patches of P C that have the topology of complex tori, cf. [15, 16]. – 12 –t turns out that the coordinates X E are already well known E E E E Figure 9:
Poisson bracket forexternal edges. Here { X E , X E (cid:48) } = { X E (cid:48) , X E (cid:48)(cid:48) } = { X E (cid:48)(cid:48) , X E } = { X E (cid:48)(cid:48) , X E (cid:48)(cid:48)(cid:48) } = (cid:126) , etc. mathematically as complexified “shear coordinates” on P C [17, 18], defined rigorously in [15] in the complex case. Byfollowing the arguments of [14], one can show that the Pois-son structure induced on these edge coordinates is { X E , X E (cid:48) } = f ( E, E (cid:48) ) , (2.16)where f ( E, E (cid:48) ) ∈ { , ± , ± } is the number of faces sharedby edges E and E (cid:48) , counted with orientation ( cf. Figure9). Expression (2.16) is precisely the Weil-Petersson Poissonstructure on P C , cf. [18]. Moreover, for each puncture p ∈ C , one finds that the sum of edgecoordinates encircling the puncture is (cid:88) E ending on p (cid:0) iπ − X E (cid:1) = 2(Λ p − iπ ) , (2.17)where exp( ± Λ p ) are the holonomy eigenvalues at p . The elements Λ p form a basis for thecenter of the Poisson algebra (2.16).Shear coordinates on P C recently featured prominently in the analysis of BPS states andwall crossing for 4-dimensional N = 2 theories associated to punctures Riemann surfaces C [16]. In particular, we note that [16] considered edge coordinates X E = exp( iπ − X E ), whichcould be identified as the exponentiated central charges for a generating set of BPS states in4d gauge theory. The electric-magnetic pairing of BPS charges was given by (2.16).The simplest example of shear/edge coordinates already ap- z z z ww w Figure 10: ∂ ∆ as a four-punctured sphere. peared above, when we described the phase space P ∂ ∆ (2.6) ofan ideal tetrahedron. If we view the boundary ∂ ∆ as a trian-gulated four-punctured sphere, we should start with six (log-arithmic) edge coordinates ( Z, Z (cid:48) , Z (cid:48)(cid:48) , W, W (cid:48) , W (cid:48)(cid:48) ) that obey aPoisson algebra { Z, Z (cid:48) } = { Z (cid:48) , Z (cid:48)(cid:48) } = { Z (cid:48)(cid:48) , Z } = { Z, W (cid:48) } = { Z (cid:48) , W (cid:48)(cid:48) } = { Z (cid:48)(cid:48) , W } = { W, Z (cid:48) } = { W (cid:48) , Z (cid:48)(cid:48) } (2.18)= { W (cid:48)(cid:48) , Z } = { W, W (cid:48) } = { W (cid:48) , W (cid:48)(cid:48) } = { W (cid:48)(cid:48) , W } = (cid:126) , according to the faces shared by these edges, with all otherbrackets vanishing. Then we impose conditions (2.17) that the holonomy eigenvalue aroundeach vertex p is Λ p = 2 πi — in other words, we require that the holonomy be unipotent: W + W (cid:48) + W (cid:48)(cid:48) = Z + Z (cid:48) + W (cid:48)(cid:48) = Z + W (cid:48) + Z (cid:48)(cid:48) = W + Z (cid:48) + Z (cid:48)(cid:48) = iπ (2.19)This forces opposite edges to have equal parameters, W = Z, W (cid:48) = Z (cid:48) , W (cid:48)(cid:48) = Z (cid:48)(cid:48) , and cutsdown the phase space to P ∂ ∆ = { ( Z, Z (cid:48) , Z (cid:48)(cid:48) ) | Z + Z (cid:48) + Z (cid:48)(cid:48) = iπ } , with Poisson structure (2.7) We thank R. Kashaev for first making us aware of this connection. – 13 – z z zz z w w w ww w yy y y y y z w z w z y z y w y w y z wy Figure 11:
Forming a bipyramid from three tetrahedra.
As an example involving a nontrivial gluing, we can consider the “bipyramid” M of Figure11. Its boundary is a 5-punctures sphere C . Here, we form the bipyramid from three idealtetrahedra, with respective shape parameters Z ( (cid:48) )( (cid:48)(cid:48) ) , W ( (cid:48) )( (cid:48)(cid:48) ) , Y ( (cid:48) )( (cid:48)(cid:48) ) . This leads to a 6-dimensional product phase space P { ∂ ∆ i } ≈ { ( Z, Z (cid:48) , Z (cid:48)(cid:48) , W, W (cid:48) , W (cid:48)(cid:48) , Y, Y (cid:48) , Y (cid:48)(cid:48) ) } with relations Z + Z (cid:48) + Z (cid:48)(cid:48) = W + W (cid:48) + W (cid:48)(cid:48) = Y + Y (cid:48) + Y (cid:48)(cid:48) = iπ . Inside P { ∂ ∆ i } there is a single gluingconstraint C ≡ Z + W + Y → πi (2.20)corresponding to the internal, vertical edge of the bipyramid; it should be used as a symplecticmoment map to reduce P { ∂ ∆ i } to the 4-dimensional phase space P ∂M = P C .Explicitly, coordinates on P C are given by the dihedral angles of the nine external edgesof the bipyramid: Z , W, Y (2.21a)for the three equatorial edges, and Z (cid:48) + W (cid:48)(cid:48) , Z (cid:48)(cid:48) + W (cid:48) , W (cid:48) + Y (cid:48)(cid:48) , W (cid:48)(cid:48) + Y (cid:48) , Y (cid:48) + Z (cid:48)(cid:48) , Y (cid:48)(cid:48) + Z (cid:48) (2.21b)for the six longitudinal edges. It is easy to check that, as functions on the product phasespace P { ∂ ∆ i } , the nine external shear/edge coordinates (2.21) all commute with C . More-over, modulo the gluing constraint (2.20), one can check using formula (2.17) that the totallogarithmic holonomy eigenvalue around each of the five punctures p of C is Λ p = 2 πi . Theresulting five relations among the nine external edge coordinates cut the dimension of P C down to four.As in the case of a single tetrahedron, the punctures on the boundary of the bipyramidcarry unipotent holonomy (with logarithmic eigenvalue 2 πi ). This is related to the fact that,upon truncating tetrahedron vertices as in Figures 5, 11, the small vertex triangles cometogether to form Euclidean 2d discs. These discs effectively cap off the punctures and forceunipotent holonomy. In general one can build 3-manifolds that have annular cusps, rather Any solid 3-ball whose boundary is an n –punctured sphere ( n ≥ – 14 –han discs, ending at the punctures of a geodesic boundary. The annular cusps will then allowany holonomy eigenvalues to be realized. Constructions of this type are extremely interestingin the context of 3d and 4d gauge theory, but will mainly be deferred to future work [13].For simple manifolds such as the tetrahedron and the bipyramid, whose boundaries carryunipotent punctures and whose interiors have the topology of 3-balls, the Lagrangian sub-manifolds L M ⊂ P ∂M are also very simple. They are always cut out by the condition thatthe puncture holonomies are actually trivial (not just unipotent) — so that a flat connectionon the boundary can be extended to the bulk of M .To conclude the discussion of geodesic boundaries, we ob- Π eq Π long Figure 12:
Two polarizationsfor the bipyramid. serve that several natural polarizations Π for a phase spaces P C can be specified by choosing maximal subsets of commut-ing edges on C . In other words, we choose a maximal set ofindependent edges that share no common faces. The corre-sponding coordinates X E then correspond to “positions” in P C . Their conjugate momenta can be constructed (not quiteuniquely) as combinations of the remaining edges.For example, in the case of the bipyramid, two such po-larizations are shown in Figure 12, one using “positions” onequatorial edges and the other on longitudinal edges. (Notethat the three equatorial edges all commute, but obey a con-straint Z + W + Y = C = 2 πi , so only two of them, say Z and W , are independent.) The respective positions X , andmomenta P , in these polarizations are summarized aspositions momentaΠ eq : X = Z , X = W P = Z (cid:48)(cid:48) + Y (cid:48) , P = W (cid:48)(cid:48) + Y (cid:48) Π long : X (cid:48) = W (cid:48) + Y (cid:48)(cid:48) , X (cid:48) = Z (cid:48) + Y (cid:48)(cid:48) P (cid:48) = Z (cid:48)(cid:48) + Y (cid:48) , P (cid:48) = W (cid:48)(cid:48) + Y (cid:48) (2.22)In equatorial coordinates ( X i , P i ), the Lagrangian L M ( i.e. the set of connections with trivialholonomy) can be shown to have the simple description L M : p + p x − , p + p x − , (2.23)while in longitudinal coordinates we have L M : p (cid:48) + x (cid:48)− − , p (cid:48) + x (cid:48)− − , (2.24)with x i = exp( X i ), p i = exp( P i ), etc.Different polarizations for a geodesic boundary phase space P C are related to one anotherby affine Sp (dim C P C , Z ) transformations. From the above discussion, it should be easy tosee that the complex dimension of P C must bedim C P C = ( C ) − ( C ) , (2.25)– 15 –hich by an Euler character argument agrees with the standard formula dim C P C = 6 g − n ,where g is the genus and n is the number of punctures of C . The affinely extended group Sp (6 g − n, Z ) is a subgroup of the full affine group Sp (2 N, Z ) of transformations on theproduct phase space P { ∂ ∆ i } = (cid:81) Ni =1 P ∂ ∆ i . Therefore, we can always choose a polarization of P { ∂ ∆ i } that is compatible with the final desired polarization of the quotient space P C . The cusp boundaries of a 3-manifold M arise from the resolution of line defects, and havethe topology of annuli or tori, depending on whether the defects are open or closed. Forsimplicity, we will only consider the closed, toroidal case in the present paper, though wenote that annular cusps share many of the the same properties, and can be analyzed in asimilar way.Suppose, then, that ∂M contains a toroidal cusp ± m ± Figure 13:
Holonomy eigenvalues ona torus boundary. boundary T . For example, M could be the comple-ment of a knot in S . To describe the associated phasespace P T , we can choose a basis of “A and B cycles”on the torus — typically called meridian and longi-tude cycles in the case of knot complements. Sincethe fundamental group π ( T ) is abelian, the SL (2 , C )holonomies along these cycles are simultaneously di-agonalizable, and P T is simply parametrized by theireigenvalues, cf. [19]: P T = (cid:8) ( m, (cid:96) ) ∈ C ∗ × C ∗ (cid:9)(cid:14) Z , (2.26)where the Weyl group Z acts by inversion ( m, (cid:96) ) (cid:55)→ ( m − , (cid:96) − ). As above, it is also convenientto take logarithms u ≡ log m and v + iπ ≡ log (cid:96) and to lift the phase space to P T = (cid:8) ( u, v ) ∈ C × C (cid:9) / Z . (2.27)Then the symplectic structure of P T becomes ω T = (cid:126) dv ∧ du [8], or { v, u } = (cid:126) / . (2.28)The logarithmic eigenvalues u and v can both be computed as linear combinations ofedge parameters Z i , Z (cid:48) i , Z (cid:48)(cid:48) i of tetrahedra in a triangulation of M . To see this, recall thata cusp boundary T is composed of small truncated-vertex triangles of the tetrahedra ∆ i .Thus, it comes with a (Euclidean) 2d triangulation, as illustrated in Figure 14. The dihedralangles of tetrahedra ∆ i become actual (complexified) angles in the 2d triangles. Logarithmicholonomies can be computed by adding and subtracting the angles subtended by a givenpath, then dividing by two [14, 20]. For example, in Figure 14 we have drawn the meridianand longitude of the figure-eight knot complement on a boundary T . The corresponding For a knot complement in S , M = S \ K , there is actually a canonical choice of cycles. The meridian isan infinitesimally small loop linking the knot K once, while the longitude intersects the meridian once and isnullhomologous in M (in particular, it has zero linking number with the knot). Presently, however, we willallow ourselves the freedom of choosing any basis of cycles whatsoever. As discussed in [12], the shift by iπ in v + iπ = log (cid:96) characterizes the correct lift from P SL (2 , C ) structures(most naturally computed by triangulation data) to SL (2 , C ). – 16 – B z zz z z z A C C A ww w w w w B D w w w z z z ww w zz z w w w z z z ww w zz z m glue Figure 14:
Gluing two tetrahedra, as indicated by calligraphic letters on the faces, to form the figure-eight knot complement. On the right is a map of the resulting torus cusp boundary, triangulated byEuclidean vertex triangles. holonomies are U ≡ u = Z (cid:48) − W (2.29a)2 v = 2( Z − Z (cid:48) ) (2.29b)As functions on the product phase space P { ∂ ∆ i } (cid:39) { ( Z, Z (cid:48) Z (cid:48)(cid:48) , W, W (cid:48) Z (cid:48)(cid:48) ) | Z + Z (cid:48) + Z (cid:48)(cid:48) = W + W (cid:48) + W (cid:48)(cid:48) = iπ } , these satisfy the expected commutation relation { v, U } = (cid:126) .Continuing with the example of the figure-eight knot complement, we find that the tri-angulation of Figure 14 has two internal edges, with corresponding gluing constraints C = 2 Z + Z (cid:48)(cid:48) + 2 W + W (cid:48)(cid:48) → πi , C = 2 Z (cid:48) + Z (cid:48)(cid:48) + 2 W (cid:48) + W (cid:48)(cid:48) → πi . (2.30)(It is easy to read these off from the map of the cusp, since every internal edge begins and endsat a “vertex” on the cusp triangulation. One just adds the angles surrounding the vertex.)Note that C and C both commute with U and v . Moreover, prior to enforcing the condition C = C = 2 πi , there is an automatic relation C + C = 4 πi , so that one of the two gluingconstraints is redundant. In general, for every closed torus cusp in a 3-manifold M , therewill be one such redundant gluing constraint. In the end, for our figure-eight example, we seethat P ∂M = P T = P { ∂ ∆ i } (cid:14)(cid:14) ( C = 2 πi ) = P { ∂ ∆ i } (cid:14)(cid:14) ( C = 2 πi ).The Lagrangian submanifold for the figure-eight knot complement is obtained by thesymplectic reduction procedure described at the end of Section 2.2 above. One starts withthe product Lagrangian L { ∆ i } = { z + z (cid:48)− − , w + w (cid:48)− − } ⊂ P { ∂ ∆ i } , (2.31)where z = e Z , z = e Z (cid:48) , w = e W , and w (cid:48) = e W (cid:48) ; rewrites the equations in terms of m = e U , (cid:96) = − e v , and one of the gluing monomials c j = e C I ; eliminates all remaining variables thatdo not commute with c j ; and sets c j = 1. The end result is L M = { (cid:96) − ( m − m − − m − + m − ) + (cid:96) − = 0 } ⊂ P M , (2.32)and this equation is the well known “A-polynomial” of the figure-eight knot [19, 8].– 17 – .5 Changing the triangulation We have explained, in principle, how to construct 3-manifolds M , phase spaces P ∂M , andLagrangians L M by gluing together ideal tetrahedra ∆ i . It would be useful to verify thatsuch constructions do not depend on a precise choice of triangulation { ∆ i } . Geometrically,once we fix the triangulation of geodesic boundaries, any two triangulations of M are relatedby a sequence of “2–3 Pachner moves,” cf. [21]. These replace two tetrahedra glued alonga common face with three tetrahedra glued along three faces and a common edge, and viceversa, as shown in Figure 15. z z z zz z w w w ww w yy y y y y rr r r r r s s s ss s Figure 15:
The 2–3 Pachner move
Invariance of phase spaces and Lagrangians under the 2–3 move was verified in detailin ( e.g. ) [12], guaranteeing the internal consistency of our present gluing constructions. Forexample, for phase spaces, the essence of the argument is that the product phase spacescorresponding to the bipyramid on the left of Figure 15 is the symplectic reduction of theproduct phase space on the right, P ∂ (bipyramid) = P ∆ R × P ∆ S = (cid:0) P ∆ Z × P ∆ W × P ∆ Y (cid:1)(cid:14)(cid:14) ( C = 2 πi ) , (2.33)where C is the gluing constraint coming from the internal edge. In fact, we already describedthe right-hand side of (2.33) in Section 2.3. The left-hand side is even easier to analyze. In thesame two polarizations Π eq and Π long of Figure 12, we now find coordinates for P ∆ R × P ∆ S :positions momentaΠ eq : X = R + S (cid:48)(cid:48) , X = R (cid:48)(cid:48) + S P = R (cid:48)(cid:48) , P = S (cid:48)(cid:48) Π long : X (cid:48) = R , X (cid:48) = S P (cid:48) = R (cid:48)(cid:48) , P (cid:48) = S (cid:48)(cid:48) (2.34)The two equivalent descriptions (2.22)–(2.34) of P ∂ (bipyramid) are related by combining orsplitting the coordinates associated to the external dihedral angles, for example splitting Z ↔ R (cid:48)(cid:48) + S (cid:48)(cid:48) . Again we note that the invariance of Lagrangians comes with a few subtle caveats, as discussed in [10, 11]and reviewed in Sections 4–5 of [12]. For sufficiently generic triangulations, these caveats can be safely ignored. – 18 –he 2–3 Pachner moves always preserve the triangulations of geodesic boundaries of M .In contrast, they do not preserve the “small” triangulations of cusp boundaries; but thetriangulations of cusp boundaries are never important for defining phase spaces here, or 3dgauge theories later on. If we want to change the triangula- p x zz z zz z x = z p pz = ∪ flip Figure 16:
Flipping an external edge by attachinga tetrahedron. tion of a geodesic boundary
C ⊂ ∂M , wemust consider another type of fundamen-tal move: a flip. The flip acts by gluing anadditional tetrahedron ∆ F onto a quadri-lateral in C , as in Figure 16, and effectively“flipping” the diagonal of this quadrilat-eral. In the process of attaching ∆ F , a newinternal edge I F is created, which imposesa new gluing constraint C I F . The flippedphase space P C (cid:48) is therefore related to P C by a symplectic reduction P C (cid:48) = (cid:0) P C × P ∂ ∆ F (cid:1)(cid:14)(cid:14) ( C I F = 2 πi ) . (2.35)Obviously P C (cid:48) and P C must be isomorphic, but the two have different “natural” polarizations.To illustrate this explicitly, if we start with a polarized phase space P C in which one ofthe canonical position–momentum pairs ( X, P ) corresponds to dihedral angles as in Figure16, then gluing on the tetrahedron ∆ F yields an internal edge constraint C I F = X + Z → πi . (2.36)Now, let us attach a new position coordinate X (cid:48) to the newly flipped diagonal, and itsconjugate momentum P (cid:48) to the same edge as P . After the symplectic reduction (in particular,imposing (2.36)), we find that X (cid:48) = 2 πi − X , P (cid:48) = − ( P + Z (cid:48) ) . (2.37)If we also keep track of Lagrangians, we would find that the flipped L M (cid:48) is related to L M bysubstituting x → x (cid:48)− , p → p (cid:48)− (1 − x (cid:48) ) in the defining equations for L M . The flip transformation, described here from a 3-dimensional viewpoint, is very familiarin 2-dimensional Teichm¨uller (and quantum Teichm¨uller) theory, cf. [22, 18, 23, 24, 15]. Thisshould not be surprising, given the above observation that shear coordinates of Teichm¨ullertheory should be identified with 3d dihedral angles.
3. Operations on 3d abelian theories
Our next goal is introduce the basic ingredients and building blocks necessary to understandthe field theory side of the correspondence ( M, Π) ↔ T M, Π . We will see a clear parallel with We suggest the verification of this statement as an exercise for the reader. – 19 –he construction of 3-manifolds in Section 2, which will lead us to the definition of the theory T M, Π in Section 4. Sp (2 N, Z ) action on 3d CFTs with U (1) N flavor symmetry3.1.1 Generalities There is a beautiful SL (2 , Z ) action on the space of 3-dimensional conformal field theorieswith U (1) flavor symmetry. This action was first described in [25] as a way to understandthe meaning of different choices of boundary conditions for an abelian gauge field in AdS inthe context of AdS /CF T .To be precise, SL (2 , Z ) acts on the space of 3d theories equipped with a specific way tocouple a U (1) flavor symmetry to a background U (1) gauge field. The SL (2 , Z ) action canbe defined by specifying the action of its generators S and T , which obey the relations S = ( ST ) = id. (3.1)The generator T does not change the underlying 3d CFT. It only modifies the prescription ofhow to couple the theory to the background gauge field A , by adding to the conserved currentfor the background flavor symmetry the Hodge dual field strength ∗ F = ∗ dA . In terms of aLagrangian, this is simply accomplished by adding a background Chern-Simons interactionat level k = 1, T : L → L + 14 π A ∧ dA . (3.2)In contrast, the S generator changes the structure of the 3d theory by making the back-ground gauge field A dynamical. The new 3d theory is then prescribed a coupling to a newbackground U (1) gauge field A new : the new flavor current is the Hodge dual field strength ∗ F of the old, now dynamical, gauge field. Equivalently, one prescribes a Lagrangian coupling S : L → L + 12 π A new ∧ dA ( A dynamical) . (3.3)It is the monopole operators for A that are charged under the new U (1) flavor symmetry;thus this U (1) is sometimes called “topological.” From the definitions of S and T , one canprove that the relations S = C and ( ST ) = id. hold, where the transformation C (chargeconjugation) just inverts the sign of the background gauge field. We will generally denote theaction of an SL (2 , Z ) group element g on a theory T as g ◦ T .There is a useful alternative interpretation of this SL (2 , Z ) action: it is the action ofelectric-magnetic duality on the space of conformally invariant boundary conditions for a freeabelian four-dimensional gauge theory. Indeed, given a three-dimensional CFT with a pre-scribed coupling to a background gauge field, we can build a boundary condition by couplingthe CFT to the value of the 4d gauge field at the boundary. This gives a generalization of One can add a Yang-Mills kinetic term at intermediate stages in the calculation. But for S to have thecorrect properties, one must flow to the IR at the end, and then g YM → ∞ and this term is removed. – 20 –eumann boundary conditions: the normal component of the 4d field strength at the bound-ary becomes proportional to the conserved current of the 3d CFT. If we denote the 3d theoryas T , we can denote the resulting boundary condition as B [ T ].Next, we can do an electric-magnetic duality transformation g ∈ SL (2 , Z ) in the four-dimensional bulk, and ask how the boundary condition B [ T ] looks in the new duality frame.This “new” boundary condition g ◦ B [ T ] turns out to coincide with B [ g ◦ T ]. This fact canbe shown readily with the help of “duality domain walls” [26]: the action of bulk dualities onboundary conditions can be interpreted as the collision (or OPE) of these domain walls withthe boundary, as illustrated in Figure 17. For an abelian gauge theory the duality walls arevery easy to construct from the definition of electric-magnetic duality. The collision with theboundary then reproduces the SL (2 , Z ) action defined above. g T g ◦ T = Figure 17:
The action of duality domain walls on boundary conditions. A duality transformation g ∈ Sp (2 N, Z ) maps a generalized Neumann boundary condition defined by coupling to a 3d theory T into a boundary condition associated with a boundary CFT g ◦ T . The SL (2 , Z ) action on boundary conditions is a little bit more general than the SL (2 , Z )action on 3d theories with a coupling to a background gauge field. For example, there existsan extra SL (2 , Z ) orbit of boundary conditions which includes the pure Dirichlet boundarycondition on the 4d gauge field. This boundary condition is invariant under T , and it is sentto the pure Neumann boundary condition by S .Now, it is rather obvious how to generalize this SL (2 , Z ) action to an Sp (2 N, Z ) actionon boundary conditions for a general four-dimensional U (1) N abelian gauge theory, or to anaction on 3d CFTs with U (1) N flavor symmetry: it is the action of the electric-magneticduality group of the U (1) N four-dimensional gauge theory. Notice that in this case, thereare several orbits of boundary conditions which involve at some point Dirichlet boundaryconditions for some of the bulk gauge fields. These orbits will look a bit singular from thepoint of view of an action on 3d CFTs. Concretely, they signal situations where the flavorsymmetry is spontaneously broken to a subgroup in the IR [26].To make this a little more explicit, suppose we are given a Lagrangian description of a3d CFT with U (1) N global symmetry, whose current is coupled to N background gauge fields (cid:126)A = ( A , ..., A N ). The generators of Sp (2 N, Z ) fall into three basic categories: “ T -type,”“ S -type,” and “ GL -type” ( cf. [27]). Representing them as matrices in N × N blocks, we find– 21 –agrangian transformations“ T -type” g = (cid:32) I B I (cid:33) , B symmetric : L [ (cid:126)A ] → L [ (cid:126)A new ] + 14 π (cid:126)A new · B d (cid:126)A new ; (3.4)“ S -type” g = (cid:32) I − J − JJ I − J (cid:33) : L [ (cid:126)A ] → L [ (cid:126)A ] + 12 π (cid:126)A new · J d (cid:126)A (3.5)(where J = diag( j , ..., j N ) with j i ∈ { , } , and we have gauged every A i for which j i = 1,replacing its U (1) with a new topological flavor symmetry); and“ GL -type” g = (cid:32) U U − t (cid:33) , U ∈ GL ( N, Z ) : L [ (cid:126)A ] → L [ U − (cid:126)A new ] . (3.6)The latter GL -type action simply redefines the flavor currents by an invertible, integral trans-formation. The Sp (2 N, Z ) action can be supersymmetrized to give an Sp (2 N, Z ) action on supersym-metric 3d theories equipped with a supersymmetric coupling to a background abelian gaugesupermultiplet. This can be done for any amount of supersymmetry, but it is important tomake a specific choice, as different choices give different group actions.In the reference [26] this was applied to theories with N = 4 supersymmetry. As auseful example of the S action in the context of N = 4 theories, we can consider a singlehypermultiplet of unit flavor charge canonically coupled to an N = 4 background gauge field.If we make the N = 4 background gauge field dynamical — performing an S operation — wehave a familiar 3d theory: N = 4 SQED with one flavor. This is the canonical setup for 3dmirror symmetry [28, 29], which provides an alternative description of the theory in terms ofa free twisted hypermultiplet that arises as a monopole operator in the original description.In particular, it carries unit flavor charge under the new N = 4 background gauge field. Sothe transformation S acts rather trivially on this simple 3d theory: it sends it back to itself[30]. On the other hand, T acts non-trivially.Any N = 4 statement can be reinterpreted as an N = 2 statement, but a little careis needed: the N = 4 3d gauge multiplet consists of an N = 2 3d gauge multiplet plus achiral multiplet. The N = 4 Sp (2 N, Z ) action is the combination of an N = 2 Sp (2 N, Z )action plus additional operations involving 3d chiral multiplets and superpotential couplings.This anticipates a central theme of this paper: the interplay between the “gauge” Sp (2 N, Z )action and a “matter” action which involves adding new chiral multiplets with appropriatesuperpotential couplings. Indeed, the 3d N = 2 theories T M associated to 3-manifolds M withboundary will be coupled both to background gauge fields and background chiral multiplets.As a first step towards understanding this statement, let us describe the “gauge” Sp (2 N, Z )action for N = 2 theories. Suppose we have a theory with U (1) N flavor symmetry, coupled to– 22 – background vector multiplets V i . Each V i , containing a real scalar field σ i and two Majo-rana fermions λ αi in addition to the gauge field A i , can also be dualized to a linear multiplet[31, 32] V i ↔ Σ i = D α D α V , (3.7)where the lowest component of Σ i is σ i . Now, in order to supersymmetrize the Sp (2 N, Z )action (3.4)–(3.6), one simply has to substitute AdA (cid:48) → V Σ (cid:48) for all relevant Chern-Simonsor FI terms:“ T -type” g = (cid:32) I B I (cid:33) : L [ (cid:126)V ] → L [ (cid:126)V new ] + 14 π (cid:90) d θ (cid:126) Σ new · B (cid:126)V new ; (3.8)“ S -type” g = (cid:32) I − J − JJ I − J (cid:33) : L [ (cid:126)V ] → L [ (cid:126)V ] + 12 π (cid:90) d θ (cid:126) Σ new · J (cid:126)V (3.9)“ GL -type” g = (cid:32) U U − t (cid:33) : L [ (cid:126)V ] → L [ U − (cid:126)V new ] . (3.10)Note that a GL ( N, Z ) linear transformation U − can be applied both to a collection of vectormultiplets (cid:126)V and linear multiplets (cid:126) Σ, wherever they occur in the Lagrangian. Z action on 3d N = 2 theories with a chiral operator The basic “matter” action on 3d N = 2 theories begins with a theory that has a coupling toa background 3d chiral multiplet φ . In practice, what we mean is a choice of chiral operator O that can be inserted in a superpotential W = φ O . (3.11)Here and elsewhere, we will not keep track of the normalization of superpotential terms. Inparticular, we will view a rescaling of O as a trivial operation.We can define an operation F that makes φ dynamical (thus, setting O effectively tozero). The new theory can be coupled to a new background chiral field φ (cid:48) by coupling to thenew chiral operator O (cid:48) = φ , namely by the superpotential W = φ (cid:48) φ . (3.12)It is easy to see that F = 1. We can simply look at the combined superpotential W = φ (cid:48)(cid:48) φ (cid:48) + φ (cid:48) φ + φ O (3.13)and integrate out φ (cid:48) .Much like the “gauge” Sp (2 N, Z ) action of the previous subsection, the operation F canbe given an interesting four-dimensional interpretation. One can consider possible boundaryconditions on a four-dimensional hypermultiplet. If we split the four real scalar fields in thehypermultiplet into two complex scalar fields, which we can denote as X and Y , then the two– 23 –asic boundary conditions are either Neumann for X and Dirichlet for Y , or vice versa. Away to understand this is that Y sits in a multiplet of the unbroken supersymmetry whichcontains the normal derivative of X .If we introduce extra degrees of freedom at the boundary, say a 3d theory with a preferredchiral operator O , then we can consider a deformed Dirichlet boundary condition Y = O . Thiswill be accompanied by a corresponding deformation of the Neumann boundary conditionsfor X , involving the corresponding piece of the supermultiplet O . This defines a certain classof boundary conditions which we denote B Y , so that the boundary condition associated to a3d theory T is denoted by B Y [ T ]. An alternative way to describe this boundary conditionis to start with the undeformed boundary condition and add the boundary superpotentialcoupling W = X O . (3.14)Naively, one can construct a completely different class of boundary conditions B X asDirichlet boundary conditions with X = O (cid:48) , where O (cid:48) is a chiral operator in a 3d boundarytheory T (cid:48) . It turns out that these two classes actually coincide, as every member B Y [ T ] ofone class has a mirror B X [ T (cid:48) ] in the other class. One simply takes T (cid:48) to be the image of T under F , with O (cid:48) = φ ; then we claim that B Y [ T ] = B X [ F ◦ T ] . (3.15)To see this, we simply follow the definition of F ◦ T to obtain an overall superpotentialcoupling W = Y φ + φ O . (3.16)Integrating out φ sets Y = −O . Furthermore, the boundary condition X = φ means that wecan simply “absorb” φ into X , thus relaxing the Dirichlet boundary conditions. It takes a bitmore work to make sure that X acquires Neumann boundary conditions, but it follows fromthe fact that the normal derivative ∂ n X plays the role of auxiliary field in the Y supermul-tiplet. In summary, if we begin with Dirichlet boundary conditions for X and perform an F transformation — adding a single boundary chiral multiplet φ and a superpotential W = Y φ — we will flow in the IR to Dirichlet boundary conditions for Y . N = 2 mirror symmetries From the above, it should be clear that the N = 4 S operation consists of a combination of N = 2 S and F operations. Indeed, the N = 4 conserved current supermultiplet contains acomplex moment map operator µ , which is a chiral operator for an N = 2 subalgebra. The N = 4 gauge multiplet contains an N = 2 chiral multiplet φ , which is coupled to the complexmoment map operator µ by the superpotential coupling W = φµ . (3.17)Thus, for example, the basic N = 4 mirror symmetry statement of Section 3.1.2 can be recastas a statement about a 3d N = 2 theory T of two chiral multiplets u and ˜ u with opposite– 24 –avor charges, and an operator O = µ = u ˜ u . This theory is invariant under the combined N = 2 S and F operations, i.e. it satisfies SF ◦ T = T .A basic consequence, pointed out in [31, 33], is that the two theories that are obtainedfrom a S operation or from a F operation on the theory of two chiral multiplets are actuallythe same in the IR, i.e. they are N = 2 mirror duals. The theory S ◦ T is just N = 2SQED with N f = 1. The theory F ◦ T , or rather CF ◦ T , is the so-called XYZ model, atheory of three chiral fields φ , u , ˜ u with a superpotential W = φu ˜ u . (3.18)These two theories are mirror to each other: SQED ( S ◦ T ) : gauged U (1) with two chirals of charge + 1 and − XYZ ( CF ◦ T ) : three chirals with superpotential W = φu ˜ u (3.19)There is actually a bit more structure to this problem. In N = 2 language, each of the twochiral multiplets in T can be rotated independently, and the theory really has U (1) flavorsymmetry. Similarly, the XYZ model has a U (1) flavor symmetry that rotates the phaseof φ , u , ˜ u and leaves the superpotential W invariant. The two U (1)’s map via the mirrorsymmetry (3.19) to an axial U (1) and a topological U (1) in N f = 1 SQED. In terms of SQED,the topological U (1) symmetry is carried by two chiral monopole operators v ± with charges ±
1. It is slightly nontrivial ( cf. [31]) to see that the monopole operators also transform withcharge − U (1). We summarize these various flavor symmetries in Table 2. N f = 1 SQED u ˜ u µ v + v − U (1) gauge − U (1) axial − − U (1) top − φ u ˜ uU (1) axial − − U (1) top − Table 2:
Correspondence of symmetries in N f = 1 SQED and the XYZ model. The designations“axial” and “topological” in the XYZ model are only introduced for comparison to SQED. Eventually, we will investigate the properties of these theories under Sp (4 , Z ) transfor-mations. For now, we would like to derive yet another useful N = 2 mirror pair by a massdeformation of this theory.We aim to understand the properties of a simple theory, consisting of a single chiralmultiplet φ of charge 1. In order to define a coupling to a background gauge field, we need toface a subtlety: a single chiral multiplet canonically coupled to a background gauge field hasan anomaly, which can be cancelled by adding a half-integral Chern-Simons coupling for thebackground gauge field. This fact is closely related to another important fact. If we integrate Notice that the coupling of the background gauge field to T is unaffected by charge conjugation C . – 25 –ut a massive chiral multiplet coupled with charge q to a background gauge field, we generatean effective (supersymmetric) Chern-Simons interaction at level k = q sign( m ).Thus we define an N = 2 theory T as a chiral field of charge 1, coupled to a backgroundfield with an extra Chern-Simons interaction at level − . We want to show that ST ◦ T coincides with T . This is certainly compatible with ( ST ) = 1. In particular, we want toshow that a U (1) CS theory at level k = coupled to a single chiral multiplet of charge +1is mirror to a free chiral multiplet of charge +1. To demonstrate this statement, we will goback to the XYZ model. (cid:43)(cid:76)(cid:74)(cid:74)(cid:86) (cid:38) (cid:82) (cid:88) (cid:79) (cid:82) (cid:80) (cid:69) (cid:38) (cid:82)(cid:88) (cid:79) (cid:82) (cid:80) (cid:69) Figure 18:
The quantum moduli space of N = 2 SQED is identical to the moduli space of vacua inthe XY Z model. It has three branches, permuted by the quantum Z symmetry. The XYZ model, or N f = 1 SQED, has a triality property. In the XYZ model this is justpermutation of the three chiral fields. The theory has three 1-complex-dimensional branchesof SUSY vacua. Indeed, the superpotential (3.18) leads to the scalar potential V = (cid:12)(cid:12)(cid:12) ∂ W ∂φ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ W ∂u (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ W ∂ ˜ u (cid:12)(cid:12)(cid:12) = | φu | + | φ ˜ u | + | u ˜ u | (3.20)which is minimized on field configurations where one of the chiral fields has a vev, while theother two vanish. The resulting three branches parametrized by the vevs of φ , u , or ˜ u meet atthe origin. In the N f = 1 SQED, on the other hand, the classical moduli space is controlledby the term σ ( | u | + | ˜ u | ) in the scalar potential V = e (cid:16) | u | − | ˜ u | − ζ (cid:17) + σ | u | + σ | ˜ u | (3.21)that forces either σ = 0 or u = ˜ u = 0. The quantum corrected moduli space of the N f = 1SQED is the same as that of the XYZ model, as shown in Figure 18. One of the branches inthe moduli space of SUSY vacua is the Higgs branch, parameterized by the vev of the meson µ = u ˜ u . The other two are halves of the Coulomb branch, where σ is real and positive, or– 26 –eal and negative. The two halves of the Coulomb branch are parameterized by the vevs ofthe corresponding vortex-creation (monopole) operators.Now, if one turns on opposite twisted mass for two of the chiral fields in the XYZmodel, it kills two branches and makes the third smooth: M SUSY = (3.22)In the N f = 1 SQED description, this statement can take three equivalent forms. The firstform is simple: an FI parameter is the same as a twisted mass for the monopole operators.It kills the Coulomb branch and smoothens the Higgs branch. The other two forms of thestatement — which are really what we need — are more subtle. We must to turn on twistedmasses for the other two choices of flavor symmetry in the XYZ description. They rotateonly one of the monopole operators, and the meson.Let us start with the XYZ model. By consulting Table 2, we see that if we turn on alarge and (say) positive twisted mass m axial for the axial U (1), and an equal mass m top forthe “topological” U (1), m axial ≈ m top (cid:29) , (3.23)we can integrate out the chirals φ and ˜ u . We are left with a single free chiral u , which stilltransforms under the difference of U (1) top and U (1) axial . Explicitly, defining a new backgroundgauge multiplet V (cid:48) top ≡ V top − V axial , which can still have a small twisted mass parameter m (cid:48) top = m top − m axial , we find that u is coupled to V (cid:48) top with charge 1. Integrating out themultiplet ˜ u generates a background Chern-Simons term k π (cid:82) d θ Σ (cid:48) top V (cid:48) top at level k = − / T . (Alternatively, we could have chosen m axial = − m top (cid:29) u and keep ˜ u . This leads to an equivalent descriptionof T .)In terms of N f = 1 SQED, the topological mass m top ≈ − m axial becomes an FI parameter24 π (cid:90) d θ Σ top V gauge = 12 π (cid:90) d θ m top V gauge . (3.24)It is this large FI term which ultimately allows us to keep the monopole v + light. This maylook a bit mysterious, but it is easily motivated by looking at what happens to the fundamentalmatter fields of N f = 1 SQED in the presence of a large axial mass. Both u and ˜ u becomevery heavy, unless we tune σ to ± m axial , so that either ˜ u is light and u is heavy, or viceversa. Let us choose σ = − m axial ; or, more appropriately, let us redefine the dynamical gaugemultiplet as V gauge → V gauge − V axial = V gauge − θ ¯ θ m axial . Then we can integrate out ˜ u , and inthe process generate a Chern-Simons term of level 1 / − V gauge + 2 V axial ,under which ˜ u is charged. Hidden in the cross-term of the supersymmetric Chern-Simons By “twisted mass” in three dimensions, we mean a mass term arising as a background value for the realscalar field in a vector multiplet. Sometimes this is also called a “real mass.” – 27 –nteraction is an FI term − π (cid:82) d θ m axial V gauge , which cancels (3.24), leaving behind a smalldifference π (cid:82) d θ m (cid:48) top V gauge ! Thus we end up with a fundamental chiral u , coupled withcharge 1 to a U (1) gauge multiplet V gauge , which has a level Chern-Simons interaction π (cid:82) d θ Σ gauge V gauge . The theory has a single light monopole operator v + , transforming withcharge 1 under the new topological U (1) (cid:48) top . This is precisely the description of ST ◦ T . Wehave therefore derived ST ◦ T (cid:39) T (3.25)as a consequence of the basic N = 2 mirror symmetry (3.19).In section 3.1, we discussed the interpretation of the Sp (2 N, Z ) action on a 3d theory T asthe action of electric-magnetic duality in 4d abelian gauge theory on a corresponding boundarycondition B [ T ]. In this interpretation, our simple 3d theory T can define a boundary conditionfor a 4d theory with gauge group U (1) — by identifying the 4d gauge symmetry with the3d flavor symmetry. If we start with a 4d duality frame in which the chiral multiplet of T carries 4d electric charge, then by acting with SL (2 , Z ) duality we obtain all other variantsof the theory T ( p,q )1 , where a distinguished chiral operator transforms as a dyon of electriccharge p and magnetic charge q . In particular, the ST element of SL (2 , Z ) acts as T (1 , ST −−→ T (0 , ST −−→ T ( − , ST −−→ T (1 , . (3.26)The mirror symmetry (3.25) actually guarantees that, just like T (1 , , the theories T (0 , and T ( − , are equivalent to theories of free chirals coupled to the appropriate (magnetic ordyonic) 4d U (1) gauge field with Chern-Simons level − . The chain of equivalences (3.26)should remind us of (2.9).This concludes our quick tour of the basic operations and mirror symmetries in 3d N = 2gauge theories. Of particular importance in the rest of the paper is the basic relation ST ◦T = T and the mirror symmetry between the XYZ model and N f = 1 SQED. These basic dualityrelations admit many generalizations in various directions (to theories that include largergauge groups and / or larger spectrum of matter fields), which have an elegant interpretationin terms of triangulations of 3-manifolds.One simple generalization, which we mention only briefly, is that the XYZ model and N f = 1 SQED appear as the first mirror pair in the infinite family of mirror abelian gaugetheories: Theory A : U (1) r with k neutral chirals and N charged hypermultiplets (3.27) Theory B : (cid:91) U (1) N − r with N − k neutral chirals and N charged hypermultipletswhere the charges of the hypermultiplets in the two theories, R ai and (cid:98) R ai , obey the “orthogo-nality” constraints N (cid:88) i =1 R ai (cid:98) R bi = 0 ∀ a, b . (3.28)– 28 –n addition, both mirror theories A and B have gauge invariant cubic superpotential of theform W = k (cid:88) α =1 N (cid:88) i =1 y αi φ α ˜ Q i Q i (3.29)with Yukawa couplings y αi (resp. (cid:98) y βi ) which obey a relation similar to (3.28): N (cid:88) i =1 y αi (cid:98) y βi = 0 ∀ α, β . (3.30)All four matrices R , (cid:98) R , y , and (cid:98) y are assumed to be of maximal rank. It is easy to see thatif we take N = r = 1 and k = 0, then Theory A is N = 2 SQED with N f = 1, whereasTheory B is the XYZ model. The next simplest case, N = r = k = 1, gives another prominentpair of mirror 3d theories that we also mentioned earlier: a free hypermultiplet and N = 4SQED. More generally, in this class of examples Theory A contains a total of 2 N + k chiralmultiplets (with charges −
1, 0, and +1), whereas Theory B contains a total of 3 N − k chiralmultiplets. For this reason, the mirror symmetry of such a mirror pair could be referred toas a “(2 N + k ) − (3 N − k ) move.”
4. Construction of T M In this section, we will now combine the ingredients of Sections 2 and 3 to provide the mapfrom a pair ( M, Π), where M is a 3-manifold and Π a polarization of its boundary phase space P ∂M , to a 3d SCFT T M, Π , with specified couplings to background gauge fields and chiralmultiplets. We will do so in two steps. First, we attach a 3d theory to any triangulation { ∆ i } Ni =1 of the three-manifold M , and then we show that different triangulations of the samethree-manifold give mirror descriptions of the 3d SCFT. In order to implement the first step, we begin by defining a theory T ∆ , Π Z that we associateto a single tetrahedron ∆ in polarization Π Z (as in (2.9)): T ∆ , Π Z = T . (4.1)Recall from Section 3.3 that T is a theory of a single chiral multiplet coupled to a background U (1) gauge field, with a level − Chern-Simons term turned on. We will say from now onthat the free chiral is associated to the edges of the tetrahedron ∆ labelled by Z , and denoteit as φ Z or O Z . It is also useful to think of the twisted mass of O Z as Re( Z ), and its R-chargeas Im( Z ) /π , where Z is the classical edge/shape parameter of ∆. We use this interpretationhere as an intuitive aid to motivate our gluing construction; it will be made much more precisein Sections 5 and 6. – 29 –e can extend the definition (4.1) to any other polarization Π obtained by an (affine) SL (2 , Z ) transformation g on Π Z : T ∆ ,g ◦ Π Z = g ◦ T (4.2)For example, in a polarization Π − Z as in (2.10), we would find T ∆ , Π − Z = T ◦ T to be the theoryof a free chiral coupled to a background U (1) with Chern-Simons level k = + . This definitionis consistent with the Z symmetry of the tetrahedron: the triality symmetry permutes threeequivalent polarizations Π Z , Π Z (cid:48) , Π Z (cid:48)(cid:48) in (2.9) which, on the N = 2 gauge theory side,correspond to the three duality frames (3.26) of the theory T permuted by the ST elementof SL (2 , Z ).The second step is the definition of T { ∆ i } , ˜Π , the theory associated to the union of N tetrahedra ∆ i , in a generic polarization ˜Π. We can always write ˜Π = g ◦ { Π i } for some g ∈ Sp (2 N, Z ), where { Π i } is a polarization defined as a product of independent polarizationsΠ i of the individual tetrahedra. We choose each Π i to be either Π Z i , Π Z (cid:48) i or Π Z (cid:48)(cid:48) i . Then wedefine M = N (cid:91) i =1 ∆ (Π) i (cid:32) T { ∆ i } , ˜Π = g ◦ N (cid:79) i =1 T ∆ i , Π i (4.3)where we regard the product of N copies of T theories as a theory with a canonical couplingto a U (1) N background gauge field. We should think of each U (1) as corresponding to an independent position coordinate in the polarization ˜Π. This definition is independent of thechoice of Π i ∈ { Π Z i , Π Z (cid:48) i , Π Z (cid:48)(cid:48) i } due to the the symmetry ST ◦ T = T .In order to define the actual SCFT T M, Π associated to the 3-manifold M , we need toimplement a field-theory version of the gluing constraints C I → πi for each internal edge I in the triangulation. The basic idea is to choose a polarization ˜Π = g ◦ { Π i } for the collectionof tetrahedra such that1) it is compatible with the final desired polarization Π of the boundary P ∂M ; and2) all the internal edge coordinates C I are “positions” in ˜Π.If we are careful, we can then construct operators O I in the theory T { ∆ i } , ˜Π , one for eachinternal edge. These operators will be charged under a subset of U (1) flavor symmetries, alsoassociated to the edges C I — or rather to independent linear combinations of them. We canthen define T M, Π by adding a superpotential to T { ∆ i } , ˜Π of the form W = (cid:88) I ∈ internaledges of M O I . (4.4)This superpotential breaks all the U (1) symmetries under which the O I are charged. It alsosets the R-charge of each O I equal to 2. We will see later that this is precisely equivalent tosetting C I = 2 πi . – 30 –n addition to the internal edge operators O I , the theories T { ∆ i } , ˜Π and T M, Π also havea set of operators O E associated to the external edgesthat are “positions” in Π ⊂ ˜Π. Theseoperators are charged precisely under the U (1) gauge symmetries that persist as symmetries of T M, Π — one for each independent position in Π. Indeed, it is easy to see that the flavor groupof T M, Π will contain exactly dim P ∂M U (1)’s. In summary, we have built a correspondence:geometry gauge theory∆ , Π Z T ∆ , Π Z = T { ∆ i } , { Π i } T { ∆ i } , { Π i } = ⊗ i T ∆ i , Π i positions, e.g. Z i operators O Z i with U (1) symmetries { Π i } → ˜Π = g ◦ { Π i } T { ∆ i } , { Π i } → T { ∆ i } , ˜Π = g ◦ T { ∆ i } , { Π i } internal edges C I operators O I external positions, e.g. X E operators O E C I → πi (symp c reduction) W = (cid:80) I O I M = ∪ i ∆ i , Π T M, Π = T { ∆ i } , ˜Π + superpotential W (4.5)The construction of operators O I (and also O E ) in the product theory T { ∆ i } , ˜Π is a littletricky. In order to describe it, we must distinguish two classes of edges. We call an edge “easy”if its classical coordinate C I (or X E ) is a sum containing at most one of the edge parameters Z i , Z (cid:48) i , Z (cid:48)(cid:48) i for any tetrahedron ∆ i ; otherwise the edge is “hard.” Thus, C I = Z + Z or C I = 2 Z (cid:48)(cid:48) + Z + Z (cid:48) would be examples of easy edges, while the internal edges (2.30) in thestandard triangulation of the figure-eight knot complement are hard.Suppose that a triangulation M = { ∆ i } Ni =1 only contains easy edges, and let us focuson the internal ones C I . For every edge I , we can define a polarization { Π Ii } so that thetetrahedron parameters appearing in C I are all position coordinates. Due to the definitionof easy edges, we can always choose Π i ∈ { Π Z , Π Z (cid:48) , Π Z (cid:48)(cid:48) } so that the product polarizationhas this property. Then, in the theory T { ∆ i } , { Π Ii } there will automatically exist an operator O I for the edge C I , constructed as a product of elementary chiral fields. For example, ifour easy edge is C I = 2 Z (cid:48)(cid:48) + Z + Z (cid:48) , we choose a product polarization { Π Ii } that includesΠ I = Π Z (cid:48)(cid:48) , Π I = Π Z , and Π I = Π Z (cid:48) . Then T { ∆ i } , { Π Ii } will have operators O Z (cid:48)(cid:48) , O Z and O Z (cid:48) , all elementary chiral fields, from which we define O I = ( O Z (cid:48)(cid:48) ) O Z O Z (cid:48) .Now, we are really interested in the theory T { ∆ i } , ˜Π , associated to the polarization ˜Π inwhich every internal edge is a position coordinate. For each individual C I , there exists an(affine) Sp (2 N, Z ) transformation g I such that˜Π = g I ◦ { Π Ii } , T { ∆ i } , ˜Π = g I ◦ T { ∆ i } , { Π Ii } . (4.6)This is not quite an arbitrary transformation. In particular, since C I is a position coordinate inboth { Π Ii } and ˜Π, the action of g I cannot gauge any of the U (1) flavor symmetries under which– 31 –he operator O I transforms. Therefore, we can easily pull O I through the transformation onthe right of (4.6) to define the corresponding internal edge operator in T { ∆ i } , ˜Π .If a triangulation only contains easy edges, we can repeatedly use this construction todefine all the operators appearing in the superpotential (4.4). Notice, however, that we defineeach O I using a different mirror Lagrangian description of T { ∆ i } , ˜Π . In any given description,one of the internal edge operators is “simple,” being a gauge-invariant product of elementarychiral multiplets. The other operators may appear more complicated, and will in general takethe form of monopole operators.Just as we defined operators for internal edges, we can also define operators O E for anyeasy external edges (or cusp holonomies) that are positions in Π ⊂ ˜Π. In various mirrorduality frames, they will appear either as products of chiral fields or monopole operators, andthey will be charged under the flavor symmetries of T M, Π that correspond to the positions X E (or U , etc.).Currently, we only have a rigorous construction of operators O I and O E for triangula-tions with easy edges. Indeed, it appears that if we try to define a theory T M, Π using atriangulation of M with hard edges, the theory will be slightly degenerate — and potentiallymissing some expected operators. We will see an example of this behavior in Section 4.6.Fortunately, it seems that we can always refine a given triangulation of a 3-manifold M sothat no hard edges are present, and then use this triangulation to construct T M, Π .One of our central claims is that the theories T M, Π constructed here are topological in-variants of a three-manifold M (and a polarization of its boundary), which do not depend onthe actual triangulation being used to define them — or on the choice of refinement, shoulda given triangulation include hard edges. In particular, we claim that different triangula-tions lead to different mirror-symmetric descriptions of the same underlying 3d SCFT. Tounderstand this, we now proceed to analyze the simplest and most important example of atriangulated 3-manifold: the bipyramid. Let’s consider the theory of the bipyramid, as constructed from two different triangulations.To keep things simple, we will focus on the “equatorial” polarization Π = Π eq for the bipyra-mid, as defined in (2.22) or (2.34). In particular, the three equatorial edges of the bipyramidare position coordinates in Π. We keep the same notation as in Section 2, and repeat Figure15 here as a visual reference.If we decompose the bipyramid into three tetrahedra, then according to our rules T { ∆ i } , Π is a theory of three free chiral multiplets, coupled to a background U (1) , with some extra CScouplings determined by our choice of momenta in Π. The operator associated to the uniqueinternal edge is simply the product of the three chiral fields. Hence T M, Π is simply the XYZmodel, with appropriate coupling to the unbroken U (1) flavor symmetry. The operatorsassociated to the external edges are the three chiral multiplets themselves.– 32 – z z zz z w w w ww w yy y y y y rr r r r r s s s ss s Figure 19:
Decompositions of the bipyramid, with labelled edge coordinates (Figure 15).
Being more explicit, we can start with a product polarization { Π i } = { Π Z , Π W , Π Y } , suchthat Z, W, Y are coordinates and Z (cid:48)(cid:48) , W (cid:48)(cid:48) , Y (cid:48)(cid:48) are momenta. In the equatorial polarization Π eq ,we know that X = Z and X = W are positions while P = Z (cid:48)(cid:48) + Y (cid:48) and P = W (cid:48)(cid:48) + Y (cid:48) are momenta; we therefore choose a compatible polarization ˜Π on P { ∂ ∆ i } with positions X , X , C and momenta P , P , Γ, where C = X + Y + Z and Γ = − Y (cid:48) . The affine symplectictransformation g from { Π i } to ˜Π is encoded as X X CP P Γ = −
10 0 0 0 1 −
10 0 0 0 0 1
ZWYZ (cid:48)(cid:48) W (cid:48)(cid:48) Y (cid:48)(cid:48) + iπiπ − iπ , (4.7)which involves a T -type transformation, a GL -type transformation, and a shift that will notbe visible at the level of Lagrangians. Thus, starting with a Lagrangian description L { Π i } [ V Z , V W , V Y ] = 14 π (cid:90) d θ (cid:16) −
12 Σ Z V Z −
12 Σ W V W −
12 Σ Y V Y (cid:17) + (cid:90) d θ (cid:0) φ † Z e V Z φ Z + φ † W e V W φ W + φ † Y e V Y φ Y (cid:1) (4.8)for T { ∆ i } , { Π i } , we construct the Lagrangian for T { ∆ i } , ˜Π = g ◦ T { ∆ i } , { Π i } simply as L ˜Π [ V X , V X , V C ] = L { Π i } [ V X , V X , V C − V X − V X ]+ 14 π (cid:90) d θ (Σ C − Σ X − Σ X ) ( V C − V X − V X ) , (4.9)in other words by adding a level 1 Chern-Simons term for V Y , and redefining V Z = V X , V W = V X , and V Y = V C − V X − V X . It is trivial to see that the elementary operator O C ≡ φ Z φ W φ Y (4.10)exists in T { ∆ i } , ˜Π , as do the individual operators φ Z , φ W , φ Y associated to the equatorialexternal edges. The bipyramid theory T M, Π eq is then defined by adding the superpotential– 33 – = O C to (4.9), which forces V C = θ ¯ θm C = 0; direct calculation then shows L M, Π eq [ V X , V X ] = 14 π (cid:90) d θ Σ X V X + (cid:90) d θ (cid:0) φ † Z e V X φ Z + φ † W e V X φ W + φ † Y e − V X − V X φ Y (cid:1) + (cid:90) (cid:0) d θ φ Z φ W φ Y + c.c. (cid:1) . (4.11)This is the promised XYZ model, with slightly redefined U (1) symmetries, and a mixedChern-Simons term.If we decompose the bipyramid into two tetrahedra instead of three, we need no superpo-tential. On the other hand, the transformation g from the polarization { Π R , Π S (cid:48)(cid:48) } for the twotetrahedra to Π eq is non-trivial: as the positions are X = R + S (cid:48)(cid:48) , X = R (cid:48)(cid:48) + S , and R (cid:48) + S (cid:48) ,it is easy to see that g involves gauging (with no CS coupling) the U (1) under which the twochiral multiplets have opposite charge. Hence with this definition T M, Π is simply N f = 1SQED, with appropriate coupling to the U (1) flavor symmetry. The operator associated tothe edge coordinate X = R + S (cid:48)(cid:48) is simply the meson operator.Again, one can go through explicit Lagrangian manipulations as above. Starting from apolarization { Π i } = { Π R , Π S (cid:48)(cid:48) } ∼ ( R, S (cid:48)(cid:48) ; R (cid:48)(cid:48) , S (cid:48) ) we reach the equatorial polarization Π eq ∼ ( X , X ; P , P ) via a symplectic transformation g = g S g T g U , with g S = −
10 0 1 00 1 0 0 , g T = , g U = − . (4.12)Therefore, we obtain L M, Π eq [ V X , V X ] by starting with L { ∆ i } , { Π i } [ V R , V S (cid:48)(cid:48) ] = 14 π (cid:90) d θ (cid:16) −
12 Σ R V R −
12 Σ S (cid:48)(cid:48) V S (cid:48)(cid:48) (cid:17) + (cid:90) d θ (cid:0) φ † R e V R φ R + φ † S (cid:48)(cid:48) e V S (cid:48)(cid:48) φ S (cid:48)(cid:48) (cid:1) , (4.13)redefining the U (1) symmetry, adding a Chern-Simons term, and gauging a U (1). A straight-forward calculation produces L M, Π eq [ V X , V X ] = 14 π (cid:90) d θ (cid:0) Σ X V X + (Σ X + 2Σ X ) V (cid:1) + (cid:90) d θ (cid:0) φ † R e V + V X φ R + φ † S (cid:48)(cid:48) e − V + V X φ S (cid:48)(cid:48) (cid:1) , (4.14)with the U (1) gauge multiplet V dynamical. This is precisely N f = 1 SQED, with a mixedChern-Simons coupling, and slightly redefined U (1) symmetry. The meson operator O X ≡ φ R φ S (cid:48)(cid:48) is obviously charged under V X . We know that SQED also has two monopole operators In the last step of the derivation of (4.14), we shifted the dynamical gauge multiplet V → V + V X ,thereby adding to the ‘ X ’ flavor current a half-integral multiplet of the gauge current. This non-integral shiftis not necessary, but can be made sense of because the multiplet V X = θ ¯ θm X is nondynamical. In the form(4.14) of the Lagrangian, the identification of V X with an axial flavor multiplet becomes immediate. – 34 – + and v − , and from the form of the FI term in (4.14) we see that they must be chargedunder the combinations V X and − V X − V X , respectively. Thus, they correspond to theremaining two equatorial edges.Thanks to the basic N = 2 mirror symmetry statement (3.19), our construction gives thesame theory T M, Π for the bipyramid, no matter how we triangulate it . By carefully comparingthe Lagrangian descriptions (4.11) and (4.14), we see that the three equatorial edge operators— elementary fields in the XYZ model and a meson/monopoles in SQED — are mapped toeach other by mirror symmetry, and their coupling to the background U (1) gauge multiplets V X and V X coincide perfectly.One can also repeat the exercise for the longitudinal polarization. The two triangulationsgive respectively N = 4 SQED with N f = 1 and the theory of a free hypermultiplet, i.e. thebasic N = 4 mirror pair. This is a useful exercise in order to show that the operatorsassociated to longitudinal edges by the two polarizations are also mapped into each other bymirror symmetry.With this result, we are in position to argue that the theories T M, Π defined by differenttriangulations of the same three-manifold M are mirror to each other. Different triangulationsare related by a sequence of 2 − Two triangulations that differ by a 2 − T M, Π that differ only by a basic mirror symmetry relation.The mirror symmetry acts on the degrees of freedom associated to the particular bipyramidthat is decomposed in two different ways in the course of a 2 − Just as 2 − T M, Π .For example, suppose that T M, Π has an operator O X , charged under a global symmetry U (1) X , that corresponds to an external edge with position coordinate X . We want to add atetrahedron ∆ Z to flip this edge, as in Figure 16. Following our gauge theory dictionary, thismeans that we form the combined theory T M, Π ⊗ T ∆ Z , Π Z , and add a superpotential coupling W = O X φ Z . (4.15)The new theory now has a chiral operator φ Z that transforms under the anti-diagonal sub-group of U (1) X × U (1) Z that is unbroken by (4.15).This transformation simply describes the F operation of Section 3.2. Just as F is a trivialoperation on a 3d SCFT, flipping a diagonal twice is a trivial operation on the boundary ofa 3-manifold. Strictly speaking, we should only consider triangulations that have easy edges, as discussed in Section 4.1.It is very plausible — although not mathematically proven — that to connect two “easy” triangulations, onecan always find a chain of 2 − – 35 – .4 T M as a boundary condition In section 3 we learned some useful facts about the relation between three dimensional theoriesand boundary conditions for four-dimensional theories. We saw that all the 3d theories in anorbit of the Sp ( N, Z ) action can be thought of as representing the same boundary condition indifferent electric-magnetic duality frames of a four-dimensional abelian gauge theory. We alsosaw that the F transformation on three-dimensional theories can be thought of as relatingtwo mirror description of the same boundary condition for one hypermultiplet.We can use these facts to try to liberate T M, Π from the dependence on the polarizationΠ, and even on the choice of triangulation of the geodesic boundary C of M . To removethe polarization dependence, we can couple T M, Π to a four-dimensional gauge theory, whosesymplectic lattice of electric-magnetic charges is modeled on the lattice generated by the edgecoordinates of the triangulation of C , the geodesic boundary of M . In order to remove thedependence on the triangulation of C , we need to couple T M, Π to a set of hypermultiplets aswell, one for each edge of the triangulation of C . In order for the flip to coincide with an F move, each hyper must be coupled by a superpotential to O E , and hence have four-dimensionalgauge charges equal or opposite to the charge associated to the edge itself.Thus we find it natural to couple T M, Π to an apparently bizarre four-dimensional theory:an N = 2 abelian gauge theory coupled to hypermultiplets of several dyonic charges, one foreach edge of the triangulation of C . This theory is less bizarre than it seems. Indeed, [34],the symplectic lattice generated by a triangulation of C coincides naturally with the latticeof IR electric-magnetic charges for the four-dimensional theory obtained from two M5 braneswrapping C . Furthermore, in a large patch of the 4d Coulomb branch, the whole spectrumof IR BPS particles can be thought of as bound states of a basis of hypermutliplet particles,each associated to an edge of the triangulation, and carrying the corresponding charges.Thus there is a sense in which the abelian gauge theory with the hypermultiplets asso-ciated to the edges of the triangulation is a complete IR description of the four-dimensionaltheory associated to C . And thus T M, Π can be thought as the description of a boundarycondition for the four-dimensional theory, in a given duality frame. This is a property whichwe surely expect to be true of T [ M, su (2)]. In later sections we will reinforce the connectionfurther. For example, the moduli space of vacua of T M, Π compactified on a circle naturallydefines a boundary condition for the four-dimensional gauge theory compactified on a circle. We include two more brief examples of three-manifold theories. The first, the octahedron,demonstrates how 2 − N = 2 SCFT dualities. The second, the figure-eight knot complement, will illustratehow potential difficulties with “hard” edges can be resolved.The simplest way to construct an octahedron is from four tetrahedra, glued togetheralong a central edge (Figure 20). Suppose we we work in an equatorial polarization Π eq as shown, with independent positions ( X, Y, Z, C ), where the internal edge has parameter– 36 – eq x y zwx y zw Figure 20:
The octahedron from four tetrahedra C = X + Y + Z + W . The resulting theory T oct , Π eq is a simple generalization of the bipyramidtheory (4.11). It starts with four chirals φ X , φ Y , φ Z , φ W and four background gauge multiplets V X , V Y , V Z , V W . The multiplet V W is redefined as V W → V C − V X − V Y − V Z − V W , and thenwe add a quartic superpotential W eq = φ X φ Y φ Z φ W (4.16)to break the global symmetry U (1) C . We are still left with U (1) X × U (1) Y × U (1) Z .To be more specific, we should fix conjugate momenta in Π eq , taking (say) ( X + W (cid:48)(cid:48) , Y + W (cid:48)(cid:48) , Z + W (cid:48)(cid:48) , − W (cid:48)(cid:48) ). This choice of momenta will add some background Chern-Simons cou-plings to the Lagrangian of T oct , Π eq , which we encourage the careful reader to work out.Now, if we change to a different polarization Π × , as in the center of Figure 21, we mustperform an Sp (6 , Z ) transformation on the theory T oct , Π eq . This transformation, call it g × ,gauges the U (1) symmetry under which ( φ X , φ Y ) transform as a hypermultiplet. Thus, weobtain a new theory T oct , Π × = g × ◦ T oct , Π eq which has a subsector that looks like N f = 1SQED. By the basic N = 2 mirror symmetry (acting on this subsector), if must be equivalentto a theory of five chirals, with no dynamical gauge group, and superpotential W × = φ T φ Z φ W + φ T φ R φ S . (4.17)From the perspective of SQED, φ T ≡ φ X φ Y is a meson, and the new fields φ R , φ S aremonopole operators; the second term in (4.17) is just the “XYZ” superpotential that wemust add during mirror symmetry.By looking at the left-hand side of Figure 21, we should immediately identify the de-scription of T oct , Π × using five chirals as arising from a five-tetrahedron triangulation of theoctahedron. The two terms in the superpotential W × come directly from the two internaledge coordinates C = T + Z + W and C = T + R + S in this triangulation.To go a bit further, we notice that there another possible triangulation into five tetrahe-dra, shown on the right side of Figure 21. In a sense, it is maximally incompatible with thepolarization Π × . If we try to use triangulation to define T oct , Π × , we will again start with fivechirals φ ˜ R , φ ˜ S , φ ˜ T , φ ˜ Z , φ ˜ W , but will have to gauge the two U (1) symmetries which treat the– 37 – × ˜ t ˜ r ˜ s ˜ z ˜ w ˜ t ˜ r ˜ s ˜ z ˜ w rs zwtrs zwt W × W × Figure 21:
The octahedron from five tetrahedra, two ways. Positions of the polarization Π × areindicated in the middle. respective pairs φ ˜ R , φ ˜ S and φ ˜ Z , φ ˜ W as hypermultiplets. What results is a mirror descriptionof T oct , Π × as a dynamical U (1) gauge theory with two hypermultiplets and a neutral chiral φ ˜ T , coupled by a superpotential W (cid:48)× = φ ˜ T φ ˜ Z φ ˜ W + φ ˜ T φ ˜ R φ ˜ S . (4.18)It is not too hard to recognize that these two descriptions of T oct , Π × correspond to the case N = 2, r = 0, k = 1 of the infinite family of mirror pairs (3.27).There are infinitely more splittings of the octahedron, all giving dual descriptions of T oct , Π × and its Sp (6 , Z ) images. We could similarly analyze triangulations of larger polyhedraor more general 3-manifolds to generate a huge class of 3d N = 2 mirror symmetries. Weexpect, in particular, that the family of dual theories mentioned in (3.27) is realized as a(small!) subset of these. As our final example, we consider the theory associated to a manifold with a torus cuspboundary: the complement of the figure-eight knot M = S \ .The minimal triangulation of M into two tetrahedra, discussed in Section 2.4, has twointernal edges and both of them are hard: C = 2 Z + Z (cid:48)(cid:48) + 2 W + W (cid:48)(cid:48) , C = 2 Z (cid:48) + Z (cid:48)(cid:48) + 2 W (cid:48) + W (cid:48)(cid:48) . (4.19)We could certainly try to write down a gauge theory from this triangulation. Indeed, startingwith T ∆ Z , Π Z (cid:48) ⊗ T ∆ W , Π W , we can change the polarization to (cid:101) Π with (positions; momenta)=(
U, C ; v, Γ), where U = Z (cid:48) − W , v = Z − Z (cid:48) as in (2.29), and Γ = − W is the conjugate to C . The resulting theory T (2) , ˜Π is a U (1) gauge theory with two chiral matter fields both ofcharge +1, and no dynamical Chern-Simons coupling. The factors in the global symmetrygroup U (1) vector × U (1) top correspond to position coordinates U and − C − U , respectively.– 38 –xplicitly, we find a Lagrangian L (2) , ˜Π [ V U , V C ] = 14 π (cid:90) d θ (cid:16) −
32 Σ U V U − (2Σ C + 3Σ U ) V (cid:17) + (cid:90) d θ (cid:0) φ † Z (cid:48) e V + V U φ Z (cid:48) + φ † W e V φ W (cid:1) , (4.20)with V dynamical. Unfortunately, we are hard-pressed to find two monopole operators O C , O C in this theory that could be added to a superpotential. Their existence is cru-cial to break the (essentially topological) U (1) C symmetry, to set V C → rs xy zw rsxyzw rs xy zw rsx yzw z z m Figure 22:
The torus cusp for the figure-eight knot complement, triangulated into six tetrahedra. Thecyclic order of edge parameters ( z, z (cid:48) , z (cid:48)(cid:48) ), etc., is always the same, so we only indicate one parameterper vertex triangle.
To resolve this problem, we must resolve the triangulation. For example, we have found adecomposition of the figure-eight knot complement into six tetrahedra, such that all internaledges are easy. We sketch a developing map of the resulting cusp neighborhood in Figure 22,from which we read off the six internal edge coordinates C = X + W + 2( R (cid:48) + S (cid:48) + Z (cid:48)(cid:48) ) , C = R + Y + 2( Z (cid:48) + W (cid:48) + S (cid:48)(cid:48) ) ,C = S + W + 2( R (cid:48)(cid:48) + X (cid:48)(cid:48) + Y (cid:48) ) , C = R + Z + 2( Y (cid:48)(cid:48) + W (cid:48)(cid:48) + X (cid:48) ) ,C = X + Y , C = S + Z . (4.21)We also find eigenvalues for the meridian and longitude cycles on the boundary T , U = S (cid:48) + R (cid:48) − X (cid:48)(cid:48) + Y (cid:48)(cid:48) − W (cid:48) + Z (cid:48)(cid:48) , v = X + R (cid:48) − S − R (cid:48)(cid:48) . (4.22)Using the combinatorial data for this gluing, it is straightforward (if tedious) to follow therules of Section 4.1 to define the actual figure-eight knot theory T , Π , where Π has position U and momentum v . This theory has six operators O C , ..., O C that can be added to thesuperpotential to break the U (1) symmetries corresponding to the internal edges. We invite the reader to check that this triangulation produces the same A-polynomial as in (2.32). – 39 – . Moduli space on R × S One simple way to test the correspondence M ←→ T M is to associate a moduli space toeach side. In the analogous construction [1] of the 4d N = 2 superconformal theory froma Riemann surface C , there is a similar test of the correspondence C ←→ T ( C ) based oncomparing the moduli space of complex (equivalently, conformal) structures on C with themoduli space of marginal couplings of the theory T ( C ).In the present case, there is a similar test of the correspondence M ←→ T M basedon comparing moduli spaces of complex flat connections on M and the moduli space ofsupersymmetric vacua of the theory T M . To be more precise, the space of complex flatconnections on M can be identified with the space of SUSY moduli in the theory T M on R × S [4]: M flat ( M, SL (2 , C )) = M SUSY ( T M ) . (5.1)While the definition of the moduli space M flat ( M, SL (2 , C )) is clear (and was reviewed insection 2) we need to properly interpret the right-hand side of (5.1).Upon compactification on R × S , the N = 2 theory T M becomes effectively two-dimensional. Supersymmetry then requires that the vevs of chiral and twisted chiral fields,whether dynamical or not, are complex valued. For example, 3d real mass parameters as-sociated to a background U (1) gauge multiplet V become complexified by the holonomiesof the background photon on S . Therefore, moduli spaces parametrized by vevs of chiraland twisted chiral fields are always complex manifolds. Here, we are mostly interested in themoduli space parameterized by vevs of twisted chiral fields — the descendants of 3d gaugemultiplets — and denote this space M SUSY .For example, if M is a closed 3-manifold without boundaries or cusps, the correspondingfield theory T M on R × S has the moduli space of supersymmetric vacua M SUSY ( T M )obtained by minimizing the twisted superpotential (cid:102) W . Since the twisted superpotential is aholomorphic function, the variety defined by the equations ∂ (cid:102) W = 0 is a complex variety, justlike the moduli space of flat SL (2 , C ) connections on M .More generally, if M is a 3-manifold with boundary C = ∂M , it is natural to project themoduli space M flat ( M, SL (2 , C )) onto the moduli space of flat connections on C , i.e. considerthose flat connections on C which can be extended to all of M . In Section 2, this projectionwas cut out by the Lagrangian submanifold L M ⊂ P ∂M = M flat ( C , SL (2 , C )) . (5.2)Correspondingly, in the N = 2 gauge theory T M , it is natural to ask for which values of theparameters v i (= vevs of non-dynamical fields) the theory T M has SUSY vacua on R × S . Inorder to answer this question, we introduce the effective twisted superpotential (cid:102) W eff obtainedby minimizing (cid:102) W with respect to all dynamical fields, and then define [35]: M (param)SUSY ( T M ) : u i = ∂ (cid:102) W eff ∂v i . (5.3)– 40 –n the the case where v i is the twisted mass in a background U (1) gauge field, the coordinate u i should be thought of as the background FI parameter for this field; then it is clear that(5.3) is the condition for unbroken supersymmetry. As we illustrate in a number of examplesbelow, (cid:102) W eff is a transcendental function, generically a sum of dilogarithm functions. However,after taking the derivatives in (5.3) and introducing the new coordinates (cid:96) i = e v i , m i = e u i (5.4)(which are natural, because the complexified vevs u i and v i are periodic), one finds a nicealgebraic variety that is identical to L M .Geometrically, it should be clear that the Lagrangian submanifold L M cannot dependon the coordinates and polarization used to describe the phase space P ∂M when M hasa boundary. Changing coordinates will simply re-parametrize L M . Similarly, the space M (param)SUSY ( T M, Π ) should not depend on the polarization Π (or boundary triangulation, etc.)used in previous sections to define a theory T M, Π . One way to see this is to interpret T M, Π on R × S as describing a boundary condition B [ T M, Π ] for a 4d N = 2 theory T [ C ]( C = ∂M ) compactified on R × S , as in Section 4.4. With a little bit of work, one canshow that the coordinates (cid:96) i , m i become boundary values of natural coordinates ( e.g. X E of [2, 16]) on the moduli space of the compactified 4d theory. From this point of view, M (param)SUSY ( T M ) = L M becomes a complex Lagrangian submanifold in the four-dimensionalmoduli space M SUSY ( T [ C ]) (cid:39) P ∂M . This Lagrangian characterizes the boundary conditionitself, rather than any specific realization of it via a 3d SCFT. In particular, changing thepolarization Π merely shifts the duality frame of the combined 4d-3d system, and must map M (param)SUSY ( T M ) to an isomorphic space.The present discussion of supersymmetric vacua, particularly as given by equations (5.3)with (cid:102) W a sum of dilogarithm functions, is highly reminiscent of recent work relating effective2d field theories to quantum integrable systems [36, 37, 38]. For example, 3d N = 2 theoriesmuch like T M compactified on a circle are related to the XXZ spin chain. A precise connectionbetween our present constructions and integrable systems would be very interesting, but hasyet to be established. Now, let us illustrate this in a few concrete examples, starting with the theory T ∆ , Π Z that weassociate to a single tetrahedron. The theory T ∆ , Π Z is a single chiral multiplet φ Z coupled toa U (1) background gauge field that also has a (supersymmetric) Chern-Simons interaction atlevel − . On a circle of finite radius β , this theory has the effective twisted superpotential( cf. [39, 40, 36, 35]) T ∆ , Π Z : (cid:102) W eff ( Z ) = Li ( e − Z ) = Li ( z − ) , (5.5) As discussed ( e.g. ) in [2, 16], this 4d moduli space actually has the structure of a hyperkahler manifold.The space M (param)SUSY ( T M ) is then embedded into M SUSY ( T [ C ]) as a brane of type ( A, B, A ). – 41 –here Z := β ˜ m Z (5.6)is proportional to the twisted mass in the 2d background gauge multiplet (which contains thereal mass m Z = Re( ˜ m Z ) of the 3d chiral φ Z ). Note that the superpotential (5.5) includes aninfinite tower of Kaluza-Klein modes on the circle S , which have been re-summed.According to (5.3) the effective complexified FI parameter in the IR is given by Z (cid:48)(cid:48) = ∂ (cid:102) W eff ∂Z (cid:48) = log(1 − e − Z ) (5.7)The relation between Z and Z (cid:48)(cid:48) can be conveniently written as M (param)SUSY : e Z (cid:48)(cid:48) + e − Z − z (cid:48)(cid:48) + z − − , (5.8)and, as promised, describes a nice algebraic curve in the variables (5.4). This is precisely thecurve (2.8) that describes the space of SL (2 , C ) structures on a tetrahedron. Hence, we justverified (5.1) in a basic example of a tetrahedron and its gauge theory counterpart T ∆ , Π Z : L ∆ = M (param)SUSY ( T ∆ , Π Z ) . (5.9)Equation (5.8) appears to allow any value of the twisted mass Z (given appropriate FIparameter Z (cid:48)(cid:48) ) except Z = 0. At Z = 0, we hit a singular point, where it looks like theFI parameter must run off to infinity to preserve supersymmetry. This can be understooddirectly in the gauge theory: at Z = 0 the chiral field φ Z is massless, and hence we were notsupposed to integrate it out. The effective description of a gauge theory theory with massivevacua breaks down there.Had we chosen any other polarization for the tetrahedron theory, say Π (cid:48) = g ◦ Π Z withposition X and momentum P such that (cid:32) XP (cid:33) = (cid:32) a bc d (cid:33) (cid:32) ZZ (cid:48)(cid:48) (cid:33) , (5.10)the Lagrangian (5.8) would be mapped to the isomorphic curve p a x − c + p b x − d − . (5.11)As a beautiful example of this behavior, we can consider the particular transformation σ : (cid:32) ZZ (cid:48)(cid:48) (cid:33) (cid:55)→ (cid:32) Z (cid:48) Z (cid:33) = (cid:32) − −
11 0 (cid:33) (cid:32) ZZ (cid:48)(cid:48) (cid:33) + (cid:32) iπ (cid:33) , (5.12)which is an affine extension of ST = (cid:0) − −
11 0 (cid:1) ∈ SL (2 , Z ) that generates the triality symmetry(3.26). (Note that, just like ST itself, σ satisfies σ = id .)From the general Sp (2 N, Z ) action on theories T M, Π (3.8)–(3.10), it is easy to see howthe twisted superpotentials (cid:102) W eff on R × S should transform. For example, the element T – 42 –dds a level 1 Chern-Simons term π (cid:82) d θ Σ Z V Z to the Lagrangian of T ∆ , Π Z , which descends(with proper normalization) to T : (cid:102) W eff ( Z ) (cid:55)→ (cid:102) W (cid:48) eff ( Z ) = (cid:102) W eff ( Z ) + 12 Z . (5.13)Similarly, S adds a mixed Chern-Simons term π (cid:82) d θ Σ Z (cid:48) V Z and makes V Z dynamical. Sincewe now should extremize with respect to Z , this must act as a Legendre transform, S : (cid:102) W eff ( Z ) (cid:55)→ (cid:102) W (cid:48) eff ( Z (cid:48) ) = (cid:104) (cid:102) W eff ( Z ) + Z (cid:48) Z (cid:105) ∂∂Z =0 . (5.14)Finally, we have affine shifts. While these were unimportant for defining Lagrangians on R ,the do show up in the theory on R × S . Namely, shifts by iπ in “position” and “momentum”coordinates appear as half-integral shifts in Wilson loops and theta angles, respectively. Thus,for the tetrahedron theory on R × S , it is the affine σ in (5.12) that implements mirrorsymmetry, σ ◦ T ∆ , Π Z (cid:39) T ∆ , Π Z , (5.15)rather than simply ST Putting together the above ingredients, we find that σ : (cid:102) W eff ( Z ) (cid:55)→ (cid:102) W (cid:48) eff ( Z (cid:48) ) ≡ (cid:20) (cid:102) W eff ( Z ) + 12 Z + ( Z (cid:48) − iπ ) Z (cid:21) ∂∂Z =0 . (5.16)Setting Z = ∂ (cid:102) W eff ( Z (cid:48) ) /∂Z (cid:48) and exponentiating, we obtain M (param)SUSY ( T ∆ , Π Z (cid:48) ) : z + z (cid:48)− − . (5.17)As expected, this transformation leaves the moduli space invariant. To find the moduli space for the bipyramid theory, let us work in the equatorial polarizationΠ eq , as discussed in Section 2 and Section 4.2. We closely follow the notation in thosesections. We can start with the decomposition into two tetrahedra, and use the Lagrangiandescription (4.14) of T M, Π eq as N f = 1 SQED, with a shift V → V − V X /
2, to obtain atwisted superpotential (cid:102) W ( X , X ; σ ) = Li ( e σ ) + Li ( e − σ + X ) + 12 σ + ( X − iπ ) σ . (5.18)Here we have extended the symplectic transformation (4.12) with an affine shift by − iπ forthe twisted mass X . By requiring ∂ (cid:102) W /∂σ = 0 (because σ is the vev of a dynamical field),and setting P = ∂ (cid:102) W /∂X and ∂ (cid:102) W /∂X , it is straightforward to derive the moduli space M (param)SUSY ( T M, Π eq ) : p + p x − , p + p x − . (5.19)– 43 –his is the same as the Lagrangian L M appearing in (2.23). An easier way to derive (5.19)would be to begin with the product of moduli spaces for two tetrahedra r (cid:48)(cid:48) + r − − , s (cid:48) + s (cid:48)(cid:48)− − , (5.20)and simply apply the affine Sp (4 , Z ) transformation r → x p , r (cid:48)(cid:48) → p , s → p , s → − p x p .Equivalently, we can take the decomposition of the bipyramid into three tetrahedra, andthe corresponding XYZ model. The twisted superpotential corresponding to the Lagrangian(4.11) is (cid:102) W eff ( X , X , C ) = Li ( e − X ) + Li ( e − X ) + Li ( e C − X − X ) + iπ ( X + X − C ) . (5.21)Note that, according to the shifts in the symplectic transformation (4.7), we have turned on ahalf-integral theta angle for the combination Σ X + Σ X − Σ C . Setting P = ∂ (cid:102) W /∂X , P = ∂ (cid:102) W /∂X , Γ = ∂ (cid:102) W /∂C and exponentiating, we find equations γp + 1 x − , γp + 1 x − , − γx x c + x x c − . (5.22)Now, however, the (ordinary) cubic superpotential of the XYZ model tells us that we mustset the twisted mass C = 0 (modulo 2 πi ), or c = e C = 1. By appending this to equations(5.22) and eliminating γ , we then obtain( x − (cid:16) p + p x − (cid:17) = 0 , ( x − (cid:16) p + p x − (cid:17) = 0 . (5.23)These are equivalent to (5.19) as long as x (cid:54) = 1 and x (cid:54) = 1. We recall, however, that x , = 1(or X , = 0) are precisely the analogues of the singular points in moduli space discussedbelow (5.8). There, either supersymmetry is broken or new Higgs branches of dynamicalvacua open up. Away from this singular locus, equations (5.23) reduce to (5.19). S b partition functions In the previous section, the correspondence ( M, Π) ↔ T M, Π was tested by comparing modulispaces attached to each side of the correspondence. A more refined test could be obtainedby associating certain functions to each side. For example, on the gauge theory side one canassociate either an equivariant partition function or an index (an analog of the elliptic genus)to the 3d N = 2 theory T M , by analogy with what was done in [3] or [41, 42] in the contextof 4d N = 2 gauge theory. Then, these functions are expected to match the correspondingtopological invariants of M .In this section, we discuss one such test based on comparing the partition function ofthe 3d N = 2 theory T M, Π on a squashed three-sphere (or “ellipsoid”) S b with the SL (2)Chern-Simons partition function of the 3-manifold M : Z SL (2)CS ( M ) = Z S b ( T M, Π ) , (6.1)– 44 –here the squashing parameter b is related to the Chern-Simons coupling coupling strength (cid:126) as (cid:126) = 2 πib . (6.2)This relation is a direct generalization of the AGT correspondence [3] to three dimensions.In fact, it is fully consistent with the AGT correspondence, which corresponds to taking M = R × C to be a product of the “time” direction and a Riemann surface C (possibly withpunctures), through a somewhat lengthy chain of correspondences [43], [35], reviewed e.g. in[44] Various aspects of partition functions in SL (2) Chern-Simons theory are discussed in[8, 9, 45, 12]. Given a 3-manifold M with boundary phase space P ∂M , as defined here inSection 2, Chern-Simons theory should promote P ∂M to a Hilbert space P ∂M (cid:32) H ∂M , (6.3)and the partition function Z SL (2)CS ( M ) can be thought of as a distinguished wavefunctionin H ∂M . In particular, Z SL (2)CS ( M ; X , X , ... ) is a function of half the coordinates on P ∂M ,the “positions” in a given polarization Π. An affine Sp (2 N, Z ) change of polarization actson Z SL (2)CS ( M ; X , X , ... ) in the standard Weil representation [46, 47]; for example, S -typeelements act as Fourier transform, and T -type elements act as multiplication by quadraticexponentials ∼ exp X i (cid:126) .Similarly, the S b partition function of T M, Π depends on the twisted masses m O of variouschiral operators that transform under U (1) flavor symmetries. These real masses are naturallycomplexified by the R-charge, due to the background curvature of the ellipsoid [48, 5]. Indeed,if we describe S b geometrically as b | z | + b − | z | = 1 , z , z ∈ C , (6.4)then Z S b ( T M, Π ) depends holomorphically on the combinations ˜ m O ≡ m O + iQ R O , with Q = b + b − . These complexified masses become identified with the “positions” in P ∂M or H ∂M , as X = 2 πb ˜ m O X = 2 πb m O X + (cid:16) iπ + (cid:126) (cid:17) R O X , (6.5)where O X is (say) the operator we associated to a boundary position X in Section 4.1. We willsee that the ellipsoid partition function Z S b ( T M, Π ; ˜ m X , ˜ m X , ... ) transforms as a wavefunctionunder changes of the polarization Π, in exactly the same way as Z SL (2)CS ( M ; X , X , ... ).Both sides of (6.1) are eminently computable. In fact, [12] developed a general stateintegral model for SL (2) Chern-Simons theory that directly quantizes the semi-classical con-struction of flat connections from ideal tetrahedra, as described in Section 2. Similarly, [5]derived a prescription for ellipsoid partition functions of Chern-Simons-matter theories, using Throughout this section, we work in units such that the “average” radius of the ellipsoid is ρ = 1.Otherwise, it would appear on the right-hand side of (6.4), and would multiply m O X in (6.5). – 45 –quivariant localization. It is not hard to see that the two constructions become equivalentwhen applied to our theories T M, Π . We proceed to study a few aspects of this equivalence,starting with basic T ∆ building blocks and then forming more general theories/manifolds. Consider a free chiral multiplet φ Z with twisted mass m Z for a U (1) flavor symmetry, andR-charge R Z . This R-charge assignment enters in a fundamental way when putting the chiralon an ellipsoid. We set ˜ m Z = m Z + iQ R Z , and find a partition function [5] Z S b (chiral multiplet) = s b (cid:0) iQ − ˜ m Z (cid:1) , (6.6)where s b ( x ) = (cid:89) m,n ∈ Z ≥ mb + nb − + Q − ixmb + nb − + Q + ix = e − iπ x ∞ (cid:89) r =1 e πbx +2 πib ( r −
12 ) e πb − x +2 πib − ( 12 − r ) (6.7)is a variant of the noncompact quantum dilogarithm function [49, 50] commonly used inLiouville theory.Two of the properties enjoyed by the function s b ( x ) are s b ( x ) s b ( − x ) = 1 , (6.8a) s b ( x ) ∼ (cid:40) e iπx / as x → + ∞ e − iπx / as x → −∞ , (6.8b)which have a nice interpretation in 3d N = 2 gauge theory. According to (6.6), the firstproperty (6.8a) implies that the partition function of two chiral fields φ , φ (cid:48) of opposite flavorcharge and R-charge adding to 2 is trivial. Indeed, this R-charge assignment allows one toadd a marginal superpotential W = M φφ (cid:48) (6.9)which makes both fields arbitrarily massive and decouples them. The second property (6.8b)agrees with an important fact: a Chern-Simons action of level k for the background gaugefield gives a contribution e − iπk ˜ m (6.10)to the partition function. Therefore, we see that at large positive σ the chiral multipletcontributes as a Chern-Simons coupling of level + , while at large negative σ as a Chern-Simons coupling of level − , as expected [31] ( cf. our discussion of such couplings in Section3.3). In a similar way, many beautiful identities obeyed by the special function (6.7) — inturn related to the combinatorics of 3-manifolds triangulations — find physical interpretationas dualities among 3d N = 2 gauge theories. Here and in the following, we will ignore overall numerical constants in front of the partition function. – 46 –he actual theory T ∆ , Π Z associated to a tetrahedron has an extra level − Chern-Simonscoupling for the background gauge field, leading to a partition function Z S b ( T ∆ , Π Z ; ˜ m Z ) = e b (cid:0) iQ − ˜ m Z (cid:1) ≡ e iπ (cid:0) iQ − ˜ m Z (cid:1) s b (cid:0) iQ − ˜ m Z (cid:1) . (6.11)With the identification (6.5), this is equivalent to the Chern-Simons partition function of asingle tetrahedron, found in [12].In order to consider other polarizations for T ∆ , we should analyze how the SL (2 , Z ) actionon gauge theories affects partition functions. It is already clear from (6.10) that the T -movesends T : Z S b ( ˜ m ) (cid:55)→ Z (cid:48) S b ( ˜ m ) = e − iπ ˜ m Z S b ( ˜ m ) . (6.12)Similarly, the S -move adds a factor e − πi ˜ m ˜ m (cid:48) to the partition function, and dictates that weintegrate over the vev m , since its gauge multiplet has become dynamical. In other words, S acts as a Fourier transform: S : Z S b ( ˜ m ) (cid:55)→ Z (cid:48) S b ( ˜ m (cid:48) ) = (cid:90) d ˜ m e − πi ˜ m ˜ m (cid:48) Z S b ( ˜ m ) . (6.13)Note that this an integral along the real line, which could be deformed to a contour in thecomplex plane. In addition to S and T , affine shifts in polarization also act nontrivially onthe ellipsoid, by redefining the R-charge used to couple a theory to background curvature.For example, a classical shift by ± iπ in a position coordinate Z corresponds to sending R Z (cid:55)→ R Z ±
1, or ˜ m Z (cid:55)→ ˜ m Z ± iQ .The above action of the affine symplectic group shows that the ellipsoid partition functiontransforms as a wavefunction under changes of polarization, precisely as claimed. In partic-ular, the above transformations are identical to those that appear in SL (2) Chern-Simonstheory. As a simple example, we can consider the affine ST action that sends the polarizationΠ Z to Π Z (cid:48) for the tetrahedron theory. This affine action was called σ in (5.12). We find Z S b ( T ∆ , Π Z (cid:48) ; ˜ m Z (cid:48) ) = σ ◦ Z S b ( T ∆ , Π Z )= (cid:90) dm Z e − iπ ˜ m Z ( ˜ m Z +2 ˜ m Z (cid:48) − iQ ) e b (cid:0) iQ − ˜ m Z (cid:1) = e b (cid:0) iQ − ˜ m Z (cid:48) (cid:1) , (6.14)up to a constant factor. The last equality follows from a standard functional identity for e b ( x )[51], and verifies the prediction from mirror symmetry that the transformation σ leaves thetetrahedron theory invariant. In Section 3.3, we derived ST -invariance of the tetrahedron theory T (cid:39) T ∆ , Π Z by startingwith N = 4 mirror symmetry, translating to N = 2 mirror symmetry for the XYZ modeland SQED with N f = 1, and and then reducing further to the theories T and ST ◦ T via– 47 – mass deformation. It is somewhat instructive to now do the same at the level of partitionfunctions. In the process, we will see how gluing of partition functions should work.Let’s begin with the partition function of a hypermultiplet, with (complex) vector twistedmass denoted by x and axial twisted mass by y : Z S b ( hypermultiplet ) = s b (cid:0) iQ − x − y (cid:1) s b (cid:0) iQ + x − y (cid:1) . (6.15)The N = 2 R-charge and axial charge are a linear combination of the Cartan generators ofthe SU (2) H × SU (2) C R-charges of the N = 4 theory. We are using a convention where inthe N = 2 language the R-symmetry of chiral multiplets in the standard hypermultiplet isabsorbed in their axial twisted mass y . Then the scalar field in the vectormultiplet has “bare”R-charge 2, and axial charge −
2, i.e. complex twisted mass iQ − y . This is also required forthe basic superpotential coupling required by an N = 4 gauging.Hence if we add a full N = 4 gauge multiplet to gauge the flavor symmetry, the chiralmultiplet in it contributes a s b (2 y − iQ/ s b (2 y − iQ/ (cid:90) s b ( iQ/ − x − y ) s b ( iQ/ x − y ) e − iπzx dx . (6.16)The basic N = 4 mirror symmetry should match this to the partition function of a twistedhypermultiplet, i.e. a hypermultiplet with the opposite axial charge [28]. The chiral fields ina twisted hypermutliplet have “bare” R-charge 1 and axial charge −
1, i.e. complex twistedmass iQ − y . Hence we should replace y with iQ − y in (6.15) and write the basic N = 4mirror symmetry relation as s b (2 y − iQ/ (cid:90) s b ( iQ/ − x − y ) s b ( iQ/ x − y ) e − iπzx dx = s b ( y − z ) s b ( y + z ) (6.17)As a check, we are supposed to obtain either the partition functions of N = 2 SQED with N f = 1 flavor or the partition function of the XYZ model by acting with S or with F on theabove relation. If we act with S , i.e. with the Fourier transform, we get s b (2 y − iQ/ s b ( iQ/ − x − y ) s b ( iQ/ x − y ) = (cid:90) s b ( y − z ) s b ( y + z ) e − iπzx dx . (6.18)The left-hand side is the partition function of the XYZ model. The real masses of the threechiral fields add to zero, and the R-charges to 2, as it should be to allow the superpotentialinteraction W = µu ˜ u , cf. Section 3.3. Equation (6.18) happens to be another well knownidentity for quantum dilogarithm functions [51, 52].Now, if we redefine x → x − y , z → z + y − iQ , and take y to be large and positivein (6.17), we replicate the mass deformation that reduces us to the theory T (cid:39) T ∆ , Π Z .Expression (6.17) becomes (cid:90) dx e − iπx (cid:0) x +2( z − iQ ) (cid:1) e b (cid:0) iQ − x (cid:1) = e b (cid:0) iQ − z (cid:1) , (6.19)– 48 –hich is precisely (6.14), expressing the mirror symmetry σ ◦ T (cid:39) T .We could also add Chern-Simons terms on both sides of (6.18) in order to reproduce theexact partition function of the bipyramid theory, as discussed in Section 4.2. Namely, we findan identity e iπ ( iQ ) ˜ m e b (cid:0) iQ − ˜ m (cid:1) e b (cid:0) iQ − ˜ m (cid:1) e b (cid:0) iQ − ˜ m (cid:1)(cid:12)(cid:12)(cid:12) ˜ m = iQ − ˜ m − ˜ m (6.20)= (cid:90) dσ e − iπσ − πiσ (cid:0) ˜ m − iQ (cid:1) e b (cid:0) iQ + σ (cid:1) e b (cid:0) iQ − σ + ˜ m (cid:1) The two sides correspond to the theories of three and two tetrahedra, respectively, both in theequatorial polarization Π eq , with external edge positions X = 2 πb ˜ m and X = 2 πb ˜ m . Forthe left-hand side, the superpotential W = O C = φ Z φ W φ Y (4.10) implements the constraint˜ m + ˜ m + ˜ m = iQ .More generally, the rules for constructing theories T M, Π in Section 4 lead to the followingrules for calculating the corresponding ellipsoid partition functions:1) Multiply together partition functions Z S b ( T ∆ i , Π i ; ˜ m Z i ) = e b (cid:0) iQ − ˜ m Z i (cid:1) , one for eachtetrahedron in the triangulation of M .2) Act with Sp (2 N, Z ) in the Weil representation ( i.e. by generalizing the quadratic expo-nentials and Fourier transforms of (6.12)–(6.13)), to transform to the polarization ˜Π inwhich all internal edges are “positions.”3) Set the complex masses ˜ m I now associated to internal edges equal to iQ .We note that the specialization in Step 3 is the only consequence of adding a superpotential W = (cid:80) I O I to the theory T M, Π . Indeed, such a superpotential sets the real masses of the O I to zero and the R-charges equal to 2. Otherwise, the ellipsoid partition function is completelyindependent of superpotential terms, and cares only about gauge and matter content.These rules for constructing Z S b ( T M, Π ) are identical to the rules presented in [12] forbuilding the SL (2) Chern-Simons partition function of M . One can see even subtle quantumeffects matching in the two descriptions. For example, in quantum Chern-Simons theory, theclassical internal edge constraints C I = 2 πi become corrected to C I = 2 πi + (cid:126) , and this followsimmediately from the dictionary (6.5) between edge parameters and complexified masses ˜ m I . We should be able to reproduce the well known Chern-Simons wavefunction for the figure-eightknot complement from the theory T , Π described in Section 4.6. The definition of the actualtheory, including internal edge operators, required a decomposition of the knot complementinto six tetrahedra. However, since ellipsoid partition functions do not depend in a crucialway on superpotential terms, we might hope to get away with the simpler decomposition intotwo tetrahedra, also discussed in Section 4.6. Indeed, this turns out to work.– 49 –rom the Lagrangian (4.20), we can immediately write down a partition function Z S b ( T , Π ; ˜ m U ) = (cid:90) dσ e iπ (cid:0) ˜ m U +(2 ˜ m C − iQ +2 ˜ m U − σ ) σ (cid:1) e b (cid:0) iQ − σ − ˜ m U (cid:1) e b (cid:0) iQ − σ (cid:1) . (6.21)Now, there are no operators in the theory to force ˜ m C = iQ , but we can put this in by hand.Up to a factor of due to a small change of polarization, the result is then identical to thefigure-eight wavefunctions described in [53, 9, 12] (see also [54, 55]). R × S Finally, we point out that our tests of the proposed duality ( M, Π) ↔ T M, Π here and insection 5 are not entirely unrelated. Indeed, in the semi-classical limit (cid:126) = 2 πib →
0, thepartition function of the theory T M behaves exactly in the same way as the partition functionof Chern-Simons theory on M , Z S b ( T M ) (cid:126) → ∼ exp (cid:16) (cid:126) (cid:102) W eff + O (log (cid:126) ) (cid:17) , (6.22)where (cid:102) W eff is the effective twisted superpotential of the theory T M on R × S . Hence, if (cid:102) W eff matches the classical SL (2) Chern-Simons action on M , (cid:102) W eff ( T M ) = S ( M ) , (6.23)then the relation between moduli spaces (5.1) follows automatically. Indeed, the moduli space M flat ( M, SL (2 , C )) is a graph of dS and, similarly, the moduli space M SUSY ( T M ) is a graphof d (cid:102) W eff . In terms of gauge theory, the reason for (6.22) is that, in the limit b →
0, thesquashed 3-sphere S b degenerates into R × S , S b (cid:32) R × S . (6.24)The relation between moduli spaces M flat ( M, SL (2 , C )) = M SUSY ( T M ) of Section 5 hasa “quantum” analog that does not require taking the limit (cid:126) →
0. Indeed, the full quantumpartition functions discussed here obey a set of q -difference equations: (cid:98) A i Z = 0 (6.25)for some operators (cid:98) A i that in the classical limit become defining polynomials of our modulispaces. In Chern-Simons theory, (6.25) is known as the generalized / quantum volume con-jecture [8] (sometimes also called the AJ-conjecture [56, 57] in the math literature), whereasin N = 2 gauge theory it expresses Ward identities for line operators. We consider these lineoperators next. – 50 – . Line operators and q –difference equations In order to understand the meaning of operator identities (6.25) in 3d N = 2 theory, we needto incorporate line operators in our correspondence (1.2).Given a triangulated 3-manifold M with nonempty boundary ∂M , each equation in (6.25)is written in terms of quantum holonomy operators that, from the viewpoint of Chern-Simons theory on M , are obtained by quantizing the space of flat SL (2 , C ) connections P ∂M on the boundary. These operators act on the Hilbert space (6.3). We illustrate this with asimple example that plays a key role in this paper, namely with the N = 2 theory T ∆ , Π Z thatwe associate with a single tetrahedron.In particular, in the previous section we identified the S b partition function of this theory(6.11) with the wave function of the SL (2) Chern-Simons theory on a tetrahedron. From theexplicit form of the partition function (6.11), it is easy to see that it satisfies the functionalequation Z ( ˜ m Z + ib (cid:1) = (cid:16) − e − πb ˜ m Z (cid:17) Z ( ˜ m Z ) . (7.1)Using ˆ Z (cid:48)(cid:48) = ib∂ ˜ m Z and ˆ Z = 2 πb ˜ m Z , we can write this equation in a more convenient form: (cid:16) e ˆ Z (cid:48)(cid:48) + e − ˆ Z − (cid:17) Z ( ˜ m Z ) = 0 , (7.2)which is clearly reminiscent of the familiar equation (5.8) that describes the space of SUSYmoduli in the theory T ∆ , Π Z (cid:48) . Indeed, for reasons that we reviewed at the end of section 6, inthe semi-classical limit (cid:126) ∼ b → M SUSY : e Z + e − Z (cid:48) − . (7.3)In terms of geometry, we know from Section 2 that Z and Z (cid:48) are the complexified “shearcoordinates” or edge parameters on the boundary ∂ ∆ of the tetrahedron; and indeed (7.2) isjust the quantization of the tetrahedron’s classical Lagrangian (2.8) [12]. More generally, ifa 3-manifold M has a triangulated geodesic boundary, it is the quantization of external edgecoordinates exp( ˆ X E ) on the boundary that appears in the operator equations (6.25).From a different perspective, the classical external edge coordinates x E = exp( X E ) ona triangulated geodesic boundary C = ∂M also correspond to vevs of line operators in the four-dimensional N = 2 theory T [ C , su (2)]; and the quantized ˆ x E = exp( ˆ X E ) correspond tothe quantum line operators themselves [58, 59, 60, 34]. To be more precise, it was shown in[16, 34] that every edge E of C determines an IR line operator exp( ˆ X E ) in the abelian N = 2theory on the Coulomb branch of T [ C , su (2)]. This operator carries the electric and magneticcharges associated to the edge E , exactly as described in Section 4.4. Using this relation, wepropose to interpret operator equations (6.25) as Ward identities for line operators in a 4dtheory coupled to the 3d boundary theory T M . For example, in the context of knot complements, these operators are often denoted as ˆ m = e ˆ u andˆ (cid:96) = − e ˆ v . We simply abbreviate Z S b ( T ∆ , Π Z , ˜ m Z ) as Z ( ˜ m Z ). – 51 – M T [ C ] H W Figure 23:
Line operators in 4d becoming identified in the boundary theory T M . In the presence of boundary conditions, not all line operators of the bulk N = 2 gaugetheory in four dimensions are independent. Indeed, one can start with a line operator L (or,more generally, a collection of line operators L i ) in the 4d N = 2 gauge theory and thenbring it to the three-dimensional boundary where the theory T M lives (Figure 23). Due tothe boundary conditions (which e.g. may identify some of the 4d fields), vevs of line operatorsthat were independent in the bulk become related on the boundary. This can be summarizedin the form of Ward identities (cid:88) c i L i = 0 . (7.4)For example, in our favorite example of the theory T ∆ the equation (7.2) can be written inthe form (7.4) as W + H − − (cid:39) Z , ˆ Z (cid:48) , and ˆ Z (cid:48)(cid:48) with the corresponding abelian Wilson / ’tHooft line operators: edge line operatorˆ z = e ˆ Z W = Wilsonˆ z (cid:48) = e ˆ Z (cid:48) Wilson-’t Hooftˆ z (cid:48)(cid:48) = e ˆ Z (cid:48)(cid:48) H = ’t Hooft (7.6)(Thus, ˆ z − = H − denotes an ’t Hooft operator of magnetic charge −
1. Similarly, W = H = 1 denotes a trivial line operator.) The above dictionary (7.6) corresponds to thepolarization Π Z for T ∆ . The triality symmetry of T ∆ (3.26), generated by the ST elementof the 4d electric-magnetic duality group SL (2 , Z ), permutes Wilson, ’t Hooft, and Wilson-’tHooft operators.To explain the origin of Ward identities like (7.5), it is instructive to simplify the theory T ∆ (which consists of a chiral multiplet and Chern-Simons coupling) even further and consideronly the Chern-Simons part of the theory. As we discussed in section 6, a supersymmetricChern-Simons interaction at level k for the background gauge field contributes to the partition– 52 –unction a factor (6.10): Z CS k = e − iπk ˜ m . (7.7)Much like the partition function of the theory T ∆ , it obeys the following q -difference equation: (cid:16) ˆ z (cid:48)(cid:48) − q k ˆ z k (cid:17) Z CS k = (cid:16) e ib∂ ˜ m − e iπb k +2 πbk ˜ m (cid:17) Z CS k = 0 . (7.8)According to (7.6), this identity should be interpreted as a statement that at a 3d boundarywith Chern-Simons term at level k a ’t Hooft operator with one unit of a magnetic flux isequivalent to a Wilson operator of electric charge k , H − e iπb k W k (cid:39) . (7.9)This is indeed correct, as one can easily verify by doing a direct path integral manipulation.Notice, it is important here that supersymmetric Chern-Simons theory lives on the boundaryof the 4d space-time where Wilson and ’t Hooft operators belong. T M T [ C ] T [ C ] S L S M C γ L p S p S C Figure 24:
Line operators in both M and T M . Most of our discussion in this section was based on interpreting T M as a boundary theoryin the 4d N = 2 theory on the Coulomb branch of T [ C , su (2)], where C = ∂M is the geodesicboundary of M . This interpretation can be easily extended to 3-manifolds with “small”boundaries ( a.k.a. cusps) and also to 3-manifolds with several boundary components. Forexample, in the latter case, each boundary component is a 2-dimensional Riemann surface C to which we associate either IR or UV limit of the 4d N = 2 gauge theory T [ C ] depending onwhether the boundary C is “big” or “small.”Within this framework, we could also look at a different class of line operators, corre-sponding to curves in a 3-manifold M itself. In general, a 1-dimensional curve γ L insidea cobordism M may have end-points on various boundary components of M , as shown inFigure 24. In order to find its interpretation in 3d N = 2 theory T M , we recall that a point p ∈ C defines a surface operator in 4d N = 2 theory T [ C ], whereas the cobordism itself definesa domain wall between two different N = 2 theories in four dimensions ( cf. Figure 1). In– 53 –our-dimensional space-time, a surface operator meets the domain wall over a 1-dimensionalcurve, which is precisely the line operator L associated to γ L ⊂ M , see Figure 24. In thisdescription of T M as a theory on a duality wall, the line operator L arises as an interfacebetween two different surface operators.The interplay between line operators on M and line operators in 3d N = 2 theory T M can be easily motivated by thinking about T M as the effective theory T [ M, su (2)] obtained byreduction of the six-dimensional (2 ,
0) theory on a 3-manifold M . This is very similar to thecorrespondence between line operators in Liouville theory on C and line operators in 4d N = 2theory T [ C , su (2)], where 6d theory again turns out to be very useful [58, 59, 60]. Indeed, six-dimensional (2 ,
0) theory contains two-dimensional surface operators. Upon compactificationon a d -dimensional manifold M d , the support of a surface operator can have the form γ L × L ,where γ L ⊂ M d is a 1-dimensional curve on M d and L ⊂ R − d is a line in the (6 − d )dimensional space-time where the theory T [ M d , su (2)] lives. Surface operators of this formgive rise to a large class of line operators in T [ M d , su (2)] labeled by curves γ L on M d . Acknowledgments
We wish to thank A. Kapustin, N. Seiberg, C. Vafa, R. van der Veen, and E. Witten for manyhelpful and enlightening discussions. The work of TD is supported in part by NSF Grant PHY-0969448. The work of DG is supported in part by NSF grant PHY-0503584 and in part by theRoger Dashen membership in the Institute for Advanced Study. The work of SG is supportedin part by DOE Grant DE-FG03-92-ER40701 and in part by NSF Grant PHY-0757647. TDand SG thank the Kavli Institute for Theoretical Physics (research supported by DARPAunder Grant No. HR0011-09-1-0015 and by the National Science Foundation under GrantNo. PHY05-51164) and the Simons Center for Geometry and Physics for their hospitalityin the summer of 2011. TD also acknowledges the Max Planck Institut f¨ur Mathematik forits hospitality and support during June, 2011. Opinions and conclusions expressed here arethose of the authors and do not necessarily reflect the views of funding agencies.
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