JHEP07(2018)145 (1) u Springer July 16, 2018 July 23, 2018 April 13, 2018 : : : Accepted Published Received -type on a 3-manifold, we 1 A Published for SISSA by https://doi.org/10.1007/JHEP07(2018)145 0) theory of , = (2 SO(3) flavor symmetry enhancement for the b / N [email protected] , . 3 and Kazuya Yonekura 1803.04009 The Authors. a Chern-Simons Theories, Conformal Field Theory, Duality in Gauge Field c
We revisit Dimofte-Gaiotto-Gukov’s construction of 3d gauge theories asso- , [email protected] Seoul 08826, Korea Kavli Institute for theUniversity Physics of and Tokyo, Mathematics Kashiwa, of Chiba the 277-8583, Universe, Japan E-mail: Center for Theorectical Physics, Seoul National University, b a Open Access Article funded by SCOAP Keywords: Theories, M-Theory ArXiv ePrint: viewpoint. Base on thestruction understanding to closed of 3-manifolds symmetry byoperation. enhancement, identifying we The the generalize generalized gauge construction theory thecalculus predicts counterpart con- in infinitely of knot many Dehn theory. 3d filling Moreover, dualitiesthat by from theories using surgery the associated symmetry to enhancementhancement all criterion, which hyperboilc we is twist show unexpected knots from have the surprising 6d SU(3) viewpoint. symmetry en- Abstract: ciated to 3-manifolds withviewpoint a of torus compactification boundary. ofpropose a a After topological 6d clarifying criterion for theirsymmetry SU(2) construction in from the a theory associated to a torus boundary, which is expected from the 6d Dongmin Gang Symmetry enhancement and closingcorrespondence of knots in 3d/3d JHEP07(2018)145 5 9 = 2 N 4 49 51 type. The 6d theory is the 1 A d 6 irred T 42 DGG 43 T and ) 19 x d 37 6 ( T qB – 1 – + 47 pA 50 N I 39 26 1 48 37 ] 25 N ) and [ 31 x SO(3) types ; d / 6 [SU(2)] irred T T ] for examples of dualities among 3d gauge theories. There could be 3 m, e ( – ) 1 A,B ( SU(2) enhancement SO(3) symmetry types of knots from six dimensions N (2,0) theory on 3-manifolds: / / I 0)-superconformal field theory (SCFT) of 1 , A = 2 dualities from surgery calculus = 2 superconformal field theories labelled by 3-manifolds -transformation of 3d SCFT S N N [SU(2)] and SU(2) In this paper, we consider a certain subclass of 3d quantum field theories with B.1 Brief reviewB.2 of The globalB.3 structure of the symmetriesB.4 and ’t Hooft SU(2) anomaly T 3.2 SU(3) enhancement 4.1 Dehn filling4.2 on Small hyperbolic manifolds 2.1 6d 2.2 Dimofte-Gaiotto-Gukov’s construction: 2.3 Relation between the two constructions 3.1 SO(3) (4 supercharges) supersymmetry whichof can the be 6d engineeredsimplest (2 by maximally a supersymmetric twisted conformalworld compactification field volume theory theory of and two describes coincident the M5-branes low in energy M-theory. The 6d theory has SO(5) dimensional case, Abelian gauge interactiongives in non-trivial 3d IR is strongly physics.the coupled same at IR Different infrared fixed (IR) gauge“duality”. point and theories Refer along to renormalization at [ group ultravioletenhanced (RG) symmetries (UV) and in could such the IR phenomena endpurely fixed is point field at called which theoretic is viewpoint, invisible these in the phenomena UV are gauge not theory. easy From to understand or predict. 1 Introduction and summary 3-dimensional (3d) quantum field theory exhibits several interesting aspects. Unlike higher 5 3d A 3d index B 4 Dehn filling in 3d/3d correspondence 3 Symmetry enhancement Contents 1 Introduction and summary 2 3d JHEP07(2018)145 ]. (1.2) (1.1) . The ). The M,K [ K d (2) flavor 6 1.1 T su M which preserves . From the non- . These theories with several compo- ] theory while all 1 K ] has a M DGG with a monodromy = 2 quantum field L T M N Using the vector SO(3) and M,K [ M,K ] theory. This point was 1 [ 3 . ) connections on the knot d µ M 6 ) is proposed by Dimoft- C M ). But there is a one-to-one = DGG T , ), and we can labell them by Z M,K T [ δW d 6 M,K of K ∂N, N,A T ] give similar integral expressions while ( 1 × ] theory generically has U(1) flavor ]. In the analogy, closed Riemann 1 S 18 is replaced by a link K 2 , – H 5 1 , × . K 4 ∈ 15 R 2 ] which says that the partition functions 2 M,K ) and ( , [ deformed by R 1 A ] give integral expressions for complex CS 10 is a knot inside ] is actually labelled by a choice of a knot R – −−−−−−−−−−−−−−−−−→ 14 6 A K – DGG M,K d T – 2 – 6 irred ] on 11 , T N,X 9 [ (2) flavor symmetry associated to the knot in the su with a partial topological twisting along for link. M,K [ DGG ) Chern-Simons (CS) theories on d L ]. However, there are two manifest differences between 6 T M C SCFT , T P × , associated to the knot 2 , M,K A [ 1 d X ] labelled by the choice of ( and a regular puncture on the surface corresponds to R M 6 appears as vacua on ]. Secondly, the = 2 theories of class S [ T 20 for knot and M K N \ M,K ], determined by the topological choice of K on a vacuum ) between them, which we reproduce here: [ ) which has more clear meaning in the 6d compactification ( M ]. Their construction guarantees that the localization integrals of the −−−−−−−−−−−−−−−→ d 2.84 19 DGG := ] theory on supersymmetric curved backgrounds are equal to the partition 6 M,K and a primitive boundary cycle [ . State-integral models [ T T M,K d 6 N K N ] theory are identical to the corresponding state-integral models. In the orig- ] is actually T (2,0)-SCFT on M,K [ 1 d is a compact (closed) 3-manifold and 6 compacitification along A M,K M,K T [ [ M −−−−−−−−−−−−−−−−−−−→ 6d with a regular co-dimension two defect along Base on the technical developments, field theoretic algorithm of constructing 3d One non-trivial task is finding field theoretical description of the 3d theory The system can be generalized to the case when a knot 1 4 supersymmetries. After the compactification, we obtain a 3d DGG DGG / nents. We use the letter symmetry. Motivated from theprecise similarity relation and ( differences of two theories, we propose the two theories. Firstly, only somecomplement subset of irreducible flatflat SL(2 connections are expectedalready to appear emphasized in as the [ symmetry, vacua denoted of as U(1) complement map between the twothe topological choice choices, of ( ( explicit map between two topologicaltrivial choices match is of explained supersymmetric aroundT partition figure functions, it is tempting to conclude that the gauge theory Gaiotto-Gukov [ T inal paper, the 3d gauge theory of the functions of purely bosonic SL(2 defect along partition functions while localizationfor techniques the supersymmetric [ partition functions of 3d field theories. surface corresponds to 6d picture predicts theresulting existence 3d of gauge theory. A hint comes from so called 3d/3d relations [ Here subgroup of SO(5) R-symmetry, we perform1 a topological twisting along theory, say are 3d analogy of 4d of this paper is as follows R-symmetry and allows a 1/2 BPS regular co-dimension two defect. The concrete set-up JHEP07(2018)145 ) ) ]. = × (2) (2) ] is = 4 1.2 1.2 (1.3) ] has M su su [ M [ N M,K [SU(2)] [ in ( d T 3 6 DGG d irred M,K . 6 [ µ irred T T T (2), and this = 4 = is a Lens-space of the vacuum DGG su N T (2) moment map ], we identify the M [SU(2)], we find a δW -reduction, the 6d su 1 ] still has the T ] appearing in ( S SCFT P M,K [ M,K [ M,K [ DGG . Each of the arrows in the d T ] for three smallest hyperbolic 6 DGG irred K T T (2) flavor symmetry. The M [ ] theory generically has only U(1) su d ] and 6 irred = 2 theory has SU(3) symmetry. T M,K [ ] has N (2) Yang-mills theory (SYM) and the co- M,K has a surprising SU(3) symmetry. As a [ d su (2) moment map operator of the 3d 6 DGG is hyperbolic. K ] might not contain the ]. By applying the Dehn filling operation, we T T M,K su [ 22 d M – 3 – ] on a particular point 6 ] corresponding to Dehn filling operation on the ] theory. The topological condition is summarized M,K T U(1) symmetry where the SU(2) rotates the two [ A ] and has similar 6d interpretation as is the (2) symmetry enhancement when × d M,K M,K M [ 6 irred [ [ µ su d T N,X 6 [ d T ], 6 irred DGG T ) which guarantees the absence of moment map operator and T . The operation is only possible when the DGG ]. One interesting aspect of our construction of T K M,K \ 23 [ N,A d 6 M ] is broken by the superpotential deformation = figure-eight knot] T = ) is a chiral operator in the triplet representation of ,K 3 symmetry associated to the knot is a linear combination of two Cartans 3 N M,K [ , µ S vector multiplet coupled to two chiral multiplets of charge +1 A d 2 6 U(1) which is expected to be enhanced to SO(3) according to the criterion 0 X = T , µ × 1 after taking the IR limit. In that case, the superpotential deformation is not M µ (2) symmetry in [ ] to the 5d theory. Then . From a group theoretical analysis, the enhancement implies that the SU(2) . For example, we expect µ su 21 = ( 1 1 DGG ] theory for all hyperbolic twist knots = A U(1) µ T Base on the proposed relation between As an application of the symmetry enhancement criterion, we show that the As an intermediate step, we introduce a 3d SCFT The proposed relation explains why the ,K . The theory is denoted as (2) flavor symmetry. The Dehn filling operation is analogous to closing of punctures on [SU(2)] theory. 3 Riemann surface in 4d/2d correspondencecan [ extend the DGG’sM construction to 3d gaugeAs concrete theories examples, labelled field theoretic by3-manifolds descriptions a of are closed given 3-manifold in [ U(1) should be enhancedconstructing to the SU(3). corresponding We conserved checked current the multiplet. symmetry enhancement byfield explicitly theoretical operation on knot complement su The theory only haschirals and manifest the SU(2) U(1) comesfield. from The the U(1) topological symmetryof of the the SU(2) dynamical abelian gauge in table S simplest example, we claim that the following 3d topological condition on ( thus the in table and do not expect the enhancement when which is themoduli IR space. fixed point Unlike operator of possible (or more precisely,symmetry. it is By irrelevant) carefully and analyzing thus the the coupled system, 5d SYM+3d which is typically atheory becomes relevant 5d deformation maximally supersymmetric indimension two the defect RG in 6d sense.theory theory [ is After realized by couplingT a copy of the 3d operator is associated theabove equation co-dimension are two nontrivial defect RG flows along which are explainedsymmetry below. associated to thesymmetry knot of while Here JHEP07(2018)145 in it (2.2) (2.1) . The ]. One K M type on a 1 ] which are on M,K [ A M T K [ . The resulting ). Since the 3d , we discuss how K d 6 5 irred 5.2 with a knot T ] (DGG) based on an where M = 2 dualities. M. 19 , 0) theory of N ]) , = 2 dualities. One way of ) which determines when the N = (2 ] (DGG) and the corresponding M,K [ 19 M,K T N is a knot in , we introduce two ways of associating M . These two theories are argued to be 2 K ) is a primitive boundary cycle . ⊂ \ Z M inside it with a 3d gauge theory ∂N, – 4 – ,K ( 1 ]. After explaining the two constructions in detail, K ]. The other is through a twisted compactification H = 2 gauge theory ∈ ]. ) between two constructions. Base on the proposed , we identify field theoretic operation corresponding to N M,K A M,K [ 4 [ d 2.84 6 ] theory is enhanced to SU(2) or SO(3). The criterion is M,K [ T ), which will be used throughout the paper. The choice can DGG (a 3d T and a knot K with a regular co-dimension two defect along T M,K
[ M M where and ) , . In section with a regular co-dimension two defect along a knot ) DGG 1 , we give a topological criterion on ( M is a closed 3-manifold T 3 M M,K ( M N,A ( has infinitely many different surgery descriptions and surgery calculus tell ↔ ]. = 2 superconformal field theories labelled by 3-manifolds M ) = 2 gauge theory ], and we will propose the more precise relation between them with supporting M [ (2,0) theory on symmetry 19 N N is a knot complement and 1 d A A 6 M,K irred X ( N Before going to detailed analysis, let us first introduce an alternative labelling for the The paper is organized as follows. In section T topological choice, ( be replaced by closed 3-manifold other way is usingideal the triangulation construction of by therelated Dimofte-Gaiotto-Gukov knot [ [ complement evidences. We describe the relation after reviewing basic aspects of two approaches. One way is through a twisted compactification of a 6d the surgery calculus in knot theory predicts infinitely many2 3d 3d In this section, weto introduce a two ways 3d of associating a 3-manifold relation, in section U(1) summarized in table Dehn filling operation inthe 3-manifold DGG’s side construction in to 3d/3d the correspondence. case when It the allows knot us is to absent. extend In section is through the construction bygauge Dimofte-Gaiotto-Gukov theory [ is denotedof as 6d 3d gauge theory is denotedwe as propose a precise relation ( theory depends on onlytheory the are topology topological of invariants3-manifold of the invariant the called 3-manifold, 3-manifold. “3d everythe index” physical As which quantities an is of example, nothing the we but introduce the a superconformal new the index choice of of 3-manifold representing closed 3-manifold is3-manifold using so calledwhen Dehn two surgery surgery descriptions representation. giveof the A same a 3-manifold. closed closed Different 3-manifoldrelated surgery give by representations 3d different dualities. field theoretical One descriptions illustrative example of is given around eq. ( that we can relate surgery calculus in knot theory to 3d JHEP07(2018)145 (2.4) (2.5) (2.6) (2.7) (2.3) . ) ) 1 ) ∂N S , K × ) in N ) 2 M A K D ⊂ knot complement. M K ], respectively. For the K is a link complement with A M ) \ [ . ). 2 N Z d A D d can be alternatively described by 6 theory on irred M hyperbolic 6 , irred . T 1 ∂N, T M ( = K A 1 N is a by closing a cycle . H ( ] N ] K and ∈ N ] and on the moduli space of vacua) \ d A M 6 [ M M,K d [ M,K T 6 [ = T SCFT d T ) is given by is the origin of 6 is a knot complement with one torus boundary P ) is determined by N T 2 – 5 – N ] = D (contractible boundary cycle of N,A ∈ M,K : Tubular neighborhood of a knot ∼ N,A p ) [ K A T M N K ( inside a closed 3-manifold , with where K , and a boundary cycle , 1 M K N ∼ K S ( N / × − )) with a regular co-dimension two defect along on a particular point 2 1 S D M × ), the corresponding ( ⊂ := 2 1 ), on the other hand, ( D S K ( ] = (Twisted compactification of 6d (2,0) ] = (The low energy limit of (2,0) theory on 3-manifolds: \ M,K := (A closed 3-manifold obtained from ∪ 1 ] to be defined, we assume that N,A } × M A (i.e. the contractible cycle in the removed tubular neighborhood p A N : A primitive boundary cycle around the knot . The choice of a knot M,K M,K { N = [ [ d = A 6 M,K N := ( := d [ 6 T irred M T K d 6 irred As a simplercase, set-up, the we resulting can 3dT also theory consider is the denoted case as when the defect is absent. In that 2.1 6d We define In most part of this paper,but we assume our that discussion can beseveral easily torus generalized boundaries. to the case when Using the map, we can use two choices interchangeably. For example, For given ( Figure 1 a choice of knot complement For a given ( JHEP07(2018)145 = 4 (2.8) (2.9) theory N 4) and 1 , A 0) ··· , , 4 , 3 ) is the vector , 1 1 = (2 , S = 0 =2 i N =0 ) × I i k ( becomes the 4d 2 iφ 2 I R , φ 1 T , A + . Only after picking a point i 2 =0 Z ] on A µ / 6 A M [ R := d i , the supersymmetry is defined 6 = ( comes from the integral of the 2-form A T M ( V 1 0) theory on . This is because we can use the + , S 1 ⊕ S ) 1 , = (2 , the × = 4 SYM is theory . However, we will be sometimes sloppy 2 compactification of the 6d / =0 2 N k R 2 , where N M R 2 T , φ Z 1 ] is as follows. The moduli space of vacua of / , ] 1 – 6 – =0 S M µ × M M,K A × [ 5 ( 1 × R 4) are twisted chiral fields. The reason that we regard The structure of the moduli space of vacua becomes d S 2 , → , which set the cutoff scale of the low energy effective 6 irred 3 R ) × T , 5 M , 2 4 , R 3 = 2 ] is not a genuine 3d theory, but should be considered more , 2 i , ( 4). After compactification on i =1 , k 0) on M,K iφ [ , d , φ ··· + 6 . However, somewhat miraculously, it is believed that 5d SYM describes even a 4 , i , 1 = 2 SYM on 3 T , S A = (2 via 5d SYM. 2 , ], we pick up a point and take the low energy limit. The low energy limit N = 0 1 = , 1 N k i ( S . 5d =0 and taking the low energy limit, the 6d 1 A I k 6d S 2 M,K × A φ [ Z ( 2 / 2 ] ] in general contains several different connected components. Then, we have to d 1 . On the other hand, the moduli space of 4d R 6 3 irred 2 S T T ] is a kind of mirror symmetry analogous to the case of 4d class S theories. × , and we split these fields as on . The moduli space of this theory is [ 2 5 In In other words, The reason that we consider M M,K 2 [ [ On general grounds, one may only expect that 5d SYM describes the moduli space only in the limit A simple example which illustrates the point is the R R d T 3 2 d d as twisted chiral fields rather than chiral fields is that the relation between 5d SYM and 6 6 6 field on on [ SYM. In this casethe the whole moduli moduli space space is of connected, vacua. but stillof there very is small no radius singlefinite of 4d radius effective of theory describing multiplet and A T on scalar fields on From the point of view of the super-algebra on The bosonic components of the 5d SYM theory are gauge fields T simpler if we compactify5 the dimensional 3d maximally spacetime supersymmetricset-up to Yang-Mills is (5d SYM) theory description. The may be described bywe a will 3d specify SCFTto which (which 5d point can SYM. on be the empty moduli or a space topological of theory). vacua we Below take, by using reduction describes the entire modulispace, space we of can vacua. obtainthat a Only point. low after energy specifying effective a field point theory on which theappropriately describes as moduli the the physics 6d near theoryand compactified call on it a 3d theory in this paper. T decide which point ofThe the typical distances moduli between spaceof different we the components consider compactification of before scale the3d on taking moduli theory. the space are Therefore, low of energy we the limit. cannot order expect that there is a single effective 3d theory which JHEP07(2018)145 C be t (2) (2.11) (2.15) (2.16) (2.10) (2.12) (2.13) as su 2.2 1 ), a part × S [SU(2)] is H T is irreducible. . of 2.13 A G (2) R / su }} ]. However, in the 3d ) to form a covariant . m ) and ( . , 29 , − 2.13 C H 3 28 is given by 2.10 A m, [SU(2)] in the context of (2) (2) { 2 3 m T can be realized as coupling su su 4 ]. The theory = . + 28 e ). ν K M A . – d ⊂ } 24 B.1 A . . The theory has ) m ) (2.14) over K 2 , e µ A − i e ν Tr m =1 . dy a 1 2 m, [SU(2)] and the 5d SYM is given by ) ) tr( i { R tr( a T 2 . This twisted superpotential corresponds M 1 ˜ A K E 2 = π Z e ν Z ds , = 4 SCFT given by U(1) vector multiplet 8 eigenvalues of R e a ν (see appendix K = , is considered as a kind of “internal manifold”. Let . This term combines with ( 1 E ˜ = 2 YM ) Z E N – 7 – → with mass parameter E is the mass. The mass of defect is related to the g K t E K 2 1 = ( 1 ˜ m 2 ∼ YM S S E∂ g e e νδ µ ] := ⊃ × mass ) A 2 = YM-defect [ 1 f W f W , and R S f W CS K × on Eigenvalues of 2 1 5 A ∧ A 2 YM ] to the fields of 5d SYM [ R g . This is a 3d + 21 ] in the limit is reducible, we can turn on the expectation values of the vector multiplets = 8 A – ] on d A 6 B.1 : := holomorphic moment map operator of := holomorphic moment map operator of YM . Then, the kinetic term along this direction comes not from the Kahler potential, , where we have used e e µ ν f W K M,K {A E [ ) ). These branches are very important in 4d class S theories [ d : 1 6 = A , e ν T + =0 ( k t is the gauge coupling of 5d SYM which is related to the radius ∂ ( , φ is the line element on 1 [SU(2)] theory [ ˜ , E vacua T 2 YM , and hence the direction along the knot =0 ] for detailed explanations of the coupling of 5d SYM to g ds 2 µ M R 28 A By solving F-term equations for the twisted superpotenal in ( We can also include (complexified) mass terms to the defect as The regular co-dimension two defect along a knot The twisted superpotential is given by complex Chern-Simons action as When the connection The gauge invariance is preserved as follows. The supersymmetry is considered in the two dimensional 5 4 = ( the coordinate along but from the twistedderivative superpotential as V theories considered in thisTherefore, paper, we we can only neglect those consider branches. points on the moduli space on which space See [ 4d class S theories.perform The a analysis detailed there analysis. may be extended to the 3d/3dof case, the but moduli we space do of not vacua where eigenvalues of Then the twisted superpotential coupling of the This means that we integrate the one-form tr( coupled two fundamental hypermultiplets ( flavor symmetry and let This is the resultsto in the [ twisted superpotential obtained in DGG’s construction discussedthe in 3d section reviewed in appendix where JHEP07(2018)145 1 α ω S A ) is ] for (2.19) (2.17) (2.18) 34 ) gauge – SCFT C P ) contains , ( 31 , M 13 SCFT . . ) connections. P e ν m C ( e , M ) of the moduli space . P ) ( 1 S M ), ]. Refer to [ -cycle) = to be zero. Then the point × 1 ) are not known and they are = 0. Namely, A , the moduli space of vacua 2 S 1 P m ( R m S × is the group of PSL(2 ]) among all flat connections M,K 2 [ M α exists. For this purpose, we define ), is a discrete set. Let us take an G R 3 ) of the moduli space of vacua on ] on A d [ ) which is determined by the unique K 6 ) is given concretely in terms of flat R irred 1 . Because of the supersymmetry, the T S P ) and ] on CS SCFT . ie . 3 , and M,K × ] on SCFT [ P R goes to a subset K − ( 2 d K ] of P 6 ( ). Then, if we compactify the theory on R ω P M,K T M 30 [ 3 ( ] on . More explicitly, using the spin-connection d M M,K = – 8 – R 6 [ d K ] on on the moduli space of vacua which is taken in the T 6 \ ( around hyp vacua T by continuity from e ν ] on ( M,K M A [ e d is defined by the condition that M,K m vacua ) holonomy matrix along SCFT 6 [ = ⊂ M d P C are determined by the mass parameter T vacua 6 ( , ) M M,K is large may be dropped since there is no phase transition N [ e ν T P d e M ( 1 SCFT ( 6 , S P ), then the point T vacua 3 ( 3 R vacua becomes a subset R M → M ) may be given concretely in terms of flat PSL(2 vacua . This is the space of flat connections of the complexifield gauge P denoted as ∈ M ] on ) := (PSL(2 M SCFT 1 A SCFT P ( S ∈ M hyp P ( hol ) with the holonomy A M,K × of the complete hyperbolic metric, the flat connection can be expressed as P [ ρ ). First, let us consider more generally. For simplicity we assume that the 2 C d e M 6 , R ] such as superconformal index, on the other hand, other flat connections in T 2.7 ) is the delta function localized on ) need not be a single point, but may have several points whose number is related ( ) may contributes. ) even for nonzero mass K P . Then, the point ( ( is defined as follows. After compactification on 1 δ S vacua M,K SCFT SCFT [ M Now we give a conjecture about how To consider a superconformal point, we set the mass The explicit forms of Now we can specify the point P P × M ( ( d 6 2 SCFT irred T M connections. First we consider theM case where a knot This flat connection haswith the parabolic greatest boundary value of holonomyto Im( and a is squashed conjectured todiscussions 3-sphere be on partition the the function only conjectureCS [ vacua from theory, contributing various holographic respects, principal state-integral and model resurgent of analysis. the To complex other physical quantities of the connection complete hyperbolic metric on and dreibein propose how P of the theoryR on to the Witten indexcondition of that the the 3d radius effective of under theory change of on the radius. defined just by the abstract field theoretical considerations as above. However, later we will on of vacua on This definition ( moduli space of vacuaarbitary on point with a radius which is large enough compared to potential barriers between different points transformations on group PSL(2 Notice that the eigenvalues of where JHEP07(2018)145 , )) N for < π ], for K 2.47 (2.23) (2.24) (2.20) (2.21) (2.22) ] A \ Z M } N,X Im[ [ = ) as hyp < G A N N } / ( . We make the DGG . χ T }} m }} m ) = 1 m − A (with 0 − ( z = 0, since we could have m, m, hol { { ρ m ) containing = = with mass N ∩ { . If the closed 3-manifold is e ν ( ν ) χ M . N . Another way of representing the ( ). We also define SCFT 0 ∼ DGG P χ , but we assume that it is independent , m / T = K ! i M A ∆ SCFT in a way that all vertices are located on the P N eigenvalues of k ( =1 eigenvalues of [ i 3 , = ) H – 9 – M
∩ { K ) = 1 is stronger than M ) ( = The construction is based on a choice of an ideal A even if its eigenvalues are zero. N ( e νδ ( N ν 0 hol : . = χ ρ , ) A T Dehn surgery component N ) = ( ) is given by 0 the connected component of A ∧ A χ { , m + SCFT := P ) = A ) of the closed manifold ( SCFT d P -th tetrahedron in the triangulation. Ideal tetrahedron can be em- , m : . ( . Notice that i M ) is proposed. Empirically, the theory associated to non-hyperbolic N SCFT M explicit by writing it as M {A P is hyperbolic. Here we give a summary of the DGG’s construction with a and both of edges and faces are geodesics. Hyperbolic structures on an ( SCFT of N,A ⊂ m m [ P 3 N ( M T H K M ) = denote the N m along a boundary cycle [ ( i χ K ], the component is called ], a combinatorial way of constructing a 3d SCFT, which we denote Next, consider the case where there is no knot on 35 19 Here ∆ bedded into a hyperbolic upperboundary half of plane ideal tetrahedron can be parameterized by a complex parameter which play a crucial role in theMechanics symmetry of enhancement of ideal the triangualtion triangulation theory. In [ given choice of ( is a trivial theory onlythe with case topological when degrees ofmodification freedom. on In superpotential this deformation subsection associated we to focus ‘hard’ on internal edges (see ( The definition of the right hand sideof contains the a knot choice a nonzero upper-right component of 2.2 Dimofte-Gaiotto-Gukov’s construction: we propose that the above equation is represented by a Dehnsome filling operation on a hyperbolic knot complement In [ dependence on This means that we considerThen all we propose flat connections with varying holonomy around the knot. just the set of vacua of the 3d effective theory near JHEP07(2018)145 , 3 H . (2.26) (2.27) (2.28) (2.25) gives a ). Right: i · , , ∞ ) π ) as equivalent 00 for all . We assign edge , z 3 0 ) and ( H , z . < π ∂ z, z ' } ] (0 ' i . , 2 ' z , Z ) 1) 0 z 1 , , . Using the isometry of 0 (0 Im[ z ¯ w , < ,Z,Z 0) ∈ { ideal tetrahedra. A solution to z 2 , 00 dwd can be obtained by gluing the y + k 00 Ii Z 2 ( N dy ' ,G ' ) = (0 → 0 Ii ' z ) z ) = ,G y, w . 3 ,Z Ii 00 H N ( 2 -th internal edge in the gluing) ,Z 0 I ds ' ,G 1 Z – 10 – ( z ) = 00 i ) with conditions 0 w → Z ) 00 Ii ' ) of an ideal tetrahedron. Left: ideal tetrahedron in , these parameters satisfy ' 00 } 00 3 2.28 G z 00 z H , z z 0 ,Z 0 + 0 = 0 with metric i z log w z, z Z } , 0 Z,Z 0 Ii C ) and ( z ' G z ∈ :( + log z 2.26 3 (sum of angles in small boundary triangle equals to i edges meeting at the = . , w z, Z Z ' ' + w iπ Ii ) to each pair of edges of ideal tetrahedron as in the figure below. z R log 00 (dihedral angle between two faces meeting on the edge) + (torsion) G = = 1 { i ( ∈ , z 00 00 i i 0 = k Z y =1 πi . Z := i X e } z, z + 00 + -internal edges in an ideal triangulation with ): 0 = = 2 i := (sum of all logarithmic edge variables associated to i k ), which is the cross-ratio of the positions of its vertices on Z . Edge parameters ( Z ,Z I z 0 ), four asymptotic vertices can be placed at ( y, w − C + ( e C i { , Z An hyperbolic structure on a knot complement Z,Z = { := log 3 There are these gluing equations ( hyperbolic (generally incomplete) structure on to impose the following conditions hyperbolic structure on each tetrahedron in a smooth way. For the smooth gluing, we need The second equation follows directly fromcross-ratios. the geometric These definition of constraints ( aretetrahedron: compatible with the following cyclic symmetry of ideal Here “torsion” is aedge. quantity which For measures an the ideal twisting tetrahedron of in hyperbolic metric around the Z parameters ( Geometrically, these edge parameters correspond to Figure 2 H PSL(2 topologically, ideal tetrahedron is a tetrahedron with truncated vertices. JHEP07(2018)145 , Ii c (2.31) (2.32) (2.29) (2.30) (2.33) is injective . T ) . . χ connections on } i = 1 iπ ). The algebraic hyp 1 00 Ii A h G ) + T 2.26 ) / (conj) 00 i ) 00 i / z Z ) is always contained in C . But it is known that ], with some integers ( ) 00 i , ] gives an irreducible flat- 0 Ii α C N 35 ). The map , G 2.21 T ) + ) 0 i of i z 0 Ii N, 2.20 ( Z [ i G T Ii ) in ( D α PSL(2 G i ) = SL(2 − ( z C , m 00 Ii → k , =1 ] for non-exotic k i X =1 ) ] is empty [ G i Y T ( , , T = N 1 Ii ( . SCFT c N, a 1 − [ For later use, we classify a primitive bound- N, P ) π [ ( k D =1 1 = 0 = , More generally, a solution to the exponenti- i D X . 00 i ] determined by gluing equations of an ideal 00 M i . A point in − z + A ) can be chosen as even-integers T – 11 – 0 i , α where m 00 i 00 i Hom i 00 i z z i S α N, ∈ z , α [ , α i + : ) ! D α 1 0 Ii } hol 2 ) = ) can be written as linear combinations of logarithmic − i 1 ρ G z a/ , { 0 N N ( 0 − ( − 0 e 1 otherwise if all ( \{ χ π Ii 2 ) := , , C ∗ ) here is equivalent to that in ( G a/ ⊂ ( N e ∈ ( N ) Ii ( ) 00 i χ
c T deformation variety χ , z χ was eliminated using the linear relations in ( ∂N 0 k i → =1 ( i 2.29 X ) = ] 1 } , z 0 i except exotic cases when i π A T + ). z ( Z 00 i i SU(2)-type SO(3)-type containing a solution of gluing eqns corresponding to T { ( , α N, hol ( [ , α ) = ρ with integer coefficients ( i T Z D ) = the connected component of α into two types, SU(2) or SO(3), depending on evenness/oddness of the linear ) from ideal triangulation. ] := : → N ( T is of A 0 T ∂N, SO(3)-type of boundary cycle ( 00 i χ χ A ] for any / N, 1 [ SCFT T , α H D i P ( α ∈ N, . Consider the algebraic variety [ Note that the lineargluing equations coefficients in are ( defined modulo the following shifts due to the last ary cycle coefficients ( We currently do notalways work have with the non-exotic field triangulations. theoreticSU(2) understanding of the exotic case and will Dependence on variety depends on thethe Dehn choice surgery of component anD ideal triangulation edge parameters where the definition of but not surjective.A Using the map, holonomy matrix along a primitive boundary cycle The variety is calledconnection a via a map ated gluing equations givesN an irreducible flat PSL(2 triangulation M JHEP07(2018)145 = } j (2.37) (2.38) (2.34) (2.35) (2.36) (2.39) (2.40) SO(3) satisfy / . Then, ,Z 1 00 i A − =1 Z k I { } and integer- . is not unique I ) such that . C = N B Z B { 1 g around . } ) torus boundaries the 00 − j 2 A a T ) even-integers. An ∂N, Z ] are linearly indepen- 00 i ( ,Z 0 i 1 6 N, , α Z H i ( { with 1 , . . . , k α ) matrix ∈ H N Z , . . . , k = B , = 1 k, } → 0 j N = 1 ) 1 of them ). The choice of Z ,Z − . i iπν . k I,J ]. An explanation of SU(2) Z T Z 2.31 ,I,J + ] { ∂N, 36 ∈ ( − 1 is of SU(2) type , is of SO(3) type , , k 00 00 00 1 2 = 0 k k 1 2 H 2 A A for all Z Z Z ...... ), only Z Z Z } : satisfying − which makes all ( I ∗ 1 The gluing equations are known to have the Γ i , = to have the properties that − =1 Ii · as in eq. ( 2.28 k I c b, kA , k } } when when { N B – 12 – I defined by B g + , , = 0 Ker Γ = a, b } { } = { B ∈ } I ), we associate a Sp(2 I ] which play a crucial role in the DGG’s construction. , } ). Let the linearly independent set as otherwise Γ → I -cycle is defined in the same way as 1 { , 1 ,A 37 Γ a, C ), are equivalent [ 1 , a, A 1 − and the boundary holonomy variable − B B B { { 2.26 k { k Γ , C ...... P X SU(2) type without relying an ideal triangulation is } 36 } Γ C = / I 2.34 I SU(2) type SO(3) type , } C C { J ( 0) theory point of view is discussed in appendix { IJ , δ ,C I SO(3)-type SU(2)-type = ) and ( C is of = (2 A, B, { } ( J as follows B N Γ 2.33 SO(3) types of ν , / I is of ] from symplectic gluing. C { -internal edge variables in eq. ( A A k -vector k is related to the holonomy along N,X is SU(2)-type if there is a choice of [ b A Among More generally, for an ideal triangulation of a knot/link complement 6 , internal edge variables DGG ij number of linearly independent internal edge variables are dent modulo linear relations inwe ( introduce their conjugate variables From the choice ofvalued 2 ( where the SU(2) where but have the following freedom of choice Using the freedom, we will always choose the followings: Further we can choose a linearly independent primitive cycle types from the 6d T following symplectic structure [ Upon a skew-symmetricδ bilinear alternative definition of SO(3) Two definitions, ( and JHEP07(2018)145 := ),” s J 2 g ~ flavor V (2.45) (2.43) (2.44) (2.41) (2.42) / ( 1 k T − L (1) u . has ! ),” “S-type ( t , t K ) 0 1 g type step I type 0 − ~ V T = 1 case. To be more , , J U i k with integer coefficients · Φ i 00 ~ 0 ( Σ i U V θ e SU(2) 4 SO(3) † i
,Z , , d i Φ Z θ := Z flavor symmetry. Field theoretic 4 d gl U k ~ π V, 1 , SO(3) . The theory SU(2) 2 i Z K V · ) action on the theory which is a gen- , g + ), we can construct the corresponding , and they are now dynamical fields. As ) + Z ~ 0 Σ 0 step I N θ T ~ i ~ V k, V 4 ! · , ν ) is of V ) is of , d J i J with U(1) = N { N J ∆ Σ 0 g , + g Z − T T 1 2 -copies of a free chiral theory − ~ ) into products of “T-type ( ) V A, B ] which corresponds to A, B ~ – 13 – π V k = 1 Z J ~ ) ⊗ V 4 − ). 38 · J J k, 1 }| times ... JI θ − − ) + − k 4 2.33 step II when ( ⊗ I I when ( ~ d V U T ( ( , (( , ∆
T T for non-dynamical background gauge field coupled to U(1) flavor symmetry) Z ) T ) z T b 2 L L L ) action [ π , b := 1 = 4 Z 2 a a, s J , k ( ( ) := ) := ⊗ k ∆ ) := =1 ( i X ~ T ~ V V ~ V , g ( ( ( = T T ) = T · · := (a free theory of single chiral multiplet Φ with CS level · ) are always linear combinations of ) = ) element. gl t s B U K J are background vector-multiplets coupled to the flavor symmetries. ! k g g g ∆ B Z 0 } T L L L ,P i )”: step I ,P k, A V I gl U T KI A { g ) case, the final theory does not depend on the decomposition and depends X ( X
,...,V Z 1 , V )) be a Lagrangian for a theory := ( only has components such that k is the field strength of the vector multiplet is a diagonal matrix whose diagonal entries are either 0 or 1. Let 0 i t K g ~ theory. As a first step, we prepare V step1 J T Now the second step of the construction is Using the symmetry, one can consider Sp(2 Using the gluing data summarized in ( L ,...,V 1 DGG V for the SL(2 only on the Sp(2 where Here ( actions of the basic types are explicit, one needs to decompose aand Sp(2 “GL-type( where Σ symmetry and eralization of Witten’s SL(2 T Notice that ( because of the even-ness condition ( where JHEP07(2018)145 0 i ,Z i step II . The Z (2.51) (2.49) (2.50) (2.46) (2.47) ] if at T N 19 as a linear I C ] A . . ! . N,X c [ step II . T is given by ) is called ‘easy’ [ + c in DGG , I . T k C 1 . For each hard internal edge, V step II } 2.28 − I in 2 k . Upon a proper choice of cyclic T C θO In the original DGG’s construc- , ) = 0 (2.48) C := 2 A 1 ). Therefore for each easy internal in Ii d , 1 X 0 step II ) in ( . . . k − I G k 00 i Z T ] has flavor symmetry of rank 1 whose C U(1) . 00 Ii 2.56 C I Z O ∈ { N U(1) G C , × Ii in = 1 [ 00 Ii i e G i i Ii 1 + G which can be written as the above form in − X Φ ... k I 00 Ii ) DGG + ,...,V ‘easy’ C C k × =1 1 0 G i T i Y ,G 1
O V 2.47 0 Ii Z for all i C – 14 – ) obtained from an ideal triangulation of = 0 Ii + G Z U(1) to its subgroup by adding chiral operators to the I := I G Ii C k C + × flavor symmetry A 2.40 e U(1) G O + X 0 Ii k step II i V × ... T k G . =1 Z i X (1) A L A Ii × Ii u X X 1 G = = G ( C is nonzero for each ( ] I A k C =1 00 Ii =1 i U(1) X k i G U(1) N,X [ P are easy, we superficially expect that × s: I i = and A DGG Z C X I T 0 Ii Superpotential deformation in ( C L G , −−−−−−−−−−−−−−−−−−−−−−−−−→ ) of edge parameters, we can make such an internal edge If all U(1) Ii G -th tetrahedron appears in 2.27 i is the symplectic matrix in ( , there is a chiral primary operator ], they proposed to use ideal triangulations with only easy internal edges. From N ) of I g 00 i 19 C Z As a final step, we break the U(1) -transformed theory still has N Cartan corresponds to the U(1) operator with non-zero spin. Thereusing is the no dyonic way local to write operators. down aHard supersymmetric deformation internal edges andtion accidental [ symmetries. superficial counting, we expect the resulting As will be explainedwhich below, are related different by cyclic a sequence labelingsedge of give basic dualities different in descriptions ( a of duality frame. Theon operator is the charged other only hand, under there U(1) may only be a corresponding gauge invariant dyonic 1/4 BPS Then, the gauge-invariant chiral primary operator and relabeling ( combination of only most one of and ‘hard’ otherwise. This condition simply means that only one of edge parameters ( superpotential An internal edge g whose background gauge fields are where JHEP07(2018)145 . one 3.2 ] and . (2.52) K 9 c . . K + c m T p -decomposition. : dynamical) K V θMT ( 2 d = 2 case reviewed above Z theories for most of knot 1). From the counting of K U 1) and a free theory with 3 − Σ + − k DGG 2 1 , T (1) symmetries except the ones m − u 1) are linearly dependent. So the ). For the construction, we choose which is large enough to span the T V W Σ (number of M5-branes) cases [ − K − N,A is sufficiently large. For higher (1) U ) K o − At first glance, the above construction K e W -decomposition, two different vertices of a − † ) K ]. 1) m + 2 A − T ) of charge (+1 2 6 U ) to each pair of two vertices of an octahedron B – 15 – ( K 00 ( + ( Φ π 1 K , p , z 4 0 N,X A T [ = (figure-eight knot complement) with 6-tetrahedra. k + W (the number of tetrahedrons in a triangulation) while z, z N e B † -decomposition also possess a symplectic structure. One k ) DGG Φ p K U T T + is parallel to the construction for V . All different ideal triangulations of a given 3-manifold are 1) copies of finer building blocks, octahedra. The construction + ( K − -decomposition which replace a single tetrahedron in an ideal N e − K † B M 2 U -decomposition case) to break all K is that there are enough number of easy internal edges (better to call e ( -linearly independent internal vertices for sufficiently large † K + Φ K internal vertices among which ( K 6 1 A M 1) ): 1) Φ m − θ V 2 4 6 e ), we can use any ideal triangulation and will argue that the resulting theory (2)) of rank 1. But the counting could be wrong as we will see below for the ,T K − K 1). A hard internal edge appears when two edges of a single tetrahedron are d ( p † A ( su K Φ Z − 2.47 k K − θ ' 1) M,T 4 − = 2 SQED with two chirals (Φ d 2 6 K For example, we show the SU(3) symmetry for all hyperbolic twist knots in section ( Z N K k chirals ( Topological invariance of seems to depend on thean various choices ideal other triangulation than ( of known to be relatedDGG’s by construction, sequence the of3d geometric a basic move local corresponds move to called a 2-3 mirror Pachner symmetry move. between In a the vertices will be atthere most are order of number of easy internal vertices are expected from 6d viewpoint. 6dof viewpoint rank expect ( that the 3dglued theory to has the a flavor internal symmetry edgessingle simultaneously. octahedron In can not meetis at located an internal nearest vertex possibly to except one when the of octahedron vertices of tetrahedrons. So the number of hard internal except tetrahedra in an ideal triangulationWe are assign replaced by 3 octahedra complex inand a parameters their ( gluing equationsdifference in in a higher internal vertices for DGG’s construction can bethere generalized is to no higher suchneed an to additional use symmetry atriangulation when into so-called of the 3d theory for higher of the consequences isthe that number rank of of independentlinearly the easy independent edges flavor easy symmetry could internal edges, be couldcomplements in we be less SnapPy’s checked than larger that census ( than have additional 1 symmetries. because The additional symmetries are accidental and unexpected from 6d viewpoint. The above symmetry ( case with an ideal triangulationThe of correct rank is alwaysproposal equal ( or greater thanis the independent of superficial the counting. choice of In ideal our triangulation regardless modified of existence of hard edges. One The counting sounds compatible with the 6d construction since the knot gives a flavor JHEP07(2018)145 . ) A . } I X Γ 2.27 , (2.56) (2.53) (2.54) (2.55) (2.59) (2.57) (2.58) B P { p m T M T : dynamical) V ( ) . 1 i , 4 , \ 3 = 1 V ! S ( Σ α 1 0 1 1 U π −
π ↔ ↔ ↔ β 1 2 1 iπ i ⊂ − + + βγ V . µ, ν ! . h Σ 1) . 00 ] = ! Z V − B Z = ∆ γ which can be identified as fundamental π 1 P T 1 Z
; 8 − · 1 0 1 1 Z · A ) ×
× βα Z 1 ST Φ + Z ST − := ( V N,X – 16 – [ = e ) = † ' αγ 2 ! : Φ ∆ 0 T DGG ,T ( T Z Z 1 θ T , we still have freedoms of choosing cyclic labeling ( 4 π !
d T B 1 = α, β, γ 004 with an ideal triangulation with 2 tetrahedra. Φ h − Z → ) A theory, we also need to choose conjugate variables m 0 1 0 1 ' Φ ) = 4 !
1 \ = 00 4 3 . The notation simply means that the figure-eight knot is the 1st U Z DGG \ Z 1 S ). To specify the background Chern-Simons coupling of the U(1) := 3 3 Σ ( 4 T theory under choice is guaranteed from a duality
S ∂ S \ U ( 3 1 π 1 1 N,A S π 8 π DGG − = T theory is invariant under the local 2-3 move and thus independent on the Φ U N e . For a given choice of † DGG T is a simplified notation, called Alexander-Briggs notation, for figure-eight knot Φ (BPS monopole operator of magnetic(BPS flux monopole +1) operator of magnetic flux T 1 + − θ 4 V V 4 d and denote the theory by Z The group contains agroup peripheral of subgroup boundary torus Example: Here which is depicted in fig (simplest) knot with 4 crossings. The fundamental group of the knot complement is topological choice ( flavor symmetry associated to theB knot, we sometimes specify the choice of boundary cycle In the construction of But these choicessymmetries. only affect So the modulo background the Chern-Simons background coupling CS coupled couplings, to the flavor theory only depends on the More explicitly, the duality is The invariance choice of of edge parameters for each tetrahedron. So the Under the duality, gauge-invariant chiral operators are mapped as follows JHEP07(2018)145 . 2 , =1 (2.64) (2.60) (2.62) (2.63) (2.61) i } = 1 2 2 ) for the flat z 1 0 2 z 4 . 2 \ . ) 3 00 2 0 1 1 S z 4 Z ( ( \ 1 00 3 1 π S + 2 ). , z 1 1 Z , ) of − flat connection on the knot 2.26 1 004 = i + 2 = − 1 m . 0 2 00 β i 1) 1 z Z 0 i h α, β, γ − ∼ − . z / 2 + i 0 2 00 ) 1 z 0 2 / z z 00 ) z 1 C , ( 2 , z , βγα Z q 00 00 2 2 1 0 2 00 1 ∆ 0 1
z z z z z − − = 0 0 1 1 1
∪ z z − 2 1 2 q 1 = z z 1 1 = 0 αγ √ √ 0 − − q 1 0 2 00 00 \ 2 2 00 ) = SL(2 − i ) = ,C 1 1 z z 2 z 0 2 00 1 00 1 2 z 2 – 17 – = (∆ z 0 0 ν 1 1 z − z C z 00 2 1 z z z z z ( 1 ) is given by 00 3 , 2 z 1 z + Z + 1+ z 1 . There are two internal edges in the triangulation 0 √ √ 1 q 4 q √ 4 − 1 1 z p ∼ \ S z \ − i 3 + 2 3 z S ) is (meridian, longitude). Upon the basis choice, the 0 1 α , i S : ( Z : 1 00 2 ) = ) = ) = π µ, λ ) = , z β γ α T + 2 0 2 µ ( ( ( 0 → ( 2 i , z hol hol hol Z 2 ) ρ ρ ρ . The simplest ideal triangulation of 1 + , z 4 00 1 00 1 \ 3 Z , z 0 1 S ( = , z ∂ 1 1 Figure 3 z 1 C { π : ] = i below for the gluing rule T 3 , 1 4 \ 3 S [ D Each point in thecomplement. variety gives The a PSL(2 holonomyconnections matrices is along the basis ( The deformation variety in this example is See figure which are linearly dependent modulo the linear equations in ( embedding The knot complement can be ideally triangulated by two tetrahedrons. Canonical choice of the basis ( JHEP07(2018)145 ) C C (1) (Σ u (2.69) (2.66) (2.65) (2.67) (2.68) C ]. The V 19 . ) 0 1 respectively. , Z ) and 2 . C . X Z − ) (1) 1 ] is given by 2 ! X − (Σ λ u Z V Φ 1 X P ) Z ; X 2 V 0 1 0 0 V µ + 3 . = 2( ,
− 0 2 1 C 2.43 ) = ) V ,X λ V V = 2 e Z µ and say 2 † 2 ( Φ − A Σ 004 2 004 V 1 2 1 X m e m with ( + Φ [ , Z † 2 + Σ(2 − 1 1 X Φ 004 V V DGG ,J m + Φ 2 = 1 X , a 2 1 , b gl / U C V T 1 − g Σ ) is given by λ ! + Σ Φ ! b ! 1 2 ) ), 1 1 1 2 V 2 004 2 V C e / − = / m 2 1 2 2 − e − 2.40 † 1 µ λ † 1 0 λ a ,V t 1 0 0 K 0 b 2.47 Φ
g − − X − θ θ (Φ in ( e e 4 4 V = θ θ 004 2 1 ( d d 2 4 4 2 ,P 1 2 00 1 1 0 0 0 – 18 – m / / d d − 004 s J 2 µ ∗ 2 λ ∗ 004 Z Z g a b ) and ( ⊗ m ∆ step2 Z e e Z Z m g T T π π 1 1 = 4 4 − L L
= + + 2.45 ] using 2 tetrahedra were already noticed in [ 1 λ = = 004 Z 004 with an ideal triangulation with 6 tetrahe- ,K 004 P ) = ) = ) = ) = ), ( m ; 2 are hard internal edges and we can not break the 1 m g = C m α µ ! g − 2 µ ,V ) matrix 1 ,V β 1 2.42 a C ,X 1 = Z − X V , − ( V = 1 1 0 1 ( ) is a dynamical U(1) vector multiplet while 004 ] µ 4 λ and
SU(2)) type and we choose ) = Hol( \ step1 m P βγα X , [ ; 3 T µ 1 1 µ = ( L S − C ¯ ,X DDV hol DGG 004 αγ ρ = 004 T m m U [ N DGG T ) = Hol( ) is of (SO(3) L ν (with Σ := The absence of chiral primary operators corresponding to hard edges in the above ( V µ, λ hol ρ Example: dra. construction of interpretation there was thattains this hard is internal due edges, to and can the be “bad” cured choice by of choosing triangulation, a which proper con- ideal triangulation which Here are background multiplets coupled to flavor symmetries,Note U(1) that both of associated to them. which can be decomposed into Following each steps in eq. ( So, ( With the choices, the Sp(4 Boundary (meridian, longitudinal) holonomies are JHEP07(2018)145 ] 19 (2.74) (2.72) (2.73) (2.70) (2.71) can be ) under Y mean that A , πi ) , 0 ) and Φ . 00 X = 2 become the same ) S 004, they propose , 00 X 0 5 0 + 2 R m N + C R 00 ( C 0 0 χ W + 2 ., W + 2 and c 00 0 . and + ⊇ + ] 0 0 00 1 W W ] , which are glued in such a T C Z + h Y < π ]. We claim that the theory Y 19 ] + 2 + 2 Y 19 N, Y [ Φ R R + 2( + 2( D X . Then, this theory is described by + + Im[ Y Z Z. and ∆ Φ , Y θ ] + + + 2 ) = W W X 1 X d + R R S S = = X Z = = = Im[ × πi πi 2 4 6 = 2 4 4 < ) + 5 R ] theory is that [ − − Y C A . The internal edges in the triangulation are [ 6 6 – 19 – Φ (1)s are broken by superpotential operators. C C u † Y . ] on ,C A Z ,C − N,X ) 0 ) [ + 2 5 + Φ 00 + 5 Y and so on, these variables C Z . Then it is clear that these fields Φ X S C 0 2 N,X + DGG [ Y Φ + Z + 2 − = 00 T 0 R,S,X,Y,Z,W † X 2 4 + 6 S are eliminated. Mathematically this corresponds to the 0-2 X → with the Lagrangian ]. The definition of the topological quantity is based on ideal DGG C C C (Φ X + + Y T θ Y 39 0 + + ( 4 00 ,R d 1 3 = R 00 R and 1 C C 5 5 = 1, where five Z Z (1) symmetry in the low energy limit which corresponds to the hard Z C and Φ and ∆ − u + 2( := := + 2( → 0 0 , ∆ 2 1 X parameter , and hence these tetrahedra are squashed to be flat. The same comment X Y,C W S C C π W W M + + + → ] S X X Y = = = theory. Intuitively, the constraints 0 1 3 5 Im[ C C C , ] At the level of edge variables, the process of integrating out the massive fields may To see it, let us focus on the two tetrahedra ∆ Our interpretation on this problem is different from [ DGG X T 2.3 Relation betweenOne the basic two characteristic constructions of the After renaming as the ones in the triangulation with two tetrahedra. Im[ also applies to ∆ be done by eliminatingwe the define variables corresponding to the massive fields. More explicitly, integrated out and thetwo theory tetrahedra becomes ∆ emptymove. in the The low invariance energythe of limit. 0-2 a move topological This is proven means quantitytriangulation in called and that [ is 3d equivalent index toof the (see localization appendix expression for the superconformal index where we have neglectedof background the fields. internal edge The superpotential is due to the presence a hidden additional edge in the triangulation with two tetrahedra. way that the system hastwo the chiral internal fields edge Φ Note that there islinearly independent. no Superficial hard counting internalsymmetry suggests of edges that rank the in 6 resulting the theory triangulation have a and flavor 5realized internal by edges six are tetrahedra tetrahedra in the is low energy actually limit. completely Therefore, the the theory same constructed by as six tetrahedra the has one realized by two the one using six tetrahedra, ∆ does not have a hard internal edge. As a “good” ideal triangulation for JHEP07(2018)145 . ) A V K x \ (2.78) (2.76) (2.75) M . (in some = ,...,N } A N = 1 X ) connections i (1) field whose for a given C ( u ) is a subset of , i B and N p ( i 0 ) can be written in 3 χ b singular loci S . ) and its decomposition \{ )
A B . X 2.45 ; gives a polynomial equation P A . The V =1 x B N i ) (2.77) T = in ( } P i A σ ) N { X = 0, which corresponds to the limit ( g A ; f W V X ) =1 , . 2 → ; N i A ) and the imaginary part corresponds to 1 } V b i πib A ) is a set of flat PSL(2 X =1 2 N i = 1 σ ; m )) which can be seen for any non-exotic X e } { ) at each of the vacua for a given parameter i V ; ( ) =1 2.29 b N i Σ V πi A =1 I } . Then, by integrating out the matter chiral { 2.32 i N i ( 1 0 X = 1 are just the condition for the vacua on } V i Σ ; S → – 20 – f N W { b Σ V ( ) A =1 { N i ( ) A X f −−−−−−→ } W ) (or ( 1 i ∂ ) is given as follows. A DGG theory in general consists X f ) A W S 1 . . . dσ Σ ; A X is the number of dynamical vector multiplets, i.e., the { 1 S V ∂ × ( 2.21 =1 X ] defined in ( N i V 2 ; × dσ , we get a twisted superpotential of Σ } f W T i V N R 1 2 k =1 be the twisted chiral field of the background N i Z Σ S R Σ } { N, A ∂ i [ ( × σ ] on X only has a torus boundary and hence of the form , the leading asymptotic behavior of the integrand is determined D 2 { f W ), and solutions of that equation in terms of A 1 ] on ( k R b B S A N Σ P I , and it depends on the details of ] ∂ × V , e ) : exp N,X 2 [ N ]. In a degenerate limit when A B 30 N,X , R k P [ X has a definite value (modulo 2 (1) e 19 , e direction. The u DGG B A 1 T DGG P X ( S T e ( ( = Re[Σ ) := ( become B k 3 b . The = theory on a curved background called squashed 3-sphere ( σ S which can be obtained from an ideal triangulation , p 2 A parameter R The relation to localization computation is as follows. The partition function of the x N . In other words, the equation parameter × A M DGG 1 where where by the twisted superpotential T following form [ S X of ( correspond to the vacua of the theory with mass parameter The conditions exp appropriate normalization), Then we define gauge group is into basic types. Also, let real part corresponds to thethe real background mass parameter flavor gaugefields field of around the theory on and may beM ignorable in ourof discussion chiral fields, as dynamical vector we fields, andbe discuss background vector later. twisted fields. chiral Let Σ Finally fieldscomponent is the constructed the left-hand from real side dynamicalfield scalar in vector of the multiplets the whose vector multiplet lowest and real the imaginary part is the gauge where our 3-manifold The deformation variety on the algebraic variety definedideal in triangulation. ( The difference between the two sets are mild, higher codimension, Recall the definition of each term of this equation. For simplicity, we only discuss the case JHEP07(2018)145 . ] ]. A d 6 X 21 but 14 T DGG d ] and )[ which (2.80) (2.79) T 6 ) with T T hyp 2.31 . Further- N, 2.29 [ ) action on A A , we get the Z D . ), and , depending on . m (2) m ) A ). Both of them 1 su =2 2.16 X S a | 2.7 ) × 1 in ( 2 or 2 S have in ( generically has U(1) d R ) of Gaiotto-Witten [ A ) are given in ( 6 × Z d d 2 T X , , 6 6 irred irred 00 i R ] on DGG T T = ) is realized in field theory as , while the SL(2 , α T i B a can not be identical to α ] on N,A ,P [ A DGG d A = 0. Field theoretically, when the T 6 . DGG d X T 6 A T where ( N,X T [ and hence X 00 i ] for an example. We may expect that α d ) 6 . vacua 19 1 Now let us compare the constructions in DGG T − ) i m to be a constant fixed value 2 1 T z ( S a − ⊂ M – 21 – (1) group on × ) u 1 (1 2 , and the integers ( i . In general, there is no reason to expect that there S R α i A might be identified with are equivalent to the gluing equations in ( z . This point has already been seen from the effective × parameter ]. So we see that 2.1 d 2 V 6 Q =1 ] on M R N i 20 ] using [SU(2)] theory on T A DGG } T 1) = 38 T − ] on . Comparing the moduli space of N,X A [ ) = 1 . See section 5.2 of [ = ( ) is realized in field theory as the SL(2 f W a f 2.2 W i e DGG N,X Σ [ ∂ A, B T ( ) action on the canonical variables ( DGG exp( Z T vacua { , as emphasized in [ ( . M (2) symmetry and N d ) action of Witten [ and section 6 vacua su can, at best, describe the low energy limit of some point of the moduli space of Z T , ) we see that M SU(2) types of boundary cycle 2.1 / ) This problem may be avoided if we only consider generic mass parameters. We DGG Then a possibility is that One crucial difference between the two theories is that The above equations may have solutions like Now, we have 2.79 N T ( 0 flavor symmetry associated tomore, the the knot SL(2 while the SL(2 the boundary cycle ( using the are genuine 3d theoriescan and be they are realized associated instill to ideal different the as triangulation. hyperbolic we now However, connection explain. it turns out that these two theories are nections on can only capture afield subsector theory of point of viewexists in a section genuine 3d theorySo which describes all components ofvacua moduli of space of vacua of in ( This is because that an ideal triangulation captures only a subset of irreducible flat con- χ assume that this is the case. Comparison of thesection two constructions. by matter chiral fields.generated Those massless by flat monopole directionsstrong operators are subtle, coupling because especially effects in when whichtwisted they that may are superpotential case not those bethose directions captured subtle appear by flat by the directions very one-loop might computation be of the the reason of the mismatch between This means that byvacua taking of the the parameter theory with the mass parameter mass parameter is zero, there could appear some continuous moduli space of vacua spanned an additional relation SO(3) The equations JHEP07(2018)145 · ] 2 µ B d ϕ 6 ; irred (2) T (2.84) (2.82) (2.83) in the su ) of the } N,A = 2, it is × 3 [ 3 d 1 µ 6 N , µ . T 2 (2) ] , µ of B su 1 , we propose the ; A 0) theory. The µ , is the holomorphic { DGG 0 (2) . As a second step, T DGG µ N,A = [ T su = (2 d µ 6 2.1 and 3 T N µ and 7 , = A ] to d 6 irred δW (2) , ϕ T [SU(2)]. In the coupling between su ) (2.81) ). The theory has T Z Z , , [SU(2) (2) to U(1). ] = T . First we put the theory on a specific S su . SL(2 means the superpotential deformation by deformed by SL(2 = 2 vector multiplet to gauge the diagonal ) 3 0 = ∈ −−−−−−−−−−−−−−−−−→ ]. ∈ µ N DGG ϕ µµ , ϕ ! T , ϕ d = as explained in section 6 , by adding the Cartan component ( – 22 – irred 3 Tr( T to r s p q is absent only for some special cases. These points will be d R δW 6 irred d µ
6 [SU(2) T [SU(2) T T = T is defined as follows. The transformed theory ϕ of SCFT d . The ] P 6 associated to the knot to the superpotential. We give more 1 A ] B T µ ; theory on of the two theories with the following superpotential coupling (2) ], we introduce a (2) qB d B A su 6 su ; + N,A T = 4 supersymmetry, the presence of the holomorphic moment map ) on [ (2) d = 4 SCFT which describe the 3d theory living on a duality domain Z . This deformation breaks pA 6 N , on a vacuum 3 su ; N,A T [ = 4 SYM associated to N µ · d −−−−−−−−−−−−−−−→ × 6 sB 1 d ϕ N T 6 , of the + = T (2) := coupling the duality wall theory (2) (2) SL(2 ] with su SCFT su ] and ] is a 3d 0) theory. Indeed, the operator su is the Cartan component of the moment map operator ⊂ P , N, rA [ is the holomorphic moment map operator associated to the , ϕ , ϕ 3 d 6 µ µ = (2 diag T In the case of 3d Thus we need two steps from Motivated by the similarities and differences between The N, A, B [ In general, the existence of this operator is guaranteed only for theories with 8 supercharges. However, N 7 d (2) moment map operator (2) [SU(2) [SU(2) 6 of higher supersymmetry ofmay the be codimension-2 empty defect depending of on the the 6d theory. However, genericallythis (but operator not often always), exists6d the due to the remnantimportant of and 8 discussed supercharges in preserved more by detail codimension-2 below. defects of the su evidence for this proposal below. operator associated to a symmetry isnot guaranteed. guaranteed. However, What when there we are call only the holomorphic moment map operator is a kind of remnant adjoint representation of the chiral operator vacuum, we deform the intermediate theory, following relation between them: Here, where which is a chiral operator inmoment the adjoint map representation operator of of wall in 4d as flavor symmetry. ForT example, su T T can be obtained by JHEP07(2018)145 (2) is a and The 3 su 1 (2.87) (2.88) (2.86) ) µ ), while Z , B associated ,P A A X X 3 (2) gauge field with . SO(3) types. type type su / ] to U(1) which can be large enough, the of the knot complement [SU(2)] almost decouple k ) 2 DGG is absent happens only in T Z ] , ] 0 T T N,A µ [ correspond with each other; B B SU(2) SO(3) , , of P P and 2 d ; ; do not exactly correspond to ) 0 6 ) A irred A Z T A 2 T A X ). Recall that they are related as , ) U(1) SL(2 SO(3) ) transformations as SL(2 SU(2) Z [SU(2)] to the in ,X a, b , Z N,X T N,X ). By taking B [ U(1) , A ) (2.85) SL(2 [ . Roughly, the relation is given by the k P Z A ) (2.89) ∈ T , − (2) X and ( DGG 2 DGG ) is of ) is of b, a S su T T T 3 / → 3 3 / − µ SL(2 µ µ ( ) and SL(2 = theory generically has only U(1) = = ) A, B ∈ A, B B and – 23 – Z 1 → is large enough, because the scaling dimension of , acts on the canonical variables ( k δW ) 1 δW δW ,P (2) global symmetry. Therefore, we conclude that ) ) ]. k Z ] 1 ] 0 A (2)-type SL(2 DGG Z , . Moreover, this argument shows that the scaling ) −−−−−−−−→ A ST ) action on the boundary su , k T B B X a, b Z Z su when ( when ( ; ; , 0 , (2) SL(2 :( :( , , d N,X (2) symmetry, and then let us perform a transformation su 1 2 ) ) action on U(1) ) SL(2 [ 6 irred b 2 S S = 4 supersymmetry. Therefore, the deformation by Z N,A T , b su N,A [ ∈ , [ 2 a a, 1 N ( ( DGG of d S d ) transformations of U(1) and SU(2) 6 irred T ( 6 irred A Z [SU(2)] theory contains the holomorphic moment map operator with T for generic T , in T ) and hence on the variables ( (2) T ) = µ A (+ the original value before B su X k A, B ,P A (2). This is done by coupling is close to 1, at least if X su ( µ -transformation acts on ( , respectively. The SL(2 S 2 2 ) ) [SU(2)] alone is 1 by Z Z T , , The We denote the U(1)-type and In particular, the deformation breaks the associated to the new (ungauged) in each other. SL(2 Therefore, generic elements of SL(2 However, there are more subtle details. SL(2 corresponds to the U(1)following SL(2 diagram; More details on SL(2 deformation explains not onlyto why the knot, but also why the SL(2 µ relevant deformation and it triggersrather RG exceptional flows. situations, and The this case will that be very importantidentified in with section U(1) gauge field is weakly coupledfrom and each hence other. the twoµ The theories the total theorydimension has of we are given a theory acting on this Chern-Simons level theory contains the moment map as chiral operators. This can be seen as follows. Suppose JHEP07(2018)145 ) ), = a . A − A B a, (2.96) (2.90) (2.94) (2.97) ( 1 2 as . One can act as f = A 2 2 2 ) S A R Z π , 1 2 ) corresponds to inside = 2 SL(2 a n SO(3) by 2.95 (2) gauge field. The ∈ ⊂ typetype (2.92) (2.93) 2 su and T f ). Then we get 2 a 1 (2). When the symmetry S − R and su ) be an π , SU(2) 1 . SO(3) 1 a, 2 ) ) A , , ( a ) (2.95) A 1 2 Z 3 = X a , 1 A = + n 2 3 . ) is not a properly quantized Chern- CS( 2 SU(2) SO(3) B 1 2 A A as n SL(2 + 1 2 2 ,P n . ) (2.91) = ∈ S = 2CS( ada. 1 2 2 A a 1 S π 1 X T AdA + 4 ( and ada ) is of ) is of ada 2 1 π π → S 4 a, b ←→ 2 tr 1 – 24 – ) = 4 ( · ) 1 a A, B A, B π . Then, take the subgroup U(1) SO(3) types of the cycles. This is natural, because 1 2 B S 2 2 4 / ) is not well defined in 3d, and only the 2CS( → → S ,P ) A ) CS( → × A on each A ) ) = 2 ). 1 X a, b when ( when ( S A A da :( :( CS( = 2.93 1 2 -transformation CS( CS( f T T T 2 2 ) T 2 T ( ↔ ). In the same way, if the gauge group is SO(3) which is broken to 2 -transformation in field theory corresponds to the shift of Chern- ) ↔ 1 T 1 2.92 T exchanges the SU(2) ( T 2 S ). Then, under this embedding, we get a = 4 theory, the gauge groups SU(2) and SO(3) are exchanged under the S-duality. − and hence the integral of CS( N a, In fact, if the group is SO(3) type, then the CS( On the other hand, the 8 To see this, consider a 4d manifold 8 2, / Simons invariant. This1 is because, in 4d, there can beand include instantons the magnetic of fluxescheck of instanton that number this configuration gives the instanton number U(1), it is natural to embed the U(1) gauge field as This equation corresponds to ( Recalling that the Simions level of background field,the we exponent can 2 see in ( that the factor of 2 in ( diag( where where the traceis is SU(2) taken and in is the broken doublet down representation to of U(1), we embed a U(1) gauge field This also has aglobal natural structure may field be theory either interpretation. SU(2) or Let SO(3). Its Chern-Simons 3-form is defined as and hence they are related as Notice that the in 4d This exchange can also be shown in purely 3d language and is explained in appendix and hence JHEP07(2018)145 - ). T → S (1). u (2.98) (2.99) 2.90 is well (2.100) (2.101) (2.102) gauge is clear. 2 ) 3 symmetry 2 (1)] µ T C u also preserves = of the moment down to × (2) 2 3 ) µ 2 W su (2) , and two pairs of T φ ) at the field theory (2) symmetry of gauge su , this condition may by 2 su T A 2.90 (1)] and ( = 2 supersymmetry, the X u ), we will determine the , and only the subgroup 1 (2) symmetry after the S N . × su U(1) 2.84 gauge , = 0 (1) type. This confirms ( [SU(2)] has a , for the summary whose details (2) u → i T 1 get nonzero expectation values as 2 (1)] . Therefore, after the deformation su (1) ˜ i -transformed theory is given by u A ] in ( φ E u h A S E [SU(2)] × = (2) = φ. T 3 . T − i su i µ with the superpontial and (2) 2 − N,X − [ E i i = , a neutral chiral field symmetry associated to the knot based on i E su ˜ h -cycle is SO(3) type, only the ( ˜ E E V . Now, this E A (2) i is SU(2) type. Under 2 A φ , ˜ ·T DGG X . These generators E , su 1 2 . See the table T B φ =1 – 25 – = ) i S A 2 ) charge = i = 1 T or (2) = [ ) becomes -transformation of W is given by ˜ T − E i u S , 1 A W C i = ˜ E = 1. Then 2.98 ) is preserved by the expectation values. Therefore, by E gauge h and ( (2) 2 i · T −→ (2) symmetry. Then ˜ ·T 1 = E E 2 for more details. The Cartan component (1) [SU(2)]. In the language of 3d su of ( i . S , 2 i S u su 2 ˜ T i B 1 S E E E 3.1 h with 2 , (2) is a gauge group which is coupled to the =1 i ) under the explicit breaking ) i su generated by ˜ E symmetry of ) are charged under , 2 i 2.93 i ) ˜ H E Z E be a theory with , acts on the index , i (2) T E H (2) which acts on ( su ) and ( (1) is the gauge group which survives the symmetry breaking [ u SL(2 (2) u , the superpotential becomes ⊂ ⊂ su 3 From the above discussion of Chern-Simons levels, the field theoretical realization 2.92 µ [SU(2)] is given by a U(1) vector multiplet (1). The right hand side is just the (1) 3 Symmetry enhancement Using the proposedsymmetry 6d enhancement interpretation pattern of of thea U(1) topological type of thewill boundary be cycle explained in section where u These expectation valuesNamely, break ( the gaugeu symmetry [ the Higgs mechanism, the theory ( Thus, an F-term condition is The at the quantum level,transformation. and See this appendix symmetry ismap operator the of new this symmetry global by where the center and the T chiral fields ( of ( Now let us also checklevel. the correspondence Let of the S-transformation ( defined. Therefore, theΓ(2) actual transformation group atthe the condition that quantum one levelnot of is be the a preserved. cycles subgroup is well defined. This means that when the JHEP07(2018)145 ] 2) N (2) , . [ 0 su ). χ ] 3.1 A both of ) whose = (2 is a flat 2.8 0 (1) vector ] theories compact- u A 10 N ,A in ( 1 , 0 ,K N,X S d µ [ 3 6 N . T S 1 SO(3) types” are / . The = DGG as twisted chiral and A T φ ), the VEV of the . After M ) and ( [ is not broken by the in [SU(2)]. K 2.17 . However, in the discussion T A A are non-closable but one φ DGG N,A X ) flat-connection in 0 that the DGG theory has T C (2) A , 9 U(1) (2) symmetry. Therefore, we SU(2) SO(3) su su → → → and A A A X A X X and a chiral operator symmetry of e µ U(1) C gets a nonzero expectation value. Then, is absent at low-energy (3.1) U(1) ), we understand the symmetry breaking U(1) ) is SU(2) type. In that case, combining 0 are given by Σ and e ν µ (2) A µ ]. The definitions of “SU(2) ). Thus the above relation implies that the become empty; 2.84 A su e µ and = 0, 2.32 if there is no PSL(2 in ( [SU(2)]), as shown in ( e – 26 – µ ) of complexified gauge field i 6 N,X µ T [ ˜ and ν A h 4) supersymmetry. In the Language of ( µ , DGG DGG hol T is expected to have an ) and ( theory. Otherwise, the ρ T = (4 have SU(3)-symmetry. We checked the enhancement for Symmetry enhancement of U(1) d and 2.21 . For the symmetry breaking to happen, we need the 6 DGG irred N A K T T X d 6 irred in T [SU(2)] theory which is put along a knot T µ does not contain Z 1 is absent. to U(1) . The definitions of “closable/non-closable” are explained in section [SU(2)] theory on S is given by log µ T A ] defined in ( × ∂N, e ν 2.2 , we will find infinitely many examples of pair ( 2 N In (2) [ 1 R 0 [SU(2)] has 2d ) is SO(3) type while the other ( H su SU(2) enhancement χ 3.2 closable T A / ∈ with a group theoretical argument, we can argue , we have to exchange the role of chiral and twisted chiral by regarding A 1 be the neutral scalar chiral field and let Σ be twisted chiral field which comes from 2.1 . Symmetry enhancement in . In the description there, the moment map operator comes from the holomorphic moment map of φ theory on non-closable, SU(2) type non-closable, SO(3) type C In the coupled system (5d SYM+ Now suppose that the Higgs branch operator This question can be answered by an inspection of the 5d picture discussed in sec- In section 2.1 We thank Y. Tachikawa forLet this argument. 9 d (2) 10 6 irred T multiplets in 2d. Then theof Cartan section components of Σ as chiral, because of the subtle mirror symmetry in the relation between 5d SYM and moment operator connection in supersymmetry, there is awhich are twisted associated chiral to operator the Coulomb branch the nonzero VEV inin Higgs the branch low makes energy the limit, the Coulomb operators branch fields massive. Therefore, need to know when tion su ification, the 3.1 SO(3) From the relation between mechanism of moment map operator mechanism and the resulting the table enhanced SU(3) symmetry. Using thefor argument, all we prove hyperbolic that twistseveral knots twist knots which gives non-trivial empirical evidence for the table explained in section corresponding DGG theories areof identical them and (say both of Table 1 JHEP07(2018)145 A A is 1), N N A (3.5) (3.6) (3.7) (3.2) (3.3) (3.4) N ] is not en- . A ∞ . < N,X [ ) are trivial ( }| A ) at low energy ( Z 1 DGG hol S T ρ . of Lens space is abelian. ∂N, . ( 1 1 π H ) is ‘closable’ Z qλ ∈ + ) = 1 . A pµ A ∂N, ( ( symmetry of is closable 1 is a Lens space, there can not be any hol is non-closable A H A ρ for any radius of X A A ∈ is nontrivial, but we assume that this step 1 N ⇒ (1) (2) symmetry at low energy 3 S A ). According to Thurston’s hyperbolic Dehn s. The Thurston’s theorem is consistent with (unknot) u ⇒ ). Such a flat connection can be thought as R \ A at low energy A × su ] with ( 3 A – 27 – 2 3 ( = 0, the eigenvalues of to S N [ hol R R 0 1 , there are only finite number of primitive boundary hol . So, we can conclude that ρ m χ ρ S ] has A . But if N A N × ] on ] on ) := to be ‘non-closable’ is that the Dehn filled manifold to be ‘non-closable’ is that the Dehn filled manifold N 2 N,A [ R A is hyperbolic is closable, by definition, there should be an irreducible p, q A ( N,A N,A [ [ A is Lens space d L A 6 irred N A d d T 6 6 irred irred If N ] with trivial T T ] with trivial , it implies that the ] = If N [ A 1 0 N [ χ 0 χ ). Let us define N,X A primitive boundary cycle if there is a point in ) is defined as [ A ) is non-closable ( Z p, q hol is absent in is absent in ( (2) except for only finite many DGG ρ L µ µ T ) may contain off-diagonal components and hence the above condition is non- ∂N, su ( which give non-hyperbolic A ). 1 Set of primitive ‘non-closable’ boundary cycles ( ⇒ ⇒ ⇒ H A |{ hol 2.85 ∈ ρ One sufficient condition for One necessary condition for generically (although not always) exists; see the discussion in the paragraph contain- A The reason is asflat follows. connection in If an irreducible flat connectionirreducible on flat connection because the fundamental group is Lens-space Lens space hanced to our field theoretical considerationµ in the previous sectioning that ( the moment map operator surgery theorem, for given hyperbolic cycles Combining with table is non-hyperbolic. This is because that thealways flat contained connection in corresponding to the hyperbolic structure on Strictly speaking, the stepholds. from trivial. Then, We remark that inbut the massless case with trivial JHEP07(2018)145 ) ) A is A (2) 2.64 (3.8) 2.31 ) after 1 su x charges S to cover A × The index 2 X ) holonomy is longitude. A R ) with ( C 11 λ , ] on 2.63 ) ] as a topological x ( for the case when 1 0 M,K 40 A ] and expected to be ] in ( [ 2 N T M d I ] theory after removing knot [ 6 irred ] after taking the closing divergent N, T M [ ( [ d M d 6 [ D irred 6 T is meridian and T , and the PSL(2 d vacua 6 A . That a primitive boundary µ irred M X A T non-trivial power series in N is non-closable and of SU(2) type, ]. The ) is closable (non-closable) ) and the A . The has properly quantized U(1) Z 1 is neither hyperbolic nor a Lens space, M,K 4 3.2 [ \ d (2) ] is embedded into the enhanced A 3 6 A ∂N, su S T N ( 2 1 = in a supersymmetric way after sitting on the . When H N,X ] and A – 28 – [ N ∈ M is hyperbolic) A A M,K DGG [ on N T . This is manifest from the relations given in eq. ( . Namely, closable closable , associated to the U(1) d A 6 A K irred A ⇐ Closability X non-closable T ) = 0) . X disappears in the moduli space (2) x ] will be a theory with supersymmetry broken. ( U(1) M su A 1) SU(2) type of N / I ( M,K [ closable ( symmetry of → d ) is ‘non-closable’ means that we can not ‘close’ (or eliminate) the . 6 6= 0 ( irred ) A A Z 5) 4) ) T Z can not be closable. When X x (2) = 4) < ( 004. SCFT A su | ≥ | | A ∂N, P . As we will study in the next section, there is an operation in SCFT side p p p m 2 ( ∂N, N | | | ( 1 ( ( ( µ I is non-closable and thus supersymmetry is broken. This is a heuristic argument 1 ) is the 3d index on a closed 3-manifold = H λ λ λ x A primitive boundary cycle if , associated to the H A 1 SCFT ( ) computes the superconformal index of the theory a ) between the variable 4 ∈ ) + + + P A ∈ x . Closability of boundary cycles of \ ( N 3 A A A I pµ pµ pµ 2.41 S 2.65 N We can further determine the global structure of enhanced symmetry, whether SO(3) An alternative definition of closable/non-closable cycle, which seems to be equivalent Here notice the difference between = = . The trivial flat connection on 11 M variables still have a supersymmetric vacuumM because there is always trivialput flat on connection the on vacua any closed 3-manifold N or SU(2), from thethe SO(3) compact U(1) symmetry via and ( I zero when supporting the equivalence between the twois definitions and known. no rigorous mathematical Weto proof checked hold the possibly equivalence except for for various exotic examples cases. and the As equivalence an seems example, see table co-dimension two defect alongvacuum a of 3d/3d correspondence whichnon-closable corresponds cycle, to we the expect operation thatknot the of operation resulting ‘closing 3d on the theory knot’. If Here cycle no simple criterion to determine the closability has beento found. the aboveinvariant definition, of is 3-manifolds with usingclosed torus 3d 3-manifolds. boundaries index and is which generalized is in introduced appendix in [ Table 2 We determine the closabilityand using ( the definition in ( Thus the cycle JHEP07(2018)145 : A ). is in λ ∗ for µ µ 664) i (2) . (3.9) X 3 B su (1) ' . u µ means that 1 λ 2 − 1 µ 3 is of SU(2)-type by ) ) and longitude ( 2 1 λ is a knot complement µ 5 \ N 3 As another example, we S ( . 2 p , p . = . Then, ) is chosen as the one induced µ 2 1 ) N 5 SU(2) type of a given primitive Z = \ / is a boundary cycle in Ker 3 ), meridian ( ). In the next section, we argue that In the case, the merdian cycle S N, A λ Z ( for odd for even 1 . , , 2.69 H µ ∂N, is of SO(3) type. See also appendix ( 2 → 1 = µ 2 A ) λ H Z is a knot complement in a homological sphere, is not unique but can be shifted by 2 A 007 and – 29 – λ m N ∂N, SU(2)-type SO(3)-type ( 007 in SnapPy’s census. The knot complement can 1 = ( are coprime) ). We always choose a particular orientation of each 1 m µ H 1 q : λ 2 004 and ∗ , the 1st link with 2 components and 5 crossings, as shown A.25 is of 1 i ). It is one of the simplest (having smallest volume 2 1 − ). When λ and 2 1 m 1 5 Z 5 µ qλ p 3 = ) Ker + 2 1 . The choice of ∂N, 1 ∈ 5 ( ) 4 pµ 1 \ λ 2 \ 3 ) is the meridian cycle and H 3 Z S Z ). Then, the Dehn filled manifolds have natural orientation induced ( S in is always of SO(3)-type. More generally, when N, ( µ ] whose Lagrangian is give in eq. ( A = 1 = ∂N, 2.42 ( µ H 1 N N H ,X in ( → . The orientation of the link complement ( ) ∆ . Whitehead link ( ∈ 004 4 Z T is actually enhanced to SU(3) which contain the SO(3) as a subgroup. -homological sphere ( m 2 µ [ µ Z In general, it is not easy to determine the SO(3) X ∂N, ( 1 DGG (1) in figure from an ideal triangulationideal in tetrahedron ( insign an of ideal triangulationfrom which the is link complement. reflected in The overline the in choice the of equation CS level u Example: consider a knot complementbe called obtained by performing DehnWhitehead filling link on is denoted one by component of Whitehead link complement. H Example: non-closable and of SO(3)-type andT we expect SO(3) symmetry enhancement of Here boundary cycle there is aMeridian canonical cycle choice is defined of to beby the the the basis circle condition around of that thedefinition knot and while longitude cycle isin determined a and the theory can have operatorswhich charged under means half that integer spin the representationsinteger of spin symmetry representation is are SU(2). allowedmore when justifications Similarly from different we arguments. can see that only operators in Figure 4 two-component hyperbolic link. JHEP07(2018)145 - 2 Z (3.14) (3.15) (3.10) (3.12) (3.13) (3.11) . ) . c . , ). The Dehn . 00 ) 3 , Z + (c 3.11 2) 3 + 2.43 − Φ 3 when the orientation , 2 2 Z 1 0 0 0 0 0 0 0 1 2 1 Φ (3 from the basis choice of 1 5 + L with ( \ 0 Φ 2 ), the Lagrangian for 3 is of SO(3) type. So from θ N Z = = S 2 . 007 A . d λ 2 + m 2 2.47 on λ ) 007 gl U is a knot complement in a 00 1 − Z i − g m µ , 2 Z 2 3 1 2 3.9 µ N 00 007 3 4 + , λ 1 1 b m + Z 2 1 ,J 1 2 − t K µ Z g + h 1 0 1 0 = 3 ] is enhanced to SO(3). Now, let us check 3 ) and ( = 007 − 2 2 Z ). The Φ µ 2 m λ 2 ) = (unknot)) iπ , s J 1 3 V − 2 \ g Z 2.45 2 3.6 1 0 1 1 1 + 2 ,X 3 − − + Σ µ 1 00 4 2 S − − = C 00 V 1 00 0 00 0 3 – 30 – Z V − 007 1 ∂N, Z ), ( e 1 21 1 01 0 10 0 0 0 00 0 0 1 0 0 0 0 0 1 0 0 ( = ( ) 007 − m † 3 1 + 2 [ ,P 2 + 2 + 2 m ,C = H 2 Z ) for this example is g 0 2 2 ,µ 1 2.42 3 1 Z µ + Φ Z Σ + Z λ = a 007 DGG ) 2), and according to ( 2 2 . From the topological fact that − 1 X 2 i 2.40 m + T + − Φ = − 2 V 00 λ Z 1 1 00 2 007 , 1 2 µ 2 in V Z 3 , λ − m + Z Z µ e (3 ) 2 1 g + 2 † 2 2 − µ 2 1 ,K X SU(2))-type, we choose L + 2 iπ µ , 1 5 , λ X + 2 0 ] is given by 1 V ] 2 \ − = Z 2 2 1 2 + Φ 3 Z λ λ (1) = 4 − Z Σ , µ 1 S A − − u 1 3 2 2 2 Φ N B = 2 = = 2( = µ µ µ 2 4 h 4 = ( 1 1 0 0 2 1 0 1 0 V 2 2 3 1 P P π ; λ µ − 1 C C A ; 2 4 1 a − µ 2 V 2 ) = N is non-closable according to ( µ e µ = Z ,X θ 4 † 1 , 4 b A ) ,X Φ d ) are of (SO(3) 007 2 1 007 m 5 [ m Z + 007 \ U 3 007 has opposite orientation to the one induced from A, B , we expect the m = S is chosen to be the one induced from an ideal triangulation in ( [ DGG m ( 1 T ∂ ( = N L 1 DGG Following eachT steps in eq. ( The matrix can be decomposed into Since ( Then, the symplectic matrix in ( data are (we choose we see that homological sphere, table the enhancement from explicit constructionknot of complement the can DGG theory. be According triangulated to by SnapPy, 3 the ideal tetrahedra and the corresponding gluing of filling also gives anH induced basis of N JHEP07(2018)145 and µ (3.17) (3.19) (3.20) (3.16) (3.18) X 0 . A (1) u ) . c (2). However, . , su ) p + (c T and SU(2) p , − . A T 2 m m A V T T 2 ( 2 p + i T SO(3) √ 2 2 2 . The theory has 3 → − 3 V . Φ + ) Φ 1 2 := , respectively. Applying the mirror → 3 2 2 2 X ,U 3 ) V φ 2 C Φ x e 1 V V † ; in 2) denotes a topological invariant + ] are identical ) is of SU(2) type θM 0 ) p + SU(3) 2 2 ) such that 2 0 0 A T 3 φ d = Φ and X m, e B A , φ ( A,B V ; ) 1 (1) vector multiplets. The superpotential + ,X ( ) N 0 X Z 0 + ( 0 p C u 1 2 I V T 2 ,B → N φ O ,A 0 [ 0 ( 0 Φ + , A 2 + ( N 1 – 31 – are non-closable cycles . N V M 0) theories, we only expect that the symmetry m I ,W e 0 C , 2 1 DGG 3 T † 2 ( A T Φ M 2 = ) = 2 1 V + Φ x √ = (2 → 2 ) and ( ; . and 2 p − ) with the following replacement B B 0 T e Σ ] and := ; A N . For such a pair, we claim that A † Φ are dynamical A C m m, e 2 , V T ( 1 A 2 M 2.53 ) V Φ + N,A SU(2) + ) + 2 N,X is of SO(3) type while 2 2 A,B [ p 1 ( → N 2 ] Φ T , φ A I V 2 and 2 1 2 A ⊕ 2 λ 2 V ) and ( 1 − DGG Φ 1) 2) 3) Both of 2 Σ + 2 V ( T µ 3 2 X 4 M V 2.52 := Φ P ; → − θ 1 2 e ⇒ 4 µ † φ d ) ,X m SU(3) Z is in the Cartan of this SO(3). T 007 3 π 1 m X [ 4 + ( whose background vector multiplets are (1) possibly modulo a topological sector The theory has enhanced SU(3)are flavor symmetry embedded where into SO(3) the SU(3) inand a way that = DGG u We consider a pair of ( C T . I. Two theories L (1) II which we call SU(3)-enhancementcalled pair. 3d index, The see appendix 3.2 SU(3) enhancement From the point ofassociated to view codimension-2 of defectswe 6d (which will are see knots that in there 3-manifolds) are are many theories which have larger symmetry enhancement. The In the dual picture,as the SO(3) symmetry is manifest after the redefinition of chiral fields we have term comes from anu easy internal edge symmetry in eq. ( In the Lagrangian, JHEP07(2018)145 2 A µ ] has = . For 0 (3.21) k 1 from A µ K ,X ] in charge 0 . The non- A x N . [ enhancement 2 2 λ ) with A λ N,X + from off-diagonal [ DGG is embedded into 3.9 − 2 T A 2 ). First, note that µ 0 A X µ X DGG -homological spheres, 2 ] = =2 = is a twist knot T 3.19 0 A Z 2 0 0 kλ + > < 2 = U(1) N,X t [ µ ) of the Whitehead link index k k s ) i A 2 1 , , w t X 5 2 under U(1) DGG \ − A.27 ,B 3 SU(2) type. Now let us check the k respectively. The only consistency T / S 2 λ y p − U(1) o . . ,B c 2 . . - 0) . . | respectively. Applying ( 2 µ , . . k µ 1 | . 2 1 µ = =2 ) 6 = -symmetry ( enhancement requires that there are conserved 0 0 is of SO(3) 8 – 32 – ). The reason is that the SU(2) 0 = ), the theory 0 N D ] shows that ( = ( A . Then the conserved currents with charge =2 41 { 3.20 k 0 2 A 3.19 A/A , the index is a non-trivial power series in λ X K k ) = ( = SO(3) N k + 3 2 − ,A t and µ above are nothing but twist knots which will be denoted s , . A minimal completion of such a situation to a Lie algebra i symmetry where the U(1) 1 1 2 A w 0 t 5 kλ and N − A − − ,A = 1 k k µ 1 2 U(1) − (4 kλ 1 under U(1) , while the SO(3) , the = +1) ). Combining the L ) is equivalent to the superconformal index of 1) + k k A 1 x 5 can be considered as a knot complement in ( (4 µ ) ) ) ,K 0 3.19 1 2 1 2 1 and and SO(3) 4 5 5 N = ( A,B 2 \ \ ( N A µ 3 3 = I ) are embedded as II in ( S S , we can conclude that 0 ( and =1 N 2 A k SU(2) λ = ( = 2 . A rational surgery calculus [ K is a doublet of SU(2) N 0 0 = N N . Let us check that the pair satisfy the 3 conditions in ( A ) = ( k λ One may wonder if there exits such a pair. Surprisingly, we can find infinitely many From 1st and 3rd conditions in ( , k K (1 As shown in figure as both of L and 2nd condition in ( SO(3) is to embed the symmetries to the SU(3) algebra. examples of these pairs. A class of examples is way of this happeningand is SO(3) that therequires theory that has there are a conservedcomponents SU(3) currents of symmetry with SU(2) charge into whichcurrents the with charge SU(2) trivial match of the superconformaltheories indices are in the actually 2nd equivalent condition possibly strongly up suggests to that a two topological sector. both of SU(2) them as example, The 3d index basis. For hyperbolic complement Figure 5 JHEP07(2018)145 . ]. ) 0 (3.25) (3.27) (3.26) (3.22) (3.23) (3.24) x ; ,A 2 0 ) ] e x N . ; − [ 42 2 , e Z )) 1 e ∈ , + DGG ) and the corre- − 2 k , T A.27 1) 2 ) , m x 3.19 − 1 ). So we confirm that ; , m in ( k 2 1 m 004 given below, their 8 ] and 2 for all . 3.6 2 m m D − m ( − , 1 ) , m − 2 1 2 ) 1 2 , m e N,A µ 003 2 e x − [ m ) 2 ; − − − , 2 m x , e 2 2 2 e ; 1 k e ,λ e 2 8 1 DGG , − m µ ) = , e , ; − + 003 and T · 1 − 2 ( 2 2 2 , λ 2 4 m L 004 2 m m \ − S m ( ( 2 3 , m 2 = 1 1 µ S = , m , 2 ) 2 , kλ kλ 1 1 ) 1 λ e e − − ) ) 2 λ , k ) = ( ( 1 1 2 2 e − ) to the above equality, 2 1 µ µ 1 2 2 1 2 1 − λ µ − (1 5 , 1 µ 5 5 4 − − \ is a generator of the 1 2 µ I I L 2 \ 2 , +1) +1) 3 e 1 k k 3 µ λ 2 (4 S ; ; − – 33 – A.21 (4 (4 ( = 2 2 S S I kλ ) ) λ λ 2 1 2 1 ) = ) = 2 → − ) − − 2 5 5 ) satisfy all the conditions in ( 1 x ,µ 2 2 2 \ \ = ( = (Sister of ; µ 1 ) = 3 3 µ µ , e ,e 2 1 1 S S 1 x 1 (2 (2 kλ λ λ ( ( e ; 3.21 +1) , e ( I I + + − 2 , e k 1 1 1 1 2 µ (4 µ µ , e , e ). Finally, from the following topological facts [ ) ) 5 ) 2 1 1 2 2 ) 2 1 2 1 1 2 ) = ) = e , m e are non-closable cycles according to ( 2 1 m 5 5 5 x x 1 , e ), in ( 0 0 5 ; ; 2 \ \ A.6 + \ 2 by explicitly constructing 1 , m + 2 2 \ m 3 3 − 3 e 1 A 2 ( − 3 2 1 ,A S S S ) , , m , e , e 0 m S ( 2 m e 1 2 2 m ( ( µ 2 2 ) ). In the above, we use the transformation rule of 3d index under N and = ( = ( 2 − m m m = 1 2 λ ( ( ( 0 ) A 1 1 A + k ,λ A 2 3.19 2 1 ) ) ) λ N kλ kλ 0 µ µ 2 2 ; + + + 2 λ λ , 2 1 1 2 N ) and ( 1 λ µ µ + + µ ( ) ) µ In the case, 2 2 2 ; − , 2 1 2 1 µ µ 2 2 1 5 5 µ ;2 ;2 µ \ \ ,µ 2 1 ; 2 2 2 3 3 N,A 1 , 2 5 ) and the following polarization transformation rules of 3d index µ µ S S 1 λ \ ( ( ( ( ,µ λ 2 1 3 − 1 I 5 ( 2 1 S = 1. − λ \ 5 1 I 3 \ − µ A.26 k ( ⇒ I 3 S ( (4 S I 2 1 003 is a knot complement called sister of figure-eight knot complement. Both 3-manifolds I 5 m have the same hyperboliccusp volume torus and boundary. areorientation the From are smallest ideal fixed. hyperbolic triangulationsare The 3-manifolds of homeomorphism equality with but in one also the that above they means have not the only same that orientation, the i.e. two the manifolds orientation of For we see that both of the pair, ( sponding DGG theory is expected toment have SU(3) explicitly flavor symmetry. for We will check the ehance- we confirm 2) inthe ( orientation reversal in ( Applying the Dehn filling formula in eq. ( we have following identity and the following matrix multiplication ( I JHEP07(2018)145 (3.34) (3.31) (3.32) (3.33) (3.29) (3.28) (3.30) X ) up to V X . . 2.69 . , , . 003 ) + Σ 2 X m 004 004 003 Φ . V is the topological ) corresponds to the m X m m 2 , whose orientation V C for 2 1 C + 3 − 5 acts on the two chiral V ,V for top for for \ V C 3 SU(3) X e V † 2 S V (1) ∈ ( u ] 2 3) λ + Φ manifest / − , 1 + Σ(2 2 2 2 µ (2) , Φ , X , P . 3 Z ; where X 2 2 V / 2 su 2 2 V λ , 1 λ Z C 0) − + − , + 2 − Σ 2 2 − 2 V λ 0 1 , µ 1 2 µ e / 2 0 1 3 − 2 Z † 1 1 2 / b − 2 Z ,X µ 1 Φ − manifest 1 2 b , ). − − − 003 θ 2 ) while boundary variables are different by a θ 1 1 = = 4 (2) m / 4 [ d ] Z Z 2 2 d 2 su λ µ A.25 – 34 – 2.62 λ + Z DGG Z 2 = 4 = 2 × T µ − π 1 2 2 := diag( 2 + 4 L 2 λ µ 8 top µ − T := diag(1 = 2 +2 2 will be enhanced to SU(3). 2 µ , ) = 3 gauge symmetry, and (1) λ = T C is coupled to the system as follows. Let u 003 which is irrelevant in symmetry enhancement. We reproduce and will be embedded to SU(3) as ,V top X m X V gauge X [ ,P 3)). manifest V (1) top ( / 2 u ] (1) , b λ 2 2 (2) u DGG 2 (1) λ − ) are provided by monopole operators with monopole , b = − 2 + u Z T ( su ,P 2 µ 2 2 C 2 µ 2 Z 2 − = A U(1) vector multiplet coupled to 2 chirals of charge +1 × a µ − P ] = ; 3 (1) 1 2 a − 2 1 2 / u manifest µ top 1 Z flavor symmetry associated to the background field = = , µ (1 Z ,X 2 2 2 (2) (1) C λ µ / = = 2 u 004 004 − su X 2 2 2 (1) m m [ λ µ µ u 1 = [ 2 a − × 2 X µ DGG 2 DGG T On the other hand, the The The theory has manifest According to SnapPy, both can be ideally triangulated by two tetrahedra and have C a L T 004 is same as the orientation induced from a Dehn filling on (1) u charge topological symmetry This is because the “off-diagonal components” of SU(3) (which are not in monopole charge of the fields. This So the theory is Thus, both DGG theoriesbackground CS are level for identical U(1) andthe described Lagrangian by here the with Lagrangian the in modified ( background CS level; For DGG’s construction, we choose is induced from an ideal triangulation in ( common internal edge variables givenfactor in 2 ( or 1 m JHEP07(2018)145 5 x ) 0 (3.39) (3.35) (3.37) (3.38) (3.36) , ). The 3 χ consists of . + m, n 2 2 e 2 F , must be en- . 2 3 u a 1 x χ e 1 ) 1 u , ) symmetry. There- 1 + 2 x χ 1 ; , manifest 2 (0) ∆=2 1 − 2] corresponds to the e χ denote a spin of space- / . 0 [0] is coupled to the chiral charge of Φ , (2) − j 0 . In the above expression, manifest ⊕ 1 + 2 χ su X C 3 1 are related to the R-charges of V , e , (2) gauge − 0 × 2 (1) SU(3) 2 e (0) ∆=2 χ , su u (1) 2 where − ∈ u . top [1] ( χ ) + ( r 2 3 -cycle is non-closable. ∆ ( ∆ 1) 1. Then the index show the SU(3) 4 . The ] and (1) , I ˜ A + ( Q : j x ) 1 u symmetry respectively. The index de- is the R ) 2 of charge x Q, F is coupled to the monopole current Σ ) = 0 − x , ; a −−−→ C , − ) 0 q 2 (1) 1 3 2 X e χ V , u ,V 1) manifest (1) ( 1 V + − u − χ ( ∆= V + 2 (2) 3 , + 1 0 1 2 is embedded in SU(3) as ). All these operators have quantum numbers e ,R su 0 )) where and χ , – 35 – = diag(0 because the a 1 3 − 0 X ... , then the , 3 X + denote the scalar and fermionic fields in chiral field χ ⊕ (Φ 2 X ( T X 2 e ) denote a gauge invariant BPS monopole operator 3 2 a , R (1) ) = ) (1) 2 − + u − a (2) − u (2) χ ( ... (1) ∆= 8 x ( ψ ∆ ) su (1 (Φ T su + 1 a I , 1 2 2 3 q R V 0 2 1 , e 3 (2) in a normalization such that [1 χ ) P = ( is different from the manifest and ( χ 2 1 2. 1 2 ... ˜ su , Q . Also, notice that a x X − X 3 φ 12 + + ( Q, − T ) = −−−→ ( -charge mixing between (1) = 1 2 (2) ) u Z R x V a ∈ ( ) = 1 su ; ) ) and form descents of conserved current multiplet for SU(3) flavor = 2 SCFT, a conserved current multiplet of flavor group 2 (0) ∆=1 x R 2 2. Therefore the 3 2 ; X ,e / 2 , , u [0] 1 as N 1 e 1 2 ) is the character of SU(3)-representation with Dynkin labels ( is enhanced to ( a 2 , u u , fugacity variable for ( 1 2 = , u u X u ( 1 1 dressed by matter fields ( must be enhanced to 004 u ( (1) 004 X /m denote a BPS monopole operator of charge u and /m m,n ∆) = (1 003 1 (1) , V χ u u m 3 003 Notice that this The superconformal index of theory is I respectively with In general, R-charge of BPS monopole operators m 12 a I R, j chiral multiplets Φ Here operators are denotedtime by its rotational quantum symmetry numberfundamental [ representation. of charge ( symmetry. In 3d following operators in the adjoint representation of correct IR R-charge mixingappearance should be of the determined SU(3) bycorrect F-maximization, in R-charge a but assignment. particular the The choice non-trivial firsttable strongly non-trivial below. suggests terms comes In that form the the operators table, Φ choice listed gives in the the Here we in particularly choose, Here structure: Here pends on the choice of fore, if hanced to SU(3).SU(3) This symmetry. agrees with our general discussion that this theory has enhanced fields via this generator with coefficients 3 This be the Cartan generator of the manifest JHEP07(2018)145 . ... (3.44) (3.40) (3.42) (3.43) (3.41) , . 82812 . 015 017 , . , m m 0 3 2 3 Z 015 017 0 0 0 0 − − for for m m + 017) = 2 3 m Z for for = x x x , ) + ) 2 ) 1 2 0 2 017 1 u 00 u u 3 , 1 2 1 Z 1 Z , /m u 00 3 u + + Z + 1 + + 0 015 017 1 , u 2 2 + 2 m 015; Z 2 015) = vol( 2 Z /m u u 2 1 / λ 2 1 m 2 1 + m u Z , − u 015 1 2 (2 + 2 − ( , ν = 2 . µ m Z µ − 00 − 2 2 1 SCI contribution − − − − 5 Z = 00 2 1 b iπν \ 1 1 λ 017 2 3 Z 2 1 − b 1 2 2) + − − S m 1 X − = = − 2) 1 Z . 1 , , = = 2 2 − 1 2 1 (1) 00 00 00 1 2 3 1 2 3 Z λ 1 1 µ 00 3 (1 – 36 – u λ λ Z Z Z − − Z Z Z 1 1 0 1 Z 2 ( 2 − − µ − +2 = 2(2 = 2 ,C + 1 1 2 · 2 2 1 0 0 0 µ µ − 3 00 3 7 ) λ µ C − Z 2 1 1 Z − − − 1 0 017 2 ) 2 5 1 0 2 0 − (1) 2 1 + µ λ \ + 2 u 3 /m 5 − − 0 00 3 2 − \ S 1 11 0 2 0 0 0 Z Z 3 ( 0 2 0 00 0 1 0 0 0 0 0 0 b . Descents of SU(3) flavor current multiplet. 0015 ) ) S − − ) + ( a a ,P m ∗ + 2 + 2 φ φ g 2 0 ( ( 2 ψ , b Z λ ( + − Z = = 2 a − , b V V 2 Z φ + 2 + ,P 2 Table 3 µ 2 017 00 00 Z 2 1 1 1 2 1 2 a µ + 2 P X Z Z )+(affine-shifts) are Γ Γ C C /m 1 2 + a 1 Z In the case, 1 + + , Z 015 = = 0 Z 2 1 1 m 2 2 Z Z − − g λ µ 2. − X = = = = 2 − µ 3 1 2 2 2 λ µ = C C a X − 2 k µ 2 a Then, the Sp(6 We choose For while boundary variables are different by a factor 2 or 1 Both manifolds have theAccording same to hyperbolic SnapPy, both volume canmon vol( be internal ideally edge triangulated variables by three tetrahedra and have com- JHEP07(2018)145 ∈ ) ) (4.1) (3.45) (3.46) (3.47) qB 2.43 ( + 13 . . 015 . . 2 is a Lens space, pA ]) M m C ] for closed 3- gl U V g M qB M [ ] theory. As discussed 015 + m d M ] for previous discus- [ 0 0 0 0 0 0 0 0 1 t 6 K irred ) + 2Σ g T 46 d V 6 – irred ] and references therein. . 015 N, pA T [ = − m 43 22 s J 3 are hard internal edges and 1 d g ). Φ C 6 2 015 irred 1 V = T C m C 1.2 V ) to obtain a 3d effective theory. Their of 3 − Σ)( 017 in ( R X and ,J µ − V /m 1 ] + 1 ) 2 C ] on C 2 V SCFT 015 λ e C P ] theory. For example, when M m † 3 [ (Σ g − d ,V M 1 2 6 [ 1 2 2 0 0 T C ], labelled by Seifert manifolds µ + Φ 0 0 0 0 0 1 ( − d − 2 6 2 irred 1 ,V , M T [ C Φ – 37 – ) X c vacua and a primitive boundary cycle ( V V 2 other than V 017 e = C ( X ] † 2 P N m 2 d [ ∈ M ,V µ 6 (irred) 015 1 − + Σ T P C 2 + Φ m λ 1 ] X DGG P ,V ; V Φ T 2 X N λ to superpotential. So, the DGG theory is X X [ V ,K − V ( 2 2 ] below. ] = − µ 3Σ C 2 d vector multiplet coupled to 3 chirals of charge +1 2 2 V λ 4 6 irred O 2 e − − 1 ,X (1) vector multiplet. Since / , µ T † 1 ] = (Giving a nilpotent vev to 2 ] is different from our can be decomposed into 1 u − µ Φ 2 − 017 M qB θ [ − and 015 4 and hence it is not possible to do the above operation while preserving c m + θ 017 P [ d ; 4 1 is standard in 4d class S theories. See e.g., [ 1 1 0 1 2 (1) m µ 2 0 1 0 [ pA d C u /m µ − − d µ Z 6 N DGG O (irred) [ ,X Z T T 015 π 1 DGG 4 by incorporating Dehn filling operation. Refer to [ ] is a non-trivial SCFT while supersymmetry is broken in our L m + d 015 = A = g T 6 irred , we need to specify a point M m [ = = [ T M c ), 015 Z 2.1 is a dynamical m DGG d 6 T U (irred) V T L ∂N, Their theory ( 1 13 cycles have empty supersymmetry. See table their in section theory may correspond to different choice of The fact that thenilpotent closing vev of to codimension-2 defectsThis (i.e., is knots) also corresponds in to accord giving with the our terminology ‘closable/non-closable’, because non-closable 4.1 Dehn fillingFor on a hyperbolicH knot complement 4 Dehn filling inIn 3d/3d this correspondence section,manifolds we generalize thesions DGG’s on Dehn construction filling to operation inof obtain the 3d context of theory, 3d/3d which correspondence we and will the denote construction The theory has manifest SU(3) flavor symmetry rotating 3 chrials as expected. Here we can not add Using the decomposition, we have The matrix JHEP07(2018)145 , ]. ]. µ . B 21 ; (4.3) (4.4) (4.2) , , induced 2 . N,A 5 , [ \ ) 3 ) p 015 by per- p S symmetry of DGG /m H T by the vev [ (2) 004 is called exceptional , C ) of the ] = 2 su [SU(2)] associated to B /m (2) qB , λ ; 2 T + ] by the transformation su µ 14 003 ) without sign change. Note ] ] -cycle. This is the reason [SU(2)] B ) pA 6 ; 2 m 2 N,A B T qB − [ λ / µ, λ p + ]. +2 − − 2 p ] and the +1 d B pA p p µ 6 N,A irred ; B [ N T ; [ )+( (2) SO(3) d ), we have λ d 6 irred SO(3) 6 − su irred N,A SU(2) [ T 2 SUSY broken T N,A 4.3 ] with additional CS level µ [ − ] with additional CS level non-trivial SCFT +[original value], where [original value] d (2 ] B ] ] 6 B p d irred ; λ λ ; B 6 , p irred T + + ; 1 T µ µ λ vector coupled to 3 chrials of charge +1 )=(meridian, -(longitude)) of the knot complement. ) and ( 2) − [SU(2)] disappears and hence we get N,A 2 λ 1 [ N,A +2) − / -cycle, but also the T [ vector coupled to 2 chrials of charge +1 vector coupled to 2 chrials of charge +1 µ − p p N,A 1 [ ( 5 ( – 38 – A 3.46 ), the boundary 1-cycle basis ( 0 0 − ) ) ) µ, d 2 1 1 2 ) can be identified with ( ] to the Chern-Simons level. To specify this con- 6 DGG d irred 2 5 4 5 6 (1) (1) (1) irred B ), ( T T 3.40 \ \ \ , λ ]. A primitive boundary cycle ; u u u T ). 3 3 3 2 = 1, the above relation can be simplified as follows. µ S S S Gapped theory (possibly with decoupled free chirals) qB A A A ( 3.31 q + 3.27 ] = N,A = = ( = = ( = = [ (2) of (2) of pA B is non-closable and hence ) ) ) (2) symmetry of N 2 2 2 su su [ d + A λ λ µ 6 irred ] is related to the theory su Z − − d T 2 2 +2 6 B irred 2 µ µ comes from the moment map operator of closable) T 2 λ + ∂N, µ − N, pA )+( explicitly in the notation +( ⇒ [ 1 2 2 +( can be identified with ( λ 2 H B d pµ − 2 1 6 irred 2 5 pµ ∈ N, pA µ \ T = (Gauging [ 3 (2 ] = (Gauging 004) S qB p non-closable d B ) as 015) is non-closable and 6 is gauging the m irred + + Z m T [( p , symmetry which is not gauged. If we give a nilpotent vev to this operator A pA 003) [( is non-hyperbolic. pA case, on the other hand, the ( C d exceptional and closable N (2) m 6 [ irred d 1 . Basic property of [( 6 qB SL(2 non-exceptional ( T 4 irred su (2) \ d + [SU(2)] becomes massive and flows to an empty theory in the low energy limit up to T 3 6 ∈ d irred As examples, we consider closed 3-manifolds obtained from Now, the operator If the su T 6 S pA Taking account of orientation reversal in ( irred p T T N 14 [SU(2)]. The Chern-Simons level of this group is that there is no orientation reversal in ( from the basis of For forming a Dehn filling. Combing ( Neglecting those Goldstone multiplets, the where we have assumed that why we are writing the the the Goldstone multiplets associated to the symmetry breaking of where T means the contribution of tribution, we have to specify not only the The theory ST Table 4 if JHEP07(2018)145 . ), 3. X − (2). (4.9) (4.7) (4.8) (4.5) (4.6) (1) su = (2). A u can be ) manifest = 3 and p 1 2 su p is embedded . ] and here we ) = ( manifest 2 ) with X 23 with the Chern- / gauge symmetry 5 (2) , 2 4.4 (1) 6 for the − G ) / su u . 2 1 − 2 λ − . − 2 ) ), µ (1) SU(2) 2 and SU(2) u µ 6 2 (resp. 2) for the is large enough where the rotating two chirals. But × )+( / (resp. | (1) symmetry for 2 X +2 0 ), 2 λ p u = Weeks | λ X − 2 4.4 2 to be the manifest SU(2) λ +( λ µ (1) 2 + u µ manifest X (1) flavor symmetry +2 µ 2 theories. The parity operation cor- 5 1)(2 4) u 1) µ ) − − 5 2 p p d − 5 − − 6 irred \ ( ( 3 = ( ] has T =Weeks, a oriented hyperbolic closed 3- . ) S 2 003) 5 λ (2) 003) 004) M m − is small, the theories could have accidental ] has accidental = ( 2 su m m ) corresponds to CS level 2 | µ ( ( 2 λ λ SU(2) subgroup. Since the p = ( – 39 – | + + / (1) flavor symmetry symmetry which is a subgroup of the SU(3). The = = 2 )+( 2 µ 3.46 u µ λ λ 5 ) ) X 2 2 − − λ µ 2 +2 ) +2) 2 in ( ] has no flavor symmetry µ 1 − p µ 2 ) corresponds to CS level 1 2 +2 ( 4 (2 ] has 5 2 λ ) µ X \ λ λ 1 − + 3 , , p 2 + 4 1 1 )+( S +( 3.31 (1) µ µ \ λ λ 2 ( 2 3 u λ 2) − − 1 1 pµ S − +2) − in ( µ µ 2 p p [( 5 5 ( ( µ ) ) ) ) X 2 1 (2 2 1 1 2 3 for d 004) p 5 6 5 4 5 irred means that we couple a vector multiplet of group − \ \ \ \ m (1) 003, T SO(3) subgroup of SU(3), the resulting theory generically has follow- 3 ( 3 3 3 k u / m S S S S 003) ( [( [( [( /G [Weeks] = (Two chiral fields coupled to U(1) m ( d d d d . The theories in the numerator has SU(3) symmetry at IR as argued in 6 6 6 irred irred irred 6 irred k T T T T and we are gauging its SO(3) symmetry is different from the manifest SU(2) 3.2 7. We will come back to this point in sec X (1) for the 3rd case comes from the topological symmetry of u − = SU(2) using the Weyl symmetryexchanged of with each the other. SU(3), Therefore, weWe the can will take symmetry just SU(2) SU(2) denote it as SU(2) in the following. Then by ( filling operation on Corresponding 3d gauge theoryThe is theory the in theory the numerator in has the SU(2) second line of eq. ( 4.2 Small hyperbolicLet manifolds us discuss themanifold case with of smallest closed hyperbolic 3-manifolds volume.supply This a was little already bit discussed more in [ details. The Weeks manifold is obtained by performing a Dehn symmetries. Actually from following topological fact (see figure. we can conclude that p The of the theory insemiclassical the analysis numerator. is The reliable. above is correct When when After gauging SU(2) ing flavor symmetry parity operation filps the signsresponds of to CS orientation levels reversal of on the topological the facts internal 3-manifold. It is compatible with following Simons level section to the SU(2) (resp.the SO(3)) CS in level +1 aSimilarly way for the that CS level Here the notation JHEP07(2018)145 2 ]. / 3 23 gives (4.12) (4.13) (4.14) (4.10) (4.11) be the σ . ) 2 bkg / σ +3 U(1) acting on ∈ 1 . The vev of − D gauge transformation, = 2 gauge theory at the 2 / is the background Chern- 5 . This is because the only N . − 2 bkg 0 n − bkg n ] are completely decoupled. Here 47 U(1) because the Lagrangian contains SU(2) U(1) − 1 ) is to give the dynamical U(1) an 2 bkg 1 ∈ is the 3d . bkg − − bkg × 3 1 1 σ 1 . Therefore, the low energy limit is 2 − bkg − / 4.11 topo CS − − 3 (2) U(1) ∨ su ]. In the AF duality, the U(1) charge acting ⊕ U(1) h and U(1) U(1) is coupled to the topological current of U(1) 23 ⊕ 3 two chirals − 3 = SU(2) . This can be done as follows. Let – 40 – − − bkg n 1 ) = (One chiral field gauged by U(1) n 3 vector multiplet, or in other words, the real mass 2 topo CS − 2 − / / 5 5 later. Then, the D-term condition implies that the − − bkg SU(2) SU(2) SU(2) 1 single chiral SU(2) topo CS − ⇐⇒ gets a vev proportional to on the right-hand-side of ( σ and we need to impose stationary condition for bkg 2 means that the U(1) σ × bkg gauge field multiplied by 2. The reason is that the Now, we can further simplify this theory to a much simpler theory [ is the global symmetry with background field, 2 / Dσ 2 +3 bkg · means that the two factors SU(2) + 2 3. This theory contains the gaugino, and by integrating out the gaugino, we get a ⊕ − The effect of We want to determine the value of Here we supply the details promised in [ Dσ 2 (Two chiral fields coupled to SU(2) / can match it withdynamical the real scalar SU(2) 3 the chiral field amake mass the term. low The energy sign CS of level the of mass the is dynamical anticipated field by as the U(1) fact that it must Namely, the gaugino reduces the level by 2 = FI parameter. We want the dynamical gauge group U(1) to be not Higgsed so that we we need to remark thelevel important point. The SU(2) pure topological Chern-Simons theory as two chiral fields on the left hand side, the left hand side flows to where where U(1) Simons level, and multiplied by two. real scalar for theassociated background to U(1) this symmetry. We choose the sign of it such that after integrating out the relation is on two chiral fields with chargeof 1 on the the U(1) left hand sidethe corresponds two to the chiral topological fields charge and can hence be all compensated gauge by invariant the operators have even charge on the left hand side. So the There is a duality found by Aharony and Fleischer (AF) [ To apply this duality, we needboth to sides know of the this relation equation between and the their flavor background U(1) Chern-Simons symmetries levels. of AF duality. JHEP07(2018)145 ) 1 is is = = ].). 1 2 − − 4.11 49 bkg bkg (4.22) (4.18) (4.19) (4.20) (4.15) (4.17) (4.21) (4.16) , or in ), we get bkg U(1) V [Weeks]. × 4.11 2 d 6 irred T ) is given by the . ) 2 / 4.11 5 − . . . . Σ bkg topo CS 2 bkg V Σ Σ 5 2 . . bkg 2 bkg U(1) and neglecting the decoupled U(1) ] for the corresponding statement in bkg − because these theories have only one Σ − bkg n 1 V = 1 ∼ 48 V nV is precisely the theory bkg π n + 1 4 + V ]. + 2 U(1) ⇒ bkg 2) is more precisely given by U(1) = bkg 2 23 by charge 2. If we integrate out (but we use the same name for simplicity). bkg ⊕ topo CS − Σ bkg − 2 2 = Σ / V V n Σ 3 bkg – 41 – V 2 − 2 2 V · n · → topo CS − [Weeks] is given by topo CS 2 4 = U(1) Σ + ( d bkg 1 = − 6 V irred U(1) Σ + 2 2 V − T U(1) 3 2 Σ + 2 V ∼ 1 V 3 2 π topo CS − 2 1 4 π 1 π π ) after gauging U(1) 1 4 1 4 4 topo CS 2 SU(2) [Weeks] = (One chiral field coupled to U(1) 4.11 ] and we have neglected U(1) d 52 6 irred Now let us gauge U(1) – to get T 50 [ = 1 is given by bkg 1 n V . First equality is well-known (see, e.g., [ + U(1) V × 2 2 → topo CS − − We conclude that the theory Let us see the right hand side. The Chern-Simons action of the right-hand-side of ( After integrating out the chiral field, the right-hand-side of ( V This is one of the small theories discussed in [ where in the second term,coupled the to factor the of topological 2other current have words, of taken by U(1) into making accounttheory, the the we shift fact get that U(1) The left hand side of ( after putting Therefore by comparing the low energy limit of the left and right hand side of ( Weeks theory. as This means that the low energy theory is given by Lagrangian where in the second term,coupled to the the factor topological of current 2 of have U(1) taken by into charge account 2. the We fact shift that the U(1) dynamical gauge field The second equality U(1) U(1) state in the Hilbertour space argument on is any not careful space enough (and to they detect are those called invertible field invertible field theories. theory), and theory given by where we have usedU(1) the fact that theWess-Zumino-Witten models gaugino which plays are no related to role topological in Chern-Simons U(1) theories [ and hence U(1) consistent way for the duality to work in low energy is to use the duality of topological CS JHEP07(2018)145 . ) = } N -th N i i λ = 2 (5.2) (5.1) , q + i µ 1 1 p 5 − N 1 { 1 − ] theory − ) L. 1 N . Combin- L, . next to [ 4 y λ i and p \ r o + 3 t µ o S s I 5 DGG I , we use a Dehn ) e I can be obtained v T 2 e o M v 5 o M M \ M 3 5 5 S − ∞ ∞ = ( λ + I I µ I 5 e I v e − o v ) o M of components of a link
1 M ]. The rational number 1
] related by a 3d duality. A rational ] 4 5 5 1) denotes the linking number between 5 \ = 41 : − − 3 M i, [ | ( S , with a hyperbolic knot complement L lk | A d SO(3) . 6 irred ] for a closed 3-manifold 3 N = SO(3) T S M = [ in , l d M – 42 – 6 1 L irred l ! r ) λ T l | r q a K vector coupled to 3 chrials of charge +1 | s ! e r + t p 2 l s ), we have a 3d duality between following twos / i p µ vector coupled to 2 chrials of charge +1 , topologically ( l 1 2 a ! r w s . The construction can be straightforwardly generalized ], every closed orientable 3-manifold t i 0 − 4.4 6 d . The basic moves may corresponds to basic 3d A ( τ ,...,p 2 2 (1) (1) 54 6 ) , λ ) u u 2 1 q , 53 i A A , + ( I 2 I k I τ µ l ]. Identifying the basic dualities would be interesting and we leave e ( 2 e + v ] = ] = v τ ,p 1 1 1 o r M λ λ o 1 q [ + / / λ + + can be given by Dehn fillings on a link complement. According to the M i ] studies the equivalence relation among the choices. Every pair of M 1 represent Dehn filling slope and 1 1 r µ µ q p d 5 5 = = 6 41 + ) irred M K = − i 1 1 : 2 ! ! r r ) T 1 µ r 5 ) which give a same closed 3-manifold. Different Dehn surgery presentations 1 1 \ | } p 4 ∞ 0 i K ) 3 as depicted in figure | = \ r 0 3 = 2 dualities from surgery calculus S L . , q / 0 i 1 \ λ S [( 2 p 3 . . . + r . Left : basic moves in rational surgery calculus [ [( { N µ S d , 5 0 6 ) irred d give different gauge theory descriptions of 2 6 L T irred = ( 5 T \ M 3 M S -th component and 1st component. Right: a sequence of basic moves showing ( ing the topological fact with eq. ( local moves depicted indualities among figure it as future work. Example: The Dehn surgery representation is notand unique ( and there are different choicesof of ( surgery calculus [ equivalent Dehn surgery representations are known to be related by a sequence of basic to the case when Lickorish-Wallace theorem [ by performing Dehn surgery along a link The DGG’s constructionDifferent is ideal based triangulations on giverelated by an different a duality. ideal field In triangulation theory ourfilling construction description descriptions of of the of a closed 3-manifold, and knot a complement primitive boundary cycle component of link i ( 5 3d Figure 6 JHEP07(2018)145 ] - k N , , µ 0 1 of (A.3) (A.1) (A.2) 4 → \ 1 T are the 3 I> S 3 [ j e, m . ). Then, the . i → DGG e 1 and m i T ) ) 2.40 e 1 2 R N m, e ), we can associate 5. ν + ] = → ( n − e 1 ( and a choice of basis , , µ + − m − 1 A, B
n i 1 ) 4 N − ] ! x \ m +1) i u u ( as in eq. ( 3 n ( 40 n + 2 x m S n k ) k +1 [ ; computes the superconformal 1 2 i ) − x 2 m r ( + x N d x k − 6 n ν irred M ) r ( T γ − x 1) 1 k =1 1 − i − − of size 2 N X ( ) g . ] 1 ( e N x 3 , := ν j i ( . After reviewing the definition based on ∞ =[ =0 ) i ∞ X + Y n r T γ qB 2 R 1 , γ + ). The theory in the first line is obtained by = x − N N ) = e g R ν pA ( x u 3.2 h N ) 1) ; – 43 – x ∆ I − ; I For given choice of an ideal triangulation, with and m, e Tr( k ( =1 and the basis boundary cycle ( T i Y ) m, e ∆ i ( k I N ) in charge basis is given by [ ,γ ∆ ) and x I N x ; ν Z h ; ) ∈ , . . . , e e 1 2 X 1 m, e x ( ] , e − m, e ∆ k ( ( 40 I )
1 and an integer-valued vector − ). It is defined with respect to a choice of an ideal triangulation A,B k N ( N Z Z g , . . . , m ] theory, which is defined as follows ∈ I 1 ) ) k M m X x [ ∂N, ; ] is an invariant associated to an knot complement (1) R-symmetry and SO(3) Lorentz spin respectively. ( or more explicitly ,...,e 1 d u 2 6 40 := ( irred e H ( m, e T ) matrix γ ( ) = Z ) of k, A,B ( N where I A, B The tetrahedron index 3d index is defined by [ Here the trace isCartans of taken over all local3d operators index in on the knottetrahedra, 3d of complements. a SCFT knot and complement Sp(2 an ideal triangulation, weincorporating generalized Dehn 3d filling. index The to 3dindex index be of for applicable a to 3-manifold closed 3-manifolds by A 3d index 3d index [ ( with positive angle structure.and But believed it to is be invariant independent under on the the local choice 2-3 of move of triangulation part of this project.Foundation The under work Project of Number DG SSTBA1402-08.by was supported the The by work WPI Samsung ofKAKENHI Research Science KY Grant-in-Aid and is Center (17K14265). Technology supported Initiative in (MEXT, part Japan), and also supported by JSPS which is claimed to havegauging SU(3) SO(3) in subgroup section of ( the SU(3) flavor symmetry withAcknowledgments CS level We are grateful to Yuji Tachikawa for very helpful discussions and for collaboration during The theory in the numerator of the first line is the theory JHEP07(2018)145 N . , i 2 (A.9) (A.4) (A.5) (A.6) (A.7) (A.8) ¯ α/ (A.11) (A.10) e ) 2 ] = 0 . 2 2 ¯ ) β/ b/ ˆ ¯ b/ ˆ ¯ e x 1 2 ; , e , e 3 2 x 2 ¯ ) CS ptn on A ˆ ¯ σ a/ b/ ˆ . = 2 3 C e , m, m , e , 2 . R 2 + 4 ¯ − m e 4 β/ is parameterized by and b/ ˆ ∈ ¯ e ] = [ e x x A 2 2 | | k 2 , e σ d ), Hilbert space of the ) cycle respectively and . − N 2 T ... 2 ¯ ˆ ¯ ( α/ a/ , e ˆ 1 a/ T + 1 ∆ + , e ( e ¯ , e I and 2 A ∧ ( 6 A, B − ) to 2 in the index as follows m =0 x b/ . ). For example, ˆ Z ) k α/ e n m, e m → ) x 1 2 e h h + ∂N Tr x ∈ H x ) 2 ( x 4 ; 2 = = k − e x β/ − Z ] = [ b/ ¯ =0 e 2 2 2 2 k − ˜ ~ 1 2 (1 i ) = , e b/ ) ˆ ¯ ˆ ¯ b/ a/ ˆ ¯ 2 − 2 − e , e | m, | ) = ( 3 , e x ... ¯ a/ ( + ∂N Z 2 where x x ∂N ) ( ∈
( i ∈ H ; 2 , e = 2 ˆ a/
i 2 m, e m, e ∈ 3 A,B x e =0 N 2 m N h =0 − h x . | m,e k , b/ I k N A β/ 2 | − − ] i} H 2 3 = ) along boundary ( e , e H x , e 2 2 iσ 2 ] = [ 2 , ~ , 40 – 44 – 2 ) = + πi )(1 b/ − e 4 a/ α/ m
− e ) 4 x 4 − x m, e ˆ ¯ a/ e e ; x , e x A e, x k | x | {| ( ; 2 d ∩ , e : − 1 N = 1 (completeness relation) 2 ), ( ∆ a/ , e m = − ∂N | 2 2 I m, e e The index can be thought as a SL(2 ˆ − a/ ( e − ˜ ) = ~ T e b/ ˆ ¯ A ∧ [ ( 2 − B + 1 e, := (1 ) , e T m, e − 1 2 =0 , ( 2 − m, e ) = 1 n m k ih A, ( h N Tr ) x h ) as follows [ x ˆ ¯ a/ =0 2 I x ∆ H iσ − k = 0; = , e T Z I πi , ( 2 m, e + H 2 2 4 | ~ i b/ ˆ 2 (0 k 2 ˆ a/ b/ ˆ =0 is real and is related to the variable ) = ) = ( Z = 0. The complex CS theory has two levels, e e k ∆ , e | | ∈ x x 2 ) and ( = I ˜ ~ ) k H ; ; e ) = X ˆ a/ ~ − ˜ ~ e m,e m, e m, e h ( , h h Basis of = | − m, e m, e ~ e ( ( ; | ~ ( ¯ ∆ ∆ A 1 2 )] = I I , 2 A T ( ( ] := e CS =0 k = 0, the S H k [ Triality : O Quantum mechanically, we associate a vector The algebra acts on exponentiated holonomy variables ( their complex conjugates. Quantum mechanically theseacting variables are on promoted the to Hilbert-space operators For The index is acomplex wave CS function theory in on a a Hilbert torus. Classically, the phase space on with quantized level Under the orientation change, the index transforms as follows Index as Chern-Simons ptn. The index satisfies following identities where [ JHEP07(2018)145 ] , . . 55 i ! , ) ) N (A.19) (A.16) (A.17) (A.18) (A.12) (A.13) (A.14) (A.15) | 32 Z Z = 0 1) along i ), − 1 = 0 which m, e ∂N, ∂N, S h ( ( = 2 1 1 A.10 × 2 2 H H , 2 b/ 1 qe, e ∈ + 1) ∈ D S + | 2 2 ) A ) CS wave function B = 1 case in [ − 2 × pm b/ e, C δ b/ e ˆ ¯ k 2
δ , e − D 1 2 ϕ − 2 . x − 2 ∼ . e, + b . qe, ˆ Z δ ! . 2 q , ,e + 0 ) ) ) ) − b/ ˆ ¯ 1 + Z qe Z e Z pm ) , , S , + δ 1 2 ˆ a ) 2 0 ) i m p 1 1 − − × is of SU(2) type is of SO(3) type − N ) x pm S S | 2 x = 0 δ Mimicking the A A SL(2 se × D × ˆ = 0). The SL(2 + ϕ , e + 2 ,m ( 0 0 2 ∈ 2 b ˆ i have trivial holonomy ( ∂ k 1 2 rm ). m se m D D 1 + 1 + N ! − ( 1 ( − x + x | 0 ih when when ˆ S 2 ( ( ϕ ∂ 0 ∂ x H 0 ( ( b/ ˆ ¯ rm r s p q 1 1 + × e, , e e qB 0 , , δ m, e ⊂ δ 2 H H se
+ i i – 45 – m ih b , ˆ + 2 = | D = ( s ∈ ∈ Z is given by N N m ˆ = ) with the completeness relation in ( pA i ϕ qe | | , rm i 0 | e ) + I 1 A B 1) x 2 + 2 qB , e ( ˆ S a 0 −
0 + r D m, e ( A.16 m, e ( , ϕ m × 2 m, 1 2 | qe, 1, the operator equation become ( pA h h ∂ ih = i 2 (unknot) is annihilated by following operators (a pair of ˆ , ϕ + | \ N ( ˆ = 1 ϕ D N 3 se, pm | | → ∼ ˆ pm a i ) H ˆ S m, e ϕ 1 δ 1 2 | / ~ corresponds to the shrinkable cycle in | + m, e ) = 1 − e S 1 ⊂ b/ ˆ h = ˆ ϕ S e S B se N × 1 2 1 rm = B = 0 on + | × m, e 2 × ∪ S − 1 ( k x 2 . A solution for the difference equations is 2 ) ) x rm D S | 1 × D B D 1) + h S × 2 h A,B 4 2 ( − N 2 Z D × ( m, e satisfies I b/ ˆ ∈ h D 2 ) 1 2 e ) = 0 h ϕ 1 2 2 2 x ,e D 0 ( Z Z ( x ) X ∈ ∈ ) ) x qB ( + X X = m,e,m m,e m,e qB pA ( ( ( + + 1 + qB N = = = b ˆ + pA e N ( pA I ⇒ I N Plugging the matrix element into ( The matrix element of the operator is given by The operator ˆ Then, the CS ptn for In the classical limit reflects the fact that flat-connectionsthe on boundary cycle on a solid torus quantum A-polynomial for unknot) Here the boundary cycle Quantum Dehn filling on, index, we give a Dehn filling operation on the index ( Then, the 3d index can be interpreted as JHEP07(2018)145 , , ) ) ) ), we x ; x x (A.24) (A.21) (A.22) (A.23) ; ; m 2 A.12 m, e m, e − ( ( ) ) 1 e ) ) se se A,B A,B − ... ( ( N N + 2 2 2 +2 I I e + rm 2 2 rm 9 2 2 , − − x − ) . , qe, qe, x x x e ) since the expression is + ) ) ; +2 + + − 1 x x + 12 e ; ; pm r, s se se 2 pm 8 m, − δ δ + x 2 2 , +2 ) 1 + m, e m, e rm − − x e 2 rm ( ( 1 ; + 8 2 2 ) ) x x e . − − − 7 ( ( ) (A.20) ( ). Combining with eq. ( 0 0 ) x e A,B A,B ∆ qe, qe, ( x m, e ( ( N N I qe, qe, ( + ; p, q p, q I ∆ I ) +2 ( 1 + + 7 ) ) ) I x 4 m +2 Z pm ) x x 6 ; \ ) 2 pm ; pm ; 3 2 x µ,λ x x + δ δ The indices for two knot complements are 2 pm ( ; S ; δ δ 2 , e 1 ) + I − − e p, q 1 p, q e ) . + 2 ; e ; x λ ( se se 5 + r, s − m, e ( – 46 – ( x + + ∆ 1; e +2 2 , µ I ) m, e m, e 5 ) + 5 5 rm ( ( → + \ ) 2 rm m, e m 4 3 µ,λ − ) 2 ( S x + ; ) can be straightforwardly extended to the case when 1) 1) 5 1 I , m e + 2 \ ) − − r, s SO(3) SU(2) 1 − ( ( ( ( 3 depends only on ( 1 e x − 3 , we can check that K K m, e e 1 2 1 2 ) S A.21 ( x ( x 2 2 1; − 1 2 2 2 Z Z , qB ∆ Z Z x − ∈ ∈ + m ∈ ∈ ) ) = ( ; 5 ) ) 2 ( − X X SO(3) ( pA x × I X X ∆ λ m,e m,e Z 2 K m,e m,e ( ( I ) and ( N + ∈ ( ( m, e 2 2 Z µ − ( Z ,e ∈ 5 X := := ∈ A.2 1 1 x ) X , , − Conjecture : the 3d index is topological invariant e e ) = ) = ) X x x − A A 1 SO(3) ( ( m,e ( 4 ) = ) = K qB qB \ 2 x x = = 1 + + 3 ; ; Z ∈ S pA pA ) N N X m, e m, e I I m,e ( ( ( 1 2 ) ) 4 5 \ \ 3 3 µ,λ µ,λ ( ( S S I I For SU(2) type For SO(3) type is a link complement with several components and the case when performing dehn filling Then, from series expansion in index and the conjectureevidence says for that the conjecture. they are allExample: equivalent. ( Let us give some non-trivial The formulae in eq. ( N along several components. Different Dehn surgery representation of a 3-manifold gives different expressions for the 3d invariant under which is expected since finally have Note that the final expression is dependent on the choice of ( JHEP07(2018)145 . ) 4 2 , n 00 . 4 ) and ) and + (A.27) (A.25) (A.26) ) ) Z 1 2 2 2 λ n n + m , b A.26 00 3 1 + + µ + Z 1 1 a 1 n n ). We checked + 00 2 , m + 2 + 2 1 Z A.2 2 n 1 2 , . n + ) ) m m + 0 2 3 00 +2 3 2 1 Z + + Z . Z e n 2 2 , − e e − − +2 + 2 2 , 00 0
2 3 1 1 2 − − 00 1 symmetry in addition to the 1 2 e m Z Z 1 1 e e Z + m m 8 − e e 1 = 1 ( + − 1 2 D 1 2 + − − m 8 0 ∆ 2 1 , [SU(2)] theory. This theory plays e e , , ) 0 m m 1 + I Z Z 1 1 2 2 1 T ) Z e n n S 2 C 1 1 1 − ) 1 − − − − S , n e x = 2 = = 2( = 2( ( 1 2 1 ; − , 1 2 2 4 2 n e e ) λ λ ). The other is using an ideal triangulation ( ( ( 2 , e / ∆ ∆ ∆ denotes Whitehead link depicted in figure 1 1 = 1 1 0 0 0 2 0 1 0 x 2 1 0 1 0 0 0 0 2 2 , e 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 − − A.21 × I × I × I − 2 S ( ,C – 47 – ): 2 i = 00 Z 4 , m e ) enjoys following ). 5 ∈ 1 2 1 2 → → Z ) k − 2 S , m , e i + K ( : X ,n 1 2 1 1 \ 0 4 1 2 1 2 2 m 5 1 2 1 2 n 3 , e ( Z e e e e I 2 − m m m m S ,S ( 1 SO(3) types := + 2 S -expansion for several examples. , m / ) = → . First one is using the index for Whitehead link ( 1
x = ( 00 ,C 3 x k ) ; m : : i 1 Z 4 -twist knot. There are two ways of computing the index for = K ( 2 k Z 2 1 \ 8 , e 1 2 kλ + i 5 3 , e ,, b b S S D 0 3 + 1 + I S m 1 2 00 Z 3 3 , e µ Z :( 2 Z Z ) 2 1 2 + 2 − + − 5 , m Z denotes 00 0 2 1 0 2 2 \ 1 3 Z Z k Z Z m S ( + + + − ) 2 00 00 1 1 3 1 ,λ Z Z Z Z 1 ) are SO(3) type basis. Corresponding index is λ 2 ; [SU(2)] and SU(2) = = = = 2 λ 2 1 T 1 2 1 3 5 ,µ , b \ µ µ 1 2 C C 3 µ a a µ ( S a I B Here we discuss some propertiestwo of different the roles: Gaiotto-Witten Recall that K the knot complement applying the Dehn filling prescription in eq.for ( the twist knotthey complement give and the apply same the index 3d in index formula in eq. ( The Whitehead index Weyl-symmetry Note that only( two internal edges are linearly independent and both of ( The corresponding link complement candatum be are (from triangulation SnapPy) can be 4 tetrahedra. The gluing Example: ( JHEP07(2018)145 H (2) (B.2) (B.3) (B.4) (B.1) su is enhanced arises at the . The C C C 1. In terms of 3d (1) (2) (2) u and monopole oper- su su φ × = 2 supersymmetry. . has U(1) charge +1 and H symmetry with the cur- ) i ). N C C (2) E Z , one neutral chiral multiplet , , µ su (1) V C u . at the quantum level. i j is the field strength of the gauge δ = 4 supersymmetric field theory SL(2 (2) C ) (2) type uses duality wall theories, are off-diagonal components of the k ∈ da su N (2) ( E 2) where su S , k = v = 4 conserved current supermultiplets . ˜ . su ) E i f ( N = 1 ←→ E 1 2 i i ) ˜ E − φ, v H 0) theory, the codimension-2 defect becomes φ )( , while j , i ˜ C , µ ˜ E – 48 – = = ( E i , where H , f i 2). On the other hand, the C E (2) = (2 , W π 1 µ comes from the topological symmetry of the gauge E (2) 2 = su C N i j su = = 1 ) 1 under the topological j i H (2) µ ) transformations of ( su )( Z i , ˜ E ⊂ , [SU(2)] i [SU(2)] theory. It is a 3d C T E T mirror : ( = 4 supersymmetry, the (1) u of ( N have charge , the holomorphic moment map operator is given by i H 1. The superpotential is given by v , the holomorphic moment map operator is given by (2) − C . su compactification of 6d [SU(2)] corresponds to the operation of f = 2 chiral operators which are in the adjoint representation of the symmetry. (2) 1 T π 1 . 2 S , N C su [SU(2)] coupled to the 5d gauge field, which gives properties of knots in complex corresponds to the Cartan of . These = . This topological symmetry is enhanced to v and T Chern-Simons theory on 3-manifolds. C (2) a φ j In In 3d SCFTs, the SL(2 There is a mirror symmetry which exchanges the Higgs branch and Coulomb branch For the Because of the has charge = 2 supersymmetry, there are one U(1) vector multiplet 2. 1. su (2) i , and two pairs of chiral multiplets ( ˜ of the theory. Under the mirror symmetry, the symmetries and operators are exchangedThis as mirror symmetry alsoto guarantees that the topological symmetry The su rent For the ators contain They are called (holomorphic)moment moment maps map of operators hyperkahler becausescaling moduli they dimensions spaces are are of associated protected the to tothese Higgs the be operators and 2. as Coulomb moment In branch, map this and operators paper their even we if abuse there the is terminology only and call At the level of Lieacts algebra, on this theory the has index quantum global symmetry level. The U(1) symmetry whose current is field N φ E 3d/3d correspondence. B.1 Brief reviewLet of us first reviewobtained by the a U(1) gauge multiplet with two hypermultiplets of charge Therefore, the properties of this theory are important in both sides of JHEP07(2018)145 . ∈ H as C 1 H (2) (B.5) (B.6) − which . Now su C H and hence C as we now see. (1) . We claim that u C 1), but this can = SO(3) C ) transform under − and SO(3) . This fact forbids i . 2 C ˜ (2) E Z H 2) which is obtained by , / i must also be of SO(3) / su 2 1 H E C ) by ( S − i , and SO(3) ˜ E and also has nontrivial magnetic 2 (2) , / i H H H su is defined such that half of it, E 2 (2) . w becomes SU(2) su 0). This is separated into gauge and is the spacetime on which the theory . , C H X (2) = U(2) 2) + diag(+1 , respectively. However, we will also show . The SO(3) / 2 su C H S +1 , multiplies the ( 2 – 49 – 2 / 2 on SU(2) H / Z . × 2 bundle, where and SO(3) Z is of SO(3) type. The fields ( . SU(2) H H U(1) C H ∈ . Thus, all operators are in integer spin representations of , the symmetry 1 C (2) H 0) = diag(+1 − su , (2) (SO(3)) = 1 su π ⊂ ) of the SO(3) 2) which is measured by a nontrivial value of the second Stiefel-Whitney 2 C diag(+1 / Z 1 (1) − u X, , ( 2 2 / H ∈ to all gauge invariant operators is trivial. This shows that the symmetry which . Now, the center 2 , meaning that it is SO(3) H mod 1, is the fractional part of the magnetic flux of SO(3). The integer part is not C H w The above argument implies the following. Suppose we gauge the group SO(3) However, there is a subtle mixed anomaly between SO(3) By mirror symmetry, it is obvious that the symmetry It is easy to see that 2 (2) (2) (2)). We denote them as SO(3) w 1 2 which has nontrivial Stiefel-Whitneysense, class. we However, for also thisthese have monopole monopole to operator operators turn have to half-integral on make they charges under a are the topological half-integral in magneticTherefore, half-integer by flux spin gauging of representations SO(3) the of gauge the U(1). Coulomb branch Then, symmetry is placed. Roughly speaking, the Stiefel-Whitney class topological invariant in non-abelianfluxes are SO(3) classified group. by The topologically invariantan magnetic SO(3) gauge group. Then it is possible to consider a monopole operator of SO(3) So, the gauge U(1) has magneticflux flux diag(+1 +1 class Then, gauge invariant operatorslet are us in consider representationsembedding a a of monopole U(1) SU(2) magnetic background fluxflavor into of parts U(2) the as as diag(+1 above U(2) on su Including the gauge group U(1), the operators are in representations of SU(2) acts faithfully to gauge invariant operators is SO(3) type. More directcharges way under to see this is to notice that all monopole operators have integer forbid gauging both of themwe as gauge SO(3) one groups. of In them other as words, SO(3) this gauge anomaly group,su implies then that the if other symmetrybe becomes cancelled SU(2). by a U(1) gauge transformation. Therefore, the action of the center Now we study theboth global of structure them of are the SO(3)of type symmetries in background the fields) sense are thatsu all in gauge representations invariant operators of (in SO(3)that the there (i.e, absence is integer a spin mixed representations ’t of Hooft anomaly between these groups SO(3) B.2 The global structure of the symmetries and ’t Hooft anomaly JHEP07(2018)145 , S X has ) be (B.9) (B.7) (B.8) = C T (B.10) (B.11) (B.12) ). H ∂Y , (SO(3) X 2 . Then, the w T (SO(3) ) = 0, and hence 2 ] where nontrivial H πi. w ) together along their 64 be the gauge field of – 0 C F. ) and B Y be the field strength of 57 H tr (SO(3) 2 ∧ 2 A and w ) mod 2 G (SO(3) , and + 2 2 H Y Z (SO(3) π π, w Z Z 2 1 2 F. / dA ] w whose boundary is π i . On the other hand, if the tr 4 in the Lagrangian is given by H = ]. (See also [ C Y ∧ (SO(3) = F F 2 56 B [SU(2)] and gauge the diagonal SU(2) w F F , respectively. We have ) mod 2 ) ) mod 2 T SU(2) X C and C tr tr C H Z . Then we get . Then glue × [SU(2)]. Then, 0 ∧ ∧ Z π B i T Y 4 G G of 0 [SU(2)] theory lives. To make the definition = Y – 50 – symmetries as SO(3) type symmetries, and this (SO(3) Y (SO(3) (SO(3) T H Z 2 2 2 f Z C w π i and SO(3) π ∧ i 4 πw πw Z 4 (2) H B Z symmetry is SO(3) = = − π su i iπ 2 C F G F 2 1 = X tr (2) tr Z and 1 2 F ∧ su and the SU(2) H tr G T ∧ (2) Y Z G su π i Z 4 Z . Choose another extension π i 4 Y . However, this depends on the extension of the manifold and the gauge has symmetry SU(2), then we add to dB ) transformation of 3d SCFT. We start from some 3d theory T Z X is a 3-manifold in which our = , -transformation of 3d SCFT be the field strength of the U(1) gauge field, X G S . Then, the coupling of the background SL(2 da If the More formally, the anomaly is shown as follows [ C ∈ = (1) transformation is performed as follows. subgroup of the SU(2) of the anomaly vanishes. The the symmetry SO(3), then we can (and choose to do) gauge the diagonal SO(3) subgroup of This represents the anomaly. The anomaly polynomial is B.3 Now it is clear whyS the SU(2) and SO(3) types are exchanged under the Gaiotto-Witten’s and hence This shows the dependencethe on Stiefel-Whitney the classes of extension. SO(3) Now, let field from common boundary to make a closed manifold and define the above coupling as where u where manifestly gauge invariant, we consider a 4-manifold means there is an anomaly. mixture of the centerf of gauge andthe mixed flavor gauge-flavor symmetries symmetry lead U(2) = to [U(1) ’t Hooft anomalies.) Let to gauge both of the JHEP07(2018)145 . as C be ·T 2 = 0, B S w SO(3) (B.14) (B.15) (B.13) i ]. Here / 0 66 and a, a is “internal h , we require A ˆ 3 , we have to A 6 ) (B.16) 2 M coefficients. By ∈ X Z 2 becomes SU(2) a , )] 3 Z 2 C Z M , ( . Then we get 5d SYM (2) 3 3 . 5 . ) ) M 2 H su X 2 ( ]. For simplicity we focus Z -transformed theory 1 Z × ⊕ , , 1 S 65 5 H 5 )] S 2 X X ⊗ ( ( Z = 2 ) . For a 2-cycle ˆ 3 , 2 5 6 3 are non-degenerate. The splitting H H Z have zero intersection X is “space-time” and X , M ∩ B ⊕ ( 3 3 B. A 2 ) B X X 2 ⊕ ( H ∈ 2 Z = and (2). , A 0 ⊗ H 5 ˆ [ B ) , we sum over all gauge configurations with su A 2 X ˆ B a, a ⊕ ( ) = Z 2 2 ) , where , ∈ 2 3 3 , Z – 51 – H and consider ) be the third homology with ) b (2) bundle on ˆ , Z 2 X 1 2 M 6 , ( ∼ = Z 3 S su Z 1 X × , ) , ( X 6 2 5 H 3 0) theory, the symmetry type is determined as follows. 3 ( [SU(2)]. Because of the anomaly, or more explicitly by [ , has SO(3) symmetry, the Z 3 X X X , T H ( ( ⊕ 5 H 3 2 T = of X H = (2 H 6 (2). Then, the above splitting determines the SU(2) H ) = × is not unique, but we have to choose one to define partition ∩ X N 2 1 su Z A B S , ( 3 = 3 in the path integeral. ˆ M 2 H theory corresponding to A and w × 1 b , but also on the polarization. ˆ A , and the pairings between 3 A 6 R On the other hand, for B X X (2) theory has the following properties. Roughly speaking, the theory is ( and the SO(3) 3 15 SO(3) symmetry types of knots from six dimensions su / H T ) into be a six manifold. To determine the partition function on 2 = 0. 6 Z , 2 X 6 w SO(3) types of the flavor symmetry associated to a knot. X ˆ a / ( -transformation. R 3 Applications of the above framework to 4d class S theories were studied inTake the [ 6d manifold as Let us compactify the 6d theory on Let In summary, if the original S In the presence of background fields for 1-form center symmetry, it can take nonzero but fixed values H 15 space” which is closed. The holomogy is given by determined by the background field. different values of we want to doSU(2) it for 3d/3d correspondence. More specifically, we want to determine the Then, the 5d SU(2) type for A-cycles and SO(3)the type second for Stiefel-Whitney B-cycles, class respectively. of the that More precisely, let theory with gauge algebra types of the 5d gauge theory. First, notice that the cohomology isUnder given this as isomorphism, we define of functions of the 6dA-cycles and theory. B-cycles, respectively. We Theon call partition the this function manifold of splitting the as theory polarization, not only and depends call Poincare duality, it is possible to split this homology as This is chosen suchand similarly that for any two elements From the point of view ofFirst 6d we have toon recall the some case of of the the properties of this theorygive [ some additional data. Let the consideration of the monopole operators discussed above, the has SU(2) symmetry andthe vice versa. In this way, SU(2)B.4 and SO(3) are SU(2) exchanged under the SO(3) of JHEP07(2018)145 , . . ) C 2 3 3 = of 16 2 K )). , T 2 Z N 3 3 M , we , × H 3 Z 3 X M ∼ = , 2 (B.17) (B.18) (B.19) N 3 , which (2) 3 X K is SO(3) ); su and ∂N , , ∂N ∂N ( ), then the . There, we 3 3 C ∗ 2 of the 2 N X Z H B.17 ( 1 , (2) because the split- 2 3 S 6 H su M X ( as a chain of ) and 1 2 B H is a torus, be a knot, and take a Z . , , 3 )] 3 )] 3 2 2 N M by the anomaly explained Z Z , , ), then the knot is of SU(2) , ∂N 3 ∈ C 3 2 ) which have nonzero values 3 2 , . Z M M ) ) ] is nonzero, then the N , K ( Z ), then the type of the symmetry ( 2 2 ( 3 1 , 2 2 ∗ ), then the type of the symmetry K Z Z 3 ] is zero in 2 N Z H , , H H ( , SO(3) given in section 3 3 Z M K 1 3 / ( ⊗ , ⊗ ), 1 3 M M H ) 2 ) M is trivial. Hence, the ( ( homology of the knot complement 2 ( 2 H 2 1 ). If [ Z M 1 2 Z 2 , ( Z K H H , 3 1 , Z ⊗ Z H 3 ). Poincare-Lefschetz duality implies that 3 , N × ) H ⊕ ⊗ becomes SU(2) 2 3 ( X 2 X 2 ) ) . The boundary of ). There is no unique way to do it. However, ( Z ∗ ( Z 2 2 C 2 M , . After the reduction on the 1 X , ( H K 3 6 Z Z – 52 – 2 H 1 \ H , , in the [ (2) ) (or more precisely, this map is a connection ho- N [ 3 X 2 2 ) in the relative homology group B.13 X ( H 2 2 ( 1 ⊕ A ] is trivial in ⊕ su X X M Z 1 Z ⊂ ( ( ) , we get , ) H A , 2 1 2 2 3 [SU(2)] theory coupled to the 5d gauge theory along H 2 3 := Z H H T Z K X , , on them. Then, by coupling the symmetry 3 ∂N 3 3 ( × = = × , ∂N N 2 1 1 X M 3 3 ˆ ˆ A w ( ( B H S 3 3 N X ( H H 2 = is that if [ 3 B ∈ H ) restricted to the = = 2 ∂ 2 X A Z B ] is nonzero in , ) be the image of B ∈ has a trivial homology in has intersection number 1 mod 2. By regarding 3 2 A has a nontrivial homology in Z B M K , ( K 3 1 ] is nontrivial, it is of SO(3)-type. N ], is nontrivial or not in H ( A ] and 1 K ⊗ be the A-cycle of the torus which is contractible on the ambient manifold A ) H 2 ) breaks it. , the defect becomes the A ∈ Z . Then, there are two cases, depending on whether the homology class of 2 , ] If the knot associated to the knot is SO(3). If the knot associated to the knot is SU(2). 2 X K A B.13 Suppose that [ Let us summarized the above result. Under the choice of polarization ( One possible choice is to take X There is no way to preserve the diffeomorphism invariance of the full 6d space • • × × ( 1 2 16 [SU(2)] to the 5d gauge group, the 1 such that [ can take the boundary momorphism in the long exact sequence of ting ( Let [ The proposal in section type, and if [ there is a dual cycle Let us compare this resultconsider with the the knot criterion of complement and SU(2) let type. can contain nontrivial elementsof of the Stiefel-WhitneyT class in the previous subsections.H On the other hand, if [ One of the consequencescodimension-2 of defect this along choice isS as follows. Let X we denote [ In this case, by taking there are only a fewand choices we which preserve assume the that diffeomorphism one invariance of of those choices is realized in 3d/3d correspondence. First we have to choose a polarization ( JHEP07(2018)145 1 M S (2). × su (B.21) ) type. 1 X . C 3 S , ), as far as M on ) is nonzero. is homotopic C and return to 2 , then we get , 2 3 Z B 1 w X , ) representations must happen on M 3 S is the B-cycle on C ) representations if = 0 on B , M and B ( C 2 1 , 3 w ) has intersection num- H = PSL(2 representation of M 2 ) is zero. (B.20) ) type. Indeed, because 2 ] and C ∈ Z J , Z C A ] 3 is concerned. However, this , , go around 3 1 M (2) gauge algebra around the ], where 3 p S ) reproduces the rules for the ∂N M M ( su ( pA 1 1 , H + H B.17 ) ). Thus we get 2 ∈ 2 B w ] Z , we get the following conditions; , 1 M 3 K 3 [ S is contractible in M × N ( 1 A X 1 into the ambient manifold S ⇐⇒ Z H are defined only for PSL(2 B – 53 – 3 representation of Jπi . If we let the point SO(3) types obtained in this appendix is the same M J 1 X / . One of them ( ) bundles instead of SO(3) is of the form [ S ∈ 3 ) because C 2 , the holonomies have periodicities of PSL(2 , X B exp(2 1 M Z ˆ ∂ ) is realized in 3d/3d correspondence. B . This is possible only if [ S , ) is nonzero 3 3 regarded as elements of 2 and Z M M B ] is nonzero in , ( B.17 ) is zero, then holonomy may be well-defined for half-integer . ∂ 3 3 1 A 2 = SL(2 ⊂ 2 N . This means that only the representations of SO(3) type are H Z M ) is in ( C , 2 Z ) is nonzero, while they may be defined for SL(2 1 M 1 3 is a point on 2 . This means that the complex Chern-Simons theory on the 3- Z S H ] and 3 , Z ] is trivial. Conversely, it is also true by Poincare-Lefschetz duality ∈ p ] in (2). M means that the 5d gauge bundle does not have a nontrivial Stiefel- 3 , ( A ∈ 3 M K 1 . Therefore, K J su ˆ M ] 3 A and ( H M A 1 ( [ 2 1 ∂N ∈ ] = [ H , the [ X H , where ] is an integer. If we embed 2 ] is zero. ⊗ pA 1 T 1 ∈ M M ⊂ p ) is in ) 1 M ] is trivial, then [ ] S S 2 2 + ∼ = S 1 X Z 1 M K Z × SO(3) symmetry types of knots. This is a nontrivial check of our proposal. Thus 3 , S B , S 3 / We only consider gauge bundles of zeroHolonomies Stiefel-Whitney around class cycles [ if [ 2 p . Therefore, [ Thus, in complex Chern-Simons theory on Remember that there are only a few polarization choices which preserve the diffeo- There are other consequences of the above choice of polarization. The fact that , and ∂N M X = [ • • 3 K ( ( 1 2 B with integer spin consistent. This is ahand, version if of [ the Diracrepresentations quantization of condition argument. On the other where we have assumed thatThus, the the holonomy well-definedness is of the taken in holonomy requires the that spin we only consider representations This fact can bewhere seen as follows. ThereNow can consider be holonomies a in nontrivialcycle magnetic the flux spin the same point, the value of the holonomy changes as H Whitney class on manifold has the SU(2) the Stiefel-Whitney class (i.e.,does discrete magnetic not flux) mean on H that the periodicity of holonomy is of SL(2 morphism invariance of SU(2) we assume that the choice ( that if [ This means that theas result the for proposal SU(2) in section ber 1 modthe 2 boundary because, roughly∂N speaking, the intersection∂ of [ to On JHEP07(2018)145 ]. B Phys. , , ]. SPIRE Commun. , IN , Geom. ][ (2007) 1895 ]. , ]. SPIRE (2013) 035 Nucl. Phys. IN 57 , ][ 12 SPIRE SPIRE IN IN (2016) 062 ][ ][ ambiguity. For exam- ]. JHEP correspondence , 10 Adv. Theor. Math. Phys. d , 3 ]. -branes on the squashed hep-th/9611063 (tangent) = 0. The second - 5 2 d [ SPIRE 3 w IN J. Geom. Phys. M JHEP , ][ , ]. , and so on, which represent this ]. SPIRE arXiv:1305.0291 2 [ IN z Chern-Simons theory ][ via the √ ]. ]. (1997) 101 SPIRE k , arXiv:0904.2715 SPIRE arXiv:0903.4172 = 6 1 [ ) connections by ideal triangulations. If IN IN z C N ][ ][ √ , SPIRE (2013) 017 SPIRE B 493 IN ]. IN ]. Mirror symmetry in three-dimensional gauge correspondence 08 Wall-crossing, Hitchin systems and the WKB ][ ][ d ), then there may be – 54 – A TQFT from quantum theory teichm¨uller arXiv:1109.6295 3 (2012) 034 - C [ (2012) 884] [ , d SPIRE supersymmetric gauge theories in three-dimensions SPIRE 3 08 arXiv:1305.2891 IN IN JHEP The quantum content of the gluing equations [ [ , Mirror symmetry in three-dimensional gauge theories ][ = 2 Nucl. Phys. 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