Symmetry enhancement and closing of knots in 3d/3d correspondence

We revisit Dimofte-Gaiotto-Gukov’s construction of 3d gauge theories associated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d N=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(2,0\right) $$\end{document} theory of A1-type on a 3-manifold, we propose a topological criterion for SU(2)/SO(3) flavor symmetry enhancement for the u(1) symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the construction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising SU(3) symmetry enhancement which is unexpected from the 6d viewpoint.


Introduction and summary
3-dimensional (3d) quantum field theory exhibits several interesting aspects. Unlike higher dimensional case, Abelian gauge interaction in 3d is strongly coupled at infrared (IR) and gives non-trivial IR physics. Different gauge theories at ultraviolet (UV) could end at the same IR fixed point along renormalization group (RG) and such phenomena is called "duality". Refer to [1][2][3] for examples of dualities among 3d gauge theories. There could be enhanced symmetries in the IR fixed point which is invisible in the UV gauge theory. From purely field theoretic viewpoint, these phenomena are not easy to understand or predict.
In this paper, we consider a certain subclass of 3d quantum field theories with N = 2 (4 supercharges) supersymmetry which can be engineered by a twisted compactification of the 6d (2, 0)-superconformal field theory (SCFT) of A 1 type. The 6d theory is the simplest maximally supersymmetric conformal field theory and describes the low energy world volume theory of two coincident M5-branes in M-theory. The 6d theory has SO (5)

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Here µ = (µ 1 , µ 2 , µ 3 ) is a chiral operator in the triplet representation of su (2), and this operator is associated the co-dimension two defect along K. Each of the arrows in the above equation are nontrivial RG flows which are explained below.
The proposed relation explains why the T DGG [M, K] theory generically has only U(1) symmetry associated to the knot while T 6d [M, K] has su(2) flavor symmetry. The su (2) symmetry of T 6d [M, K] is broken by the superpotential deformation δW = µ 3 in (1.2) which is typically a relevant deformation in the RG sense. After S 1 -reduction, the 6d theory becomes 5d maximally supersymmetric su(2) Yang-mills theory (SYM) and the codimension two defect in 6d theory is realized by coupling a copy of the 3d N = 4 T [SU (2)] theory [21] to the 5d theory. Then µ is the su(2) moment map operator of the 3d N = 4 T [SU (2)] theory.
As an intermediate step, we introduce a 3d SCFT T 6d irred [M, K] appearing in (1.2) which is the IR fixed point of T 6d [M, K] on a particular point P SCF T of the vacuum moduli space. Unlike T 6d [M, K], T 6d irred [M, K] might not contain the su(2) moment map operator µ after taking the IR limit. In that case, the superpotential deformation is not possible (or more precisely, it is irrelevant) and thus the T DGG [M, K] still has the su(2) symmetry. By carefully analyzing the coupled system, 5d SYM+3d T [SU (2)], we find a topological condition on (N, A) which guarantees the absence of moment map operator and thus the su(2) symmetry in T DGG [N, X A ] theory. The topological condition is summarized in table 1. For example, we expect su(2) symmetry enhancement when M is a Lens-space and do not expect the enhancement when M is hyperbolic.
As an application of the symmetry enhancement criterion, we show that the T DGG [M = S 3 , K] theory for all hyperbolic twist knots K has a surprising SU(3) symmetry. As a simplest example, we claim that the following 3d N = 2 theory has SU(3) symmetry. The theory only has manifest SU(2) × U(1) symmetry where the SU(2) rotates the two chirals and the U(1) comes from the topological symmetry of the dynamical abelian gauge field. The U(1) X A symmetry associated to the knot is a linear combination of two Cartans of the SU(2) × U(1) which is expected to be enhanced to SO (3) according to the criterion in table 1. From a group theoretical analysis, the enhancement implies that the SU(2) × U(1) should be enhanced to SU (3). We checked the symmetry enhancement by explicitly constructing the corresponding conserved current multiplet. Base on the proposed relation between T 6d [M, K] and T DGG [M, K], we identify the field theoretical operation on T DGG [M, K] corresponding to Dehn filling operation on the knot complement N = M \K. The operation is only possible when the T DGG [M, K] has su(2) flavor symmetry. The Dehn filling operation is analogous to closing of punctures on Riemann surface in 4d/2d correspondence [22]. By applying the Dehn filling operation, we can extend the DGG's construction to 3d gauge theories labelled by a closed 3-manifold M . The theory is denoted as T 6d irred [M ] and has similar 6d interpretation as T 6d irred [M, K]. As concrete examples, field theoretic descriptions of T 6d irred [M ] for three smallest hyperbolic 3-manifolds are given in [23]. One interesting aspect of our construction of T 6d irred [M ] is JHEP07(2018)145 that we can relate surgery calculus in knot theory to 3d N = 2 dualities. One way of representing closed 3-manifold is using so called Dehn surgery representation. A closed 3-manifold M has infinitely many different surgery descriptions and surgery calculus tell when two surgery descriptions give the same 3-manifold. Different surgery representations of a closed 3-manifold give different field theoretical descriptions of T 6d irred [M ] which are related by 3d dualities. One illustrative example is given around eq. (5.2). Since the 3d theory depends on only the topology of the 3-manifold, every physical quantities of the theory are topological invariants of the 3-manifold. As an example, we introduce a new 3-manifold invariant called "3d index" which is nothing but the superconformal index of the T 6d irred [M ]. The paper is organized as follows. In section 2, we introduce two ways of associating the choice of 3-manifold M and a knot K inside it with a 3d gauge theory T [M, K]. One is through the construction by Dimofte-Gaiotto-Gukov [19] (DGG) and the corresponding gauge theory is denoted as T DGG [M, K]. The other is through a twisted compactification of 6d A 1 (2,0) theory on M with a regular co-dimension two defect along K. The resulting 3d gauge theory is denoted as T 6d [M, K]. After explaining the two constructions in detail, we propose a precise relation (2.84) between two constructions. Base on the proposed relation, in section 3, we give a topological criterion on (M, K) which determines when the U(1) X A symmetry T DGG [M, K] theory is enhanced to SU(2) or SO (3). The criterion is summarized in table 1. In section 4, we identify field theoretic operation corresponding to Dehn filling operation in 3-manifold side in 3d/3d correspondence. It allows us to extend the DGG's construction to the case when the knot is absent. In section 5, we discuss how the surgery calculus in knot theory predicts infinitely many 3d N = 2 dualities. 2 3d N = 2 superconformal field theories labelled by 3-manifolds In this section, we introduce two ways of associating a 3-manifold M with a knot K in it to a 3d N = 2 gauge theory T [M, K]. One way is through a twisted compactification of a 6d N = (2, 0) theory of A 1 type on a closed 3-manifold M with a regular co-dimension two defect along a knot K on M . The other way is using the construction by Dimofte-Gaiotto-Gukov [19] (DGG) based on an ideal triangulation of the knot complement M \K. These two theories are argued to be related [19], and we will propose the more precise relation between them with supporting evidences. We describe the relation after reviewing basic aspects of two approaches. Before going to detailed analysis, let us first introduce an alternative labelling for the topological choice, (M and K), which will be used throughout the paper. The choice can be replaced by  In most part of this paper, we assume that N is a knot complement with one torus boundary but our discussion can be easily generalized to the case when N is a link complement with several torus boundaries. As a simpler set-up, we can also consider the case when the defect is absent. In that case, the resulting 3d theory is denoted as T 6d [M ] and T 6d irred [M ], respectively. For the T 6d irred [M, K] to be defined, we assume that N = M \K is a hyperbolic knot complement.

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The reason that we consider T 6d irred [M, K] is as follows. The moduli space of vacua of T 6d [M, K] in general contains several different connected components. Then, we have to decide which point of the moduli space we consider before taking the low energy limit. The typical distances between different components of the moduli space are of the order of the compactification scale on M , which set the cutoff scale of the low energy effective 3d theory. Therefore, we cannot expect that there is a single effective 3d theory which describes the entire moduli space of vacua. Only after specifying a point on the moduli space, we can obtain a low energy effective field theory which describes the physics near that point. 2 In other words, T 6d [M, K] is not a genuine 3d theory, but should be considered more appropriately as the 6d theory compactified on M . However, we will be sometimes sloppy and call it a 3d theory in this paper.
In T 6d irred [M, K], we pick up a point and take the low energy limit. The low energy limit may be described by a 3d SCFT (which can be empty or a topological theory). Below we will specify which point on the moduli space of vacua we take, by using reduction to 5d SYM.
T 6d on R 2 × S 1 via 5d SYM. The structure of the moduli space of vacua becomes simpler if we compactify the 3d spacetime to R 2 × S 1 . This is because we can use the 5 dimensional maximally supersymmetric Yang-Mills (5d SYM) theory description. The set-up is 3 The bosonic components of the 5d SYM theory are gauge fields A I (I = 0, · · · , 4) and scalar fields φ k (k = 0, · · · , 4). After compactification on M , the supersymmetry is defined on R 2 , and we split these fields as From the point of view of the super-algebra on R 2 , the V = (A µ=0,1 , φ k=0,1 ) is the vector multiplet and A i = A i + iφ i (i = 2, 3, 4) are twisted chiral fields. The reason that we regard A as twisted chiral fields rather than chiral fields is that the relation between 5d SYM and T 6d [M ] is a kind of mirror symmetry analogous to the case of 4d class S theories. 2 A simple example which illustrates the point is the T 2 compactification of the 6d N = (2, 0) A1 theory on T 2 . The moduli space of this theory is [R 5 × S 1 ]/Z2, where S 1 comes from the integral of the 2-form field on T 2 . On the other hand, the moduli space of 4d N = 4 SYM is R 6 /Z2. Only after picking a point on [R 5 × S 1 ]/Z2 and taking the low energy limit, the 6d N = (2, 0) theory on T 2 becomes the 4d N = 4 SYM. In this case the moduli space is connected, but still there is no single 4d effective theory describing the whole moduli space of vacua. 3 On general grounds, one may only expect that 5d SYM describes the moduli space only in the limit of very small radius of S 1 . However, somewhat miraculously, it is believed that 5d SYM describes even a finite radius of S 1 .

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The twisted superpotential is given by complex Chern-Simons action as where g 2 YM is the gauge coupling of 5d SYM which is related to the radius R of S 1 as 1 This is the results in [6][7][8] in the limit S 2 → R 2 . This twisted superpotential corresponds to the twisted superpotential obtained in DGG's construction discussed in section 2.2 The regular co-dimension two defect along a knot K ⊂ M can be realized as coupling the 3d T [SU (2)] theory [21] to the fields of 5d SYM [24][25][26][27][28]. The theory T [SU (2)] is reviewed in appendix B.1. This is a 3d N = 4 SCFT given by U(1) vector multiplet coupled two fundamental hypermultiplets (E a ,Ẽ a ) a=1,2 . The theory has su(2) H × su(2) C flavor symmetry and let µ := holomorphic moment map operator of su(2) C , ν := holomorphic moment map operator of su(2) H . (2.12) Then the twisted superpotential coupling of the T [SU (2)] and the 5d SYM is given by This means that we integrate the one-form tr( νA i )dy i over K. 4 We can also include (complexified) mass terms to the defect as where ds is the line element on K, and m is the mass. The mass of defect is related to the eigenvalues of ν: See [28] for detailed explanations of the coupling of 5d SYM to T [SU (2)] in the context of 4d class S theories. The analysis there may be extended to the 3d/3d case, but we do not perform a detailed analysis. By solving F-term equations for the twisted superpotenal in (2.10) and (2.13), a part of the moduli space of vacua 5 on R 2 × S 1 with mass parameter m is given by The gauge invariance is preserved as follows. The supersymmetry is considered in the two dimensional space R 2 , and hence the direction along the knot K is considered as a kind of "internal manifold". Let t be the coordinate along K. Then, the kinetic term along this direction comes not from the Kahler potential, but from the twisted superpotential as W ⊃Ẽ∂tE. This term combines with (2.13) to form a covariant derivativeẼ(∂t + A)E, where we have used µ ∼ EẼ (see appendix B.1). 5 When the connection A is reducible, we can turn on the expectation values of the vector multiplets V = (Aµ=0,1, φ k=0,1 ). These branches are very important in 4d class S theories [28,29]. However, in the 3d theories considered in this paper, we only consider points on the moduli space on which A is irreducible. Therefore, we can neglect those branches.

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where δ(K) is the delta function localized on K, and G is the group of PSL(2, C) gauge transformations on M . This is the space of flat connections of the complexifield gauge group PSL(2, C) with the holonomy e ν around K.
ρ hol (A) := (PSL(2, C) holonomy matrix along A-cycle) = e ν . (2.17) Notice that the eigenvalues of e ν are determined by the mass parameter m. Now we can specify the point P SCFT on the moduli space of vacua which is taken in the definition (2.7). First, let us consider more generally. For simplicity we assume that the moduli space of vacua on R 3 , M vacua (T 6d [M, K] on R 3 ), is a discrete set. Let us take an arbitary point P ∈ M vacua (T 6d [M, K] on R 3 ). Then, if we compactify the theory on S 1 with a radius which is large enough compared to potential barriers between different points on M vacua (T 6d [M, K] on R 3 ), then the point P goes to a subset M(P ) of the moduli space of vacua on This M(P ) need not be a single point, but may have several points whose number is related to the Witten index of the 3d effective theory on P . Because of the supersymmetry, the condition that the radius of S 1 is large may be dropped since there is no phase transition under change of the radius. The explicit forms of M vacua (T 6d [M, K] on R 3 ) and M(P ) are not known and they are defined just by the abstract field theoretical considerations as above. However, later we will propose how M(P SCFT ) may be given concretely in terms of flat PSL(2, C) connections.
To consider a superconformal point, we set the mass m to be zero. Then the point P SCFT is defined as follows. After compactification on S 1 , the moduli space of vacua of the theory on P SCFT becomes a subset M(P SCFT ) of the moduli space of vacua on R 2 × S 1 . Then, the point P SCFT is defined by the condition that M(P SCFT ) contains the connection A hyp ∈ M vacua (T 6d [M, K] on R 2 × S 1 ) which is determined by the unique complete hyperbolic metric on N = M \K. More explicitly, using the spin-connection ω and dreibein e of the complete hyperbolic metric, the flat connection can be expressed as This flat connection has the greatest value of Im(CS[A α ]) among all flat connections A α with parabolic boundary holonomy and is conjectured to be the only vacua contributing to a squashed 3-sphere partition function [30] of T 6d irred [M, K]. Refer to [13,[31][32][33][34] for discussions on the conjecture from various respects, state-integral model of the complex CS theory, holographic principal and resurgent analysis. To other physical quantities of T 6d irred [M, K] such as superconformal index, on the other hand, other flat connections in M(P SCFT ) may contributes. Now we give a conjecture about how M(P SCFT ) is given concretely in terms of flat connections. First we consider the case where a knot K exists. For this purpose, we define M(P SCFT ) even for nonzero mass m by continuity from m = 0. Namely, M(P SCFT ) is

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just the set of vacua of the 3d effective theory near P SCFT with mass m. We make the dependence on m explicit by writing it as M(P SCFT , m). We also define χ(N ) as This means that we consider all flat connections with varying holonomy around the knot. Then we propose In [35], the component is called Dehn surgery component. Another way of representing the above equation is M(P SCFT , m) = χ 0 (N ) ∩ {eigenvalues of ν = {m, −m}}.
Next, consider the case where there is no knot on M . If the closed 3-manifold is represented by a Dehn filling operation on a hyperbolic knot complement N = M \K for some K along a boundary cycle A, we propose that the M(P SCFT ) is given by The definition of the right hand side contains a knot K, but we assume that it is independent of the choice K ⊂ M . Notice that ρ hol (A) = 1 is stronger than m = 0, since we could have a nonzero upper-right component of ν even if its eigenvalues are zero.

Dimofte-Gaiotto-Gukov's construction: T DGG
In [19], a combinatorial way of constructing a 3d SCFT, which we denote T DGG [N, X A ], for given choice of (N, A) is proposed. Empirically, the theory associated to non-hyperbolic N is a trivial theory only with topological degrees of freedom. In this subsection we focus on the case when N is hyperbolic. Here we give a summary of the DGG's construction with a modification on superpotential deformation associated to 'hard' internal edges (see (2.47)) which play a crucial role in the symmetry enhancement of the theory.

Mechanics of ideal triangualtion
The construction is based on a choice of an ideal triangulation T of N .
Here ∆ i denote the i-th tetrahedron in the triangulation. Ideal tetrahedron can be embedded into a hyperbolic upper half plane H 3 in a way that all vertices are located on the boundary of H 3 and both of edges and faces are geodesics. Hyperbolic structures on an ideal tetrahedron can be parameterized by a complex parameter z (with 0 < Im[Z] < π, z ' z '' z Figure 2. Edge parameters (z, z , z ) of an ideal tetrahedron. Left: ideal tetrahedron in . Using the isometry of H 3 , PSL(2, C), four asymptotic vertices can be placed at (y, w) = (0, 0), (0, 1), (0, z) and (∞, ·). Right: topologically, ideal tetrahedron is a tetrahedron with truncated vertices.
Z := log z), which is the cross-ratio of the positions of its vertices on ∂H 3 . We assign edge parameters (z, z , z ) to each pair of edges of ideal tetrahedron as in the figure below. Geometrically, these edge parameters correspond to {Z, Z , Z } := {log z, log z , log z } = i(dihedral angle between two faces meeting on the edge) + (torsion) . (2.25) Here "torsion" is a quantity which measures the twisting of hyperbolic metric around the edge. For an ideal tetrahedron in H 3 , these parameters satisfy  The second equation follows directly from the geometric definition of (z, z , z ) as equivalent cross-ratios. These constraints are compatible with the following cyclic symmetry of ideal tetrahedron: An hyperbolic structure on a knot complement N can be obtained by gluing the hyperbolic structure on each tetrahedron in a smooth way. For the smooth gluing, we need to impose the following conditions C I := (sum of all logarithmic edge variables associated to edges meeting at the I-th internal edge in the gluing) (2.28) There are k-internal edges in an ideal triangulation with k ideal tetrahedra. A solution to these gluing equations (2.26) and (2.28) with conditions 0 < Im[Z i ] < π for all i gives a hyperbolic (generally incomplete) structure on N .
The variety is called a deformation variety. where the definition of χ(N ) here is equivalent to that in (2.20). The map χ T is injective but not surjective. Using the map, holonomy matrix along a primitive boundary cycle A ∈ H 1 (∂N, Z) = π 1 (∂N ) ⊂ π 1 (N ) can be written as linear combinations of logarithmic edge parameters (2.32) We currently do not have the field theoretic understanding of the exotic case and will always work with non-exotic triangulations.
SU(2)/SO(3)-type of boundary cycle A. For later use, we classify a primitive boundary cycle A into two types, SU(2) or SO(3), depending on evenness/oddness of the linear coefficients (α i , α i ).
A is of SU(2)-type , if all (α i , α i ) can be chosen as even-integers Note that the linear coefficients are defined modulo the following shifts due to the last gluing equations in (2.29)

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and A is SU(2)-type if there is a choice of c Ii which makes all (α i , α i ) even-integers. An alternative definition of SO(3)/SU(2) type without relying an ideal triangulation is Two definitions, (2.33) and (2.34), are equivalent [36]. An explanation of SU(2)/SO(3) types from the 6d N = (2, 0) theory point of view is discussed in appendix B.
T DGG [N, X A ] from symplectic gluing. The gluing equations are known to have the following symplectic structure [36,37] which play a crucial role in the DGG's construction. where b is related to the holonomy along B as in eq. (2.31). The choice of B is not unique but have the following freedom of choice Using the freedom, we will always choose B to have the properties that , we associate a Sp(2k, Z) matrix g N and integervalued 2k-vector ν as follows  Notice that (X A , P B ) are always linear combinations of Z i , Z i with integer coefficients because of the even-ness condition (2.33).
Using the gluing data summarized in (g N , ν N ), we can construct the corresponding T DGG theory. As a first step, we prepare k-copies of a free chiral theory T ∆ := (a free theory of single chiral multiplet Φ with CS level −1/2 for non-dynamical background gauge field coupled to U(1) flavor symmetry) , where Σ i is the field strength of the vector multiplet V i . The theory T step I has u(1) k flavor symmetry and {V i } are background vector-multiplets coupled to the flavor symmetries.
Using the symmetry, one can consider Sp(2k, Z) action on the theory which is a generalization of Witten's SL(2, Z) action [38] which corresponds to k = 1 case. To be more explicit, one needs to decompose a Sp(2k, Z) into products of "T-type (g t K )," "S-type (g s J )," and "GL-type(g gl U )": Here J is a diagonal matrix whose diagonal entries are either 0 or 1. Let L T ( V := (V 1 , . . . , V k )) be a Lagrangian for a theory T with U(1) k flavor symmetry. Field theoretic actions of the basic types are where V only has components such that J V = V , and they are now dynamical fields. As for the SL(2, Z) case, the final theory does not depend on the decomposition and depends only on the Sp(2k, Z) element. Now the second step of the construction is

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where g N is the symplectic matrix in (2.40) obtained from an ideal triangulation of N . The g N -transformed theory still has u(1) k flavor symmetry As a final step, we break the U(1) k to its subgroup by adding chiral operators to the superpotential [19] if at most one of G Ii , G Ii and G Ii is nonzero for each i, and 'hard' otherwise. This condition simply means that only one of edge parameters (Z i , Z i and Z i ) of i-th tetrahedron appears in C I for all i = 1 . . . k. Upon a proper choice of cyclic relabeling (2.27) of edge parameters, we can make such an internal edge C I as a linear combination of only Z i s: Then, the gauge-invariant chiral primary operator O C I in T step II is given by (2.50) As will be explained below, different cyclic labelings give different descriptions of T step II which are related by a sequence of basic dualities in (2.56). Therefore for each easy internal edge C I , there is a chiral primary operator O C I which can be written as the above form in a duality frame. The operator is charged only under U(1) C I . For each hard internal edge, on the other hand, there may only be a corresponding gauge invariant dyonic 1/4 BPS operator with non-zero spin. There is no way to write down a supersymmetric deformation using the dyonic local operators.
Hard internal edges and accidental symmetries. In the original DGG's construction [19], they proposed to use ideal triangulations with only easy internal edges. From superficial counting, we expect the resulting T DGG [N ] has flavor symmetry of rank 1 whose Cartan corresponds to the U(1) X A .
If all C I are easy, we superficially expect that

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The counting sounds compatible with the 6d construction since the knot gives a flavor symmetry (su(2)) of rank 1. But the counting could be wrong as we will see below for the case with an ideal triangulation of N = (figure-eight knot complement) with 6-tetrahedra. The correct rank is always equal or greater than the superficial counting. In our modified proposal (2.47), we can use any ideal triangulation and will argue that the resulting theory is independent of the choice of ideal triangulation regardless of existence of hard edges. One of the consequences is that rank of the flavor symmetry could be larger than 1 because the number of independent easy edges could be less than (k − 1). From the counting of linearly independent easy internal edges, we checked that T DGG theories for most of knot complements in SnapPy's census have additional symmetries. For example, we show the SU(3) symmetry for all hyperbolic twist knots in section 3.2. The additional symmetries are accidental and unexpected from 6d viewpoint. The above DGG's construction can be generalized to higher K (number of M5-branes) cases [9] and there is no such an additional symmetry when K is sufficiently large. For higher K one need to use a so-called K-decomposition which replace a single tetrahedron in an ideal triangulation into 1 6 K(K 2 − 1) copies of finer building blocks, octahedra. The construction of the 3d theory for higher K is parallel to the construction for K = 2 case reviewed above except tetrahedra in an ideal triangulation are replaced by octahedra in a K-decomposition. We assign 3 complex parameters (z, z , z ) to each pair of two vertices of an octahedron and their gluing equations in a K-decomposition also possess a symplectic structure. One difference in higher K is that there are enough number of easy internal edges (better to call internal vertices for K-decomposition case) to break all u(1) symmetries except the ones expected from 6d viewpoint. 6d viewpoint expect that the 3d theory has a flavor symmetry of rank (K − 1). A hard internal edge appears when two edges of a single tetrahedron are glued to the internal edges simultaneously. In K-decomposition, two different vertices of a single octahedron can not meet at an internal vertex possibly except when the octahedron is located nearest to one of vertices of tetrahedrons. So the number of hard internal vertices will be at most order of k (the number of tetrahedrons in a triangulation) while there are k K(K 2 −1) 6 internal vertices among which (K − 1) are linearly dependent. So the number of easy internal vertices are k K(K 2 −1) At first glance, the above construction seems to depend on the various choices other than (N, A). For the construction, we choose an ideal triangulation of N . All different ideal triangulations of a given 3-manifold are known to be related by sequence of a basic local move called 2-3 Pachner move. In the DGG's construction, the geometric move corresponds to a mirror symmetry between a 3d N = 2 SQED with two chirals (Φ A , Φ B ) of charge (+1, −1) and a free theory with 3 chirals (M, T p , T m ):

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Under the duality, gauge-invariant chiral operators are mapped as follows So the T DGG theory is invariant under the local 2-3 move and thus independent on the choice of T . For a given choice of T , we still have freedoms of choosing cyclic labeling (2.27) of edge parameters for each tetrahedron.
The invariance T DGG theory under choice is guaranteed from a duality More explicitly, the duality is (2.56) In the construction of T DGG theory, we also need to choose conjugate variables {P B , Γ I }.
But these choices only affect the background Chern-Simons coupling coupled to flavor symmetries. So modulo the background CS couplings, the theory only depends on the topological choice (N, A). To specify the background Chern-Simons coupling of the U(1) X A flavor symmetry associated to the knot, we sometimes specify the choice of boundary cycle B and denote the theory by Here 4 1 is a simplified notation, called Alexander-Briggs notation, for figure-eight knot which is depicted in fig 3. The notation simply means that the figure-eight knot is the 1st (simplest) knot with 4 crossings. The fundamental group of the knot complement is The group contains a peripheral subgroup Z × Z which can be identified as fundamental group of boundary torus Canonical choice of the basis (µ, λ) is (meridian, longitude). Upon the basis choice, the embedding i : The knot complement can be ideally triangulated by two tetrahedrons.
See figure 3 below for the gluing rule ∼. There are two internal edges in the triangulation which are linearly dependent modulo the linear equations in (2.26).
The deformation variety in this example is Each point in the variety gives a PSL(2, C) = SL(2, C)/ ±1 flat connection on the knot complement. The holonomy matrices along the basis (α, β, γ) of π 1 (S 3 \4 1 ) for the flat connections is

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Boundary (meridian, longitudinal) holonomies are (2)) type and we choose With the choices, the Sp(4, Z) matrix g m004 in (2.40) is given by Following each steps in eq. (2.42), (2.45) and (2.47), T DGG [m004, X µ ; P λ ] is given by (2.69) are background multiplets coupled to flavor symmetries, U(1) X A and say u(1) C respectively. Note that both of C 1 and C 2 are hard internal edges and we can not break the u(1) C associated to them.
Example: N = S 3 \4 1 = m004 with an ideal triangulation with 6 tetrahedra. The absence of chiral primary operators corresponding to hard edges in the above construction of T DGG [m004, X µ ; P λ ] using 2 tetrahedra were already noticed in [19]. The interpretation there was that this is due to the "bad" choice of triangulation, which contains hard internal edges, and can be cured by choosing a proper ideal triangulation which

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does not have a hard internal edge. As a "good" ideal triangulation for m004, they propose the one using six tetrahedra, ∆ R,S,X,Y,Z,W . The internal edges in the triangulation are [19] Note that there is no hard internal edges in the triangulation and 5 internal edges are linearly independent. Superficial counting suggests that the resulting theory have a flavor symmetry of rank 6 − 5 = 1, where five u(1)s are broken by superpotential operators. Our interpretation on this problem is different from [19]. We claim that the theory realized by six tetrahedra is actually completely the same as the one realized by two tetrahedra in the low energy limit. Therefore, the theory constructed by six tetrahedra has a hidden additional u(1) symmetry in the low energy limit which corresponds to the hard edge in the triangulation with two tetrahedra.
To see it, let us focus on the two tetrahedra ∆ X and ∆ Y , which are glued in such a way that the system has the internal edge C 5 = X + Y . Then, this theory is described by two chiral fields Φ X and Φ Y with the Lagrangian where we have neglected background fields. The superpotential is due to the presence of the internal edge C 5 = X + Y . Then it is clear that these fields Φ X and Φ Y can be integrated out and the theory becomes empty in the low energy limit. This means that two tetrahedra ∆ X and ∆ Y are eliminated. Mathematically this corresponds to the 0-2 move. The invariance of a topological quantity called 3d index (see appendix A) under the 0-2 move is proven in [39]. The definition of the topological quantity is based on ideal triangulation and is equivalent to the localization expression for the superconformal index of T DGG theory. Intuitively, the constraints 0 < Im[X], Im[Y ] < π and C 5 = 2πi mean that Im[X], Im[Y ] → π, and hence these tetrahedra are squashed to be flat. The same comment also applies to ∆ S , ∆ Z and C 6 = S + Z. At the level of edge variables, the process of integrating out the massive fields may be done by eliminating the variables corresponding to the massive fields. More explicitly, we define (2.72) After renaming W → Z 1 , R → Z 2 and so on, these variables C 1 and C 2 become the same as the ones in the triangulation with two tetrahedra.

Relation between the two constructions
One basic characteristic of the T DGG [N, X A ] theory is that [19] M parameter (T DGG [N,

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Recall the definition of each term of this equation. For simplicity, we only discuss the case where our 3-manifold N only has a torus boundary and hence of the form N = M \K.
The deformation variety D[N, T ] defined in (2.29) is a set of flat PSL(2, C) connections on N which can be obtained from an ideal triangulation T . The χ 0 (N ) is a subset of the algebraic variety defined in (2.21) (or (2.32)) which can be seen for any non-exotic ideal triangulation. The difference between the two sets are mild, higher codimension, and may be ignorable in our discussion as we discuss later. Finally the left-hand side is given as follows. A DGG theory in general consists of chiral fields, dynamical vector fields, and background vector fields. Let Σ i (i = 1, . . . , N V ) be twisted chiral fields constructed from dynamical vector multiplets whose lowest real component is the real scalar of the vector multiplet and the imaginary part is the gauge field in the S 1 direction. The N V is the number of dynamical vector multiplets, i.e., the gauge group is u(1) N V , and it depends on the details of g N in (2.45) and its decomposition into basic types. Also, let X A be the twisted chiral field of the background u(1) field whose real part corresponds to the real mass parameter m and the imaginary part corresponds to the background flavor gauge field around S 1 . Then, by integrating out the matter chiral fields of the theory on R 2 × S 1 , we get a twisted superpotential of Σ i and X A (in some appropriate normalization), ; X A ) = 1 are just the condition for the vacua on S 1 ×R 2 . The P B has a definite value (modulo 2πi) at each of the vacua for a given parameter X A . In other words, the equation gives a polynomial equation of (x A , p B ) := (e X A , e P B ), and solutions of that equation in terms of p B for a given x A correspond to the vacua of the theory with mass parameter x A .
The relation to localization computation is as follows. The partition function of the T DGG theory on a curved background called squashed 3-sphere (S 3 b ) can be written in following form [19,30] In a degenerate limit when b → 0, which corresponds to the limit where S 3 b become R 2 × S 1 , the leading asymptotic behavior of the integrand is determined by the twisted superpotential are equivalent to the gluing equations in (2.29) with an additional relation e a = (−1) (2) types of boundary cycle A, and the integers (α i , α i , ) are given in (2.31) [14]. Now, we have This means that by taking the parameter a to be a constant fixed value 2m, we get the vacua of the theory with the mass parameter m.
The above equations may have solutions like X A = 0. Field theoretically, when the mass parameter is zero, there could appear some continuous moduli space of vacua spanned by matter chiral fields. Those massless flat directions are subtle, especially when they are generated by monopole operators because in that case those directions appear by very strong coupling effects which may not be captured by the one-loop computation of the twisted superpotential W. See section 5.2 of [19] for an example. We may expect that those subtle flat directions might be the reason of the mismatch between D[N, T ] and χ 0 (N ) This problem may be avoided if we only consider generic mass parameters. We assume that this is the case.
Comparison of the two constructions. Now let us compare the constructions in section 2.1 and section 2.2. Comparing the moduli space of T 6d in (2.16), and T DGG in (2.79) we see that This is because that an ideal triangulation captures only a subset of irreducible flat connections on N as emphasized in [20]. So we see that T DGG can not be identical to T 6d but can only capture a subsector of T 6d . This point has already been seen from the effective field theory point of view in section 2.1. In general, there is no reason to expect that there exists a genuine 3d theory which describes all components of moduli space of vacua of T 6d . So T DGG can, at best, describe the low energy limit of some point of the moduli space of vacua of T 6d . Then a possibility is that T DGG might be identified with T 6d irred in (2.7). Both of them are genuine 3d theories and they are associated to the hyperbolic connection A hyp which can be realized in ideal triangulation. However, it turns out that these two theories are still different as we now explain.
One crucial difference between the two theories is that T DGG generically has U(1) X A flavor symmetry associated to the knot while T 6d and hence T 6d irred have su(2) A . Furthermore, the SL(2, Z) action on the canonical variables (X A , P B ) is realized in field theory as the SL(2, Z) action of Witten [38] using u(1) group on T DGG , while the SL(2, Z) action on the boundary cycle (A, B) is realized in field theory as the SL(2, Z) of Gaiotto-Witten [21] using the su(2) symmetry and T [SU (2)] theory on T 6d .

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The su(2) SL(2, Z) on T 6d is defined as follows. The transformed theory ϕ · can be obtained by (2.82) T [SU (2), ϕ] is a 3d N = 4 SCFT which describe the 3d theory living on a duality domain wall in 4d su(2) N = 4 SYM associated to ϕ ∈ SL(2, Z). The theory has su(2) 1 × su (2)  where µ is the holomorphic moment map operator associated to the su which is a chiral operator in the adjoint representation of su(2) A , 7 and µ is the holomorphic moment map operator of su (2) Motivated by the similarities and differences between T 6d irred and T DGG , we propose the following relation between them: Here, µ 3 is the Cartan component of the moment map operator µ = {µ 1 , µ 2 , µ 3 } in the adjoint representation of su(2) A . The δW = µ 3 means the superpotential deformation by the chiral operator µ 3 . This deformation breaks su(2) to U(1). Thus we need two steps from T 6d to T DGG . First we put the theory on a specific vacuum, P SCFT , of the T 6d theory on R 3 as explained in section 2.1. As a second step, we deform the intermediate theory, T 6d irred , by adding the Cartan component (µ 3 ) of the su(2) moment map operator µ associated to the knot to the superpotential. We give more evidence for this proposal below.
In the case of 3d N = 4 supersymmetry, the presence of the holomorphic moment map operator associated to a symmetry is guaranteed. However, when there are only N = 2, it is not guaranteed. What we call the holomorphic moment map operator is a kind of remnant of higher supersymmetry of the codimension-2 defect of the 6d N = (2, 0) theory. The µ may be empty depending on the theory. However, generically (but not always), the T 6d irred 7 In general, the existence of this operator is guaranteed only for theories with 8 supercharges. However, this operator often exists due to the remnant of 8 supercharges preserved by codimension-2 defects of the 6d N = (2, 0) theory. Indeed, the operator µ is absent only for some special cases. These points will be important and discussed in more detail below.

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theory contains the moment map as chiral operators. This can be seen as follows. Suppose we are given a theory T with su(2) symmetry, and then let us perform a transformation acting on this su(2). This is done by coupling T and T [SU (2)] to the su(2) gauge field with Chern-Simons level k(+ the original value before T k ). By taking k large enough, the su(2) gauge field is weakly coupled and hence the two theories T and T [SU (2)] almost decouple from each other. The T [SU (2)] theory contains the holomorphic moment map operator µ associated to the new (ungauged) su(2) global symmetry. Therefore, we conclude that the total theory has µ for generic k. Moreover, this argument shows that the scaling dimension of µ is close to 1, at least if k is large enough, because the scaling dimension of µ in T [SU(2)] alone is 1 by N = 4 supersymmetry. Therefore, the deformation by µ 3 is a relevant deformation and it triggers RG flows. The case that µ is absent happens only in rather exceptional situations, and this will be very important in section 3 In particular, the deformation breaks the su (2)  Notice that the S 2 exchanges the SU(2)/SO(3) types of the cycles. This is natural, because in 4d N = 4 theory, the gauge groups SU(2) and SO (3) are exchanged under the S-duality. This exchange can also be shown in purely 3d language and is explained in appendix B.
On the other hand, the T -transformation T 1 ∈ SL(2, Z) 1 and T 2 ∈ SL(2, Z) 2 act as and hence they are related as This also has a natural field theory interpretation. Let A be an su(2) gauge field. The global structure may be either SU(2) or SO(3). Its Chern-Simons 3-form is defined as where the trace is taken in the doublet representation of su (2). When the symmetry is SU (2)

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is well defined. This means that when the A-cycle is SO(3) type, only the (T 2 ) 2 is well defined. Therefore, the actual transformation group at the quantum level is a subgroup Γ(2) ⊂ SL(2, Z) 2 generated by S 1 and (T 2 ) 2 . These generators S 1 and (T 2 ) 2 also preserves the condition that one of the cycles A or B is SU(2) type. Under T 2 , this condition may not be preserved. From the above discussion of Chern-Simons levels, the field theoretical realization of (2.92) and (2.93) under the explicit breaking su(2) A → U(1) X A by W = µ 3 is clear. Now let us also check the correspondence of the S-transformation (2.90) at the field theory level. Let T be a theory with su(2) symmetry. Then S-transformed theory is given by where the center su(2) is a gauge group which is coupled to the su(2) symmetry of T and the su(2) H symmetry of T [SU (2)]. In the language of 3d N = 2 supersymmetry, the T [SU (2)] is given by a U(1) vector multiplet V , a neutral chiral field φ, and two pairs of The su(2) H acts on the index i of (E i ,Ẽ i ) i=1,2 . Now, this T [SU (2)] has a su(2) C symmetry at the quantum level, and this symmetry is the new global su(2) symmetry after the Stransformation. See appendix B for more details. The Cartan component µ 3 of the moment map operator of this symmetry su(2) C is given by µ 3 = φ. Therefore, after the deformation by µ 3 , the superpotential becomes Thus, an F-term condition isẼ i E i = 1. ThenẼ i and E i get nonzero expectation values as These expectation values break the gauge symmetry [su(2) × u(1)] gauge down to u(1). Namely, (E i ,Ẽ i ) are charged under u(2) = [su(2) × u(1)] gauge , and only the subgroup u(1) ⊂ u(2) which acts on (E 2 ,Ẽ 2 ) is preserved by the expectation values. Therefore, by the Higgs mechanism, the theory (2.98) becomes

Symmetry enhancement
Using the proposed 6d interpretation of T DGG [N, X A ] in (2.84), we will determine the symmetry enhancement pattern of the U(1) X A symmetry associated to the knot based on a topological type of the boundary cycle A. See the table 1 for the summary whose details will be explained in section 3.1. In section 3.2, we will find infinitely many examples of pair (N, A) and (N , A ) whose corresponding DGG theories are identical and both of A and A are non-closable but one of them (say A) is SO(3) type while the other (A ) is SU(2) type. In that case, combining the table 1 with a group theoretical argument, we can argue 9 that the DGG theory has enhanced SU(3) symmetry. Using the argument, we prove that T DGG [M = S 3 , K] theories for all hyperbolic twist knots K have SU(3)-symmetry. We checked the enhancement for several twist knots which gives non-trivial empirical evidence for the table 1.

SO(3)/SU(2) enhancement
From the relation between T 6d irred and T DGG in (2.84), we understand the symmetry breaking mechanism of su(2) A to U(1) X A . For the symmetry breaking to happen, we need the su(2) moment map operator µ in T 6d irred theory. Otherwise, the su(2) A is not broken by the mechanism and the resulting T DGG is expected to have an su(2) symmetry. Therefore, we need to know when µ is absent.
This question can be answered by an inspection of the 5d picture discussed in section 2.1. In the description there, the moment map operator comes from the holomorphic su(2) C moment map of T [SU (2)] theory which is put along a knot K. After S 1 compactification, the T [SU (2)] has 2d N = (4, 4) supersymmetry. In the Language of N = (2, 2) supersymmetry, there is a twisted chiral operator µ and a chiral operator µ, 10 both of which are associated to the Coulomb branch su(2) C symmetry of T [SU (2)]. Now suppose that the Higgs branch operator ν gets a nonzero expectation value. Then, the nonzero VEV in Higgs branch makes the Coulomb branch fields massive. Therefore, in the low energy limit, the operators µ and µ become empty; In T [SU (2)] theory on ν = 0, µ is absent at low-energy  9 We thank Y. Tachikawa for this argument. 10 Let φ be the neutral scalar chiral field and let Σ be twisted chiral field which comes from u(1) vector multiplets in 2d. Then the Cartan components of µ and µ are given by Σ and φ. However, in the discussion of section 2.1, we have to exchange the role of chiral and twisted chiral by regarding φ as twisted chiral and Σ as chiral, because of the subtle mirror symmetry in the relation between 5d SYM and T 6d in (2.8).

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with trivial ρ hol (A). Let us define We remark that in the massless case m = 0, the eigenvalues of ρ hol (A) are trivial (±1), but ρ hol (A) may contain off-diagonal components and hence the above condition is nontrivial. Then, has su(2) symmetry at low energy .

(3.3)
Strictly speaking, the step from R 2 × S 1 to R 3 is nontrivial, but we assume that this step holds.
One necessary condition for A to be 'non-closable' is that the Dehn filled manifold N A is non-hyperbolic.
This is because that the flat connection corresponding to the hyperbolic structure on N A is always contained in χ 0 [N ] with trivial ρ hol (A). According to Thurston's hyperbolic Dehn surgery theorem, for given hyperbolic N , there are only finite number of primitive boundary cycles A which give non-hyperbolic N A . So, we can conclude that |{Set of primitive 'non-closable' boundary cycles A ∈ H 1 (∂N, Z)}| < ∞ . Combining with table 1, it implies that the u(1) X A symmetry of T DGG [N, X A ] is not enhanced to su(2) except for only finite many As. The Thurston's theorem is consistent with our field theoretical consideration in the previous section that the moment map operator µ generically (although not always) exists; see the discussion in the paragraph containing (2.85). One sufficient condition for A to be 'non-closable' is that the Dehn filled manifold N A is Lens-space Lens space L(p, q) is defined as The reason is as follows. If

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A ∈ H 1 (∂N, Z) Closability pµ + λ (|p| ≥ 5) closable (⇐ N A is hyperbolic) non-trivial power series in x pµ + λ (|p| = 4) closable divergent pµ + λ (|p| < 4) closable 1 µ non-closable 0 Thus the cycle A can not be closable. When N A is neither hyperbolic nor a Lens space, no simple criterion to determine the closability has been found. An alternative definition of closable/non-closable cycle, which seems to be equivalent to the above definition, is using 3d index which is introduced in [40] as a topological invariant of 3-manifolds with torus boundaries and is generalized in appendix A to cover closed 3-manifolds.

A primitive boundary cycle
Here I N A (x) is the 3d index on a closed 3-manifold N A . That a primitive boundary cycle A ∈ H 1 (∂N, Z) is 'non-closable' means that we can not 'close' (or eliminate) the co-dimension two defect along a K on M in a supersymmetric way after sitting on the vacuum P SCFT . As we will study in the next section, there is an operation in SCFT side of 3d/3d correspondence which corresponds to the operation of 'closing the knot'. If A is non-closable cycle, we expect that the resulting 3d theory T 6d irred [M ] after taking the closing knot operation on T 6d irred [M, K] will be a theory with supersymmetry broken. 11 The index I M =N A (x) computes the superconformal index of the theory T 6d irred [M ] and expected to be zero when A is non-closable and thus supersymmetry is broken. This is a heuristic argument supporting the equivalence between the two definitions and no rigorous mathematical proof is known. We checked the equivalence for various examples and the equivalence seems to hold possibly except for exotic cases. As an example, see table 2 for the case when N = S 3 \4 1 = m004.
We can further determine the global structure of enhanced symmetry, whether SO(3) or SU(2), from the SO(3)/SU(2) type of A. When A is non-closable and of SU(2) type, the compact U(1) . This is manifest from the relations given in eq. (2.31) and (2.41) between the variable X A , associated to the U(1) X A , and the PSL(2, C) holonomy variables a, associated to the su(2) A . Namely, 2 su(2) has properly quantized U(1) X A charges 11 Here notice the difference between T  In general, it is not easy to determine the SO(3)/SU(2) type of a given primitive boundary cycle A in H 1 (∂N, Z). When N is a knot complement in a homological sphere, there is a canonical choice of the basis of H 1 (∂N, Z), meridian (µ) and longitude (λ). Meridian cycle is defined to be the circle around the knot and longitude cycle is determined by the condition that λ ∈ Ker i * : H 1 (∂N, Z) → H 1 (N, Z) . Then, λ is of SU(2)-type by definition while µ is always of SO(3)-type. More generally, when N is a knot complement in a Z 2 -homological sphere (p and q are coprime) pµ + qλ is of SU(2)-type , for even p , SO(3)-type , for odd p . (3.9) Here µ ∈ H 1 (∂N, Z) is the meridian cycle and λ is a boundary cycle in Ker i * : . The choice of λ is not unique but can be shifted by 2µ.
Example: N = S 3 \4 1 = m004 and A = µ. In the case, the merdian cycle µ is non-closable and of SO(3)-type and we expect SO(3) symmetry enhancement of u(1) Xµ in T DGG [m004, X µ ] whose Lagrangian is give in eq. (2.69). In the next section, we argue that u(1) Xµ is actually enhanced to SU(3) which contain the SO(3) as a subgroup.
Example: N = (S 3 \5 2 1 ) 3µ 1 −2λ 1 = m007 and A = µ 2 . As another example, we consider a knot complement called m007 in SnapPy's census. The knot complement can be obtained by performing Dehn filling on one component of Whitehead link complement. Whitehead link is denoted by 5 2 1 , the 1st link with 2 components and 5 crossings, as shown in figure 4. The orientation of the link complement (S 3 \5 2 1 ) is chosen as the one induced from an ideal triangulation in (A.25). We always choose a particular orientation of each ideal tetrahedron in an ideal triangulation which is reflected in the choice of CS level sign of T ∆ in (2.42). Then, the Dehn filled manifolds have natural orientation induced from the link complement. The overline in the equation N = (S 3 \5 2 1 ) 3µ 1 −2λ 1 means that JHEP07(2018)145 N = m007 has opposite orientation to the one induced from S 3 \5 2 1 when the orientation of N is chosen to be the one induced from an ideal triangulation in (3.11). The Dehn filling also gives an induced basis of H 1 (∂N, Z) = µ 2 , λ 2 on N from the basis choice of H 1 (∂(S 3 \5 2 1 ), Z) = µ 1 , µ 2 , λ 1 , λ 2 . From the topological fact that we see that A is non-closable according to (3.6). The N is a knot complement in a Z 2homological sphere, N A = L(3, −2), and according to (3.9) A is of SO(3) type. So from table 1, we expect the u(1) Xµ 2 in T DGG [m007, X µ 2 ] is enhanced to SO(3). Now, let us check the enhancement from explicit construction of the DGG theory. According to SnapPy, the knot complement can be triangulated by 3 ideal tetrahedra and the corresponding gluing data are (we choose B = 4µ 2 − λ 2 ) Since (A, B) are of (SO(3), SU (2))-type, we choose The matrix can be decomposed into g m007 = g s J m007 g t K m007 g gl U m007 with (2.43) (3.14) Following each steps in eq. (2.42), (2.45) and (2.47), the Lagrangian for T DGG [m007, X µ 2 ; P 4µ 2 −λ 2 ] is given by

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In the Lagrangian, V 1 and V 2 are dynamical u(1) vector multiplets. The superpotential term comes from an easy internal edge C 1 , The theory has u(1) Xµ and u(1) C whose background vector multiplets are V X and V C respectively. Applying the mirror symmetry in eq. (2.52) and (2.53) with the following replacement (3. 16) we have (3.17) In the dual picture, the SO(3) symmetry is manifest after the redefinition of chiral fields as The u(1) X is in the Cartan of this SO(3).

SU(3) enhancement
From the point of view of 6d N = (2, 0) theories, we only expect that the symmetry associated to codimension-2 defects (which are knots in 3-manifolds) are su(2). However, we will see that there are many theories which have larger symmetry enhancement. We consider a pair of (N   Figure 5. A rational surgery calculus [41] shows that (S 3 \5 2 1 ) µ2+kλ2 is a twist knot K k . For example, K k=1 = 4 1 , K k=−2 = 5 2 and K k=2 = 6 1 .
As shown in figure 5, the N above are nothing but twist knots which will be denoted as K k . Let us check that the pair satisfy the 3 conditions in (3.19). First, note that both of N and N can be considered as a knot complement in Z 2 -homological spheres, L(1, k) = (N ) µ 2 and L(4k − 1, −k) = (N ) µ 2 respectively. Applying (3.9) with µ = µ 2 and λ = λ 2 , we can conclude that A/A is of SO(3)/SU(2) type. Now let us check the 2nd condition in (3.19). Combining the we have following identity

(3.24)
Applying the Dehn filling formula in eq. (A.21) to the above equality, (3.25) we confirm 2) in (3.19). In the above, we use the transformation rule of 3d index under the orientation reversal in (A.6). Finally, from the following topological facts [42] N A = (S 3 \5 2 1 ) µ 1 +kλ 1 ,µ 2 = L(1, k) , For k = 1. In the case, m003 is a knot complement called sister of figure-eight knot complement. Both 3-manifolds have the same hyperbolic volume and are the smallest hyperbolic 3-manifolds with one cusp torus boundary. From ideal triangulations of m003 and m004 given below, their orientation are fixed. The equality in the above means not only that the two manifolds are homeomorphism but also that they have the same orientation, i.e. the orientation of

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m004 is same as the orientation induced from a Dehn filling on S 3 \5 2 1 , whose orientation is induced from an ideal triangulation in (A.25).
According to SnapPy, both can be ideally triangulated by two tetrahedra and have common internal edge variables given in (2.62) while boundary variables are different by a factor 2 or 1/2 For DGG's construction, we choose Thus, both DGG theories are identical and described by the Lagrangian in (2.69) up to background CS level for U(1) X which is irrelevant in symmetry enhancement. We reproduce the Lagrangian here with the modified background CS level; The theory has manifest u(1) top × su(2) manifest where u(1) top is the topological monopole charge of the u(1) gauge gauge symmetry, and su(2) manifest acts on the two chiral fields. This u(1) top × su(2) manifest will be enhanced to SU (3).
The u(1) C flavor symmetry associated to the background field V C corresponds to the topological symmetry u(1) top and will be embedded to SU(3) as This is because the "off-diagonal components" of SU(3) (which are not in u(1) C × su(2) manifest ) are provided by monopole operators with monopole charge ±1 = ±(1/3 − (−2/3)).
On the other hand, the V X is coupled to the system as follows. Let

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be the Cartan generator of the manifest su(2) manifest . The V X is coupled to the chiral fields via this generator T 3 . Also, notice that V X is coupled to the monopole current Σ with coefficients 3/2. Therefore the u(1) X is embedded in SU (3) as This u(1) X must be enhanced to su(2) X because the A-cycle is non-closable. Notice that this su(2) X is different from the manifest su(2) manifest symmetry. Therefore, if u(1) X is enhanced to su(2) X , then the u(1) top × su(2) manifest must be enhanced to SU(3). This agrees with our general discussion that this theory has enhanced SU(3) symmetry.
The superconformal index of theory is (3.36) Here u 1 and u 2 fugacity variable for u(1) X and u(1) C symmetry respectively. The index depends on the choice of R-charge mixing between u(1) R and u(1) C . In the above expression, we in particularly choose, 12 Here V ± denote a BPS monopole operator of charge ±1. Then the index show the SU(3) structure: Here χ m,n (u 1 , u 2 ) is the character of SU(3)-representation with Dynkin labels (m, n). The correct IR R-charge mixing should be determined by F-maximization, but the non-trivial appearance of the SU(3) in a particular choice strongly suggests that the choice gives the correct R-charge assignment. The first non-trivial terms comes form operators listed in the table below. In the table, φ a and (ψ ± ) a denote the scalar and fermionic fields in chiral field Φ a respectively with a = 1, 2. V ± (. . .) denote a gauge invariant BPS monopole operator of charge ±1 dressed by matter fields (. . .). All these operators have quantum numbers (R, j 3 , ∆) = (1, 1 2 , 3 2 ) and form descents of conserved current multiplet for SU(3) flavor symmetry. In 3d N = 2 SCFT, a conserved current multiplet of flavor group F consists of following operators in the adjoint representation of F :  Table 3. Descents of SU(3) flavor current multiplet.
For k = −2. In the case, (3.40) Both manifolds have the same hyperbolic volume vol(m015) = vol(m017) = 2.82812 . . .. According to SnapPy, both can be ideally triangulated by three tetrahedra and have common internal edge variables while boundary variables are different by a factor 2 or 1/2 We choose Then, the Sp(6, Z)+(affine-shifts) are

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The matrix g m015/m017 can be decomposed into g m015/m017 = g s J m015 g t K m015 g gl U m015 (2.43) Using the decomposition, we have Here V is a dynamical u(1) vector multiplet. Since C 1 and C 2 are hard internal edges and we can not add O C 1 and O C 2 to superpotential. So, the DGG theory is = A u(1) −1/2 vector multiplet coupled to 3 chirals of charge +1 . The theory has manifest SU(3) flavor symmetry rotating 3 chrials as expected.

Dehn filling in 3d/3d correspondence
In this section, we generalize the DGG's construction to obtain T 6d irred [M ] for closed 3manifolds M by incorporating Dehn filling operation. Refer to [43][44][45][46] for previous discussions on Dehn filling operation in the context of 3d/3d correspondence and the construction of 3d theory, which we will denote T 6d  The fact that the closing of codimension-2 defects (i.e., knots) corresponds to giving the nilpotent vev to µ is standard in 4d class S theories. See e.g., [22] and references therein. This is also in accord with our terminology 'closable/non-closable', because non-closable cycles have empty µ and hence it is not possible to do the above operation while preserving supersymmetry. See  If the A is non-closable and q = 1, the above relation can be simplified as follows.
where su (2)  To specify this contribution, we have to specify not only the A-cycle, but also the B-cycle. This is the reason why we are writing B explicitly in the notation T 6d irred [N, A; B]. Now, the operator µ comes from the moment map operator of T [SU (2)] associated to the su(2) C symmetry which is not gauged. If we give a nilpotent vev to this operator µ, the T [SU (2)] becomes massive and flows to an empty theory in the low energy limit up to the Goldstone multiplets associated to the symmetry breaking of su(2) C by the vev [21]. Neglecting those Goldstone multiplets, the T [SU (2)] disappears and hence we get A u(1) −1/2 vector coupled to 3 chrials of charge +1 SO(3) p−6 .

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Here the notation /G k means that we couple a vector multiplet of group G with the Chern-Simons level k. The theories in the numerator has SU(3) symmetry at IR as argued in section 3.2 and we are gauging its SO(3)/SU(2) subgroup. Since the u(1) X is embedded to the SU(2) (resp. SO(3)) in a way that 2 su(2) = (±1) u(1) X (resp. 2 su(2) = (± 1 2 ) u(1) X ), the CS level +1 for u(1) X in (3.31) corresponds to CS level 1/2 (resp. 2) for the su (2). Similarly the CS level −3 for u(1) X in (3.46) corresponds to CS level −6 for the su(2). A parity operation filps the signs of CS levels of the T 6d irred theories. The parity operation corresponds to orientation reversal on the internal 3-manifold. It is compatible with following topological facts The u(1) for the 3rd case comes from the topological symmetry of u(1) −1/2 gauge symmetry of the theory in the numerator. The above is correct when |p| is large enough where the semiclassical analysis is reliable. When |p| is small, the theories could have accidental symmetries. Actually from following topological fact (see figure. 6), we can conclude that T 6d irred [(S 3 \4 1 ) (p+2)µ 2 +λ 2 ] has accidental u(1) symmetry for p = 3 and p = −7. We will come back to this point in sec 5.

Small hyperbolic manifolds
Let us discuss the case of closed 3-manifolds M =Weeks, a oriented hyperbolic closed 3manifold with smallest hyperbolic volume. This was already discussed in [23] and here we supply a little bit more details. The Weeks manifold is obtained by performing a Dehn filling operation on m003, Corresponding 3d gauge theory is the theory in the second line of eq. (4.4) with p = −3.
The theory in the numerator has SU(2) X symmetry which is a subgroup of the SU(3). The SU(2) X symmetry is different from the manifest SU(2) manifest rotating two chirals. But using the Weyl symmetry of the SU(3), the symmetry SU(2) X and SU(2) manifest can be exchanged with each other. Therefore, we can take SU(2) X to be the manifest SU(2) manifest . We will just denote it as SU(2) in the following. Then by (4.4), AF duality. Now, we can further simplify this theory to a much simpler theory [23].
There is a duality found by Aharony and Fleischer (AF) [47] (Two chiral fields coupled to SU(2) −5/2 ) = (One chiral field gauged by U(1) +3/2 ). (4.10) To apply this duality, we need to know the relation between the flavor U(1) symmetries of both sides of this equation and their background Chern-Simons levels.
Here we supply the details promised in [23]. In the AF duality, the U(1) charge acting on two chiral fields with charge 1 on the left hand side corresponds to the topological charge of the U(1) +3/2 gauge field multiplied by 2. The reason is that the −1 ∈ U(1) acting on the two chiral fields can be compensated by the −1 ∈ SU(2) −5/2 gauge transformation, and hence all gauge invariant operators have even charge on the left hand side. So the relation is where U(1) bkg is the global symmetry with background field, n is the background Chern-Simons level, and ×2 means that the U(1) bkg n is coupled to the topological current of U(1) 3/2 multiplied by two.
We want to determine the value of n. This can be done as follows. Let σ bkg be the real scalar for the background U(1) bkg vector multiplet, or in other words, the real mass associated to this symmetry. We choose the sign of it such that after integrating out the two chiral fields on the left hand side, the left hand side flows to where ⊕ means that the two factors SU(2) 3 and U(1) bkg 1 are completely decoupled. Here we need to remark the important point. The SU(2) −3 is the 3d N = 2 gauge theory at the level −3. This theory contains the gaugino, and by integrating out the gaugino, we get a pure topological Chern-Simons theory as SU(2) −3 = SU(2) topo CS −1 (4.13) Namely, the gaugino reduces the level by 2 = h ∨ su (2) . Therefore, the low energy limit is The effect of σ bkg on the right-hand-side of (4.11) is to give the dynamical U(1) an FI parameter. We want the dynamical gauge group U(1) to be not Higgsed so that we can match it with the SU(2) topo CS −1 later. Then, the D-term condition implies that the dynamical real scalar σ gets a vev proportional to σ bkg because the Lagrangian contains 3/2Dσ + 2 · 2Dσ bkg and we need to impose stationary condition for D. The vev of σ gives the chiral field a mass term. The sign of the mass is anticipated by the fact that it must make the low energy CS level of the dynamical field as U(1) −2 . This is because the only JHEP07(2018)145 consistent way for the duality to work in low energy is to use the duality of topological CS theory given by where we have used the fact that the gaugino plays no role in U(1) and hence U(1) −2 = U(1) topo CS −2 . First equality is well-known (see, e.g., [48] for the corresponding statement in Wess-Zumino-Witten models which are related to topological Chern-Simons theories [49].). The second equality U(1) topo CS [50][51][52] and we have neglected U(1) ±1 because these theories have only one state in the Hilbert space on any space (and they are called invertible field theory), and our argument is not careful enough to detect those invertible field theories.
After integrating out the chiral field, the right-hand-side of (4.11) is given by the Lagrangian where in the second term, the factor of 2 have taken into account the fact that U(1) bkg is coupled to the topological current of U(1) by charge 2. We shift the dynamical gauge field as V → V + V bkg to get This means that the low energy theory is given by Therefore by comparing the low energy limit of the left and right hand side of (4.11), we get Weeks theory. Now let us gauge U(1) bkg (but we use the same name for simplicity).
The left hand side of (4.11) after gauging U(1) bkg is precisely the theory T 6d irred [Weeks]. Let us see the right hand side. The Chern-Simons action of the right-hand-side of (4.11) after putting n = 1 is given by 1 4π (4.20) where in the second term, the factor of 2 have taken into account the fact that U(1) bkg is coupled to the topological current of U(1) 3/2 by charge 2. If we integrate out V bkg , or in other words, by making the shift V bkg → V bkg + 2V and neglecting the decoupled U(1) 1 theory, we get 1 4π We conclude that the theory T 6d irred [Weeks] is given by T 6d irred [Weeks] = (One chiral field coupled to U(1) −5/2 ). (4.22) This is one of the small theories discussed in [23]. Example: as depicted in figure 6, topologically (S 3 \4 1 ) −5µ+λ = (S 3 \5 2 ) 5µ+λ . Combining the topological fact with eq. (4.4), we have a 3d duality between following twos

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where γ := (m 1 , . . . , m k , e 1 , . . . , e k ) T and ν N , The tetrahedron index I ∆ (m, e; x) in charge basis is given by [40] e∈Z Index as Chern-Simons ptn. The index can be thought as a SL(2, C) CS ptn on N with quantized level k = 0. The complex CS theory has two levels, k and σ , where k ∈ Z and σ ∈ R . The algebra acts on H k=0 (T 2 ) as follows [40] Basis of H k=0 (T 2 ) = {|m, e } m,e∈Z , Quantum Dehn filling on index, I pA+qB (x). Mimicking the k = 1 case in [32,55] , we give a Dehn filling operation on the index (k = 0). The SL(2, C) CS wave function on a solid torus D 2 × S 1 = S 3 \(unknot) is annihilated by following operators (a pair of quantum A-polynomial for unknot) Here the boundary cycle B corresponds to the shrinkable cycle in D 2 × S 1 , (A.14) In the classical limit x = e → 1, the operator equation become (e b/2 + 1) 2 = 0 which reflects the fact that flat-connections on D 2 × S 1 have trivial holonomy (e b/2 = −1) along the boundary cycle B. A solution for the difference equations is Then, the CS ptn for k = 0 on N pA+qB is given by

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Note that the final expression is dependent on the choice of (r, s) since the expression is invariant under (r, s) → (r, s) + Z(p, q) (A. 20) which is expected since N pA+qB depends only on (p, q). Combining with eq. (A.12), we finally have For SU(2) type A,
Therefore, the properties of this theory are important in both sides of 3d/3d correspondence.

B.1 Brief review of T [SU(2)]
Let us first review the T [SU(2)] theory. It is a 3d N = 4 supersymmetric field theory obtained by a U(1) gauge multiplet with two hypermultiplets of charge ±1. In terms of 3d N = 2 supersymmetry, there are one U(1) vector multiplet V , one neutral chiral multiplet φ, and two pairs of chiral multiplets (E i ,Ẽ i ) (i = 1, 2) where E i has U(1) charge +1 and E i has charge −1. The superpotential is given by At the level of Lie algebra, this theory has global symmetry su(2) H × su(2) C . The su(2) H acts on the index i of (E i ,Ẽ i ) (i = 1, 2). On the other hand, the su(2) C arises at the quantum level. The u(1) C ⊂ su(2) C comes from the topological symmetry of the gauge U(1) symmetry whose current is j = 1 2π f , where f = da is the field strength of the gauge field a. This topological symmetry is enhanced to su(2) C at the quantum level.
Because of the N = 4 supersymmetry, the N = 4 conserved current supermultiplets contain N = 2 chiral operators which are in the adjoint representation of the symmetry. They are called (holomorphic) moment map operators because they are associated to the moment maps of hyperkahler moduli spaces of the Higgs and Coulomb branch, and their scaling dimensions are protected to be 2. In this paper we abuse the terminology and call these operators as moment map operators even if there is only N = 2 supersymmetry.
For the su(2) H , the holomorphic moment map operator is given by For the su(2) C , the holomorphic moment map operator is given by φ and monopole operators v ± , The φ corresponds to the Cartan of su(2) C , while v ± are off-diagonal components of the su(2) C . These v ± have charge ±1 under the topological u(1) C symmetry with the current j = 1 2π f . There is a mirror symmetry which exchanges the Higgs branch and Coulomb branch of the theory. Under the mirror symmetry, the symmetries and operators are exchanged as mirror : (su(2) H , µ H ) ←→ (su(2) C , µ C ) . (

B.4)
This mirror symmetry also guarantees that the topological symmetry u(1) C is enhanced to su(2) C .

B.2 The global structure of the symmetries and 't Hooft anomaly
Now we study the global structure of the symmetries su(2) H and su(2) C . We claim that both of them are SO(3) type in the sense that all gauge invariant operators (in the absence of background fields) are in representations of SO(3) (i.e, integer spin representations of su (2)). We denote them as SO(3) H and SO(3) C , respectively. However, we will also show that there is a mixed 't Hooft anomaly between these groups SO(3) H and SO(3) C which forbid gauging both of them as SO (3) groups. In other words, this anomaly implies that if we gauge one of them as SO (3) gauge group, then the other symmetry becomes SU (2). It is easy to see that su(2) H is of SO(3) type. The fields (E i ,Ẽ i ) transform under su(2) H . Now, the center −1 ∈ SU(2) H multiplies the (E i ,Ẽ i ) by (−1), but this can be cancelled by a U(1) gauge transformation. Therefore, the action of the center −1 ∈ SU(2) H to all gauge invariant operators is trivial. This shows that the symmetry which acts faithfully to gauge invariant operators is SO(3) H .
By mirror symmetry, it is obvious that the symmetry su(2) C must also be of SO(3) type. More direct way to see this is to notice that all monopole operators have integer charges under u(1) C ⊂ su (2)  So, the gauge U(1) has magnetic flux +1/2 on S 2 . The SO(3) H also has nontrivial magnetic flux diag(+1/2, −1/2) which is measured by a nontrivial value of the second Stiefel-Whitney class w 2 ∈ H 2 (X, Z 2 ) of the SO(3) H bundle, where X is the spacetime on which the theory is placed. Roughly speaking, the Stiefel-Whitney class w 2 is defined such that half of it, The above argument implies the following. Suppose we gauge the group SO(3) H as an SO(3) gauge group. Then it is possible to consider a monopole operator of SO(3) H which has nontrivial Stiefel-Whitney class. However, for this monopole operator to make sense, we also have to turn on a half-integral magnetic flux of the gauge U(1). Then, these monopole operators have half-integral charges under the topological u(1) C and hence they are in half-integer spin representations of the Coulomb branch symmetry su(2) C . Therefore, by gauging SO(3) H , the symmetry su(2) C becomes SU(2) C . This fact forbids JHEP07(2018)145 to gauge both of the su(2) H and su(2) C symmetries as SO(3) type symmetries, and this means there is an anomaly.
More formally, the anomaly is shown as follows [56]. (See also [57][58][59][60][61][62][63][64] where nontrivial mixture of the center of gauge and flavor symmetries lead to 't Hooft anomalies.) Let f = da be the field strength of the U(1) gauge field, F = dA + A 2 be the field strength of the mixed gauge-flavor symmetry U(2) = [U(1) × SU(2) H ]/Z 2 , and B be the gauge field of u(1) C . Then, the coupling of the background B and F in the Lagrangian is given by where X is a 3-manifold in which our T [SU(2)] theory lives. To make the definition manifestly gauge invariant, we consider a 4-manifold Y whose boundary is X, ∂Y = X, and define the above coupling as where G = dB. However, this depends on the extension of the manifold and the gauge field from X to Y . Choose another extension Y . Then glue Y and Y together along their common boundary to make a closed manifold Z. Then we get  (2)]. Because of the anomaly, or more explicitly by the consideration of the monopole operators discussed above, the su(2) C becomes SU(2) C .
In summary, if the original T has SO(3) symmetry, the S-transformed theory S · T has SU(2) symmetry and vice versa. In this way, SU(2) and SO (3) are exchanged under the S-transformation.

B.4 SU(2)/SO(3) symmetry types of knots from six dimensions
From the point of view of 6d N = (2, 0) theory, the symmetry type is determined as follows. First we have to recall some of the properties of this theory [65]. For simplicity we focus on the case of the A 1 theory corresponding to su (2).
Let X 6 be a six manifold. To determine the partition function on X 6 , we have to give some additional data. Let H 3 (X 6 , Z 2 ) be the third homology with Z 2 coefficients. By Poincare duality, it is possible to split this homology as This is chosen such that any two elements a, a ∈ A have zero intersection a, a = 0, and similarly for B, and the pairings between A and B are non-degenerate. The splitting of H 3 (X 6 , Z 2 ) into A and B is not unique, but we have to choose one to define partition functions of the 6d theory. We call this splitting as polarization, and call A and B as A-cycles and B-cycles, respectively. The partition function of the theory not only depends on the manifold X 6 , but also on the polarization. Let us compactify the 6d theory on S 1 and consider X 6 = S 1 ×X 5 . Then we get 5d SYM theory with gauge algebra su (2). Then, the above splitting determines the SU(2)/SO(3) types of the 5d gauge theory. First, notice that the cohomology is given as H 3 (S 1 × X 5 , Z 2 ) ∼ = H 2 (X 5 , Z 2 ) ⊕ H 3 (X 5 , Z 2 ). (B.14) Under this isomorphism, we definê A = A ∩ H 2 (X 5 , Z 2 ),B = B ∩ H 2 (X 5 , Z 2 ). (B.15) Then, the 5d su(2) theory has the following properties. Roughly speaking, the theory is SU(2) type for A-cycles and SO(3) type for B-cycles, respectively. More precisely, let w 2 be the second Stiefel-Whitney class of the su(2) bundle on X 5 . For a 2-cycleâ ∈Â, we require that â w 2 = 0. 15 On the other hand, forb ∈B, we sum over all gauge configurations with different values of b w 2 in the path integeral. Applications of the above framework to 4d class S theories were studied in [66]. Here we want to do it for 3d/3d correspondence. More specifically, we want to determine the SU(2)/SO(3) types of the flavor symmetry associated to a knot.
Take the 6d manifold as X 6 = X 3 × M 3 , where X 3 is "space-time" and M 3 is "internal space" which is closed. The holomogy is given by

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First we have to choose a polarization (B.13). There is no unique way to do it. However, there are only a few choices which preserve the diffeomorphism invariance of X 3 and M 3 , 16 and we assume that one of those choices is realized in 3d/3d correspondence. One possible choice is to take In this case, by taking X 3 = S 1 × X 2 , we get

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The first condition is consistent with the connection coming from hyperbolic metric, because the tangent bundle of any orientable 3-manifold has w 2 (tangent) = 0. The second condition is consistent with constructing PSL(2, C) connections by ideal triangulations. If we try to uplift the holonomy to SL(2, C), then there may be ± ambiguity. For example, the holonomies (2.64) contain square roots √ z 1 , √ z 2 , and so on, which represent this ambiguity.
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