Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence

By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d $(2,0)$ theory on a three-manifold $M_3$. This generalization is applicable to both the 3d $\mathcal{N}=2$ and $\mathcal{N}=1$ supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by $M_3$. This is carried out in detail for $M_3$ a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on $M_3$, which matches the Witten index computation that takes the higher-form symmetries into account.


Introduction
The compactification of higher dimensional Quantum Field Theories has led to a deeper understanding of the physical properties of the lower dimensional theories, especially their dualities and symmetries. One well-studied example is the compactification of the 6d N = (2, 0) SCFT with ADE Lie algebra g on a three-manifold M 3 , with a topological twist along M 3 . The choice of topological twist determines the amount of preserved supersymmetry in 3d, and we find either a 3d N = 2 [1][2][3][4] or a 3d N = 1 theory [5]. These theories are often referred to as T [M 3 ]; more explicitly, we can write T N =1,2 [M 3 , g], when we need to specify the additional data. These theories depend on the topological manifold M 3 , and many of their detailed properties can be understood in terms of the topology of M 3 .
For each of these theories, there exists a 3d-3d correspondence with a 'dual' 3d topological theory, which is obtained by considering the 6d theory on the space (1.1) The partition function of this topological theory on M 3 is conjectured to compute the partition function of the supersymmetric theory, T [M 3 ], on W 3 . The most prominent choices for W 3 for which such 3d-3d duals have been discussed are the squashed three-sphere S 3 b [1,2,6], the superconformal index on S 2 × S 1 [3], and the twisted index on Σ g × S 1 [7]. The special case W 3 = T 3 was considered in [5,7] and computes the (regularized) Witten index [8] I = Tr(−1) F . 1 However, an important characteristic of the theories has so far been largely ignored: their higher-form symmetries [12]. Namely, the theory T [M 3 ] has a higher-form symmetry, which, as with other properties of T [M 3 ], is determined by the topology of M 3 . In fact, the theory T [M 3 ] is not fully specified by the manifold, M 3 , but requires additional topological data. This is related to the fact that the 6d N = (2, 0) theory itself is a relative QFT, i.e., it is only well-defined as the boundary of a 7d TQFT [13,14]. Equivalently, its observables depend on a choice of polarization, i.e., a choice of maximal isotropic subgroup of H 3 (M 3 , Z G ), where Z G is the center of the simply connected group, G, with Lie algebra g. Naturally, we expect that T [M 3 ] also depends on the polarization, as is the case for 4d theories [15,16]. In fact, we will see that this additional information translates into the residual 0-and 1-form symmetry of the 3d theory. We propose therefore a refined definition of the theories, which specifies this in (1.2) for M 3 a graph manifold [17], a class of three-manifolds we review in section 2.1. These manifolds, which also occur as the boundary of plumbed four-manifolds, are sometimes called plumbed three-manifolds, a special case of which are Seifert manifolds. Similar Lagrangians for the 3d theories associated to these manifolds were studied in [6,[18][19][20][21][22][23].
In the following we point out new features related to the global structure of the gauge groups and higher-form symmetries of these theories, as well as the explicit computation of the Witten index and related observables. The approach we take is as follows: 1. Graph manifolds can be cut along disjoint embedded tori into degree k circle fiber bundles, M g,k , over genus g Riemann surfaces with boundary. This allows us to assign to each graph manifold, M 3 , a graph, Ω, as follows. Each copy of M g,k is represented by a vertex, dressed by the degree k and the genus g. These are connected by edges representing gluing along boundary tori by S-transformations, which have the effect of exchanging the cycles of the boundary tori. 3 From the graph we define theories T [Ω, G] with the building blocks vertex g,k : N = 2 G k CS theory + N = 4 g adjoint hypermultiplets edge : T (G) S-wall theory , where T (G) was defined in [24]. In addition to (1.3) we need to couple the theory to N = 2 adjoint scalar multiplets via holomorphic moment maps for the G symmetries.
Note that T (G) has flavor symmetry group G × G ∨ , but here we take all gauge fields in G.
2. The graph Ω representing M 3 is not unique. Instead, it represents a four-manifold M 4 with boundary M 3 . There are various operations that leave M 3 topologically invariant but change Ω. We can check that T [Ω, G] is only invariant under these operations up to decoupled topological sectors [25].
The physical theory T [M 3 , g] should only depend on M 3 itself. Inspired by [26], we propose that we can decouple the additional topological factors on which the physical 1form symmetry acts trivially. Then, different choices of Ω correspond to dual descriptions of T [M 3 , g], confirming various dualities [27,28]. 3. We denote the theory obtained in this way by T [M 3 , g, 1]. It has no discrete 0-form symmetry and a global anomaly-free 1-form symmetry with group H 1 (M 3 , Z G ). Any subgroup of this can be gauged and we can always find a subgroup such that the new 0-form symmetry is H, giving a theory we denote T [M 3 , g, H].
Having established a Lagrangian description for T [M 3 , g, H], we can exploit the Bethe/gauge correspondence [29,30] to compute the Witten index. The twisted superpotential of T [Ω, G] is obtained from its quiver description, where the building blocks, corresponding to (1. 3), have been determined in [7,9,[30][31][32]. From this we can determine the Bethe equations, whose solutions determine the "Bethe vacua," forming a special subsector in the Hilbert space of T [Ω, G] on T 2 . With this method we can in principle compute them for any choice of Ω and G. We will do this explicitly for M 3 a Seifert manifold with b 1 = 0 and G = SU (2). 3 Gluing by more general large diffeomorphisms can be achieved by decomposing these into products of S and T generators, as we discuss in section 2.1. The 3d-3d correspondence is one of the original motivations for this work. Taking the setup in (1.1), we can either reduce along M 3 to obtain T [M 3 ], or along W 3 , which yields a 3d topological field theory on M 3 . This TFT depends on the choice of W 3 as well as the choice of topological twist [1,2,5]. This is summarized in table 2. Similar conjectures have also appeared for M5-branes on Σ × W 4 [33] and M 4 × W 2 [18,34,35]. In our setup, we expect that the Witten index of T N =2 [M 3 , g, H] counts complex flat connections on M 3 , as discussed also in [36]. However, we find the choice of H, and more generally the action of the higher-form symmetries, has an important impact on the precise 3d-3d dual observable. As given by a trifundamental chiral multiplet of SU (2) 3 , we gauge the three SU (2) symmetries with CS levels k i , i = 1, 2, 3; the quiver is shown in figure 1. 4 More precisely, depending on the k i , this theory may contain decoupled topological sectors which must be projected out.
This simple description passes many non-trivial consistency checks. For example, it possesses the expected 0-form and 1-form symmetries. Moreover, we compute the refined Witten index of this theory and find that it matches the detailed properties of flat P SL(2, C) and SL(2, C) connections on M 3 .
The remainder of this paper is structured as follows. In section 2 we review the description of graph manifolds in terms of a graph, Ω. We prescribe how to translate Ω into the theory T [Ω, G], and mention the possible operations on the graph, which also serves to set our notation. In section 3 we briefly review some relevant aspects of higher-form symmetries.

T [M 3 ] from Graphs
In this section we begin our study of the theories T [M 3 ] obtained by compactification of the 6d (2, 0) theory on a Seifert, or more generally, a graph manifold, M 3 . We introduce some of the basic concepts, such as the topological twist, the quiver description of the resulting 3d theories, and dualities. In particular, as we describe below, we may associate a graph to a decomposition of a three dimensional graph manifold along disjoint embedded tori, and this graph directly determines a quiver gauge theory description.
In this section, we will treat the global structure of the gauge group in a naive way by taking the gauge group to be G for every node in the quiver gauge theories below. As we will see, this prescription means that the quiver theories we associate to a graph, Ω, are not, in general, the theories T [M 3 ] obtained by compactification, but rather some closely related theories we define as When G is equal to its Langlands dual G ∨ (i.e., the lattice of weights of G is self-dual), then we may define where Ω is any graph decomposition of M 3 . However, in general, two graphs, Ω and Ω , which give rise to the same three-manifold do not always give the same 3d theory. In the following sections we will describe how to pass from the theory T [Ω, G] to the more physical T [M 3 , G] theories, which involves a more careful treatment of the global structure of the gauge groups.
The 6d N = (2, 0) SCFT has 16 supercharges and exists for any ADE Lie algebra. Here, we will focus on g = A N −1 , as this is related to the SCFT that is conjectured to be the effective theory on a stack of N M5-branes, which has gauge algebra u(N ). Now consider a threemanifold M 3 , and compactify the 6d theory on M 3 down to 3d. Supersymmetry is preserved for a non-flat M 3 only if a suitable R-symmetry background field is turned on, i.e., the theory is partially topologically twisted. There are essentially two choices, which result in 3d N = 2 and 3d N = 1 theories, respectively. The latter will be discussed in section 7. These theories arise by wrapping M5-branes on supersymmetric cycles: either special Lagrangian three-cycles inside Calabi-Yau three-folds or associative three-cycles inside seven-manifolds with holonomy preserving four supercharges. We refer to this twist as the N = 2 twist and the 3d theory is , where here we take G = U (N ), or more generally, any self-dual ADE Lie group. The conjectured 3d-3d correspondence relates this theory to a 'dual' 3d topological theory, which depends on the transverse three-dimensional spacetime. In particular if the 6d geometry is M 3 × T 3 , the partition function of the topological theory that is obtained by reducing first along the T 3 is conjectured to compute the Witten index of the theory T [M 3 ] [38].
For the N = 2 twist the topological theory is a complex BF-model, and the Witten index of T [M 3 , G] is conjectured to be computed by flat G C -connections. We will return to this interpretation in section 6.
A useful point of view that was already observed in [1] is to consider the theory on M 3 as arising from a 4d N = 2 class S theory T [Σ g,n ] with boundary conditions. Geometrically this means that locally we embed the curve Σ g,n of genus g and n punctures, into T * Σ g,n , which is a local K3. The BPS equations for the class S theory are the Hitchin equations [39] where F zz is a (1, 1)-form on Σ and Φ is a section of K Σ ⊗ Adj(G C ). The connection between these equations and the complex flat connections is well-known, by considering M 3 = Σ × R.
In this paper the main focus will be on a class of three-manifolds, M 3 , that are naturally associated to graphs, a special case of which are Seifert manifolds. In this section, we define these graph manifolds, and associate quiver gauge theories to them. Geometric identities become dualities in the quivers. We illustrate these in the case of the simplest non-abelian groups, U (2) and SU (2).

Seifert and Graph Manifolds
The class of geometries M 3 that we will consider here include Seifert manifolds (see e.g., [40]), which are circle bundles over a Riemann surface Σ g,r of genus g with r marked points The fibration is specified by the Seifert data where (d, q i ) ∈ Z and p i ∈ Z + . Furthermore, p i and q i are coprime. Away from the punctures, M 3 is a smooth degree d circle fibration over Σ g . The exceptional fibers are located above the marked points. Around each puncture, excise a tubular neighborhood which together with the fiber forms a solid torus, S 1 × D 2 . On each of these boundaries we attach a mapping class torus, i.e., a T 2 fibered over an interval x 3 ∈ [0, L]. We view the torus as a complex manifold with complex structure τ and SL(2, Z) action τ → (rτ + p)/(sτ + q), where 5 and r, s encode the framing, i.e., the choice of trivialization of the tangent bundle. The integers k j are given by the continued fraction (2.8) This representation ensures that the (0, 1)-cycle at x 3 = 0 is identified with the (p, q)-cycle at Here and in the following we denote the A-and B-cycle of the boundary of a solid 5 Here S and T are the generators of SL(2, Z), for which we take the explicit representations S = 0 −1 1 0 and T = 1 1 0 1 .
torus as the S 1 and the ∂D 2 respectively and denote general cycles in the (A, B)-basis. To close the boundary we then glue back in a solid torus.
We will also consider a generalization of Seifert manifolds to so-called graph manifolds, which are glued together from Seifert manifolds building blocks. More formally, they can be defined as three-manifolds, which can be cut along tori, where the resulting building blocks are circle-bundles over Riemann surfaces. There are various equivalent ways to characterize graph manifolds, however here we will make use of the formulation in terms of a plumbing graph. This construction will first of all produce a four-manifold, whose boundary is the graph manifold.
The graph will have vertices v i corresponding to disc-bundles over a Riemann surface, where e(M i ) is the Euler number of the disc-bundle and g is the genus of the base curve. We include edges between v i − v j if there is a gluing between two disc bundles as follows: consider Then gluing the discs across (i.e., gluing the fibral disc with the disc within the base, and vice versa) D i × B i with B j × D j corresponds to an edge along which the two vertices v i and v j are glued. The boundary of such a four-manifold is automatically a graph manifold.
There are redundancies in the definition of the plumbing data which leave the boundary -i.e., the graph manifold -invariant. For example, there are operations which change the 4-manifold by taking the connected sum with P 2 and with P 2 , but change the 3-manifold by a connected sum with S 3 , which is a trivial operation [18]. We will discuss these identities and their implications on the graph shortly. For a general discussion of graph manifolds see [41].
The case of Seifert manifolds is a specialization, where the plumbing graph is given by figure 2, and the Seifert data is encoded in the continued fractions p i /q i given by (2.8). In general there is a choice of genus for the central node, which we generally take to be g = 0 in the following.
The simplest class of examples are the Lens spaces, M 3 = L(p, q), which are obtained by gluing two solid tori, (S 1 × D 2 ) ± , identifying the (0, 1) + cycle with the (p, q) − cycle. This is known as the Heegaard splitting of the Lens space, see e.g., [42,43]. We can easily see that this definition corresponds to Seifert manifolds with g = 0 and r ≤ 2. First consider r = 0, i.e., M 3 is a degree d circle fibration over S 2 . This is a definition of L(d, 1) in terms of a generalization of the Hopf fibration. We thus assign to L(d, 1) the element ϕ d,1 = T d . Now add a puncture with exceptional fiber (p 1 , q 1 ). Excising the tubular neighborhood of the puncture leaves us with a circle-fibration over the disk. We now glue this to the solid torus as explained above.  Figure 2: Plumbing graph for Seifert manifolds with r special fibers. For r = 3 this graph corresponds to the Seifert quivers in the text. We consider here g = 0 for the central node. This also gives the quiver for the theory After the gluing we obtain (2.10) The generalization to two exceptional fibers is obvious. By exchanging the two fibers one can see that L(p, q) ∼ = L(p, q −1 mod p).
Introducing a third marked point fundamentally changes the geometry and the resulting Seifert manifold is no longer a Lens space (for generic (p i , q i )). The description in terms of a linear chain of S and T in SL(2, Z) breaks down but we will present a natural way to couple the central node to three fibers when discussing the quiver description in section 2.2.3. In a similar fashion we can translate the plumbing graph of a general graph manifold into a web of vertices M g,k , degree k circle bundles over Σ g , connected by S-transformations. In what follows, we will primarily be concerned with M 0,k , with g = 0, and will label the vertices only by k.
The continued fraction presentation of p/q in (2.8) is not unique, which implies a nonuniqueness of the Seifert data or, equivalently, the ambient four-manifold. These relations are summarized in figure 3, and extend straightforwardly to more general graph manifolds: This corresponds to adding a factor of T ST on the right hand side of ϕ p,q . This leaves (p i , q i ) invariant but affects the framing of the manifold and corresponds to taking the connected sum of M 3 with an S 3 . We can also attach this to the central node, adding a new fiber with p r+1 , q r+1 = (1, 1), while increasing the degree by one. Thus, ii) [k 1 , . . . , k j , . . . , k n ] ∼ = [k 1 , . . . , k j − k, 0, k, . . . , k n ] for any k: This corresponds to including a factor of S 2 = C. This sends (p, q) → (−p, −q) leaving p/q invariant.  Figure 3: The relations T ST , S 2 and (T S) −3 acting on the plumbing data associated to the three-manifold.
This follows from including a factor of (T S) −3 = C. Inserting this element between the central node and a fiber decreases the degree by one and acts on the fiber element as ϕ p,q → ϕ p,q−p . This generalizes to Some examples of interesting Seifert manifolds are given in appendix B.
Before moving on, we note that an efficient way to characterize the graph, Ω, associated to a graph manifold is by its linking matrix, which we denote by Q. Its entries are defined as Here the choice of taking off-diagonal terms to be −1 is a convention, but will be useful in the definition of T [M 3 ] below.

T [Ω, G]
We now define the theories T [Ω, G], which will be the building blocks for all the physical theories that we consider in later sections. It is useful to define this theory in terms of boundary conditions on N = 2 * gauge theory with gauge group G that preserve a U (1) t flavor symmetry. For the N = 2 twist, the construction in terms of boundary conditions has been extensively studied in e.g., [1,6,7,19,21,24,44,45], which we briefly summarize below. For example, the Lens spaces can be constructed from the Heegaard splitting with one-punctured torus boundaries, which corresponds to the 4d N = 2 * theories on an interval with half-BPS boundary conditions. In this picture, the puncture on the boundary torus induces a network of line defects in M 3 , which correspond to the circle actions in the Seifert fibration (2.5).

Boundary Conditions for 4d N = 4 SYM
The basic building blocks can be constructed from the 1 2 -BPS equations of the mass deformed 4d N = 2 * G gauge theory, as studied in [6]. The bosonic degrees of freedom of the 4d theory can be decomposed into that of an N = 2 vector multiplet and a hypermultiplet in the adjoint representation, (A µ , φ) and (X, Y ). In the N = 2 language, the theory has a U (1) t global symmetry, under which the complex scalars (φ, X, Y ) have charges (0, 1, −1) respectively.
The basic boundary conditions that can be used to build a T [Ω, G] theory are 14) where N and D(V) denote the Neumann and Dirichlet boundary conditions for the vector multiplet respectively N : where the Dirichlet boundary condition depends on the choice of background 3d N = 2 vector multiplet V whose lowest component is a. For the Neumann boundary condition, the vector multiplet V remains dynamical.
The overlaps between these boundary conditions can be obtained by studying the low energy limit of the 4d N = 2 * theory on an interval with two boundary conditions. First of all, one can check where we integrate over the dynamical 3d N = 2 vector multiplet V. In addition to this, we have the N = 2 chiral multiplet valued in the adjoint representation with U (1) t charge ±1, which we denote by adj ± (V). If both types of adjoints are present in the theory, the pair can be integrated out in the low-energy effective theory, which leads to the relation adj + (V)adj − (V) = 1. We also need the following relations interface in the 4d gauge theory, which defines an operator acting on the space of boundary conditions (2.14). We now list the operators and the associated boundary conditions for the building blocks: 1. The simplest example is the mapping cylinder for the identity element ϕ 0,1 = I. This corresponds to inserting the identity operator in the space of boundary conditions This can be equivalently expressed in terms of the multiplet V − . Note that the measure includes the contribution from the adjoint multiplet that is compatible with the inner product (2.17).
2. The mapping cylinder implementing the T operation corresponds to adding a background N = 2 Chern-Simons level to the theory where the last factor denotes an N = 2 CS theory at level k.
3. The S operation corresponds to the T (G) theory defined in [45], which has G×G ∨ flavor (2.20) Using the relation V + | = adj(V ) − V − |, we can write the expression more symmetrically in the two background gauge fields as where F T (G) is the so-called "flipped" T (G) theory defined by adding an adjoint field in one of the flavor symmetries to the original description of the T (G) theory [46].
Finally, let us consider the building blocks with more than two torus boundaries. Similarly to the identity operator (2.18), we claim that Σ 0,r × S 1 with the A-cycle along the S 1 identifies all the global symmetries associated to the r punctures. For the three-punctured sphere, we which can be equivalently expressed in terms of the multiplet V − .

Definition of T [Ω, G]
A description of T [Ω, G] can be obtained by a combination of the above building blocks according to the surgery procedure discussed in section 2.1.
We start with the plumbing graph Ω of a graph manifold M 3 , which corresponds to a web of vertices M g,k connected by S-transformations. From this we construct T [Ω, G] in the following way: 6 1. To each vertex M g,k we assign an N = 2 gauge multiplet at Chern-Simons level k together with an N = 2 adjoint chiral multiplet, which we assign U (1) t charge +1. We also couple this to a set of g N = 4 adjoint hypermultiplets [21,37], although in this paper we will mostly consider the case g = 0. This M g,k vertex has r boundaries along which it is glued to neighboring vertices. To each of these boundaries we assign a V + |.

2.
To each edge we assign the S-transformation (2.21), i.e., the T (G) theory. More precisely, it is convenient to use the second line of (2.21) to write this more symmetrically using the F T (G) theory, which is the T (G) theory with one flavor symmetry coupled to an adjoint chiral multiplet of negative U (1) t charge. To the two flavor groups we assign |V − and |V − respectively.
3. We glue these building blocks together using the overlap For Ω the graph associated to a Seifert manifold, we will show that the theories defined in the above way are closely related -and in the case of self-dual G, identical -to the theories In this paper we use the following depiction of the quivers for the T [Ω, G], where we henceforth assume g = 0 for all vertices: gauge field V at CS level k ←→ round node dressed with k flavor group ←→ square node adj ± (V) ←→ node dressed with upward/downward arc It consists of two copies of M 0,k j connected by an edge and therefore (2.24) Using the delta functions we can integrate out V 2 and V 3 after which two adjoints with opposite charge cancel. After these simplifications This theory is depicted in figure 4, illustrating the conventions of (2.23). More generally, we , G], where p/q = k 1 − 1/k 2 and the dictionary in (2.23) is used. On the left we use the T (G) theory, and on the right the F T (G) theory, which is related by flipping an adjoint field. This flipping operation can be represented pictorially by contracting an upward arc on a node and an adjacent downward arc on an edge to leave an upward arc on the edge. Further equivalent descriptions of this theory are shown in figure 5 below.

General Gauge Group
The non-uniqueness of the Seifert data discussed in section 2.1, should map to dualities of the theories T [Ω, G]. However, we will see below that for non-self-dual G, the SL(2, Z) actions and equivalence relations of three-manifolds discussed in section 2.1 holds only up to a decoupled topological sector at the level of the T [Ω, G] theory, indicating an undesired dependence on the decomposition graph Ω. We will discuss how to cure this problem in the following sections.
We now summarize the dualities: i) [k 1 , . . . , k n ] ∼ = [k 1 , . . . , k n + 1, 1]: The equivalence of the quiver under this operation was discussed in [7,18,20]. In the case of the G = U (N ) theory, this is related to the following duality of [47] U (N ) k=1 + adjoint chiral ↔ N free chirals . (2.26) We discuss this relation and the role of this duality in more detail in the N = 2 case below. For G = SU (N ), we use instead the following duality, a special case of which is the duality appetizer of [27] SU (N ) k=1 + adjoint chiral ↔ (N − 1 free chirals) ⊗ (topological U (1) N CS theory) .

(2.27)
The duality implies the relation 7 : [k 1 , ..., k n + 1, 1] quiver ↔ ([k 1 , ..., k n ] quiver) ⊗ (topological U (1) N CS theory) . (2.28) Note the presence of the decoupled topological sector, which is directly related to the fact that SU (N ) is not self-dual. This sector will play an important role in what follows. 7 Up to a relative gravitational CS term which does not affect our discussion. Figure 5: Sequence of moves leading to the duality between the F T (G) theory and a trinion theory with one G symmetry gauged with level 1 CS term. In the last step with use the star-shaped quiver description of the trinion of [37].
ii) [k 1 , . . . , k i , . . . , k n ] ∼ = [k 1 , . . . , k i − k, 0, k, . . . , k n ]: This corresponds to two factors of T (G) coupled at a common node and gauged at level zero. However, since we take the gauge field in G (rather than G ∨ ) this operation adds a topological factor, which for Using the relation (2.28) for the theory on the right hand side, we find that this theory precisely corresponds to the trinion theory with the three flavor symmetries gauged at levels (k 1 − 1, k 2 − 1, 1) respectively.
In fact, we may make a stronger statement. Rather than taking these nodes with level k 1 and k 2 CS terms as gauge symmetries, we may retain them as flavor symmetries.
Then, in the former case, we obtain the T N theory with one of its G flavor symmetries gauged with a level 1 CS term, while in the latter case we obtain the T (G) theory, or more precisely, the flipped theory, F T (G). Both have G × G flavor symmetry. We also notice that the levels of the CS terms are shifted by one in the latter description, which we may attribute to relative contact terms appearing in the duality. In summary, this yields the following duality T G trinion theory with one flavor symmetry gauged at level 1 This is illustrated in figure 5. This observation was also made in [28] for G = SU (N ).
We will discuss this in more detail in the case G = SU (2) below. Let us illustrate some of the properties discussed in the previous subsection in more detail in the special cases G = SU (2) and G = U (2), and also point out some additional subtleties that arise.
To start, let us briefly recall some relevant features of the T (SU (2)) theory. The usual description of the T (SU (2)) theory is as a 3d N = 4 U (1) gauge theory with N f = 2 hypermultiplets. This theory has SU (2) F × SU (2) J flavor symmetry, the former acting manifestly on the hypermultiplets, and the latter being an infrared enhancement of the U (1) J symmetry visible in the UV gauge theory Lagrangian. This is the theory living on a domain wall interpolating between S-dual instances of the 4d N = 4 SU (2) SYM theory, with each bulk theory coupling to one of the SU (2) flavor symmetries of the domain wall, as discussed in [24]. Now let us consider the relation i) above in the case of G = SU (2). Consider a linear quiver associated to the RHS of this relation: this ends with an S-wall, followed by a SU (2) gauge node with a CS level 1. Let us take the latter SU (2) symmetry to couple to the SU (2) F symmetry of the T (SU (2)) theory, and let us denote the gauge symmetry of T (SU (2)) as U (1) g . Thus the SU (2) F node has a single fundamental flavor, also charged under U (1) g , and CS level 1. Then this theory has a dual description [48,49] as a single chiral multiplet, and importantly, a level 2 contact term for the U (1) g symmetry. Thus this part of the quiver is replaced by singlet chiral multiplet along with a U (1) g level 2 topological CS theory, leading to G = SU (2) : [k 1 , ..., k n + 1, 1] quiver ↔ ([k 1 , ..., k n ] quiver)⊗(topological U (1) 2 CS theory) , and so the relation only holds up to a decoupled topological sector, as noted above. A special case of this, noted in [20], is the case n = 0, where this relation formally gives the "appetizer duality" of [27], which indeed contains such a decoupled U (1) 2 topological sector.
As described in [50], this is simply the T (SU (2)) theory, whose symmetry we denote by SU (2) F − × SU (2) J − , along with a background level 2 FI parameter coupling to an U (1) F + × U (1) J + symmetry. This exhibits the flavor symmetry as SU but one can check that the true symmetry acting on the matter content is As above, this interpolates between S-dual instances of the 4d N = 4 U (2) SYM theory. Now let us consider the relation i) in this case. The analysis for the T (SU (2)) factor goes through as above, but now the final node also contains a U (1) gauge group with level 2 CS term. Now when we take the Z 2 quotient, these two U (1) 2 topological sectors are eliminated, 8 and so there is no residual topological sector, and we find G = U (2) : [k 1 , ..., k n + 1, 1] quiver ↔ ([k 1 , ..., k n ] quiver) . (2.33) As above, the n = 0 case formally leads to the U (2) case of the duality of [47] U (2) 1 + adjoint chiral ↔ 2 free chirals , (2.34) which also holds without any extra decoupled sectors.
Next consider the relation iii). Recall this follows from the duality of the T G theory with one G flavor symmetry gauged at level 1 and the T (G) theory. In general, the T G theory is non-Lagrangian, and so this duality is not of immediate practical use. However, for the case of G = SU (2) (the U (2) case is a straightforward extension) the T 2 theory is simply a free bifundamental hypermultiplet. Thus, we arrive at the duality 9 3d N = 4 U (1) with N f = 2 hypermultiplets ↔ SU (2) k=1 with N f = 2 hypermultiplets with level −1 contact terms for SU (2) × SU (2) ∼ = SO(4) symmetry .

(2.35)
This duality was noticed earlier in [51], and also discussed in [28]. Importantly, note that in this dual description, the SU (2) × SU (2) symmetry is manifest in the Lagrangian, unlike in the usual description of T (SU (2)). This observation makes this description of T (SU (2)) more convenient for constructing Lagrangians for general Seifert manifold theories, and it will play an important role in what follows.

Higher-Form Symmetry in QFT
In this section we briefly review higher-form, or q-form symmetries of QFTs [12]. These turn out to play an important role in understanding the precise definition of the theories, T [M 3 ], obtained by compactification of the 6d N = (2, 0) theory on a general three-manifold, M 3 , which we will discuss in the next section.  For q > 0, it can be shown that Γ must be abelian.

Higher-Form Symmetries
When q = d−1 2 , the charge operators have the same dimension as the charged operators, and so they may themselves be charged under the symmetry. When this happens we say the symmetry has an 't Hooft anomaly, and this is an obstruction to gauging it. 10 When the symmetry is non-anomalous, the expectation value of a set of charge operators depends only on the cycle ω ∈ H d−q−1 (M d , Γ) determined by the choice of σ and γ of the operators.
By Poincaré duality, this is the same as a choice of cocycle, ω ∈ H q+1 (M d , Γ), and we may equivalently interpret this as computing the partition function in the presence of a background (q + 1)-form gauge field, labeled by ω, and define This gives a refinement of the ordinary partition function, which keeps track of the response of the system to sources for the higher-form symmetry.
When a higher-form symmetry is non-anomalous, we may gauge it to obtain a new theory.
This means we promote the background (q + 1)-form gauge field to be dynamical, and so the partition function of the gauged theory is given by summing over all such background gauge fields. It is natural to also introduce a coupling to a new background gauge field, and define where ω ∈ H d−q−1 (M d , Γ), Γ is the Pontryagin dual group, where Z P is the contribution to the partition function from a principal G/Γ-bundle, P , over spacetime, and we sum over all principal bundles with a specified value for their second Stiefel-Whitney class, w 2 (P ) ∈ H 2 (M d , Γ), which measures the obstruction to lifting a G/Γ bundle to a G bundle. Then the G/Γ theory is simply given by gauging this symmetry, i.e., summing over all G/Γ bundles The G/Γ theory naturally comes with a d − 3 form Γ symmetry which we call the magnetic symmetry, and whose charged operators are 't Hooft operators, which have dimension d − 3.
We may also gauge a subgroup of Γ, or include additional local actions for the background gauge fields, to obtain other versions of the gauge theory.

1-Form Symmetries in 3d Gauge Theories
Let us now review some general features of 1-form electric symmetries in 3d gauge theories, which will arise in the study of T [M 3 ] below.
Recall that d = 3 is the case where a 1-form symmetry may have an 't Hooft anomaly, and such anomalies turn out to arise from CS terms for the gauge groups. To describe these anomalies more carefully, we follow [52]. Consider a 3d gauge theory with gauge and flavor groups, G g and G f , which we take to be the simply connected. However, there may in general be some subgroup, Γ, of the center of these groups which does not act on the matter content.
Then in principle, we might try to couple the system to a ( G g × G f )/Γ bundle. However, anomalies may cause this observable to be ill-defined.
To see this, let us exhibit the 3d spacetime as the boundary of a 4d manifold, and extend the gauge fields into the bulk. Then the 3d theory may include CS terms for the gauge and flavor symmetries, which we assume are properly quantized for G g × G f connections. However, they may not be well-defined for ( G g × G f )/Γ connections. To cure this, we include appropriate 4d topological terms, which are also not well-defined on a four-manifold with boundary, but such that the combined 3d-4d system is well-defined. Thus we obtain an action where c is some characteristic class of a 4d ( G g × G f )/Γ gauge bundle. We refer to this as the anomaly polynomial. If we consider two different ways of extending the 3d system into a 4d bulk, the difference is given by the integral of the anomaly polynomial over a closed fourmanifold. This characterizes the extent to which the 3d partition function is not uniquely defined. While this is not necessarily a problem for the flavor symmetries, the theory must be well-defined with respect to the gauge connection, and so we demand that the anomaly polynomial vanishes for any allowed choice of gauge bundles.
Below we will be mostly interested in theories with SU (2) n gauge and SU (2) m flavor symmetry. The center is Z 2 n ⊕ Z 2 m , and let us suppose there is a subgroup, Γ, which acts trivially on the matter. Let us further include Chern-Simons levels k i , i = 1, ..., n, and K j , j = 1, ..., m, for the gauge fields. Then for these to be well-defined, we extend the gauge fields into a bulk four-manifold, M 4 , and include the term Here w 2 is the second Stiefel-Whitney class, valued in H 2 (M 4 , Z 2 ), and is the Pontryagin square operation, which satisfies P(w) = w 2 (mod 2). On a spin manifold, where the intersection form is even, P/2 is well-defined as an element of H 4 (M 4 , Z 2 ).
For example, consider the trinion theory, given by a N = 2 chiral multiplet transforming in the trifundamental representation of an SU (2) 3 flavor symmetry. Then the subgroup acting trivially on the matter is generated by (1, 1, 0) and (1, 0, 1) in Z 2 3 . This means the Stiefel-Whitney class of the three groups must satisfy Now we gauge one of the SU (2) symmetries, say the third, with a level k CS term, and we also include level K 1,2 CS terms for the flavor symmetries. Note we cannot sum over arbitrary 2 , but it is fixed by the classes of the flavor gauge fields by (3.10). The anomaly polynomial in this case is Using (3.10), we may rewrite this independently of the dynamical gauge field as In particular, for k = −K 1 = −K 2 = 1, which gives the SU (2) 1 dual description of the T (SU (2)) theory, as in (2.35), we have This result was also found in [53] using the usual description of T (SU (2)) as U (1) with two hypermultiplets, and this gives another check of the duality between these two descriptions.
More generally, using the fact that the T N theory has a Z N × Z N symmetry, which acts on any two of the three SU (N ) flavor nodes, as above, we may use the description of the T (SU (N )) theory in section 2 to argue analogously that it has the same anomaly polynomial as Note that this means that we may not gauge both flavor symmetries as G ∨ = G/Γ gauge groups, but instead, if we take one as G ∨ , the other must be G. This reflects the fact that T (G) exchanges these two versions of the theories.
In general, given a theory with Z N n 1-form symmetry, we may encode the anomalies in a symmetric n × n matrix, A ij , with entries valued in Z N , such that the anomaly polynomial is given by (3.14) Then one finds that non-anomalous symmetries are ones labeled by a vector γ = (γ 1 , ..., γ n ) ∈ Z N n such that More generally, for a subgroup Λ of the set of 1-form symmetries to be non-anomalous, we impose also that all mixed anomalies vanish, i.e., Note that when A is trivial, every subgroup is non-anomalous. In this case we refer to the group Γ as being "anomaly-free."

Representation of 1-Form Symmetries on H T 2
One important property of 1-form symmetries we will exploit below is that they act on the Hilbert space of the theory on a non-trivial manifold. Below we will consider 3d theories with 1-form symmetries on the spatial manifold T 2 . Then we may define linear operators on the Hilbert space associated to 1-form generators wrapping the two cycles of the torus Then the Hilbert space must fall into representations of the group of these 1-form generators.
Although the 1-form symmetry group itself is abelian, in the presence of anomalies this group need not be abelian since we have in general where A : Γ × Γ → U (1) ∼ = R/Z is the anomaly form. Then the group acting on H T 2 is the central extension, E, of Γ × Γ associated to the anomaly form, A. It fits into a short exact We define the "center" 11 of the 1-form symmetry, Z ⊂ Γ, to be the set of elements z such that A(z, γ) = 0 for all γ ∈ Γ. Then we may simultaneously diagonalize U A z and U B z for z, z ∈ Z, and so refine the Hilbert space into characters is the group of characters of Z. In fact, one can show [54] that the irreducible representations, R χ A , χ B , of E are labeled by such a pair of characters, and have dimension More explicitly, let us suppose that the exact sequence splits, so that we may identify Then the representations R χ A , χ B can be written as where P χ A , χ B is a 1d space only the U z A,B , z ∈ Z act on, and R Γ/Z is the unique irreducible representation of the quotient 1-form symmetry Γ/Z, of dimension |Γ/Z| = |Γ|/|Z|. Thus, in this case, the Hilbert space on T 2 can be factorized as where H χ A , χ B is not acted on by the 1-form symmetry and where all information about the anomaly is encoded in the R Γ/Z factor, and the center, Z, only acts on the second factor. 12 This factorization property will be important below.
In the case where the 1-form symmetry, Γ, is anomaly-free, we may also consider "twisted sectors" in the Hilbert space. These are sectors where we let a 1-form charge operator, U γ , lie along the time direction, and may be labeled by the element γ ∈ Γ where the untwisted sector corresponds to γ = 1, and we may also simultaneously grade these sectors into characters of the U A,B operators, as above. These twisted sectors are important when gauging the 1-form symmetry. Namely, the Hilbert space of the gauged theory is given by H gauged

T [M 3 , g] and Higher-Form Symmetries
In this section we define the theories obtained by compactification of the 6d N = (2, 0) theory of type g on a three-manifold M 3 . Most of what we discuss in this section is applicable to both N = 1 and N = 2 versions, so we will drop the subscript in the following. What will emerge from this section is that to fully define the theory, we must specify, in addition to the gauge algebra and three-manifold, a subgroup H of the cohomology group H 2 (M 3 , Z G ), and so we will actually define theories Here Z G is the center of the simply connected Lie group G with Lie algebra g. For ease of notation, we will often refer to To describe the procedure of obtaining the former theory from the latter, it turns out to be natural to use the language of higher-form symmetries, reviewed in the previous section. In particular, the theories T [M 3 , g, H] in general have several interesting higher-form symmetries, and the theories corresponding to different subgroups, H, can be obtained from each other by gauging these symmetries in a suitable way. This structure can be traced to a higher-form symmetry present already in the 6d theory, and in particular, to its nature as a relative QFT.

6d theory as a Relative QFT
Let us first review some properties of the 6d theory relevant to the compactification to 3d, which we consider in the next subsection. The role of discrete topological data in compactifications of the 6d N = (2, 0) theory has been discussed extensively in the literature; see, for example, [13,14,16,35,53,55] for related discussions.
First we recall that the 6d N = (2, 0) theory is a relative QFT, meaning it is not welldefined by itself, but naturally lives on the boundary of a 7d topological theory [13,14,55].
Then the notion of a partition function is replaced by a partition vector, which is an element in the Hilbert space of this TQFT. This is analogous to the case of a chiral CFT in 2d, which lives on the boundary of a 3d Chern-Simons theory. In the 6d case, this 7d TQFT can be described as a Wu-Chern-Simons (WCS) theory [13,56].
In more detail, let us first consider the A N −1 type 6d N = (2, 0) SCFT. Then, roughly speaking, the WCS theory can be thought of as a level N 3-form Chern-Simons theory, and has 3d Wilson surface operators, O ω , supported on 3-cycles, ω ∈ H 3 (M 7 , Z N ). These may be thought of as charge operators for a 3-form symmetry of the Chern-Simons theory, with group Z N . This symmetry has an 't Hooft anomaly, which means the charge operators do not commute. Rather, when we consider the theory on a spacetime M 6 × R, they obey the appropriate version of (3.1), specifically 13 This can also be described by saying the operators generate a Heisenberg group, W , which is a central extension of H 3 (M 6 , Z N ), and fits into the short exact sequence The Hilbert space of the TQFT on M 6 is given by an irreducible representation of W , which can be described as follows. First we must choose a polarization of H 3 (M 6 , Z N ), which is a maximal isotropic subgroup, Λ, i.e., with λ ∩ λ = 0 for all λ, λ ∈ Λ. Let us assume that we can write where it follows that Λ is also maximal isotropic. Then we pick a basis in which the operators, O λ , λ ∈ Λ, are diagonal. Namely, one can find a unique vector |0 , which is fixed by all O λ , and for λ ∈ Λ we define states Then one can check that and so these states form a basis for the irreducible representation of W , and so for the Hilbert space of the 7d theory. The choice of polarization picks out a corresponding basis of the Hilbert space as above. E.g., if we had chosen Λ rather than Λ, the basis states would be related by a discrete Fourier transform. Note that a choice of polarization will typically break some of the diffeomorphism symmetry of M 6 .
For other ADE Lie algebras g, we expect a similar description, where in general the polarization is determined by a decomposition where Z G is the center of G, the simply connected Lie group with Lie algebra g.
After picking the polarization, we may write the partition vector of the 6d N = (2, 0) theory on M 6 in the corresponding basis as We refer to Z M 6 λ , λ ∈ Λ, as the components of the partition vector. We emphasize that they depend not only on λ, but also on the choice of polarization, Λ.

5d Reduction
As an example, let us consider M 6 of the form M 5 × S 1 . Then we may write, by Künneth, Here there are two natural choices of polarization which preserve diffeomorphism invariance on M 5 , which are to take Λ = A or Λ = B. In the first case, the components of the partition vector can be written as The 6d N = (2, 0) theory on M 5 × S 1 is believed to be described by 5d N = 2 SYM with Lie algebra g on M 5 . Then Z M 5 ×S 1 ω is the contribution to the path-integral of this theory from principle g bundles with second Stiefel-Whitney class equal to ω. In particular, Z M 5 ×S 1 0 is the partition function of the theory with simply connected gauge group, G, and more generally, is the refinement by the electric 1-form symmetry of this theory, as in (3.5).
On the other hand, in the B-polarization the states are labeled by and this corresponds to the G/Z G theory, refined by its 2-form magnetic symmetry. Indeed, this change in polarization is implemented on the representation by a discrete Fourier transform, and we obtain which agrees with (3.6). Note these two choices preserve diffeomorphism invariance on M 5 , which is natural in the context of describing a 5d QFT. Other such choices of polarization give rise to the different global forms of the group with Lie algebra g. Thus we see the different choices of polarization gives rise to different 5d theories.

Abelian case
In addition to the simple Lie algebras considered above, we may define the 6d N = (2, 0) theory in the case g = u(1), where it corresponds to a free N = (2, 0) tensor multiplet. Although this theory is free, there are still several subtleties that arise, as discussed, e.g., in [55,57].
A useful analogy to this theory is the chiral periodic scalar theory in 2d. This theory depends on the period R, of the scalar, where the self-dual radius R = 1 corresponds to a free chiral fermion. For all other choices of R, the theory is not modular invariant by itself, but instead, the partition function is mapped by modular transformations among a space of conformal blocks. If we focus on R = N ∈ Z >0 and spacetime Σ, these blocks are labeled by elements of an isotropic subgroup Λ ⊂ H 1 (Σ, Z N ). For example, for Σ = T 2 , |Λ| = N , and there are N conformal blocks on the torus. As modular transformations act on H 1 and so on the choice of Λ, they also act on the space of conformal blocks. The space of conformal blocks may also be naturally interpreted as the Hilbert space of 3d U (1) level N Chern-Simons theory on Σ. Indeed, the chiral CFTs are not defined on their own, but must live on the boundary of this 3d TQFT, and so their partition functions on Σ are naturally elements in the Hilbert space of these TQFTs.
The self-dual 2-form in 6d is completely analogous. Here we also must specify a periodicity, N ∈ Z >0 , and rather than a partition function, the observables on a spacetime M 6 are given by conformal blocks or a partition vector, as we saw in the ADE case above, namely These may also be understood as vectors in the Hilbert space of a 7d TQFT on M 6 , a 3-form version of the Chern-Simons theory. In particular, for N > 1, the theory is not an ordinary QFT, but a relative QFT, as we saw in the ADE case above.
We note for later reference that the self-dual 2-form with periodicity N has a U (1) 2-form symmetry [12], whose charge operators are where H = dB is the self-dual 3-form field strength. This symmetry has an 't Hooft anomaly, which implies that when we place the U α operator on a manifold with boundary, ∂Σ 3 = Σ 2 , this configuration has a non-trivial 2-form charge supported on Σ 2 . We will return to the consequences of this statement below.

Self-dual G
Let g be a semi-simple, simply-laced Lie algebra, and suppose we can find a subgroup, L, of Z G , such that L ∼ = Z G /L. In this case, the group G ≡ G/L, is isomorphic to its Langlands dual group, and we call such groups "self-dual (4.14) One can check that this is a maximal isotropic subgroup. Note that this choice does not break any diffeomorphism symmetry of M 6 , and so we may naturally interpret these observables as partition functions of an ordinary QFT. Specifically, since the partition function is labeled by an element in H 3 (M 6 , L), we see this theory has a 2-form symmetry with group L. We may naturally refer to the 6d theory of type g with the polarization Λ G as the "6d N = (2, 0) theory of type G," which is an ordinary rather than relative QFT. We emphasize this terminology only makes sense for G self-dual.
Similar considerations apply in the non-semi-simple case. For example, G = U (1) is selfdual, and we saw above that the N = 1 case of this theory is indeed an ordinary QFT. More generally, we may define the "6d N = (2, 0) theory of type U (N )" as follows. We start with the tensor product of the theory of type g = su(N ), along with the u(1) theory with periodicity N . These are separately relative QFTs, and so we must choose a maximal isotropic subgroup of However, as above, a natural choice of polarization that preserves diffeomorphism symmetry is given by This gives rise to an ordinary 6d QFT, which is acted on by a Z N 2-form symmetry. In fact, this Z N sits inside a larger U (1) 2-form symmetry coming from the u(1) factor, as in (4.13).
This theory corresponds to the low energy theory of N M5-branes, without decoupling the center of mass motion.
In the above cases, the reduction to 5d gives rise to the 5d N = 2 SYM theory with gauge group G, which has both 1-form and 2-form symmetries with group Z G .

Compactification on M 3
In this subsection we describe some general features of the compactification of the 6d theory on M 3 , which follow from properties of the 6d theory described above. 15 In this context, we are interested in the case where M 6 is a product manifold of the form which describes a 3d theory, T [M 3 , g], on the spacetime W 3 . As we will see in a moment, additional data is necessary to fully specify the theory.
Here the Tor Z G (X, Y ) are the torsion groups over the rings Z G .
Below we will mostly be interested in the case W 3 = T 3 , and assume from now on that W 3 has no torsion. 16 This implies that all the Tor groups vanish. Under this assumption, we may decompose the third cohomology group as with the notation where we have used the universal coefficient theorem to rewrite these groups (using the assumption that W 3 is torsionless), and have also defined which are naturally Pontryagin dual.
We now have to make a choice of polarization. 17 One possibility is to pick the Apolarization. In this case, the observables of the theory will be labeled by a choice of element in H 2 (W 3 , Υ). This naturally suggests that this 3d theory has a 1-form symmetry with group Υ. Another choice is the B-polarization, in which case the observables are labeled by an element in H 1 (W 3 , Υ), which gives a theory with 0-form Υ symmetry. These choices are analogous to the choices which led to the 5d G and G/Z G theories, and we should interpret these as leading to different versions of the T [M 3 , g] theories.
More generally, we may choose a subgroup and correspondingly define Then we may take our polarization, Λ, to correspond to a subgroup Here the A and B polarizations above correspond to H = 1 and H = Υ, respectively. These give rise to a general class of polarizations preserving the diffeomorphism symmetry of W 3 , which is natural in the context of defining a 3d QFT.
Thus we find there is not a unique theory T [M 3 , g], but rather, different choices corresponding to the choice of H above. We define the theory to be the 3d theory corresponding to the compactification of the type g 6d N = (2, 0) theory with the polarization labeled by H above. We see that the observables are labeled by which we may interpret as corresponding to symmetries 0-form symmetry:

T [M 3 , g, H] for M 3 a Graph Manifold
Let us turn to an explicit description of the T [M 3 , g, H] in the case of M 3 a graph manifold.
Here we will use the theories T [Ω, G] defined in the previous section, but we will see that, in general, some additional operations are necessary to produce the T [M 3 , g, H] theory. In this section we will mostly consider the N = 2 twist, and return to the N = 1 twist in section 7.

U (1) Case
Let us start in the abelian case, and consider the reduction of the 6d theory of type u(1).
Recall we must also choose the periodicity, N , and we first consider the case N = 1 which gives rise to an ordinary QFT in 6d, which we refer to as the U (1) theory. Then we denote the 3d theory obtained by compactification of this theory on M 3 , with the appropriate twist, We focus on the case of M 3 a graph manifold, for which we take a particular graph decomposition, Ω. This graph gives a prescription for forming M 3 by gluing simple pieces along boundary tori by S and T transformations. Then, as discussed in [18], the theory may be built by iteratively applying the S and T operations of Witten [25], or more precisely, their supersymmetric completions. Namely, we assign a U (1) gauge multiplet to each node in the graph associated to M 3 , a level k i CS term for the gauge group in the ith node, where k i is the label of the node, and an off-diagonal CS term to two nodes connected by an S-gluing. In summary, this results in where Q ij is the linking matrix, defined in (2.13), and the dots denote additional fields in the (N = 1 or N = 2) vector multiplet, which we suppress. This is precisely the description of T [Ω, U (1)] given in the previous section, where we recall the duality wall theory, T (U (1)), is simply a mixed BF-term for a U (1) × U (1) flavor symmetry. Thus we have, in this case As a consistency check, we must verify that this theory is independent of the choice of graph decomposition Ω of M 3 . This is almost true, but there is a slight subtlety, which is that the SL(2, Z) operations used above do not quite close onto a representation of SL(2, Z), but rather form a projective representation, as noted in [6,25]. For example, applying the (ST ) 3 operation does not completely leave the theory invariant, but introduces a tensor factor with an invertible TQFT, the U (1) level 1 CS theory, and correspondingly the partition function is not invariant, but incurs a phase. This can be attributed to a gravitational anomaly of the 6d theory. We will work modulo such invertible TQFTs in what follows.
More generally, we may consider the theory, obtained by the compactification of the free tensor multiplet with periodicity N . The case above corresponds to N = 1, and more generally, we claim we should now consider the action Let us denote this theory by T [Ω, U (1) N ], as we have not yet checked independence on Ω.
Indeed, once again we find that the SL(2, Z) relations are not satisfied on the nose, but now the discrepancy is more severe. For example, the (ST ) 3 relation now introduces a tensor factor of the U (1) level N CS theory, which is not invertible. This means the partition function is multiplied by a more complicated factor than the phase above, and even the Hilbert space of the theory will in general be modified. Thus the theory T [Ω, U (1) N ] is not independent of the graph decomposition of M 3 , and so is not yet equal to the theory T [M 3 , u(1) N ]. We will see below that a similar issue arise in the ADE case, and will describe the resolution in that context.

1-form Symmetries
Let us restrict our attention to the case with periodicity N = 1. As mentioned above, the 6d self-dual tensor theory has a 2-form U (1) symmetry, and upon compactification, this descends to a higher-form symmetry of the T [M 3 , U (1)] theory. Namely, we find 1-form symmetry: This holds for general M 3 , but in the case of M 3 a graph manifold, we may see this 1-form symmetry explicitly in the Lagrangian, as follows. To each U (1) factor in the gauge group, there is naively a U (1) ∼ = R/Z 1-form symmetry. However, the Chern-Simons terms break this symmetry to the subgroup where Q is the n × n linking matrix. Then one may check that the above group is indeed Moreover, as a result of the anomaly of the 6d 2-form symmetry, one finds that this 1-form symmetry can have an anomaly. Similarly to the discussion in [12,58], suppose we wrap a charge operator of the 6d theory on a Z k torsion cycle in M 3 . Topologically, this cycle may have a non-trivial boundary, but one which is a multiple of k, so that it vanishes with Z k coefficients. However, due to the anomaly, this boundary may render the charge operator itself to be charged under the higher-form symmetry. As a result, the 1-form symmetry in has an anomaly associated to the torsion subgroup, T H 2 (M 3 ), of H 2 (M 3 ), and the anomaly form gives a topological invariant of M 3 , which is the "linking form" : We may see this explicitly in the Lagrangian for T [M 3 , U (1)] above, as one computes that the anomaly form of the Chern-Simons theory is which indeed reproduces the linking form on M 3 .

Self-dual G
The case G = U (1) above is a special case of the more general situation where G is self dual.
Then, as discussed in section 4.1.3, there is a natural choice of polarization which preserves all diffeomorphism symmetry of the spacetime, M 6 , and may naturally be interpreted as an ordinary QFT, which we call the 6d theory with group G. Then we claim that, in this case, compactification on M 3 gives a theory In particular, this is independent of the choice of graph decomposition of M 3 . Indeed, one can check that, in the self-dual case, the SL(2, Z) relations discussed in section 2.3 are satisfied on the nose (or at least up to invertible TQFTs). We saw this in section 2.3.2 for the case The higher-form symmetry of this 3d theory may be inferred by dimensional reduction of the 2-form Z G -symmetry of the 6d theory, and the theory has 0-form symmetry: (4.37) In the case of G = U (N ), the 1-form symmetry has a contribution from the U (1) sector, and is given as above by Γ 1 = H 2 (M 3 , Z). However, due to the periodicity of the U (1) theory being N , the anomaly form is now N times the linking form. Then one can check that the non-anomalous subgroup is given as in (4.37), but with Z G → Z N .
In the language of section 4.2, this theory corresponds to where we recall L is defined by G = G/L. We describe the construction of general T [M 3 , g, H] next.

General g
Let us now consider the general case.
where we recall G is the simply connected group with Lie algebra g, and Ω is some graph decomposition of M 3 . However, one basic problem we run into is that this theory is not independent of the graph. Indeed, as observed in section 2.3.2, for g = su (2), the theory can pick up decoupled tensor factors of topological theories upon operations on Ω which preserve the topology of M 3 . In order to find a definition of T [M 3 , g] which depends only on M 3 , we must somehow eliminate these decoupled topological sectors.
First, let us review the result of [26], which states that, for a 3d TQFT T with anomalous 1-form symmetry, Γ, one finds that the theory factorizes where One way to isolate the sector T is by coupling to another topological theory, T Γ,−A , which has the opposite anomaly, and gauging the non-anomalous diagonal subgroup We conjecture that a similar decoupling takes place in the theories T [Ω, G]. Namely, we conjecture that we may write Here Γ is the 1-form symmetry of T [Ω, G], with center Z, so that Γ/Z is the anomalous part.
We conjecture that this anomalous symmetry acts only on a decoupled topological sector, and the physical theory, T [M 3 , g], appears as a subsector. Then one way to isolate the latter is as in (4.43), namely However, any procedure which eliminates this decoupled sector will lead to an equivalent result. For example, this can sometimes be achieved by gauging a suitable subgroup of the symmetry Γ in the T theory, as we will see in examples below.
It would of course be desirable to find a first principles derivation of the conjecture (4.44).
However, let us mention several pieces of evidence in favor of it. First note that with this assumption, the dependence on the graph, Ω, due to the appearance of decoupled sectors, as discussed in section 2.3, is cured. Namely, as we will describe in more detail below, such sectors contribute only to T Γ/Z,A , and so will be removed in (4.45).
One tempting interpretation of this decoupling is as follows. Recall from section 2.1 that the graph, Ω, can naturally be interpreted as describing a plumbed four-manifold, Next, from (3.24) we see that the anomalous part of the 1-form symmetry can be taken to act only on a decoupled tensor factor of the Hilbert space of the theory on T 2 . Then we claim that the Hilbert space of the theory T [M 3 , g] is given by the second tensor factor, which has an anomaly-free 1-form symmetry, and the procedure above is defined so as to isolate this factor. We will see this explicitly when we study the T 2 vacua of these theories in the next section.
Finally, we will see in section 6.2 that this decoupling arises naturally when we study the topological side of the 3d-3d correspondence, and is necessary to achieve a precise matching to the counting of flat connections.
An important feature of this procedure is that it removes the anomalous 1-form symmetry, This will leave a theory with 1-form symmetry Υ H and a new 0-form symmetry H, as expected.
As in the 5d example above, this gauging of 1-form symmetry is precisely the same as changing the polarization of the 6d theory. While this is applicable to general M 3 , here and below we focus on the case of M 3 a Seifert manifold, and moreover a rational homology sphere, and we use the Lagrangians discussed in section 2.2.

Lens Space Quiver
We start with the case of the Lens space, M 3 = L(p, q). Then, This theory has a Z N 1-form symmetry, however, due to the Chern-Simons term, it has an 't Hooft anomaly with coefficient p, and so only a Z = Z (N,p) subgroup is non-anomalous.
Comparing to (4.48), we see we may identify this with as expected from (4.46).
In the case where p is a multiple of N , there is no remaining anomalous symmetry, and we have simply  (4.57) To identify this symmetry, first, using (3.13), we find that the anomaly matrix agrees with the linking matrix (2.13), i.e., Thus this theory has a Γ = Z N n 1-form symmetry. However, many of these symmetries are anomalous, and we are interested in the non-anomalous subgroup, Z. This can be identified as the kernel of A acting on Γ. One may compute that det A = p mod N . This implies the anomaly matrix has a (unique, up to scalar multiplication) null-vector δ precisely when the RHS of (4.57) is non-trivial. We call δ the central element, and in general it generates the central subgroup Z ∼ = Z (N,p) . Thus we have identified the expected 1-form symmetry in (4.57).
Next let us discuss the remaining, anomalous 1-form symmetry. For simplicity, we consider the case of N prime. Then, since Γ ∼ = Z N n is freely generated, we find that it splits into a direct sum of the center, Z, and the remaining, anomalous part, Γ/Z  because T (G) squares to the identity [24], when the global structure is taken into account.

Seifert Quivers
Next we look at more general quivers associated to Seifert manifolds. For simplicity, we focus on the gauge algebra g = su (2), which already reveals much of the interesting structure. We also restrict ourselves to graphs Ω (k i ) with intersection matrix with the k i chosen to correspond to the manifolds S 3 /Γ ADE in (B.8). If any k i = 1, M 3 is a Lens space, which were discussed in the previous section, so we will consider k i = (2, −2, n − 2) for D n and k i = (3, −2, m − 3) for E m . The theory T [Ω (k i ) , SU (2)] has a Γ = Z 2 4 1-form symmetry, which in the notation of (4.60) splits as depending on the choice of M 3 . Concretely, We then define T [S 3 /Γ DE , g] in analogy to (4.62).
Let us now focus on the D n case. The kernel of the anomaly matrix A n 18 has dimension two if n is even and dimension one if n is odd, in agreement with (4.65). For n odd the 1-form (2)] is generated by δ 1 = (0, 0, 1, 1). Gauging this symmetry gives us the theory T [S 3 /Γ Dn , su(2), Z 2 ]. For n even there is an additional Z 2 1-form symmetry generated by δ 2 = (0, 1, 0, 1). Thus there are multiple choices for H, namely 2 is the diagonal subgroup generated by δ 3 = δ 1 + δ 2 . The E m case does not unveil any additional structure with E 7 being equivalent to D n with n odd and for E m , m even, there is no 1-form symmetry.

Geometric Interpretation of Global Structure
Here we comment on a geometric picture for the 1-form symmetry above in terms of compact- Then for this to be contractible on the right side, so that we can consistently close the 2-cycle, we see that we must impose p = 0 mod N (recalling we are working with Z N coefficients).
This is the geometric origin of the fact that the Z N homology is equal to Z N only in this case. Then this maps precisely to the generator of the Z N 1-form symmetry that is present in T [L(p, 1), su(N )] in this case.
Let us now consider a more general Lens space. Recall from the previous subsection that, in this case, the 1-form symmetry is Z N when p is a multiple of N , and is generated by an 18 Recall that this is given by A = Q mod 2. We have p 0 = 1, p 1 = k 1 , p 2 = k 1 k 2 − 1, · · · , p n−1 = q, p n = p .   For N not prime this is more involved as we have to also take into account the phases where a subgroup of Z N is gauged. The precise prescription can be found in [15].
If we look at the transformation behavior of (4.74) we see that it agrees with the geometric behavior of (p i , q i ), i.e., More generally, for N not prime, we can define any other H = Z k version, where k divides N , provided k is a multiple of (N, p). To this end we assign the charge vector (0, N k ) to (SU (N )/Z k ) 0 , i.e., we let the generating 2-cycle wrap the disk N k times. Then, we can go through the Heegaard splitting as before, but at each step the charges are multiples of N k and thus non-anomalous.

Bethe Vacua for T [M 3 , g]
In this section we study the set of supersymmetric vacua of the theory T [M 3 , g] for M 3 a Seifert manifold with g = 0. This provides a detailed test of several of the general features described above. For example, we will see that the set of 1-form symmetries of T [M 3 , g] act on the supersymmetric vacua, and constrain the states that can appear. Furthermore, in the next section, we compare the number of vacua to the expectation from the 3d-3d correspondence, which maps this observable to the counting of certain flat connections on M 3 , and find agreement. Applications to the N = 1 twist are considered in section 7.
We will focus on the cases of g = u(N ) and su(N ). The first case that will be considered is u(1), where the theory is free. In the other cases, and specializing to the N = 2 twist, we will use the Bethe ansatz equation approach to studying supersymmetric vacua of 3d N = 2 gauge theories of [30], which we briefly review in section 5.2.

Abelian Case
The theory with G = U (1) is free and has featured in [18]. As discussed in section 4.3.1 the theory is independent of the specific choice of graph Ω for M 3 , so we can immediately define

Twisted Superpotentials and Coulomb Branch Vacua
To compute the vacua for non-abelian gauge groups, we will use the framework developed in [30], i.e., we compute the vacua using the twisted superpotential and counting the solutions to the resulting Bethe equations.
Let us consider a 3d N = 2 gauge theory of gauge group G on R 2 × S 1 . The dynamics of the Coulomb branch is governed by the twisted superpotential, W(u, m), where u is the Coulomb branch variable where t C is the complexified Cartan subalgebra, and β is the radius of the S 1 . Here a 0 is the holonomy of the abelianized gauge field along the S 1 The large gauge transformation along S 1 is defined by where λ is an element in the lattice Λ co-char .
The twisted superpotential receives a contribution from the classical CS level as well as the chiral multiplets of the theory. The N = 2 CS action contributes where x = e 2πiu . We adopt the U (1) −1/2 quantization of the CS level as discussed in [9]. For example, the chiral multiplet with charge ±1 under the Cartan U (1) contribute as follows

(5.17)
We will use the notation where t is the fugacity for the distinguished flavor symmetry in N = 4 theories, which corresponds to the N = 2 * mass. If G contains a U (1) factor, we turn on the FI parameter ζ for the associated topological symmetry. 19 In what follows we will ignore all the numerical constants in W, which are irrelevant for the vacuum count.
where W G is the Weyl group. Note that we removed the solutions which are fixed under the Weyl group action. We call such solutions degenerate vacua. We provide further details on the twisted superpotential and Bethe equations in appendix C.

The Ω [k] Theory
As a simple example, let us first consider T [Ω [k] , SU (2)], which can be described as the N = 2 * Chern-Simons theory at level k. The twisted superpotential for the N = 2 * CS theory is given by

23)
20 Note this is the same description one obtains from the quiver T [Ω, U (2)], as follows from the more general fact that Then, T (U (N )) can essentially be regarded as a tensor product of T (SU (N )) and T (U (1)) with level N BF coupling, with an additional operation to account for the ZN quotient.
where W CS (u) = ku 2 (5.24) is the contribution from the N = 2 CS action and is the contribution from the N = 2 adjoint multiplet, where here and below we use the convention that each choice of sign in '±' is summed over. More explicitly, this is given by The Coulomb branch vacua are then determined as solutions to the Bethe equations It is convenient to split this into two polynomial equations Since the Weyl symmetry acts as

The Linear Quivers Ω [k 1 ,··· ,kn]
The graph for a general Lens space L(p, q) is linear and we denote it by Ω [k 1 ,··· ,kn] . The corresponding quiver is shown in figure 8. To simplify the computation, we recall the duality (2.35) which relates the T (SU (2)) theory to the trinion T 2 . 22 The effect of this duality on the quiver is shown in figure 5. In principle this duality could be used for any SU (N ), however for N > 2 the T N theory is more complicated than T (SU (N )).

Seifert Quivers
For general Seifert manifolds the counting of vacua becomes computationally difficult. However for a subclass we are able to perform the analysis explictly. These are Seifert manifolds, which have base S 2 and three special fibers, and are discussed in appendix B and summarized in table 3. We can simplify the quiver using the trinion theory T N , using the following duality: couple three copies of T (SU (N )) by identifying one of their flavor groups each and gauging it  Figure 10: Quiver for two coupled trinions and an adjoint at the central node indicated by the arc. We denote this quiver by Ω (k 1 ,k 2 ,k 3 ,k 4 ,1) .
at Chern-Simons level zero, together with an adjoint scalar. This theory is dual to the trinion theory [37], see also (2.22). This means that if the degree in the Seifert data vanishes, we can write the quiver using the trinion description, as in figure 9. The vanishing of the degree can always be assumed using (2.11) and (2.12). Then, the graph Ω (k i ) 23 for the Seifert manifolds that we are interested in is given by figure 10. In order to guarantee that this corresponds to a Seifert manifold we have to impose k 5 = 1. and q > 0 for simplicity. The graphs for Prism manifolds are characterized by We find that the number of vacua of the quiver is given by  Table 3: Three-manifold, quiver data and vacuum count in terms of the Witten index I for various Seifert manifolds M 3 with G = U (2).
Note that for q = 1 this reduces to  where : T H 2 (M 3 , Z) × T H 2 (M 3 , Z) → Q/Z is the linking form. As a check, we may verify the set of vacua found above falls into representations of the 1-form symmetry operators. For example, for L(p, q), Γ = Z p . When p is odd, the center, Z, of the 1-form symmetry is trivial, and so the irreducible representations have dimension p, while when p is even, the center is Z 2 , and so they have dimension p/2. In both cases, we find the number of vacua, given by p(p+1) 2 , is indeed divisible by the size of this irreducible representation, which is compatible with this decomposition. For an example involving a non-linear quiver, in the case M 3 = S 3 /Γ Dn , the symmetry is Z 2 × Z 2 for n even, and Z 4 for n odd. In the former case, this symmetry is anomaly-free, so the irreducible representations have size 1, while in the latter, it has an anomaly with coefficient 2, and so the irreducible representations have size 2. Thus the number of vacua must be even in the latter case, and one can verify from (5.37) that this is always true.

T [M 3 , U (N )]
In principle, the considerations above carry over to the case of general U (N ). That is, we From the discussion above, the 1-form symmetry of this theory is Z k , with an anomaly with coefficient N (mod k). The center of this symmetry therefore is Z (N,k) , and so the irreducible representations have size k/(N, k). One may check that the index (5.43) is indeed divisible by this integer, giving another check that the 1-form symmetry acts as expected.

Bethe Vacua for T [M 3 , su(2), H]
In this section we construct the physical theories T [M 3 , su (2), H] from T [Ω, SU (2)] and count the number of vacua. Recall that the unphysical theory T is obtained by gauging all nodes of a graph Ω corresponding to the manifold M 3 at gauge group SU (2). However, as discussed in section 4.4 the theory has anomalous Z 2 1-form symmetries associated to decoupled topological sectors, which need to be eliminated to obtain the physical theories.

The Ω [k] Theory
First, consider the case M 3 = L(k, 1) using the minimal graph with a single node. The Bethe equation is where x = e 2πiu , which has k + 1 solutions, after accounting for the Weyl symmetry u → −u.
For k even, the counting of vacua is therefore the same as above, while for k odd, the decoupling procedure removes a tensor factor of dimension 2, so that The above correspond to the T [L(k, 1), su(2), H = 1] theories. When k is even, we may also consider the case H = Z 2 , which is obtained by gauging the non-anomalous Z 2 1-form symmetry of this theory. Thus, it will be important to understand how the 1-form symmetries act on the Bethe vacua of these 3d N = 2 theories. As discussed in section 3.3, the space of T 2 vacua is naturally acted on by two 1-form charge operators, U A,B , wrapping the two cycles of the torus. The action of these charge operators on the Bethe vacua is discussed in [54], and reviewed in appendix D.1. In the present case, we find these operators act on the Bethe vacua as 24 Note that in general, U A U B = (−1) k U B U A , so for k even this symmetry is indeed nonanomalous. The irreducible representations are then 1-dimensional, and the Hilbert space decomposes into the eigenspaces, H A B , introduced in (3.24). We will be interested in the space of Bethe vacua V , which is a subspace of the full Hilbert space, that also decomposes as V A , B under the 1-form charge operators wrapping the A and B cycles of the torus which we refer to as the refined Witten index. Gauging the Z 2 symmetry then amounts to projecting onto the V ++ subspace, and including the twisted sectors, as in (3.26). To analyze the decomposition (5.48) it is useful to split the common eigenbasis of the U into where | u B-inv. is invariant under U B after using the Weyl-symmetry. Clearly To determine v ++ we still need to find the action of U A on the | v ± j . Alternatively, to count the solutions we can directly solve the equation U A = ±1, and one finds the result always
The theory T [Ω [k 1 ,··· ,kn] , SU (2)] has n 1-form symmetries, with anomaly matrix given by (4.58), which we reproduce here When p is even we may gauge an additional Z (δ) 2 1-form symmetry generated by δ, the central element in (4.68). This amounts to projecting onto the V ++ eigenspace and by studying a few low n cases explicitly we find and so the answer for the full Witten index is (5.60)

Gauged Trinion
As a final example consider the manifolds S 3 /Γ ADE also discussed in section 4.4.2 and appendix B. Recall that the Seifert data can be written as with the k i given by: The T [Ω (k i ) , SU (2)] theory is the gauged trinion theory: a trifundamental chiral multiplet of SU (2) with the three SU (2) flavor symmetries gauged at levels k 1 , k 2 , and k 3 and the three Bethe equations are given by This set of equations is discussed in appendix C.2 and the number of solutions is given by 27  The Hilbert space spanned by these vacua splits into the eigenspaces of the charge operators where the δ j generate the Z r 2 1-form symmetry. Let us now study this explicitly. If two of the k i , say k 1 and k 2 , are even then the theory has a physical non-anomalous Z 2 1-form symmetry, as argued for in section 4.4.2. Thus, the theory has a 1-form symmetry generated by δ = (1, 1, 0). The charge operators act as, see (D.3), We proceed as for the T k theory and determine the solutions fixed under U B δ , which are given by By plugging in these solutions into the Bethe equations (5.62) we find that they reduce to This is nothing but the T k 3 theory with an adjoint and thus has |k 3 | + 1 vacua. 28 These As before, the remaining vacua | v ± split into pairs with eigenvalues ±1 under U B . Modular invariance fixes the dimension of the subspaces with A,B = ±1. We can repeat this for all choices of δ j , j = 1, · · · , r to determine the refined Witten index The following discussion depends heavily on the rank of the maximal 1-form symmetry, Υ = Z r 2 : 1. Lens spaces L(p, q), with p odd, and S 3 /Γ Em even have no 1-form symmetry, i.e., r = 0.
This means there is no further decomposition of V .
where we take the first cohomology to be generated by δ 1 and δ 2 . 30 We note the quiver has a symmetry which means that we can exchange the two Z 2 s, i.e., Together with modular invariance we only need to specify the dimensions in table 5.

(5.78)
We also find n − 1 twisted sector states in this case, and so 2 ]) = 2n .

Flat Connections on M 3 and the 3d-3d Correspondence
In this section we use the results for the refined Witten index of the previous section to test the 3d-3d correspondence [1]. This correspondence predicts that the Witten index of the theory , g] counts the flat g C connections on the 3-manifold M 3 , which has been extensively studied in the literature [19,21,23,36,60,61]. However, a more careful consideration of the global structure of the gauge group leads to a refinement of this statement, which we derive below. Recall that the T 2 Hilbert space of the T [M 3 , g] theory is acted on by an anomaly-free 1form symmetry with group Υ = H 1 (M 3 , Z G ). Then we may refine the Hilbert space into eigenspaces of the two 1-form operators, U γ A and U γ B , wrapping the two cycles of the torus On the other hand, we consider flat G C connections on M 3 . More precisely, in addition to ordinary G C connections, we may consider connections which are not strictly G C connections, but rather connections for the quotient group, G C /Z G . These may be classified according to their second Stiefel-Whitney class, ω ∈ H 2 (M 3 , Z G ), which measures the obstruction to lifting these to well-defined G C connections. In addition, the connections are acted on by large gauge transformations (LGTs), which are determined by elements of H 1 (M 3 , Z G ). We may refine the flat connections into eigenspaces of these LGTs, 31 which are labeled by χ LGT ∈ H 2 (M 3 , Z G ).
Then we claim 32 We will derive this formula by reduction from the 6d N = (2, 0) theory, and verify it in explicit examples below. One interesting feature of this formula is that modular invariance of the torus exchanges the two arguments on the RHS, and so we expect a symmetry under exchange of these arguments. This implies the geometric quantity on the LHS also exhibits this symmetry under exchange of ω and χ LGT , which is not obvious, but which we will verify in examples.
This can also be understood as a consequence of S-duality of the 4d N = 4 SYM theory.

Self-dual Case
Let us first consider the choice of polarization of the 6d theory associated to a self-dual group  In appendix E.1 we find the same result for the number of flat GL(2, C) connections on these spaces.

General Case
Now let us consider the case where G is not necessarily self-dual. For concreteness, we will focus on g = su(N ), but similar arguments apply in general. Consider the 6d A N −1 theory where we have singled out one of the circles of T 3 as the "time" circle. The partition vector of the 6d theory on this space depends first of all on a choice of polarization, Associated to the two compactification orders on this geometry, as discussed in the introduction to this section, there are two natural polarizations, which we consider in turn.
We first take a choice which is natural in the context of the reduction to 4d SYM. We can write (suppressing the coefficients, which are all Z N ) where the superscripts denote that we take the cup product with the A and/or B cycles of the T 2 . Then we take the polarization As discussed in [16], this choice corresponds to considering the N = 4 SU (N ) SYM theory on M 3 × S 1 t . The first factor corresponds to a refinement by a Z N 0-form symmetry of this theory, which we will ignore in what follows, and the second by its 1-form Z N electric symmetry. Specifically, we have, as in (3.5) Now let us consider the effect of specifying non-zero λ ∈ H 2 (M 3 × S 1 t ). First, if we take λ to be contained in M 3 , then we expect the flat connections on M 3 to not be genuine SL (N, C) connections, but rather to be a P SL(N, C) connection with second Stiefel-Whitney class equal to λ. Thus we find Here we must be careful with how we treat large gauge transformations. Namely, although In that case we would be taking a sum over all P SL(N, C) connections weighted according to their Stiefel-Whitney class. E.g., for λ with a leg along S 1 and one along γ ∈ H 1 (M 3 ), we have Then the polarization which gives the theory 14) The partition function labeled by a choice of λ ∈ Λ 3d is just the expectation value of the 1form charge operators along the corresponding cycles, i.e., writing λ = (0, α, β, τ ) for α, β, τ ∈ Here we recall that a charge operator lying along the time direction corresponds to taking the trace in a twisted sector of the Hilbert space. Thus we may write this in terms of the decomposition in (3.25) as

Comparison
Now to compare these, let us first rewrite the 4d polarization (6.7) as Comparing to (6.14), we see they differ in the factor H 2 (M 3 ) A , which appears here, and the dual factor, H 1 (M 3 ) Bt , which appears in (6.14). Thus to relate the 3d result to the 4d SYM partition function, and so to the flat connection counting, we should perform a Fourier transform over the variable α in (6.16). For example, taking λ ∈ Λ 4d to have component where χ A , χ B run over the characters of Z N . This is a generalization of (5.48) with the prescription that Without loss we may write the image of the generator of Z p as diag(α, α −1 ), (written as an where we identify by Weyl symmetry . As described in appendix E.2, we can classify the connections according to their class in H 2 (L(p, 1), Z 2 ) = Z 2 , which in in this case is determined by α p ∈ {±1}. These solutions are also acted on by LGTs, which act as multiplication by the Z 2 center of SL(2, C), and take α → −α .  One interesting feature of this calculation is that modular invariance of the field theory implies a relation between the Stiefel-Whitney class and LGT behavior of the flat P SL (2, C) connections. We do not know of a simple derivation of this fact.

Example: Gauged Trinion
As a final example, we consider the theory associated to the S 3 /Γ DE discussed in section 5.4.3.
Note that the analysis for L(p, q) is independent of q and is thus covered by section 6.3. We again focus on N = 2, where the results for the vacuum counting, refined by 1-form charges, are summarized in tables 4 and 5.
Let us compare this to the flat connection counting, which is reviewed in appendix E.2.
The Seifert data of a Seifert manifold with three exceptional fibers fixes its fundamental group (E.9). In the general case the number of P SL(2, C) representations is given by (E.15) and their Stiefel-Whitney classes can be computed explicitly, see appendix E.2. We find for the flat P SL(2, C) connections with w 2 = ω ∈ H 2 (M 3 , Z 2 ) D n , n = 0 mod 4 : , D n , n = 2 mod 4 :   Table 6: Counting of flat P SL (2, C) connections on S 3 /Γ Dn refined by Stiefel-Whitney class, ω, and χ LGT , for n = 0 (mod 4) on the top and n = 2 (mod 4) below. Here we show the contribution from the trivial connection in blue, the other reducible connections in green, and the irreducible connections in red.
we see v ++++ = n 4 , agreeing with the ω = (0, 0) entry above, and one may similarly compare the remaining entries.
We can also check the flat connections with non-trivial χ LGT . Let us analyze the case of  In this final section we extend our discussion to 3d N = 1 theories. The analysis of higherform symmetries will generalize to this case naturally. We focus on a special class of N = 1 theories which have an N = 2 enhancement point in the space of mass parameters. We study the action of higher-form symmetries in the space of Bethe vacua on these supersymmetry enhancement loci. In particular, starting with a known N = 2 enhancement point of the T N =1 [L(p, 1)] theory [63], we propose a quiver description for the T N =1 [L(p, q)] theory. We show that this description reproduces the expected number of vacua for g = u(2) and su (2) at the supersymmetry enhancement point.
where S is the spin bundle of M 3 and V is a rank two SU (2) -bundle. The 3d theory depends on the choice of this bundle -in the local G 2 , the existence of sections of this bundle will depend on the local metric that is induced from the ambient G 2 . The simplest choice is when V is the trivial bundle, which will be referred to as the minimal twist.
The 3d-3d correspondence for this twist was discussed in [5], where the BPS equations along M 3 were determined to be the generalized Seiberg-Witten equations (gSW) Here φ α α are sections of the normal bundle (7.2), where α, α are indices labeling the spinor representations. It was conjectured in [5], that the number of "abelian" flat (real) G-connections reproduces the Witten index, whenever the three-manifold does not admit any twisted harmonic spinors (in the above equations, this corresponds to trivial φ α α ). Here the "abelian" flat connection 33 is defined as a representation π 1 (M 3 ) → G such that all the irreducible representations contribute at most once. We will confirm this for U (2) by counting the Bethe vacua for these theories anon.
The minimal twist can also be thought of as arising from 4d N = 1 class S theories on an interval, i.e., T N =1 [Σ g ], where Σ g is a genus g curve, which now is embedded inside a local Calabi-Yau three-fold [64]. This is given by the total space of the rank two bundle where the L i are line bundles with deg L 1 + deg L 2 = 2g − 2. The BPS equations of these theories are given by a generalized Hitchin system [65] where the Φ i are adjoint-valued sections of the L i .
In order to study 3d N = 1 theories we can put this 4d N = 1 class S theory on an interval. The most general boundary conditions for T N =1 [Σ] have not been studied, however a nice observation is that the BPS equations of T N =1 [M 3 ] can be related to those of the Hitchin system (7.5). Geometrically, we can view the setup as a consequence of the Heegaard splitting of M 3 , see e.g. [42,43]. Each three-manifold can be obtained by gluing two manifolds H g ± along its boundaries ∂H g ± = Σ g . The gluing map is given by an element of the mapping class group of Σ g .
This also means that the Hitchin system (7.5) should be related to the generalized SW equations (7.3). Indeed, locally, the geometry of M 3 is given by Σ g × R with coordinates (z,z, x 3 ) and the gSW equations become where we identify Φ 1 = φ 11 = (φ 22 ) * and Φ 2 = φ 12 = −(φ 21 ) * . We see that this agrees with the generalized Hitchin system in [65] after imposing Neumann boundary conditions. Clearly this relation between generalized Hitchin and generalized Seiberg-Witten equations deserves further study.

Higher-Form Symmetries for
The analysis of sections 4.1 and 4.2, i.e., the higher-form symmetries of 3d theories obtained by dimensional reduction of the 6d (2, 0) theory, carries over directly to the N = 1 case, as it does not explicitly depend on the choice of topological twist. In particular, we expect that for any subgroup, we may define a theory with suitable modifications. For example, we expect that these will involve N = 1 rather than N = 2 Chern-Simons terms, and the coupling of the nodes to the S-walls must also be suitably modified. However, one important point is that the higher-form symmetry structure of this theory should be the same as their N = 2 cousins. This is because these theories have the same gauge groups and Chern-Simons couplings, and differ only in the couplings of additional adjoint-valued matter fields, which do not affect the higher-form symmetry structure.
It would be interesting to develop a detailed Lagrangian description of these theories for general graph manifolds. For the remainder of this section, we consider this description in the special case of M 3 = L(p, q), where we will argue that a deformation of T [M 3 ] leads to a theory with N = 2 supersymmetry, which we can study more explicitly.  , 1), which can be associated to the quiver Ω [k] , corresponding to T k ∈ SL(2, Z).
Extending our notation in the natural way, we first define the T N =1 [Ω [k] , G] theory. Following [5,63], we find that this consists of an N = 1 G vector multiplet at Chern-Simons level k and a massless adjoint scalar multiplet. 34 Then it is known that the associated 3d N = 1 theory has an N = 2 enhancement point in the space of mass parameter. Tuning on the scalar mass where g is the 3d gauge coupling, yields a pure N = 2 G k CS-theory. The mass deformation from m = 0 to m N =2 preserves the number of vacua [67]. At the enhancement point, we can define and count Bethe vacua of T N =1 [Ω [k] , G], which can be described as an N = 2 quiver consisting of a single gauged node at Chern-Simons level k, but now without an adjoint chiral multiplet. We will return to this computation below.
We conjecture that we can similarly define a more general N = 2 quiver which computes the number of vacua of the N = 1 theory for general [k 1 , · · · , k n ], which are the Lens spaces L(p, q). This can be done by taking each T k i theory, enhancing it to N = 2 supersymmetry and coupling them together with the T (G) theory. This quiver description can also be obtained from a procedure similar to section 2.2.1, with the adjoint multiplets adj(V ± ) removed from all the basic building blocks and gluing rules. The quiver for T N =1 [Ω [k 1 ,··· ,kn] , G] is then given in figure 11. It is obtained from the quiver in figure 8 by removing the adjoint attached to the left-most node. It would be interesting to show that this is indeed the same as the prescription that is obtained from the direct dimensional reduction on L(p, q) with the N = 1 twist. This prescription is supported by the vacuum count and the compatibility with the dualities of section 2.3, which will also apply for these N = 2 quivers.
The physical theory T N =1 [M 3 , g] is then obtained by decoupling various topological sectors, precisely as in the earlier discussions for the N = 2 case. 34 There are no fields associated to the zero-sections of the normal bundle as the Lens space has positive scalar curvature [5,66]. The solutions are u j = j 2k , with j = 0, . . . , 2k − 1, with the Z 2 Weyl symmetry acting as u j → u −j . Except for the two degenerate solutions u 0 , u k they appear in Weyl pairs, so The anomaly is exactly as for the N = 2 case in section 5.4, so If k is even we can gauge the Z 2 1-form symmetry of this theory. Since the Bethe vacua are known explicitly we can do this by hand. The Hilbert space is spanned by the physical vacua | u j , j = 1, . . . , k − 1 and the two generators of the 1-form symmetry (D.3) act as 35 Thus, the common eigenvectors of the U are | u k/2 and | v ± j = 1 √ 2 (| u j ± | u k−j ), j = 1, . . . , k/2 − 1. The eigenvalues of the U are given by The overall sign of U A is fixed by modular invariance.
From this we easily see that the subspace with eigenvalues (+1, +1) has dimension k 4 . We can explicitly check this structure for general linear quivers. This is analogous to the N = 2 theories discussed in section 5. We find from which we can deduce that (7.18) Furthermore, we can gauge the Z 2 1-form symmetry if p is even. Doing this explicitly for small values of n and p we find as for the N = 2 twist, there is also a single twisted sector, leading to The interesting point to note here is that the Bethe vacua counting is -as expected, although it is not manifest -independent of q. It would be interesting to extend this to higher rank and more general, non-linear quivers and to test the N = 1 3d-3d correspondence for these theories.

Acknowledgments
We thank C. Closset

A Notations and Nomenclature
As there are several notational intricacies in this paper, we summarize these in the following table. where (z 1 , z 2 ) ∈ C 2 . The first homology group is with p = |p 2 p 3 + p 2 q 3 + p 3 q 2 |.
The last manifold is also called the Poincaré homology sphere, and is a famous example of an integer homology sphere, i.e., with the same homology as S 3 but different fundamental group. For simplicity we will not consider the general case with non-trivial q i .
These three series are connected to the semi-simple Lie algebras by the McKay correspondence.
In particular, if all the q i = 1 they are known as the S 3 /Γ ADE

C Twisted Superpotentials and Bethe Equations
In this appendix we review some aspects of the twisted superpotential and associated Bethe  (2)]. To compute the twisted superpotential for this theory, we first need to compute the contribution from the building blocks: ϕ = T k and ϕ = S.
Let us start with the T (SU (2)) theory, i.e., ϕ = S. It has a UV Lagrangian description as a U (1) gauge theory with two hypermultiplets. This theory has a manifest SU (2) m flavor symmetry acting on the hypers, and the U (1) J topological symmetry enhances in the IR to another SU (2) ζ flavor symmetry. We consider the N = 2 * version of the theory by turning on a coupling to the diagonal combination of the R-symmetry where U (1) C and U (1) H are a maximal torus of the SU (2) C and SU (2) H factors of the Rsymmetry. The global symmetry of the theory in the infrared is then The twisted superpotential can be written as where the first term is the contribution from the FI parameter and the contribution from the matter can be written as where W Φ is defined in (5.16). Here and below we use the convention that each choice of sign in '±' is summed over.
As mentioned in section 2.3.2 this theory has a dual description as an SU (2) theory with a k = 1 CS term and N f = 2 fundamental flavors, see figure 5. This dual theory has twisted superpotential W S (u, ζ, m; τ ) = W Φ (τ ± u ± m ± ζ) + u 2 − W adj (ζ; τ ) − ζ 2 − m 2 , (C. 6) where the first term comes from the trinion coupling. We have added an adjoint chiral multiplet (the "flip" field) charged under the SU (2) ζ symmetry, as well as background CS terms ("contact terms") for the SU (2) m ×SU (2) ζ symmetries. These are both necessary for a precise duality to the usual description of T (SU (2)) above. Here, and in general, coupling an adjoint chiral multiplet of positive U (1) t charge corresponds to whereas an adjoint of negative charge contributes with −W adj (up to τ -dependent contact terms, which we ignore). In addition, we have equipped the flavor groups in T (SU (2)) with a background Chern-Simons term. The corresponding contribution to the twisted superpotential where k is the background level.
We may use these ingredients to construct the twisted superpotential of a general quiver theory, T [Ω, SU (2)]. Including a copy of the twisted superpotential, W S , of the T (SU (2)) theory for each edge, and an appropriate Chern-Simons term for each node, we may write this conveniently in terms of the linking matrix Q ij of (2.13), as where N adj i is the number of adjoints attached to the ith node, which can be determined following the prescription in section 2.2.3, and I runs over the connecting T (SU (2))s. Here u i is the SU (2) fugacity for the ith node.
The number of vacua is given by the number of solutions to the vacuum equations using the definition (5.19) modulo the action of Z 2 Weyl group As a concrete example we will go through the explicit computation of the index for the Lens space with p q = k 1 − 1 k 2 . The quiver for this theory is shown in figure 4.
We first compute the vacua for T [Ω [k 1 ,k 2 ] , SU (2)]. The linear quiver with ϕ = T k 1 ST k 2 has twisted superpotential where the . . . only depend on τ . However, we can also use the dual trinion description, discussed in section 2.3.2. Then, the theory is described by a T 2 trinion with the three flavor groups gauged at levels ( k 1 , k 2 , k 3 ) = (k 1 − 1, k 2 − 1, 1) respectively and no additional adjoints.
The superpotential for general k i is Using the description in terms of the trinion with SU (2) 3 gauge group, the Bethe equations are given by where (i, j, k) ∈ {(1, 2, 3), (2, 3, 1), (3, 1, 2)}. This gives a system of three couple polynomial equations in the x i .
Next we must count the solutions to these equations, modulo Weyl invariance. We first factor out the Weyl invariant solutions at x i = ±1, which are unphysical, and we may then change to Weyl invariant variables, y i = x i + x −1 i . Note that an arbitrary Weyl-invariant polynomial in x i can be written as a linear combination of the polynomials For n non-negative, this is a degree n polynomial in y i , which has only even or odd degree terms. This definition extends also to negative n, and satisfies p n (y i ) = −p −2−n (y i ) .
(C. 16) Then one finds the Bethe equations, (C.14), lead to the following equations for y i P 1 (y 1 , y 2 , y 3 ) ≡p k 1 +1 (y 1 ) + t 4 p k 1 −3 (y 1 ) − ty 2 y 3 (p k 1 (y 1 ) + t 2 p k 1 −2 (y 1 )) + t 2 (y 2 2 + y 3 2 − 2)p k 1 −1 (y 1 ) = 0 , (C. 17) and similarly for P 2 and P 3 . 38 This represents a system of three coupled polynomial equations, which is difficult to solve analytically. However, we may attempt to count the solutions by using the fact that their number does not jump as we vary the parameter t continuously. Thus, let us consider the behavior in the t → 0 limit. We first assume all k i ≥ 0. Then this limit is well-behaved because the leading term of each P i survives in the limit. Then the equations simplify to and we find that the number of vacua is One can check that the equations are invariant under taking all k i → − k i and also t → t −1 , so one finds a similar formula when all k i ≤ 0.
When two of the k i 's have different signs, we may not use this argument, as the leading terms of some of the polynomials vanish as we take t → 0, so some solutions run off to infinity in a way that is difficult to control analytically. In this case, the equation (C.17) can be solved using Gröbner basis methods in Mathematica or Singular. These methods are similar to solving systems of linear equations using Gaussian elimination: the system of polynomial equations is mapped to the Gröbner basis, retaining the same set of solutions.
The system of equations can however now be successively solved. For small values of k i this can be implemented, and the results are consistent with the following formula If we now take the values these reduce to the counting of vacua in terms of p to be For larger quivers, the procedure to obtain twisted superpotential and Bethe equations is completely analogous, but the resulting systems of polynomial equations quickly become difficult to solve, even using computer algebra programs like Mathematica and Singular.
We have studied some further quivers cases with up to seven gauge group factors (leading to seven coupled equations), and some of the results are quoted and utilized in the main text.
This is now a set of coupled equations. To make progress, let us first consider a closely related theory, where we have instead a CS level k 2 and k anti-fundamental chiral multiplets, which are acted on by a U (k) flavor symmetry which we assign fugacities w a , a = 1, ..., k. Then the Bethe equations become Notice that if we take all w a → 0, we recover the equations in (C.25). This reflects the fact that giving all the anti-fundamental chirals large real masses and integrating them out, the CS level is shifted from k 2 to k. Thus we may instead do the vacuum counting in this theory with anti-fundamentals.
The equations (C.26) are still coupled, so this is is not obviously an improvement. However, let us now consider these equations in the limit z → ∞. Then we can see that the remaining factor in the Bethe equations must be approximately zero, which implies Thus the approximate solutions can be organized into towers over each w a ...
where there are a ≥ 0 39 solutions in the tower above w a , and a a = N . The number of such solutions, up to permutations, is just the number of ways of distributing N elements among k boxes, which gives Next we consider the counting for G = SU (N ). For this, we recall that U (N ) ∼ = (SU (N ) × U (1))/Z N . Then each U (N ) CS theory above is a 1-form gauging of a tensor product of the corresponding SU (N ) theory with a U (1) theory. 40 This U (1) theory has a level N k Chern-Simons term, and so N k vacua, and we must gauge the diagonal of its Z N 1-form symmetry with that of the SU (N ) factor. Then we expect this gauging to simply remove a factor of N 2 from the number of vacua of the tensor product. 41 Thus we find For the N = 1 twist, this gives which is the known result for the pure CS theory. For the N = 2 twist, we find We note that the results (C.29) and (C.32) agree with [18], which were computed by different means.

D One-form Symmetries and Anomalies for Bethe Vacua
In this appendix we discuss the action of 1-form symmetries on the supersymmetric Bethe vacua of a 3d N = 2 theory. We mostly focus on Z 2 1-form symmetries in the context of SU (2) m gauge theories, but much of the discussion below can be easily generalized. 39 If a = 0, wa and its tower is absent from (C.28). 40 Note this argument would not hold if there were fundamental matter. 41 More precisely, if (N, k) = 1, these two tensor factors have a 1-form symmetry with a maximal anomaly (in the sense that the center is trivial). In this situation, we have seen that gauging the ZN symmetry removes a factor of N 2 from the vacuum counting. We have checked for some low values of N and k that this holds also when (N, k) > 1.

D.1 1-form symmetries and Bethe vacua
Let us first describe how 1-form symmetries act on the space of supersymmetric, or Bethe vacua on T 2 . We refer to [54] for a derivation of these statements, and more details. where r G is the rank of the gauge group. These are rational functions of the gauge variables, u i , i = 1, ..., r G , as well as the other flavor symmetry parameters, which we suppress from the notation. We may associate states in the Hilbert space of vacua to the solutions, u, to these equations (modulo Weyl symmetry) Recall that 1-form electric symmetries in a gauge theory are associated to a subgroup Γ of the center of the gauge group which acts trivially on the matter content. To such a 1-form symmetry, we expect that we may define operators acting on the vacua on T 2 , U A,B γ , for each element γ ∈ Γ, and associated to the two cycles of the torus, as described in section 3.3.
To see how these operators act on the Bethe vacua of (D.2), note that we may identify the center of the gauge group with a set of transformations acting on the gauge variables, u i → u i + γ i , which are defined up to large gauge transformations. Now suppose an element γ = (γ 1 , ..., γ r G ) in the center acts trivially on the matter. Then we may define the 1-form operators acting on V as above, which act on the basis of Bethe vacua as We note both operators are well-defined when acting on the space of Bethe vacua (one can show Π γ is a rational function, given the assumption that γ acts trivially), and satisfy the appropriate group law. One also finds their commutation relations correctly encode the anomalies of the theory Thus in addition to the space of vacua, we can consider the extra data of how the vacua are acted on by these symmetries. This data will be important when gauging the symmetries.
To illustrate this with a simple example, consider the T [Ω [k] , SU (2)] theory, which is an SU (2) gauge theory with level k CS term and an adjoint. The Bethe equation is Then the center acts as u → u + 1 2 , or x = e 2πiu → −x, and so we may take γ = ( 1 2 ) above, and find from (D.3) These satisfy U A γ U B γ = (−1) k U B γ U A γ , which is the expected anomaly coefficient. The sign for U A is fixed by modular invariance.
Finally, we must describe the twisted sectors, as in (3.25). As shown in [54], for each state that is fixed by the operator U B γ , this state also contributes to the twisted sector H γ . Conversely, if a solution, u, is fixed by U B γ for all γ in a subgroup, Γ u , of Γ, then this solution contributes |Γ u | states, one untwisted sector and a twisted sector for each non-trivial element in Γ u .
These twisted sector states become physical states in the theory obtained by gauging Γ.
Then we can understood the counting above physically from the fact that, in this situation, Γ u corresponds to an unbroken subgroup of the gauge group acting at u, and so there remains a low energy Γ u -gauge theory at this solution, which is known to have |Γ u | states. In the example above, we see that the non-trivial twisted sector has a single state, corresponding to the u = 1 4 fixed point.

D.2 Representations of Z 2 1-form symmetries
Now let us specialize to a gauge theory with SU (2) n gauge group, as all the T [Ω, SU (2)] theories may be taken to have this form. The 1-form symmetry group, Γ ∼ = Z 2 n , can be identified with the subgroup of the center of the gauge group, Z G = Z 2 m , that acts trivially on the matter. This 1-form symmetry may have 't Hooft anomalies, which are obstructions to gauging, as discussed in section 3. Namely, if we pick a basis for Γ as a Z 2 vector space, then we may define the "anomaly matrix," A, and the mutual anomaly of two 1-form symmetries, γ, γ ∈ Γ, is given by γ T Aγ . Then in order to gauge a subgroup, Λ, of Γ, we require The anomaly matrix defines a symmetric bilinear form on the Z 2 vector space, Γ. Such a bilinear form, B, is classified according to its dimension d its rank d − r and whether or not it is "even", meaning that every element has vanishing inner product with itself [69]. For a non-even bilinear form, we have where E is the unique non-degenerate bilinear form on Z 2 , and Z is the trivial bilinear form on Z 2 . For an even bilinear form, the rank must be even, and we find where H is the bilinear form on Z 2 2 associated to the matrix 0 1 1 0 . (D.10) The space of vacua forms a representation of the group of U A γ and U B γ , defined above, which can give strong constraints on the structure of this space, especially in the presence of anomalies. To illustrate this, let us consider a single Z 2 symmetry with a non-vanishing self-anomaly, i.e., corresponding to the space E above. Then the U A and U B operators each form a Z 2 group acting on the vacua, but the anomaly implies the full group acting on the vacua is a central extension of Z 2 × Z 2 , which is given by the dihedral group, D 4 . Then the only non-trivial 42 representation of this group is two dimensional, and we may take U A and U B to act as We will denote this basic two dimensional representation as R. More generally, if the 1-form group is isomorphic to E m , then the irreducible representations are tensor products R ⊗m , each of dimension 2 m . Similarly, for H, the basic irrep is the tensor product of two copies of R, associated to the pairs U A 1 , U B 2 and U A 2 , U B 1 . Explicitly, we may represent these as where we observe that each non-degenerate dimension of B contributes a tensor product of an R representation, for a total of 2 (d−r) in each irrep, and dim(V ) is the number of such irreps.
In particular, we see that the number of vacua must be a multiple of 2 (d−r) . Finally, V itself may be decomposed into the eigenspaces associated to the various Z factors (D.14) Note that (D.13) and (D.14) are a special case of the more general factorization and decomposition of the Hilbert space discussed around (3.24).
Gauging To gauge a non-anomalous symmetry group, Λ ⊂ Γ, as noted above, we project onto the subspace Here the superscript "untwisted" indicates that in general this is only a subset of states of the gauged theory, as there may also be "twisted sector" states, described above. where the 1-form operators act as in (D.12). We can see from this representation that the simultaneous 1-eigenspace of U A 1 and U B 1 is generated by the state (1, 0, 1, 0), and so gauging effectively cuts the dimension by a factor of 4. In particular, we find simply V (1, 0) gauged, untwisted = V .
Iterating the second example, we see that each time we gauge a non-anomalous subgroup of the non-degenerate part of B, we simply remove two tensor powers of R from the Hilbert space.

D.3 T [M 3 , U (2)]
As an example, let us consider the theory T [M 3 , U (2)]. As described in the main text, this consists of two decoupled sectors, the T [Ω, SU (2)] theory, and the T [Ω, U (1) 2 ] theory One can check that both theories admit a Z 2 1-form symmetry acting on each node, and the anomaly matrices are identical, given in both cases by, e.g., (4.58) in the case of M 3 = L(p, q). Thus if we take the diagonal sum of these two Z 2 symmetries at each node, this combination is non-anomalous, and the Hilbert space takes the form of (D.18). We may then gauge these symmetries to form the gauge group (U (1) × SU (2))/Z 2 ∼ = U (2), which simply removes a tensor factor of R ⊗2 from the Hilbert space. Thus each such gauging reduces the dimension of the Hilbert space by a factor of four, as claimed in the main text.

E Flat Connections on S 3 /Γ ADE
According to the 3d-3d correspondence we expect that the index of the theory T [M 3 , g] counts the number of flat connections on M 3 . The precise statement is discussed in section 6. In this section we will investigate this correspondence by computing the G C -connections on In the N = 1 twist we only count the flat connections with n i = 0 or 1, ∀i, which behave like fermionic states. In this case we find #("abelian" GL(N, C) reps of L(k, 1)) = k N . (E.8)

E.2 P SL(2, C)
Next we consider the P SL(2, C) connections for the Seifert manifolds M 3 = [0; 0; (p i , q i )] with three exceptional fibers. The fundamental group is given by π 1 (M 3 ) = x, y, h|x p 1 = h q 1 , y p 2 = h q 2 , (xy) −p 3 = h q 3 , (E.9) where h is a central element [71]. 43 43 Note that this group reduces to a cyclic group if any of the pi = 1.
Next we need to compute the Stiefel-Whitney (SW) class of these connections. For this, we note that a P SL(2, C) representations of a group Γ determines an SL(2, C) representation of some Z 2 central extension of Γ. Such extensions are classified by the group H 2 (Γ, Z 2 ), and this determines the Stiefel-Whitney class of the connection. To compute this we note that for the ADE groups we are interested in which implies this is an SL(2, C) representation of Z p × Z 2 for j even, and Z 2p for j odd. Thus we find the counting of flat P SL(2, C) connections with w 2 = ω is given by p = 0 mod 4 : For an example involving irreducible connections, we consider D n with even n. Here Γ Dn is the binary dihedral, or dicyclic group, Dic n−2 , with presentation Dic n−2 = x, y, h | x 2 = y 2 = (xy) n−2 = h, h central .

(E.22)
The commutator subgroup is Z n−2 , generated by (xy) 2 , and the quotient is generated by x and y. Thus