Refined 3d-3d Correspondence

We explore aspects of the correspondence between Seifert 3-manifolds and 3d $\mathcal{N}=2$ supersymmetric theories with a distinguished abelian flavour symmetry. We give a prescription for computing the squashed three-sphere partition functions of such 3d $\mathcal{N}=2$ theories constructed from boundary conditions and interfaces in a 4d $\mathcal{N}=2^*$ theory, mirroring the construction of Seifert manifold invariants via Dehn surgery. This is extended to include links in the Seifert manifold by the insertion of supersymmetric Wilson-'t Hooft loops in the 4d $\mathcal{N}=2^*$ theory. In the presence of a mass parameter for the distinguished flavour symmetry, we recover aspects of refined Chern-Simons theory with complex gauge group, and in particular construct an analytic continuation of the $S$-matrix of refined Chern-Simons theory.

Simons theory using localization has appeared in [4][5][6]. For a recent review of the 3d-3d correspondence we refer the reader to [7].
The purpose of this paper is to explore aspects of the 3d-3d correspondence for Seifert manifolds [8][9][10][11]. Seifert manifolds are circle fibrations over a Riemann surface and therefore admit a locally-free circle action. The corresponding 3d N = 2 theory has a distinguished u(1) f flavour symmetry associated to this circle action, which can be incorporated into partition functions on squashed S 3 or S 1 × S 2 by turning on a mass parameter or fugacity.
Furthermore, the construction of Seifert manifolds via surgery on a torus is expected to have a counterpart in the construction of 3d N = 2 theories using boundary conditions and interfaces implementing SL(2, Z) duality transformations in a 4d N = 2 * gauge theory.
A natural question is how the additional parameter for the distinguished u(1) f flavour symmetry manifests itself as a 'refinement' of complex Chern-Simons theory. Our goal is therefore to develop a concrete dicionary between the partition function of T (M 3 ) on squashed Chern-Simons theory. In particular, we will reproduce an analytic continuation of the Smatrix of refined Chern-Simons theory introduced in [12,13] from the partition functions of T (S 3 ) in the presence of supersymmetric loop operators.

Summary
We will focus on twisted compactifications of the six-dimensional superconformal N = (2, 0) theory of type g = su(N ) on a compact Seifert manifold M 3 . This leads to a 3d N = 2 theory As explained in [14], the twisted compactification on a 3-manifold with torus boundary should be regarded as a boundary condition in 4d N = 4 gauge theory. Choosing a metric on M 3 such that the boundary region forms a semi-infinite cylinder R + × T 2 with complex structure τ , compactification on T 2 in the asymptotic region leads to a 4d N = 4 theory with gauge algebra g on a half-line R + with holomorphic gauge coupling τ . The 3-manifold M 3 adjoined to this semi-infinite cylinder then defines a boundary condition in the 4d N = 4 theory preserving 3d N = 4 supersymmetry [15][16][17]. Turning on a mass parameter for the distinguished u(1) f flavour symmetry corresponds to adding a codimension-2 defect supporting the u(1) f flavour symmetry wrapping a curve in M 3 that intersects the boundary at a point p ∈ T 2 . In particular, in the cylindrical region R + × T 2 the codimension-2 defect is wrapping R + × {pt}. This is illustrated in the top of figure 1. This corresponds to turning on an N = 2 * deformation of the 4d N = 4 gauge theory and the boundary condition now preserves 3d N = 2 supersymmetry and flavour symmetry u(1) f . In many cases, a genuinely three-dimensional theory can be obtained from a boundary condition in the degeneration limit τ → +i∞, where the four-dimensional degrees of freedom are decoupled. In this limit, the boundary T 2 degenerates and we obtain a compact 3-manifold where the boundary is replaced by a maximal codimension-2 defect of the 6d N = (2, 0) theory supporting a flavour symmetry g. This flavour symmetry is then gauged in coupling to the 4d N = 2 * theory when the gauge coupling is turned back on.
Extending the discussion above, a 3-manifold with a pair of torus boundaries corresponds to an interface between 4d N = 2 * theories. For example, in the Dehn surgery M + 3 ∪ φ M − 3 , the mapping class element φ ∈ SL(2, Z) corresponds to a mapping cylinder implementing the modular transformation on T 2 . This corresponds to an interface implementing the corresponding SL(2, Z) duality transformation of the 4d N = 2 * theory. Such interfaces can also viewed as 3d N = 2 theories in their own right associated to compact 3-manifolds with a pair of codimension-2 defects supporting g flavour symmetries. For example, the generator φ = S corresponds to a Hopf network of codimension-2 defects in S 3 supporting flavour symmetries g, g and u(1) f . This corresponds to the three-dimensional theory T (g) introduced in [17]. This is illustrated in figure 2.  A large class of Seifert manifolds known as Lens spaces can be constructed by starting from a mapping torus implementing an SL(2, Z) duality transformation and then capping off the torus boundaries with solid tori D 2 × S 1 . This corresponds to constructing the cor-responding theory T (M 3 ) by compactification of a 4d N = 2 * theory on an interval with boundary conditions at each end corresponding to the solid tori D 2 × S 1 and a sequence of SL(2, Z) duality interfaces inserted in the intermediate region. For more general Seifert manifolds, one needs to consider boundary conditions and interfaces for a 4d N = 2 * theory with gauge algebra equal to a direct sum of several copies of g.
This setup can be further enriched by including codimension-4 defects of the 6d N = (2, 0) theory labelled by a dominant integral weight of g. We will focus on the case of codimension-4 defects labelled by the fundamental weights of g, or equivalently by the anti-symmetric In the course of this paper, we will implement the construction outlined above to compute the partition functions of theories T (M 3 ) on the squashed three-sphere S 3 b [18] (generalizing the round sphere introduced in [19][20][21]) in the presence of a mass parameter for the distinguished u(1) f flavour symmetry.

Outline
We begin in section 2 by summarizing our conventions for the 4d N = 2 * theory and describing the class of 3d N = 2 boundary conditions and interfaces that will appear throughout the paper.
In section 3, we consider the supersymmetric vacua of the 4d N = 2 * theory on S 1 × R 3 and therefore the supersymmetric vacua of the theories T (M 3 ) on S 1 × R 2 . We recall how the Coulomb branch has a description as the moduli space of SL(N, C) flat connections on T 2 /{p}, and describe the Coulomb branch images of the aforementioned 3d N = 2 boundary conditions and interfaces as holomorphic Lagrangian submanifolds.
In section 4, we consider boundary conditions and interfaces in the 4d N = 2 * theory on Having introduced the necessary tools, in section 5 we construct the partition function of N = 2 theory T (S 3 ) in a variety of ways from compactifying the 4d N = 2 * theory on an interval with appropriate boundary conditions. We then introduce codimension-4 defects labelled by anti-symmetric tensor representations of su(N ) using supersymmetric Wilson-'t Hooft loops in the 4d N = 2 * theory, corresponding to the unknot and Hopf link in S 3 . In this way, we recover an analytic continuation of the S-matrix of refined Chern-Simons theory.
Finally, in section 6 we construct the partition functions of T (M 3 ) for more general Lens spaces and Seifert manifolds, and perform further checks of our proposal in various limits.
We conclude in section 7 with directions for further study. Appendices A-C provide some conventions, background and further details of our computations.

The N = 2 * Theory
The 4d N = 2 * theory consists of an N = 2 vectormultiplet together with a hypermultiplet in the adjoint representation of the gauge algebra g, which we will assume to be su(N ) 1 . In addition to the standard R-symmetry u(1) r ⊕su(2) R , the theory has a u(1) f flavour symmetry acting on the adjoint hypermultiplet. The mass parameter for the adjoint hypermultiplet is obtained by coupling to a background vectormultiplet for u(1) f and turning on a background expectation value m for the scalar component.
We will denote the complex scalar in the dynamical vectormultiplet by φ and decompose the adjoint hypermultiplet scalars into a pair of complex scalars (X, Y ). The charges of these fields under the Cartan generators of the R-and flavour symmetries are given in table 1.

Boundary Conditions
We will consider boundary conditions preserving a 3d N = 2 supersymmetry with unbroken Rsymmetry and u(1) f flavour symmetry. We introduce a coordinate s normal to the boundary and coordinates x j = {x 1 , x 2 , x 3 } parallel to the boundary. In general there is an S 1 × CP 1 family of such boundary conditions corresponding to a choice of breaking pattern u(1) r ⊕ su(2) R → {pt}⊕u(1) R . We choose the phase such that (A j , Re(φ)) and (A s , Im(φ)) transform as a 3d N = 2 vectormultiplet and chiral multiplet respectively, and u(1) R is generated by T R from table 1 such that X and Y transform as chiral multiplets.
The basic boundary conditions for the vectormultiplet correspond to a choice of Neumann boundary conditions for (A j , Re(φ)) and Dirichlet boundary conditions (A s , Im(φ)) or vice versa [14]. In more detail, the boundary conditions are defined by and a is a valued in a Cartan subalgebra of g. Neumann boundary conditions preserve the full gauge symmetry g, whereas Dirichlet boundary conditions break the gauge symmetry but inherit a global symmetry equal to the subalgebra of g commuting with a. For Neumann boundary conditions (A j , Re(φ)) transform as a 3d N = 2 vectormultiplet at the boundary, whereas for Dirichlet boundary conditions (A s , Im(φ)) transform as a chiral multiplet.
The boundary conditions for the N = 2 hypermultiplet correspond to a choice of Neumann boundary conditions for X and Dirichlet for Y or vice versa. We will therefore consider the following 'Neumann' boundary conditions and 'Dirichlet' boundary conditions Note that X has Neumann boundary conditions in N X and D Y and becomes a chiral multiplet on the boundary, whereas Y has Neumann boundary conditions in N Y and D X and becomes a chiral multiplet on the boundary, with charges as in table 1. If we want to emphasize the dependence on the boundary expectation value a, we will write Dirichlet boundary conditions as D X (a), D Y (a).
These basic boundary conditions can be modified by coupling to boundary degrees of freedom [14]. For example, the Neumann boundary condition N X can be modified by coupling to a 3d N = 2 theory with unbroken R-symmetry u(1) R and flavour symmetry at least u(1) f ⊕ g by coupling to the dynamical vectormultiplet at the boundary. We can also add a boundary superpotential W (X|, O) depending on additional boundary chiral operators O, which modifies a right boundary condition to and a left boundary condition to In the paper we use the notation · | and | · to denote the expectation values of bulk operators at right and left boundary conditions respectively.
An important example is to deform the right Neumann boundary condition N X by a boundary chiral multiplet O Y with the same T R and T f charges as Y and a boundary super- From equations (2.4), it is straightforward to see that this boundary condition flows to N Y with Y | = O Y , and similarly one can convert the boundary condition N Y back to N X . There is an essentially identical construction for Dirichlet boundary conditions. Following [14,22], we will refer to this operation as a 'flip'.

Interfaces
We will also consider interfaces preserving a 3d N = 2 supersymmetry with unbroken Rsymmetry u(1) R and flavour symmetry u(1) f . A variety of such interfaces can be constructed by coupling the basic boundary conditions introduced above to additional three-dimensional degrees of freedom by gauging and/or adding a boundary superpotential.
An important class of interfaces are those that flow to the identity interface. For example, let us first impose Dirichlet boundary conditions D Y (a) on the left and D Y (a ) on the right of the interface. We then identify the boundary flavour symmetry on each of these boundary conditions and gauge it by coupling to a dynamical 3d N = 2 vectormultiplet. Finally, we add a boundary chiral multiplet O and a boundary superpotential The boundary superpotential requires ensuring that the interface identifies the chiral multiplets on each side. There is an identical construction starting from D X boundary conditions by exchanging the role of Y and X. Such interfaces will be used to 'cut' the path integral in our computations in section 4.
Another important class of interfaces are those that implement SL(2, Z) duality transformations 2 . SL(2, Z) duality transformations are generated by S and T satisfying where P is a central element such that P 2 = I. The corresponding interfaces were introduced in [17].  The interface generating the action of T on boundary conditions is constructed by adding an N = 2 supersymmetric Chern-Simons term at level +1. To construct an S-duality interface at s = 0, we deform a right N X boundary condition on s ≤ 0 and a left N Y boundary condition on s ≥ 0 by coupling to the three-dimensional theory T (g) at s = 0 and gauging the flavour symmetry g ⊕ g [17].
There is a description of T (g) as a triangular quiver with gauge algebras u(j) for j = 1, . . . , N − 1. The g symmetry that rotates the N pairs of chiral at the final node is manifest, while the second one is an enhancement of the u(1) N −1 topological symmetry in the infrared.
Sandwiching the S interface between Dirichlet boundary conditions D X (a) on the left and D Y (a ) on the right isolates the three-dimensional degrees of freedom in T (g). In particular, 2 Provided it is simply-laced, SL(2, Z) transformations do not change the gauge algebra g. However, there are distinct physical theories on R 4 with the same g but different sets of mutually compatible line operators, on which SL(2, Z) transformations act in an intricate way [23]. We will generally omit this distinction, mentioning it explicitly when needed. a = (a 1 , . . . , a N ) are identified with the mass parameters and a = (a 1 , . . . , a N ) with the FI parameters of T (g) -as shown in figure 3.   By an SL(N, C) transformation, we can diagonalize the holonomy matrix W and introduce the following convenient parameterization of the holonomy matrix H, where and are the traces of the holonomy matrices Tr Λ r (W ) and Tr Λ r (H) respectively in the antisymmetric tensor representations Λ r of SL(N, C) of rank r = 1, . . . , N − 1. In these expressions, For example, It is also straighforward to compute the trace of the holonomy around other cycles of T 2 /{p} in terms of these coordinates,

Boundary Conditions
The image of a boundary condition preserving 3d N = 2 supersymmetry is a holomorphic The parameters {h + 1 , . . . , h + N } and {h − 1 , . . . , h − N } are the four-dimensional lift of the abelian monopole operators introduced in [26] to describe the Coulomb branch of 3d N = 4 gauge theories and further used in [27] to find the Coulomb branch images of 2d N = (2, 2) boundary conditions. We can therefore uplift these results to compute the Coulomb branch images of 3d N = 2 boundary conditions in the 4d N = 2 * theory.

Neumann
Let us first consider Neumann boundary conditions. The holomorphic Lagrangians for right Neumann boundary conditions N X and N Y are (3.12) In terms of the original variables, the Neumann boundary condition N X is described by h i = 1 whereas N Y is described byh i = 1.
It is straightforward to check that both Neumann boundary conditions N X and N Y in fact describe the same holomorphic Lagrangian, which can be defined invariantly by fixing the eigenvalues of the holonomy matrix H to be t ρ , where is the Weyl vector.
In terms of supersymmetric non-abelian 't Hooft loops, the right N X boundary condition has the property that This expression is in fact independent of w j and sums to which is the quantum dimension of the representation Λ r with quantum parameter t. Since the quantum dimension is invariant under t → t −1 , we obtain the same result for N Y . This reproduces the localization computation of the S 1 partition function of an N = 4 gauged quantum mechanics that flows to a sigma model onto the Grassmannian Gr(r, N ) [28]. This can be interpreted as the S 1 partition function of the one-dimensional degrees of freedom supported on the 't Hooft loop.
It will also be important to note the expectation values of mixed Wilson-'t Hooft loops at the Neumann boundary condition N X , Removing the puncture by turning off the mass parameter for the u(1) f symmetry sends t → 1, and therefore the holomorphic Lagrangian for a Neumann boundary condition becomes This shows that the holonomy around the (0, 1) cycle becomes trivial. The 3-manifold corresponding to this holomorphic Lagrangian is therefore a solid torus S 1 × D 2 obtained by contracting the (0, 1) cycle.
Turning back on the mass parameter for the u(1) f symmetry, the holomorphic Lagrangian still describes a solid torus S 1 × D 2 obtained by collapsing the (0, 1) cycle, but now punctured by a monodromy defect at the origin of the disk D 2 with fixed holonomy eigenvalues t ρ . We will simply refer to this as the solid torus S 1 × D 2 obtained by collapsing the (0, 1) cycle, with the presence of the monodromy defect understood.
Finally, the boundary Ward identities for left Neumann boundary conditions are found by exchanging the roles of h + i and h − i in the above formulae, which define the same holomorphic Lagrangian in this example.

Generalized Neumann
We now briefly consider the generalized Neumann boundary conditions N X [T ] and N Y [T ] obtained by coupling Neumann boundary conditions N X or N Y to a 3d N = 2 gauge theory T with unbroken R-symmetry u(1) R and flavour symmetry at least g ⊕ u(1) f .
Let us denote the effective twisted superpotential of the three-dimensional theory T by W(w j , t, s a ), where s a are the abelian Wilson loops for the three-dimensional gauge symmetry 3 . The boundary Ward identities generalizing those for pure Neumann boundary conditions (3.12) are  (3.20) where the function f (w) satisfies It is straightforward to check using equation (3.18) that the boundary Ward identities for

Dirichlet
Let us now consider the Dirichlet boundary conditions D X . The holomorphic Lagrangian is defined by setting the eigenvalues of W equal to fixed values {w 0 1 , . . . , w 0 N }, or equivalently by fixing the expectation values of supersymmetric Wilson loops W (r) for all r = 1, . . . , N − 1.
The corresponding three-manifold is therefore the solid torus S 1 × D 2 obtained by contracting the (1, 0) cycle, punctured by a monodromy defect at the origin of the disk D 2 with

Interfaces
An interface corresponds to a holomorphic Lagrangian submanifold in the product M × M of Coulomb branch moduli spaces on each side of the interface, with holomorphic symplectic form Ω − Ω . We now describe the holomorphic Lagrangians corresponding to the interfaces generating SL(2, Z) transformations that were described in section 2.3.
In preparation for our discussion of the T interface, let us first consider a class of interfaces generalizing N X [T ], which are constructed by coupling to a 3d N = 2 gauge theory with unbroken R-symmetry u(1) R and flavour symmetry at least g ⊕ u(1) f . As above, we denote the effective twisted superpotential of this theory by W(w j , t, s a ). This interface defines the holomorphic Lagrangian where in the second equation we have cancelled a factor of on each side since w i | = |w i from the first equation. This is again supplemented by the vacuum condition The T interface is now a special case of the above construction where we couple to a supersymmetric Chern-Simons term at level +1, with effective twisted superpotential It therefore corresponds to the holomorphic Lagrangian which can be written more invariantly as In what follows, we will introduce a graphical notation where supersymmetric loop operators are always denoted acting on right boundary conditions. With this convention, the translation of supersymmetric loop operators through the T interface is shown in figure 5.
From the detailed computations in [24,25], this holomorphic Lagrangian can be written invariantly as In diagrammatic conventions, with the understanding that all operators act on right boundary conditions, the action of the S interface on supersymmetric loop operators is shown in figure   6. Figure 6. Under S duality, a Wilson loop becomes an 't Hooft loop.
The S-dual of the Neumann boundary conditions N X and N Y will play an important rôle later. We denote them by Nahm pole boundary conditions N P X and N P Y . Given that Neumann boundary conditions of all types correspond to setting the eigenvalues of H equal to t ρ , the Nahm pole boundary conditions correspond to setting the eigenvalues for Nahm pole boundary conditions.

Squashed S 3 Partition Function
In this section, we will replace S 1 × R 2 parallel to the boundary conditions and interfaces by a squashed three-sphere S 3 b . This will lead to a quantization of the Coulomb branch moduli space M of SL(N, C) flat connections on T 2 /{p}, which is captured by a Chern-Simons theory with complex gauge group SL(N, C). Such a quantization is specified by a pair levels (k, σ) where k ∈ Z is quantized and σ ∈ C is continuous [29]. From supersymmetric localization of the six-simensional N = (2, 0) theory [5], the expected levels for the complex Chern-Simons theory corresponding to S 3 b partition functions are Our approach will be to utilize results from supersymmetric localization of 3d N = 2 theories on S 3 b to construct partition functions of SL(N, C) Chern-Simons theory on Seifert manifolds by surgery on T 2 /{p}.

Setup
A 4d N = 2 theory on R × S 3 b can be viewed as an infinite-dimensional supersymmetric quantum mechanics on R with a pair of supercharges Q, Q † , which coincide with the supercharges used in the localization of 3d N = 2 theories on S 3 b . A compatible boundary condition that preserves 3d N = 2 supersymmetry in flat space can be represented as a 'boundary state' in the space of supersymmetric ground states annihilated by Q, Q † . Instead of attempting to describe this supersymmetric quantum mechanics directly, for example as in [30], we will perform computations using known localization results for 3d N = 2 theories on S 3 b . Our conventions regarding contributions to the S 3 b partition functions are summarized in appendix A. In particular, we have imaginary mass parameters (a 1 , . . . , a N ) obeying j a j = 0, in keeping with our choice of anti-hermitian Lie algebra generators, and an imaginary hypermultiplet mass parameter m associated to the T f symmetry. It will also be convenient to also introduce the combination = Q 2 − m, where Q = b + b −1 , such that * = Q 2 + m. With this notation, the contribution of a 3d N = 2 vectormultiplet is The contributions from chiral multiplets in the adjoint representation with the same T R and T f charges charges as X and Y (shown in table 1) are respectively. An important consequence of the identity S b (x)S b (Q − x) = 1 is that these partition functions obey K X (a)K Y (a) = 1. The physical reason is the existence of the superpotential Tr(XY ) allowing both chiral multiplets to be integrated out. As we will see momentarily, it also ensures consistency of the flip.
It is also convenient to introduce the notation which combine a 3d N = 2 vectormultiplet and an adjoint chiral multiplet with the same charges as X or Y . These combinations correspond to the contributions from 3d N = 4 vectormultiplets or twisted vectormultiplets, deformed to 3d N = 2 supersymmetry by the mass parameter m associated to T f .

Basic Overlaps
The basic computation we want to perform is the parition function of the 4d N = 2 * theory on S 3 b times an interval with 3d N = 2 boundary conditions at each end. This corresponds to the overlap of boundary states in the putative supersymmetric quantum mechanics. A standard but crucial observation is that the momentum generator P s ∝ {Q, Q † } is exact with respect to both supercharges, and therefore acts trivially on the boundary states that are annihilated by Q, Q † . The correlation functions of boundary conditions are therefore independent of the position on the s-axis, and we can perform computations by reducing the distance between boundary conditions to zero and applying known localization computations for 3d supersymmetric gauge theories on S 3 b . To gain some familiarity with such computations, we will compute the correlation functions of the Neumann and Dirichlet boundary conditions introduced in section 2.2.
Let us first consider the overlap of a Neumann boundary condition and a Dirichlet boundary condition. For the overlap of D X (a) with N X or D Y (a) with N Y , after sending the distance between the boundary conditions to zero, it is straightforward to see from the definitions (2.2) and (2.3) that there are no fluctuating degrees of freedom remaining on S 3 b and therefore the partition functions are '1'. We write this as However, for the boundary conditions D Y (a) and N X , the chiral multiplet X has Neumann boundary conditions at both ends and therefore contributes to the correlation function. Similarly, Y contributes to the correlation function of D X (a) and N Y . We therefore have However, this expression is singular with a pole of order N −1 from the contribution S b (0) N −1 of the neutral scalars, indicating that a more careful analysis is needed. Note that there is a simple pole for each independent parameter, since j a j = 0. Further, recall that the a j are imaginary: a j = ir j , and that the residue of S b (ir) at r = 0 is 1 2πi . We therefore replace the singular contribution by a Weyl invariant delta function, where S N is the set of permutations of {1, . . . , N }. This delta function should be considered as a contour prescription around the aforementioned pole. Using the identity This argument extends immediately to where the additional contributions come respectively from the chiral multiplets Y and X. It Finally, let us consider the correlation function of a pair of Neumann boundary conditions.
where we defined d N −1 a ≡ da 1 · · · da N δ(a 1 + · · · + a N ). For a pair of N X or N Y boundary conditions we have additional adjoint chiral multiplets X and Y on the boundary, so that These correspond to the partition functions of 'bad' theories in the terminology of [17] and therefore formally diverge due to the presence of unitarity violating monopole operators [31].
They can nevertheless be defined by analytic continuation, as explained in [32]. Since the partition functions are independent of boundary superpotential couplings, we would therefore expect correlation functions of D X (a) to be obtained from those of D Y (a) by multiplying by the contribution K Y (a) from O Y . Using the identity K X (a)K Y (a) = 1, it is straightforward to verify that this is the case in the above examples.

Cutting the Interval
Our strategy for computing a general correlation function B 1 , B 2 is to 'cut' the path integral at an intermediate point and express the result in terms of the 'wave functions' and D X (a), B 2 associated to the boundary conditions B 1 and B 2 . It is therefore convenient to introduce a shorthand notation The cutting construction can be performed using D X (a) or D Y (a) or a mixture of both, leading to considerable flexibility in notation.
Let us briefly recall the construction of the 'identity' interface from section 2.3. First, cut the interval at some intermediate point and impose the boundary condition D X (a) on the left and the boundary condition D X (a ) on the right of the cut. Next, identify the boundary flavour symmetry on each side of the cut, forcing a = a , and introduce a dynamical 3d N = 2 vectormultiplet, together with a chiral multiplet O X and the boundary superpotential which identifies the chiral multiplets X and Y across the interface.
This construction is straightforward to implement at the level of partition functions: the boundary superpotential is exact and therefore makes no contribution. Hence, the result is where we introduce the shorthand notation for the measure of integration. This is illustrated in figure 8.
Although we will mostly concentrate on cutting the path integral using D X (a) boundary conditions, it is straightforward to provide a similar construction using D Y (a) boundary conditions, leading to the following equivalent expressions where we introduce shorthand notations for the measures analogous to equation (4.16). These expressions are of course compatible since by performing a flip.
Finally, it is straightforward to check that all of the correlation functions of Neumann and Dirichlet boundary conditions in section (4.2) are compatible with this procedure. Figure 8. The construction of a general correlation function B 1 , B 2 by inserting cutting the path integral and expressing the result in terms of the wave functions Z X,B1 (a) and Z X,B2 (a).

Loop Operators
Supersymmetric Wilson-'t Hooft operators can be inserted at points in the interval and on Hopf linked circles S 1 andS 1 of length 2πb and 2π/b in the squashed three-sphere S 3 b . This corresponds to the insertion of operators in the putative supersymmetric quantum mechanics annihilated by Q or Q † . As before, their correlation functions are independent of the position on the s-axis. We will focus on supersymmetric loop operators wrapping S 1 .
It will be sufficient to determine the correlation function of a supersymmetric loop operator inserted between a Dirichlet boundary condition D X (a) or D Y (a) and a general boundary condition B. Results from supersymmetric localization imply this will act as a difference operator on the wave functions Z X,B (a) or Z Y,B (a). From these ingredients, more general correlation functions can be computed by cutting the path integral.

Wilson Loops
Let us first consider a supersymmetric Wilson loop in the representation Λ r inserted between a Dirichlet boundary condition D X (a) or D Y (a) and another boundary condition B. Moving the supersymmetric Wilson loop operator to the Dirichlet boundary condition, it is evaluated on the vacuum expectation value A j = 0 and Re(φ) = a. We therefore find where is the character of the representation Λ r and we write a I = i∈I a i . Note that if we define exponentiated variables w j = e 2πiba j this contribution concides with the expectation value of a supersymmetric Wilson loop from section 3.1. This is summarized in figure 9.
The correlation function of a supersymmetric Wilson loop between any pair of boundary

't Hooft loops
Let us now move to supersymmetric 't Hooft loops. The form of these difference operators can be determined from supersymmetric localization [33]. The result takes the following form 4 H (r) where are elementary difference operators preserving the constraint j a j = 0 and we have used the shorthand notation h I = h i 1 · · · h ir for I = {i 1 , . . . , i r }. The contributions in the numerators 4 The localization results in [33] are for supersymmetric 't Hooft loops on S 4 supported on a circle S 1 ⊂ S 3 where S 3 is the equator. In the neighbourhood of the equator, the background looks like our R × S 3 . Since the contributions to the difference operator arise from 1-loop contributions localized at the equator, we expect these expressions to be correct also for our computation. A further conjugation is required to bring these operators into the form shown here [34,35].
of these difference operators arise from 1-loop contributions from the chiral fields X and Y in the background of an 't Hooft loop, explaining the relative dependence on the combinations and * .
If we define exponentiated parameters   inverted. In what follow, we focus on constructing wave functions with D X (a), so that our formulae reduce directly to those in section 3.1 in the 'classical' limit b 2 → 0.    Finally, the difference operators acting on wave functions constructed using D X (a) and D Y (a) are intertwined by the contribution from chiral multiplets X and Y , which is a consequence of the identity . Analogously, we find dν X a (n) Q n+1,n a (n+1) , a (n) e 2πi(a n −a n+1 ) a (n) 1 +···+a  Figure 13. A quiver description of T (g) with hypermultiplet mass parameters (a 1 , . . . , a N ) and FI parameters labelled a 1 − a 2 , . . . , a N −1 − a N . parameters at the final node. The FI parameter at the n-th node is a n − a n+1 . Finally is the one-loop contribution to the partition function from the hypermultiplet in the bifundamental representation of u(n + 1) ⊕ u(n).
The integral (4.39) may be evaluated as a series expansion in e 2πi(a n −a n−1 ) by summing the contributions from the poles of the hypermultiplet contributions, see for example [25].
However, the resulting expression is rather unwieldy. An exception is the limit b = 1 and = 1, in which the partition function reduces to a product of simple trigonometric functions [36].
Nevertheless, using the integral representation (4.39), it is possible to show that the partition function obeys the following properties: • Mirror symmetry Z(a, a , ) = Z(a , a, * ) . (4.41) • It has an analytic continuation away from imaginary a, a with simple poles at for all i < j and n 1 , n 2 ∈ Z ≥0 .  The first symmetry property (4.41) reflects the expectation that T (g) is self-dual under three-dimensional mirror symmetry. This property has been proved in the case N = 2 using the integral representation in reference [37].

• It is a simultaneous eigenfunction of 't Hooft loop difference operators
The analytic structure (4.42) in the mass parameters (a 1 , . . . , a N ) can be determined from the integral representation (4.39) by analysing where the poles from the hypermultiplet contributions to the integrand collide and pinch the contour. The analytic structure in the FI parameters (a 1 , . . . , a N ) is not simple to determine directly from the integral representation (4.39) but can be determined from the analytic structure in (a 1 , . . . , a N ) using the mirror symmetry property (4.41).  (For consistency, we could also define a function S Y (a, a ) by sandwiching the interface in between boundary conditions D Y (a) and D Y (a ), although we will not need it.) The origin of the two functions is summarized in figure 15. Y (a), we find that the function S X (a, a ) has the following properties: • Mirror symmetry S X (a, a ) = S X (a , a) . (4.45) • It has an analytic continuation away from imaginary a, a with simple poles at for all i < j and n 1 , n 2 ∈ Z ≥0 .   which is expected to be the contribution of a decoupled topological sector. This is a familiar feature from the SL(2, Z) action of three-dimensional quantum field theories with abelian flavour symmetries [38].
We can prove the relation S 2 = P by inserting a supersymmetric 't Hooft loop in between the S transformation interfaces. Using the eigenfunction property (4.47) and the conjugation property (4.28) we find that

Boundary Conditions Revisited
Now that we have constructed the partition functions of interfaces generating SL(2, Z) duality transformations, we can in principle compute the partition functions involving boundary conditions in the SL(2, Z) orbits of the basic Neumann and Dirichlet boundary conditions introduced in section 2.1.
In particular, we will define the Nahm pole boundary condition such that the Neumann boundary condition N X is the S transformation of N P X . The wave functions Z X,N P X (a) = D X (a), N P X for the Nahm pole boundary condition and Z X,N X (a) = 1 are then related by and its inverse Z X,N P X (a) = dν X (a ) S X (−a, a ) . (4.59) We will not need an explicit expression for the Nahm pole wave function Z X,N P X , as we can rely the following property. By inserting a supersymmetric 't Hooft loop between the interface and the Neumann boundary condition and using the conjugation property (4.28), the eigenfunction equation (4.47) and H (r) with an identical equation for supersymmetric loop operators wrapping the circle of length 2π/b. This implies Z X,N P X (a) vanishes in the physical regime where a is imaginary. We can define the wave function by analytic continuation, although its detailed form will not be needed. The important point is that, due to (4.60), we can replace a → ρ in any invariant function f (a) multiplying the wave function Z X,N P X (a).

Case Study: T (S 3 )
We will now apply the results of the previous section to the computation of the S 3 b partition function of the 3d N = 2 theory associated to S 3 , T S 3 . In addition, we compute the

S 3 Partition Function
The simplest way to construct the three-manifold S 3 by surgery is to identify the boundaries of two solid tori D 2 × S 1 by an SL(2, Z) transformation φ = S. Using solid tori obtained by contracting the (1, 0) cycle of the boundary T 2 , this corresponds to computing the correlation function of the S interface between a pair of Nahm pole boundary conditions N P X . Equivalently, it corresponds to the correlation function of a Nahm pole boundary condition N P X and a Neumann boundary condition N X . The partition function of T (S 3 ) can therefore be expressed as = dν X (a) Z X,N P X (a) .

(5.1)
We can evaluate the integral in the second line without requiring the form of the Nahm pole wave function Z X,N P X (a). We start from the relation between the Neumann and Nahm pole wave functions (4.58) and consider the limit as a → ρ . We claim that the function S X (a , a) remains finite in this limit and is independent of a. In particular, from the eigenfunction equation (4.47), we find a) . e 2πi(a n −a n−1 ) is consistent with this argument and demonstrates that in fact The computation is performed in Appendix C. Therefore, we find Apart from the 1/ √ N factor out front, this expression coincides with the partition function of (N − 1) chiral multiplets with T R charges 2, . . . , N and T f charges 2, . . . , N .
There is an alternative surgery construction of the partition function of T (S 3 ), which is related to the computation above by following the sequence of operations shown in figure 16.
The starting point for this computation is the correlation function of the S interface between Figure 16. The sequence of moves relating the different surgery constructions of T S 3 .
a pair of Nahm poles N P X . The next step is to note that the interface T acts on the Nahm pole wave function Z X,N P X (a) by multiplying by as a consequence of equation (4.60). We can therefore insert a pair of T −1 interfaces at the expense of a framing factor T (ρ ) 2 . Next, applying the relation (4.54) and using the resulting S interfaces to convert the Nahm pole boundary condition to Neumann boundary conditions, we arrive at the final line in figure 16.
Therefore, modulo framing, T (S 3 ) can also be constructed from the T interface sandwiched between a pair of Neumann boundary conditions N X , leading to a description in terms of a supersymmetric Chern-Simons theory at level +1 and a chiral multiplet with the same charges as X. The sequence of moves shown in figure 16 translates into concrete expressions at the level of partition functions, In Appendix C, we check agreement of the asymptotic behaviour of both sides of this equation in the limit → ∞ with Im( ) > 0. In particular, this asymptotic analysis determines the framing factor T (ρ ) 2 in equation (5.6) exactly, which furthermore determines the coefficient ζ in the SL(2, Z) relations (4.49).
We therefore find that T (S 3 ) is a supersymmetric SU (N ) Chern-Simons theory at level +1 together with an a chiral multiplet in the adjoint representation, as proposed in [39]. In our construction, the adjoint chiral multiplet comes naturally with the same T R charge as X, namely +1. However, at the level of partition functions this can be modified by analytic continuation in the mass parameter m for the T f symmetry. The equivalence with (N − 1) chiral multiplets together with a decoupled topological sector is a known three-dimensional duality [40,41]. Figure 17. The computations leading to an unknot in S 3 .
Let us briefly consider the special case N = 2. The equivalence between the supersymmetric Chern-Simons and (N − 1) chiral multiplet descriptions (5.6) is equivalent to the following integral identity, where γ is a suitably deformed contour from supersymmetric localization, which satisfies γ = iR in the physical region, where Re( ) > 0, Im( ) > 0. In the limit b → 1 of a round three-sphere, this reproduces the result checked numerically in [40], with analytically continued from ∆ ∈ (0, ∞).
Finally, in the limit that we remove the mass parameter m → 0 for u(1) f and set the T R charge r to an even integer, the partition function (5.6) vanishes. This supports the expectation that, due to the absence of SL(N, C) flat connections on S 3 without monodromy defects, supersymmetry is spontaneously broken in T S 3 . This contributes W (r) (a) to the integrand, which should be evaluated at a = ρ since it multiplies the Nahm pole wave function:

Unknot in
In terms of the exponentiated variable, t = e 2πib , we have Figure 18. Sequence of moves to evaluate Z S 3 (ω r ).
which is the quantum dimension of the representation Λ r or Λ N −r , with quantum parameter t. We also recognize this result as an analytic continuation of S r,0 , where S r,s is the S-matrix of the refined Chern-Simons theory from [13].
As before, we can express the same result in the alternative framing of Thus we conclude that

Hopf Link in S 3
We now consider two codimension-4 defects labelled by anti-symmetric tensor representations of rank r and s wrapping two Hopf-linked circles in S 3 . In the first surgery construction of S 3 by gluing two solid tori S 1 × D 2 with an S transformation, this corresponds to inserting a pair of codimension-4 defects at the origin of each D 2 .
This corresponds to inserting two supersymmetric 't Hooft loops on the two sides of the interface, as depicted in figure 19. By cutting the path integral at both sides of the S interface, we find the following integral representation of this correlation function, Z T (S 3 ) (ω r , ω s ) = dν X (a)dν X (a ) Z X,N P X (a)Z X,N P X (a )H where and by a slight abuse of notation, we have defined δ I to be the vector whose elements satisfy (δ I ) j = χ I (j) − |I| N , with χ I the indicator function of I. Now we make use of the results for the Nahm pole in section 4.7 to see that we need to evaluate equation (5.16) at a, a = ρ . In where ω s = δ {1,...,s} is the highest weight of the rank s fundamental representation of su(N ).
Finally, we evaluate dν X (a)dν X (a ) Z X,N P X (a)Z X,N P X (a ) = 1 S X (ρ , ρ ) Putting everything together, we find that 21) or in terms of the exponentiated variables t = e 2πib , q = e 2πib 2 : This precisely reproduces an analytic continuation of the S-matrix S r,s for a pair of antisymmetric tensor representations Λ r and Λ s in refined Chern-Simons theory [12,13].
Again, we can make contact with the alternative framing of    Figure 20. An evaluation of Z T (S 3 ) (ω r , ω s ) This allows us to conclude that

Surgery
Closed orientable three-manifolds have the property that they can be constructed by Dehn

Lens Spaces
The Lens space L(p, 1) can be constructed by gluing a pair of (1, 0) solid tori by the SL(2, Z) transformation ST −p S, or equivalently two (0, 1) solid tori by the SL(2, Z) transformation T −p . This corresponds to the partition function of the interface T −p in between a pair of Neumann boundary conditions N X or N Y . Sending the size of the interval to zero, this leaves a supersymmetric Chern-Simons theory for g at level −p together with an adjoint chiral multiplet 5 . Applying our considerations from section 4, the partition function is given by the following integral For a general Lens space L(p, q), we expand −p/q as a continued fraction The Lens space L(p, q) is then constructed using rational surgery by gluing two solid tori (0, 1) with the SL(2, Z) transformation T r 1 ST r 2 S . . . ST rm . This corresponds to a series of SL(2, Z) duality interfaces between a pair of Neumann boundary conditions N X . The partition function of T (L(p, q)) is Z T (L(p,q)) = dν X (a 1 ) · · · dν X (a m ) T r 1 (a 1 )S X (a 1 , a 2 ) T r 2 (a 2 )S X (a 2 , a 3 ) · · · T r m−1 (a m−1 )S X (a m−1 , a m ) T rm (a m ) . and L(p, 1), which differ only by a change of orientation.

Seifert Manifolds
Seifert manifolds are S 1 -orbibundles; they can be realized using surgery on various solid tori and they are described by a collection of pairs of integer numbers (p i , q i ) ∈ Z⊕Z, as described in appendix B. To compute the partition function of the 3d N = 2 theory associated to a general Seifert manifold M ((m 1 , n 1 ), . . . , (m k , n k )) we must now consider the 4d N = 2 * theory with gauge algebra g ⊕k .
The boundary condition on the right is 'unentangled': it is a product of (m j , n j ) type boundary conditions for each factor of the gauge group separately. After expanding each m j /n j as a continued fraction: m j /n j = r j 1 , . . . , r j l j , the wave function associated to this boundary condition is given by encodes all the information down each fibre of the plumbing tree.
However, on the left we must introduce an 'entangled' boundary condition corresponding to the manifold S 2 × S 1 \ ∪ k i=1 N i , where the N i are k unlinked solid tori. This is defined by starting from Neumann boundary conditions N X for each factor g in the gauge algebra, and deforming it by coupling to the dimensional reduction of the class S theory corresponding to S 2 with k full punctures and flavour symmetry g ⊕k . This has a mirror description as a star-shaped quiver [43], leading to the wave function ψ(a 1 , . . . , a k ) = dν X (a) S X (a, a 1 ) . . . S X (a, a k ) . (6.6) The partition function corresponding to the Seifert manifold is now This expression mirrors the standard surgery construction for Seifert manifolds in regular Chern-Simons theory.
The structure of the result (6.7) is manifest in the plumbing diagram for the Seifert manifold, represented in figure 21, where to each node we associate an integral and a Tfunction, and to each edge we associate an S X -function: We can check that this reproduces the result (6.3) for a Lens space in two different ways, First, using the representation of the Lens space L(p, q) as M ((q, p)), we write q/p = [r 1 , . . . , r l ]. Then L(p, q) has the following partition function, Z T (L(p,q)) = dν X (a) dν X (a 1 ) . . . dν X (a l ) S X (a, a 1 )T r 1 (a 1 ) · · · S X (a l−1 , a l )T r l (a l ).

Special Limits and Topological Invariance
We are currently not able to compute the Seifert manifold partition function for general .
Nevertheless, in certain limits the general formula (6.7) reduces to a simpler form and we are able to calculate the partition function explicitly.
By analytic continuation, we will consider the limits → 0 and → Q which are expected to correspond to removing the puncture from T 2 . For example, in the limit → 0, it is straightforward to show that where c = (N − 1) + N (N 2 − 1)Q 2 , and up to a numerical factor that [44] . (6.13) This reproduces the modular S-and T -matrices for characters of non-degenerate representations of the W N -algebra with momentum α = Qρ − a [45], as expected once we remove the puncture from T 2 .
In this section, we will simply consider the case g = su(2) and discuss the limits → 0 and → Q. In the limit → 0, we can fully determine the partition function, while for → Q, we can expresse the integrals in terms of trigonometric functions, which can then be used to get some analytic and numerical results.
In these limits, we test the statement that the partition function of T (M 3 ) on S 3 b is a topological invariant of Seifert manifolds M 3 . We have tested in both limits the equality of partition functions of the manifolds satisfying the following homeomorphisms [46]: • L(p, q) ∼ = L(p, q ) if and only if q ≡ ±q ±1 mod p. Note that we had already established invariance when qq ≡ 1 mod p in section 6.1.
Furthermore, we will investigate the following homeomorphism: where # denotes the connected sum, and show that the relevant partition functions satisfy with M and each L(p j , q j ) in Seifert framing and S 3 in canonical framing. This suggests that the following formula from regular Chern-Simons theory [47] is valid in our construction.
In these limits, all partition functions become either 0 or infinite due to the contribution from an adjoint multiplet of T R and T f charge 0. In fact we find that the combination is regular, with an overall factor of S b ( ) in Z T (M ) ( ) from the contribution of such an adjoint chiral at the central node of the plumbing tree. In principle, one should first compute the partition function Z T (M ) ( ) explicitly for general , and then take a limit after removing the S b ( ) factor. However, since we cannot perform the relevant integrals in closed form for general , we shall assume that we can push the limits through integrals. We find that this leads to consistent results.

The limit → Q
Let us first consider the limit → Q. This limit of the partition function S X (a, a ) has been considered previously in [37]. We find that . (6.18) Specifically, note that the product ν X (a)S X (a, a ) is regular. Evaluating the integrals (6.7) in closed form for a general Seifert manifold is beyond our current capabilities. However, we checked numerically that the expression ( Let −p i /q i = [r i 1 , . . . , r i m i ] in the general formula (6.7). In the limit → Q, the partition function simplifies to Consider the singular fibre (0, 1), represented above by the t integral. The integration of √ 2 ν(it) cos(4πxt) yields a sum of delta functions This simplifies the integral to Now recognize the remaining integrals as the partition functions Z T (L(p j ,q j )) , and notice the Therefore the connected sum formula (6.15) holds, with M and L(p j , q j ) both in Seifert framing and S 3 in canonical framing.
6.3.2 The limit → 0 In the limit → 0, it is straightforward to check that Again, we note that the product ν X (a)S X (a, a ) is regular in the limit.
Now consider a general Seifert manifold M = M ((p 1 , q 1 ), . . . , (p n , q n )). Assume that each p i = 0 and, as before, write p i /q i = [r i 1 , . . . , r i m i ]. Each fibre in the plumbing diagram contributes (6.26) Unlike the limit → Q, this integral has a nice recursive structure, namely: where we used that 29) and that sign [r i j , . . . , r i m i ] = sign(r i j ). Performing the final integration over a, we then find that where φ L = −3σ(Q L )+ n i=1 m i j=1 r i j is the framing of the manifold, with σ(Q L ) the signature of the linking matrix Q L of the plumbing tree. Furthermore, recognize that [8] | det Q L | = n j=1 q j p j n i=1 p i , (6.32) to get the result This expression gives the partition function in Seifert framing; this suggests that to move to canonical framing we multiply by exp(πiφ L /12) and find which is a topological invariant. which is not covered by our previous computation due to the appearance of the (0, 1). Again, let −p j /q j = [r j 1 , . . . , r j m j ]. Then where ∆(a) = 1 2 (δ(a) + δ(−a)) is a Weyl-invariant delta function on the Cartan subalgebra of su (2). Furthermore S b ( )S X (0, a ) = 2 √ 2, so that, using the definition of φ j (a) and φ j (a j ), (6.36) simplifies to By the definition of φ j (a j ), the latter integrals are precisely the partition functions of the Lens spaces L(p j , q j ) in Seifert framing: Moreover, using the general result (6.34), we see that S 3 has the following partition function in canonical framing: Lastly, observe that M and all L(p j , q j ) are both in Seifert framing. Thus it is again true in this limit that the connected sum formula (6.15) holds.

Discussion
We have given a prescription for computing the partition functions of 3d N = 2 theories T (M 3 ) associated to Seifert manifolds M 3 by compactification of a 4d N = 2 * theory on an interval with appropriate boundary conditions and a set of SL(2, Z) duality interfaces.
Throughout, we have turned on a mass parameter for the distinguished u(1) f flavour symmetry associated to the circle action on Seifert manifolds. This construction is the analogue of Dehn surgery on the supersymmetric side of the 3d-3d correspondence.
We expect the partition functions of 3d N = 2 theories T (M 3 ) to correspond to computations in SL(N, C) Chern-Simons theory on M 3 with a network of defects supporting the mass parameter for the flavour symmetry u(1) f . In particular, we recovered an analytic continuation of the S-matrix of refined Chern-Simons theory [12,13] from the study of supersymmetric line defects in T (S 3 ). Our analysis therefore provides an insight into the structure of refined Chern-Simons with complex gauge group SL(N, C).
To develop the full 3d-3d correspondence with complex Chern-Simons theory, it is important to consider the complete spectrum of supersymmetric defects of the 6d N = (2, 0) theory. In the case g = su(N ), we could consider general combinations of codimension-2 and codimension-4 defects of the 6d N = (2, 0) theory wrapping a curve C in M 3 labelled by data Λ = (ρ, λ,λ) with • An embedding ρ : su(2) → g.
Here, λ andλ correspond to codimension-4 defects wrapping respectively the circles S 1 and S 1 inside the squashed sphere S 3 b on the supersymmetric side of the correspondence. In terms of SL(N, C) Chern-Simons theory, ρ specifies a monodromy defect on C, while the weights λ,λ correspond to Wilson loops in irreducible representations of the subgroup of SL(N, C) left unbroken by the monodromy defect [48,49].
It would be interesting to map out the full dictionary with the supersymmetric side of the correspondence. For example, it seems reasonable to construct an S-matrix S Λ 1 ,Λ 2 element corresponding to the correlation function of any combinations of defects labelled by data Λ 1 and Λ 2 supported on Hopf linked circles in S 3 . Here, we have considered only particular combinations: 1. Λ = ([1] N , 0, 0): maximal codimension-2 defects supporting a flavour symmetry g.
It would clearly be very interesting to perform the analogous computations for the superconformal index and Lens space partition functions, which should allow access to complex Chern-Simons theory at other values of the levels [50].
In addition, it has simple zeroes at and simple poles at (A.4) The following useful formula for any n, m ∈ Z ≥0 , is a consequence of the functional equations for the double sine function.
The asymptotics of the double sine function are given by Let us now summarize the contributions to the partition function of three-dimensional theories on S 3 b with these conventions:

B Surgery on three-manifolds
In this appendix we review some of the ideas in three-dimensional topology that are relevant to our constructions, specifically those relating to surgery. Excellent reviews are [52] and [53].
Consider two compact n-manifolds with boundary M 1 and M 2 , with homeomorphic boundaries, and a homeomorphism f : ∂M 2 → ∂M 1 between the latter. The operation of surgery between the two consists in the construction of a new manifold by gluing the boundaries with f . More precisely, we define where the equivalence relation is between points of the boundaries: Recall that a knot K in a closed orientable 3-manifold M is a smooth embedding of S 1 in M .
A link L is a disjoint union of a finite collection of knots in M .
A knot K ⊂ M can be thickened to a tubular neighbourhood N (K), a smoothly embedded disjoint solid torus (D 2 × S 1 ), whose core {0} × S 1 forms the knot K. Consider now the knot exterior M 1 = M \ N (K) and tubular neighbourhood M 2 = N (K), which both have a T 2 boundary, and an arbitrary homeomorphism f : (∂M 2 ∼ = T 2 ) → (∂M 1 ∼ = T 2 ). We perform surgery between the two using f , gluing the knot exterior and the tubular neighbourhood using f . This results in a new closed orientable 3-manifold We say that M is obtained from M via surgery along the knot K, and refer to the process as Another way we can describe a Dehn surgery is by determining the knot K along which it is performed and the homeomorphism up to isotopy, that is, by an element of the mapping class group of the boundary. In this specific case, the boundaries are homeomorphic to tori, and the mapping class group is therefore we can decompose the homeomorphism in terms of the generators S, T of SL(2, Z).
The Lickorish-Wallace theorem states that any closed orientable connected 3-manifold can be obtained from S 3 through an integral Dehn surgery on a link in S 3 [54,55].
Seifert manifolds are a special class of 3-manifolds that are S 1 -bundles over two-dimensional orbifolds. They can also be described using surgery in the following way.
n is a two-sphere with n discs removed. Then ∂M = n i=1 N i is a disjoint union of n solid tori. We can glue in solid tori by identifying the meridian on the i-th solid torus boundary to a curve on N i , whose isotopy class is described by (p i , q i ) ∈ Z⊕Z.
This forms the Seifert manifold M (0; (p 1 , q 1 ), . . . , (p n , q n )) ≡ M ((p 1 , q 1 ), . . . , (p n , q n )), where the 0 refers to the fact that the construction used S 2 , a genus 0 surface. The construction can be generalized by using F g = Σ g \ int D 2 1 ∪ . . . D 2 n instead of F , where Σ g is a closed orientable surface of genus g.
For a general Seifert manifold M ((p 1 , q 1 ), . . . , (p n , q n )), obtained as above, the i-th component of the link was glued back in after twisting the boundary using M i ∈ SL(2, Z), where Seifert manifolds can also be described in terms of surgery diagrams, which encode how the surgery on links in S 3 has taken place. The simplest such diagram is that of a Lens space L(p, 1):

−p
This indicates the surgery happened over a single unknot, with framing −p. For a general Lens space L(p, q) with −p/q = [a 1 , . . . , a m ], we have the following diagram: Alternatively, we can represent these as plumbing graphs, or plumbing trees, which are weighted graphs with each vertex representing an unknot, and each edge representing that two unknots are linked. For example, the diagram above translates into the plumbing tree where lk(K i , K j ) is the linking number of knots K i and K j . The intersection form is particularly simple given a plumbing graph with weights a i at vertex i, namely

C Details of T (S 3 )
In this appendix we provide some details on computations used in section 5.

S 3 Integral
In our conventions explained in the main body of the paper, the partition function for T A N S 3 is given by (5.6). For the first non-trivial case A 1 we have where γ is a suitably deformed contour coming from the localization computation, such that I( ) with Im( ) < 0 is the analytic continuation of I( ) in the physical region, where Re( ) > 0, Im( ) > 0 and γ = iR. Due to the asymptotics of the S b functions (A.6), the contour needs to close in either the second or fourth quadrant of the complex plane, and as such our integral is only defined for arg( ) ∈ [0, π), i.e. we can think of our integral defined on the half-open disk in CP 1 .
We claim that the integral above evaluates to We will check that the asymptotics and the analytic structures match as functions of . The poles are all located at arg ( ) = π, and as such the residues should be interpreted as the coefficient of −1 of the Laurent expansion of I( ), for near the pole with arg ( ) < π.
We start by considering the asymptotics at → ∞ of the two sides of the equation in the region Im( ) > 0. For the right-hand side, we immediately find that On the other hand, we assume that we can exchange limit and integral in I( ) and thus obtain the following expression for the integrand Another immediate check that we can perform is considering the behaviour near = 0.
Both sides of the equality have a simple pole there, and we can compute the residues, which should match. For the right-hand side, using (A.4), we immediately find thus obtaining a match.
In the same way, performing the integrals on the left-hand side using a computer, we can check consistency of equation (5.6). More specifically we check the matching of the asymptotic behaviour for → ∞ in the region Im > 0 of both sides of dν X (a) e −πi j a 2 which is equivalent to (5.6). This was done for N = 3, 4, as for larger N it becomes too challenging from the computational point of view.
Value of S X (ρ , a ) We would like to show that S X (ρ , a ) = S X (a , ρ ) is independent of a , and that it is proportional to with the respective eigenvalues W (r) (ρ ), W (r) (ρ ) is one-dimensional [56], this shows that Z(a, ρ , ) is constant, so independent of a.
To find the value of S X (ρ , a ), we shall evaluate Z(a, a , ) at the symmetric point a = a = ρ and use the explicit evaluation of Z(a, a , ) in [25], equation (3.37) 8 .
Let σ ∈ W ∼ = Sym(N ). Firstly, we find (C.13) 7 The operators with the tilde indicate that we are considering loops of length 2πb −1 . 8 After correcting a small error in this formula to restore Weyl invariance in the variable t.
Consider now the combination .
Note that the numerator in the product is never 0 or infinite. The denominator however causes the expression to vanish whenever 1 + σ j − σ i = 0 for some i < j, by introducing a factor of 1 S b (0) = 0. This happens for every σ ∈ Sym(N ) except σ = id, in which case this product simplifies to give .
(C. 16) Furthermore, when im = ρ , and σ = id, the vortex and anti-vortex partition functions become 1. To see this, firstly note that (in our language) η 2 = t, so that we can rewrite in our language η 2 µ σ(i) µ σ(j) = t 1+ρ σ(j) −ρ σ(i) = t 1+σ i −σ j . (C. 17) In the Weyl sum over σ ∈ Sym(N ), the only contribution to Z(a, a , ) is from σ = id, and there is always a pair (i, j) with 1 ≤ i ≤ n, 1 ≤ j ≤ n + 1 with 1 ≤ n ≤ N − 1 such that 1 + i − j = 0. Therefore, there is always such a pair (i, j) for which making any contribution to the vortex partition function with n ≥ 1 vanish, as claimed. An isomorphic calculation shows that the anti-vortex partition function becomes 1.
Putting this together, we find the result which is the required form. Multiplying by √ N gives S X (ρ , ρ ).