3d-3d Correspondence Revisited

In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d theories constructed this way recovers theories found previously by Dimofte-Gaiotto-Gukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation.

is denoted simply as T [M 3 ]. Moreover, in such cases, the theory T [M 3 ] can be thought of as the effective three-dimensional theory on the R 3 part of the fivebrane world-volume in an M-theory setup: space-time: where M 3 is embedded in a Calabi-Yau 3-fold CY 3 as a special Lagrangian submanifold. The neighborhood of every special Lagrangian submanifold always looks like the total space of the cotangent bundle, T * M 3 , which is one popular choice of CY 3 . Another popular choice -that appears e.g. in physical realization of knot homologies -is the resolved conifold geometry, i.e. when CY 3 is the total space of O(−1) ⊕ O(−1) bundle over CP 1 . Various partition functions of the 3d N = 2 theory T [M 3 ] have a nice geometric interpretation and can be realized in the setup (1.2) by replacing R 3 with a (squashed) 3-sphere, or S 2 × q S 1 , or The effective 3d N = 2 theory T [M 3 ] should exhibit all properties of the fivebrane system (1.2), including symmetries and the space of classical vacua. In particular, the fivebrane system (1.2) has at least three U (1) symmetries which are independent of the choice of M 3 : one is a Cartan subgroup of the SO(3) rotation symmetry of R 3 , another is a rotation symmetry in two transverse dimensions of R 3 ⊂ R 5 , and the third U (1) is the R-symmetry. Certain combinations of these symmetries give rise to three conserved charges which are familiar in the study of surface operators in N = 2 gauge theories, e.g. the index of such 2d-4d systems depends on three universal fugacities, whose nature is independent of the details of the theory or choice of a surface operator. From the viewpoint of the 3d theory T [M 3 ] on the R 3 part of the fivebrane worldvolume, these three U (1) symmetries have the following interpretation. The rotation along R 3 is part of the Lorentz symmetry, while (certain combinations of) the other two become the U (1) R R-symmetry group of 3d N = 2 supersymmetry algebra and a close cousin that we denote U (1) t . The non-R flavor symmetry U (1) t is not present in a generic, garden variety 3d N = 2 theory. But, since it is a symmetry of the system (1.2), theories T [M 3 ] should have it as well. This special symmetry illustrates to what extent T [M 3 ] are non-generic 3d N = 2 theories and plays a key role in the 3d-3d correspondence, as will be discussed in this paper.
Another basic aspect of the fivebrane system (1.2) that should be reflected in the physics of T [M 3 ] is the relation between G C flat connections on M 3 and the space of supersymmetric vacua of the 3d N = 2 theory on S 1 × R 2 : This basic property of the 3d-3d correspondence (1.1) was originally taken [1] as a definition 1 of the theory T [M 3 ]. Since then many attempts to construct T [M 3 ] systematically have been undertaken, including the approach [2,3] based on triangulations of M 3 . It leads to a 3d N = 2 theory T DGG [M 3 ] with many desired properties, but also presents some puzzles. In particular, as noted already in [2,3], since certain branches of flat connections are always missing. Examples of such "lost branches" include even the simplest flat connections on M 3 , namely the abelian ones (i.e. flat connections that can be conjugated to the maximal torus of G C ). At first, it was unclear how severe this problem is. However, a number of independent recent developments all lead to the same conclusion: the complete theory T [M 3 ; G] must realize all G C flat connections on M 3 . These developments include: • The recent proof [4,5] of the 3d-3d correspondence indicates that all complex flat connections on M 3 should be treated democratically and, therefore, no one should be left behind.
• Various deformations / refinements of (1.3) necessarily require taking a proper account of all branches [6][7][8][9], and can serve as a useful tool in identifying T [M 3 ] even in the undeformed case. This will be our approach in the present paper.
Our first goal in this paper is to construct some theories T [M 3 ; G] with all expected flavor symmetries and with vacua corresponding to all flat connections on M 3 , and to investigate their relation to theories T DGG [M 3 ; G] . We will mainly focus on the case G = SU (2), and on knot complements M 3 = S 3 \K. A knot-complement theory T [M 3 ] := T [M 3 ; SU (2)] is defined by compactification of the 6d (2,0) theory on S 3 with a codimension-two defect wrapping the knot K ⊂ S 3 . In this case T [M 3 ] should gain a U (1) x flavor symmetry, part of the SU (2) x flavor symmetry of the defect, in addition to U (1) t and U (1) R . What we find can be then summarized by the following diagram: In particular, the theory T DGG [M 3 ] is a particular subsector of T [M 3 ] obtained by Higgsing the U (1) t symmetry.
The left-hand side of the diagram (1.5) indicates an expected relation between T [M 3 ] and a theory T poly [M 3 ; r] whose partition functions compute the Poincaré polynomials of r-colored SU (2) knot homology for K. Indeed, our practical approach to constructing T [M 3 ] will be to identify a 3d N = 2 theory with U (1) x × U (1) t symmetry whose partition functions reduce to the desired Poincaré polynomials in a special limit. Physically this limit corresponds to another Higgsing procedure, this time breaking the U (1) x symmetry of T [M 3 ] while creating a line defect or vortex, similar to scenarios studied in [11][12][13].
An important feature of (1.5) is that the two arrows corresponding to Higgsing do not commute. In particular, while it is easy to obtain Jones polynomials of knots from the Poincaré polynomials on the left-hand side by ignoring U (1) t fugacities, it is (seemingly) impossible to do this from T DGG [M 3 ] on the right-hand side. Jones polynomials include a crucial contribution from the abelian flat connection on a knot complement M 3 , and vacua corresponding to the abelian flat connection are lost during the Higgsing of U (1) t .   [14]. It is obtained from two copies of the half-index I S 1 ×qD ± (T ± ) Z vortex (T ± ) convoluted via the index (flavored elliptic genus) of the wall supported on S 1 × S 1 eq , where D ± is the disk covering right (resp. left) hemisphere of the S 2 and S 1 eq := ∂D + = −∂D − is the equator of the S 2 .
Later, in Section 5 we discuss gluing of knot and link-complement theories to form closed M 3 , in particular 3d N = 2 theories for lens spaces, Seifert manifolds, and Brieskorn spheres. The importance of such gluing or surgery operations is two-fold. First, it will give us another clear illustration why all flat connections need to be accounted by 3d N = 2 theories T [M 3 ] in order for cutting and gluing operations to work. Moreover, it will help us to understand half-BPS boundary conditions that one needs to choose in order to compute the half-index of T [M 3 ]. As explained in [10], a large class ("class H") of boundary conditions can be associated to 4- in complex space C m parametrized by (multi-)variable s. Here, P K (q, t . . .) stands for the Poincaré polynomial of a doubly-graded [15][16][17][18] or triply-graded [19][20][21] homology theory H(K) of a link K: that categorifies either quantum sl(N ) invariant [22] or colored HOMFLY polynomial [23], respectively. Depending on the context and the homology theory in question, the sum runs over all available gradings, among which two universal ones -manifest in (2.2) -are the homological grading and the so-called q-grading. In the case of HOMFLY homology, there is at least one extra grading and, correspondingly, the Poincaré polynomial depends on one extra variable a, whose specialization to a = q N makes contact with sl(N ) invariants. The Poincaré polynomials of triply-graded HOMFLY homology theories are often called superpolynomials.
In general, such invariants are also labeled by a representation / Young diagram R and referred to as colored, unless R = in which case the adjective 'colored' is often omitted. In this section we will write the Poincaré polynomials of colored knot homologies in the form (2.1) of contour integrals, whose physical interpretation will be discussed in the later sections. Our basic examples here (and throughout the paper) will be the unknot, trefoil, and figure-eight knot complements.
In general, superpolynomials or Poincaré polynomials are expressed as finite sums of products of q-Pochhammer symbols (z; q) n := and monomials. For instance, the unnormalized superpolynomial of the trefoil 3 1 is [24] (see also [21,[25][26][27]): This is the Poincaré polynomial of the HOMFLY homology (2.2) colored by the r-th symmetric power of the fundamental representation of SU (N ) or, in the language of Young diagrams, by a Young tableau with a single row and r boxes. For our applications here, we specialize to SU (2) homology 2 by setting a = q 2 . It is further convenient to renormalize the SU (2) polynomial by a factor (−1) r , defining 3 We remark that the following steps could also be carried out for generic a, though for our applications we specialize from SU (N ) to SU (2). Let us suppose that |q| > 1 (for reasons that will become clear momentarily), and define x → q r 1 Figure 3. Possible integration contours for the trefoil, drawn on the cylinder parametrized by log s.
There are three half-lines of poles in the integrand Υ 31 (s, x, t; q), coming from (s) − ∞ , (−1/(qst)) − ∞ , (x/s) − ∞ in the denominator; and a full line of zeroes from θ − (q 3 2 sxt 3 ) in the numerator. On the right, we demonstrate a pinching of contours as x → q r .
Note that the three terms (s) − ∞ , (−1/(qst)) − ∞ , and (x/s) − ∞ each contribute a half-line of poles to Υ 3 1 . If we take q > 1 to be real, then the asymptotics of the integrand are given by so the integral along Γ I does converge in a suitable range of x and t (namely, if |xt 3 | < 1). In contrast, the integrals along the other obvious cycles here, Γ II and Γ III , always converge.
Moreover, a little thought shows that upon setting x = q r the integral along Γ I must equal the integral along Γ III ; indeed, as x → q r , some r + 1 pairs of poles in the lines surrounded by Γ I and Γ III collide, and all contributions to the integrals along either Γ I and Γ III come from the r + 1 points where the contours get pinched by colliding poles. (Such pinching would usually cause integrals to diverge, but here the divergence is cancelled by one of the s-independent theta-functions in Υ 3 1 .) Therefore, letting (with the obvious relation B I + B II + B III = 0), we find (2.14) We can repeat the analysis for the unknot U = 0 1 and figure-eight knot 4 1 . The superpolynomials of these knots are given by [24,26,28,29]: and Poincaré polynomials for G = SU (2), i.e. specializations to a = q 2 , normalized by (−1) r , are given by for the figure-eight knot. In the latter case, the integrand Υ 4 1 has four half-lines of poles in the s-plane, coming from the four factors (s) − ∞ , (−1/(qts)) − ∞ , (x/s) − ∞ , (−1/(q 2 xt 3 s)) − ∞ in the denominator of (2.20). Let Γ I , Γ II , Γ III , Γ IV be contours encircling these respective half-lines of poles. A formal sum of residues along poles in the first half-line, evaluated at x = q r , most directly gives P r 4 1 (t; q); but the actual integral along Γ I does not converge for generic x. In contrast, the integrals along Γ II , Γ III , Γ IV always converge, and where These examples indicate how the analysis may be extended to other knots and links (e.g. those whose superpolynomials are found in [8,9]), and to Poincaré polynomials of other homological invariants. In general, the required integrals will not be one-dimensional, but will require higher-dimensional integration cycles. Generalizations of some results of this paper to other knots and links are also discussed in section 4.4.

Knot polynomials as partition functions of T [M ]
In this section, we construct some examples of 3d N = 2 theories T [M 3 ] for knot complements M 3 (and G = SU (2)) with the properties outlined above. In particular, we would like the vacua of T [M 3 ] on R 2 × S 1 to match all flat connections on M 3 .
Although our strategy will be a little indirect, it is based on a simple key observation: the contour integral (2.1) for colored Poincaré polynomials has the form of localization integrals in supersymmetric 3d N = 2 theories as well as in Chern-Simons theory on certain 3-manifolds. Indeed, powerful localization techniques reduce the computation of Chern-Simons partition functions to finite dimensional integrals of the form (2.1), where the choice of the contour is related to the choice of the classical vacuum [30][31][32][33][34], as we briefly review in section 5.
Similar -and, in fact, closer to our immediate interest -contour integrals of the form (2.1) appear as a result of localization in supersymmetric partition functions of 3d N = 2 theories, such as the (squashed) sphere partition function [35,36], the index [37][38][39], and the vortex partition function [2] or the half-index [40]. Since in the last case the space-time is noncompact it requires a choice of the asymptotic boundary condition or vacuum of the theory on R 2 × q S 1 , which manifests as a choice of the integration contour in the localization calculation. (The integrand is completely determined by the Lagrangian of 3d N = 2 theory.) This has to be compared with the first two cases, where localization of 3-sphere partition function and index lead to a contour integral with canonical choices of the integration contour.
Therefore, in order to interpret (2.1) as a suitable partition function of 3d N = 2 theory in this paper we mainly focus on half-indices, vortex partition functions, and their IR variants called holomorphic blocks [41] that do not necessarily come from localization. This gives us enough flexibility to interpret (2.1) and we generically expect that the full set of blocks for T [M 3 ], labelled by a full set of vacua, corresponds to a complete basis of independent convergent contours for the integrals of Section 2. On the other hand, we also expect that a basis of convergent contours is in 1-1 correspondence with flat connections on M 3 : The reason we expect these correspondences to hold is outlined more carefully in Section 3.1. In order to capture all flat connections, it turns out to be crucial that we start with Poincaré polynomials for knot homology rather than unrefined Jones polynomials. In Section 3.2 we then demonstrate the construction of T [M 3 ] in a few examples.
In Section 3.3 we examine the physical meaning of the limit x → q r that recovers Poincaré polynomials from T [M 3 ]. We argue that it is a combination of Higgsing and creation of a line operator in T [M 3 ], as on the left-hand side of (1.5). We also show that Poincaré polynomials can be obtained by directly taking residues of S 2 × q S 1 indices and S 3 b partition functions of T [M 3 ], bypassing holomorphic blocks.

Recursion relations
One understanding of why contour integrals as in Section 2 should capture all flat connections on a knot complement follows from looking at the q-difference relations that the integrals satisfy.
Let us start with the Poincaré polynomials P r K (t; q) for colored SU (2) knot homology of a knot K. As found in [6,8,24], the sequence of Poincaré polynomials obeys a recursion relation of the form where A ref ( x, y; t; q) is a polynomial operator in which x, y act as xP r K = q r P r K and yP r K = P r+1 K . The limit q → 1 of A ref ( x, y; t; q) is a classical polynomial A ref (x, y; t), whose subsequent t → −1 limit contains the classical A-polynomial of K [42] as a factor, The physical interpretation of the classical A-polynomial A(x, y) goes back to [43]. Its roots at fixed x are in 1-1 correspondence with all flat connections on M (with fixed boundary conditions at K); but the root corresponding to the abelian flat connection is distinguished because it comes from a universal factor (y−1) in A(x, y). However, the t-deformed polynomial A ref (x, y; t) is irreducible (at least in simple examples 4 ), and none of its roots is more or less important than the others.
Alternatively, note that the t → −1 limit of A ref ( x, y; t; q) leads to a shift operator known as the quantum A-polynomial, A( x, y; q), which annihilates colored Jones polynomials [43,46]. One can also consider a-deformations of these shift operators. Such a deformation of the quantum A-polynomial was called Q-deformed A-polynomial in [47], and it agrees with the mathematically defined augmentation polynomial of [48,49]. More generally, one can consider shift operators A super ( x, y; a; t; q) depending on both a and t, which annihilate colored superpolynomials, and which were called super-A-polynomials in [24] (for a concise review see [50]). However, as mentioned above, we are only interested here in a = q 2 specializations. Now, in Section 2 we expressed for a suitable integrand Υ K and a choice of integration contour Γ P . It is easy to see that B P (x, t; q) satisfies a q-difference equation even before setting x = q r , with x, y acting as xB P (x, ...) = xB P (x, ...) and yB P (x, ...) = B P (qx, ...). More so, the integral B α = Γα ds/s Υ K for any convergent integration contour Γ α (that stays sufficiently far away from poles) should provide a solution to the q-difference equation A ref ·B = 0, and one generally expects that a maximal independent set of integration contours generates the full vector space of solutions. 5 The situation is entirely analogous to the solution of Picard-Fuchs equations for periods of a holomorphic form on a complex manifold. Here A ref plays the role of a q-deformed Picard-Fuchs operator, and B P is a fundamental period; the general integrals B α compute the remaining periods.
If we fix the values of x, t, and q, the convergent integration cycles Γ α can be labelled by the roots y (α) (x, t) of the classical equation A ref (x, y; t) = 0i.e. by the flat connections on M 3 with boundary conditions (meridian holonomy) fixed by x. The correspondence follows roughly by identifying the solutions to A ref (x, y; t) = 0 with critical points of the integrand Υ K (s, x, t; q) at q ≈ 1, then using downward gradient flow with respect to log |Υ K (s, x, t; q)| to extend the critical points into integration cycles Γ α . This is the standard construction of so-called "Lefschetz thimbles," modulo some subtleties that were discussed in [41,51,52].
We have claimed that by writing one solution of A ref ·B = 0 as a contour integral (3.4), we can actually reproduce all other solutions from integrals on a full basis of contours Γ α . This reasoning relies on an important assumption: that the quantum A ref (and hence the classical A ref ) is irreducible. Otherwise, we may only get solutions corresponding to one irreducible component. For this reason, it is crucial that we use refined knot polynomials and recursion relations rather than Jones polynomials and the quantum A-polynomial. See [50,53] for further details as well as pedagogical introduction.
To complete the chain of correspondences (3.1), we simply use [1-3, 6, 8, 41, 54-58] to translate the above observations to the language of gauge theory. Momentarily we will engineer gauge theories T [M 3 ] for which the integrals * ds/s Υ K compute various partition functions on R 2 × S 1 annihilated by A ref and labelled by vacua of T [M 3 ] on R 2 × S 1 , i.e. classical solutions of A(x, y; t) = 0. This approach is almost successful, and good enough for our present purposes, though we should mention an important caveat. In general, one must also specify relevant superpotential couplings for a UV description of T [M 3 ], which are crucial for attaining the right superconformal theory in the IR; but it is very difficult to specify such couplings just by looking at partition functions. At the very least one would like to find superpotential couplings that break all "extraneous" flavor symmetries whose fugacities don't appear in supersymmetric partition functions, and are not expected for the true T [M 3 ]. Even this is difficult, because the naive prescription (3.6) leads to theories that simply don't have chiral operators charged only under the extraneous symmetries. This problem was discussed in [2, Section 4], and solved by finding "resolved" theories with the same partition functions as the naive ones, but with all necessary symmetry-breaking operators present.
We also note that, while it is possible to construct the space of holomorphic blocks for a number of examples considered here, the construction is not always systematic, even in the simplest 3d N = 2 theories such as pure super-Chern-Simons theory. For this reason, it may be more convenient to work with other partition functions of theories T [M 3 ] that include halfindices, i.e. UV counterparts of holomorphic blocks that are labeled by boundary conditions on R 2 × S 1 . A large class of boundary conditions in 3d N = 2 theories comes from 4manifolds and will be discussed in section 5. It is expected that all holomorphic blocks can be reproduced (via RG flow) from suitable choice of boundary conditions in the UV. Other prominent examples of partition functions include the index (or, S 2 × q S 1 partition function) [37][38][39] and the S 3 b partition function [35,36], both of which will be discussed in section 3.3. Presently, we will follow the naive approach to obtain simple UV descriptions for putative T [M 3 ]'s, where some but not all symmetry-breaking superpotential couplings are present. We expect that these theories are limits of the "true" superconformal knot-complement theories T [M 3 ], where some marginal couplings have been sent to infinity. Thus, any observables of T [M 3 ] that are insensitive to marginal deformations -such as holomorphic blocks, supersymmetric indices, massive vacua on S 1 , etc. -can be calculated just as well in our naive descriptions as in the true theories, as long as masses or fugacities corresponding to extraneous flavor symmetries are turned off by hand. This is sufficient for testing many of the properties we are interested in.
Theory for unknot, The theory for the unknot that gives (2.19) was already discussed in [24] and has four chirals Φ i , corresponding to the terms ( Letting x and (−t) be fugacities for flavor symmetries U (1) x and U (1) t , we use the rules of [1-3, 6, 8, 41, 54-58] to read off the precise charge assignments and levels of (mixed) background Chern-Simons couplings (Here all background Chern-Simons couplings simply vanish. 6 ). In this case, we can add an obvious superpotential that breaks most extraneous flavor symmetries and preserves U (1) x , U (1) t , and U (1) R (note that the operator in (3.8) has R-charge two). The chiral Φ 3 is completely decoupled from the rest of the theory and rotated by an extraneous U (1) symmetry. We could break this U (1) by adding Φ 3 to the superpotential (3.8), but prefer not to do this as it would forbid T [0 1 ] from having a supersymmetric vacuum. Ignoring the Φ 3 sector, the putative unknot theory looks just like the 3d N = 2 XYZ model. Similarly, if we follow [6] and compactify T [0 1 ] on a circle turning on masses (i.e. complexified scalars in background gauge multiplets) for U (1) x and U (1) t , we find that the theory is governed by an effective twisted superpotential (We have removed from W 0 1 an infinite contribution from the massless Φ 3 ; this could be regularized by turning on a mass for the U (1) symmetry rotating Φ 3 .) The equation for the becomes the refined A-polynomial equation which further reduces to the unknot A-polynomial y −1 = 0 at t → −1. Equation (3.11) has a unique solution in y at generic fixed x, t, corresponding to the unique, abelian flat connection on the unknot complement (with fixed holonomy eigenvalue x on a cycle linking the unknot).
Theory for trefoil knot, In this case, the integrand (2.11) suggests a theory with six chirals, with charges and Chern-Simons levels CS : (3.12) This is now a gauge symmetry with a dynamical U (1) s symmetry in addition to U (1) x and U (1) t flavor symmetries. Standard analysis of [59] shows that this theory has a gauge-invariant anti-monopole operator V − formed from the dual photon, with charges as indicated in the table. Altogether we can write a superpotential that preserves all symmetries we want to keep, and breaks almost all other flavor symmetry.
There remains a single extraneous U (1), just like in the unknot theory, which plays (roughly) the role of a topological symmetry dual to U (1) s . When compactifying the theory on a circle with generic twisted masses x and (−t) for U (1) x and U (1) t , and scalar s in the U (1) s gauge multiplet, we obtain the effective twisted superpotential (3.14) The critical-point equation exp s ∂ W 3 1 /∂s) = 1, namely determines two solutions in s at generic values of x and t; plugging these into the SUSYparameter-space equation then determines two values of y. More directly, they are solutions of the quadratic has vacua corresponding to both of the flat SL(2, C) connections on the trefoil complement, one irreducible, and one abelian. The two independent holomorphic blocks B II and B III of (2.13) (or, more precisely, some linear combinations of these blocks) are in 1-1 correspondence with the two flat connections.
The net Chern-Simons couplings all turn out to vanish. This particular theory does not admit gauge-invariant monopole or anti-monopole operators. We can introduce a superpotential Thus there are two extraneous U (1)'s, including the topological symmetry of the theory. As before, we can find an effective twisted superpotential on R 2 × S 1 of the form whose critical point equation generically has three solutions in s -which in turn determine More directly, the solutions in y are roots of the cubic which deforms the standard figure-eight A-polynomial A ref ] has massive vacua on S 1 corresponding to all three flat SL(2, C) connections on the figure-eight knot complement, two irreducible and one abelian. Again, these flat connections label linear combinations of the three independent holomorphic blocks

Vortices in S
Having obtained a theory T [M 3 ] whose vacua on R 2 × S 1 match flat connections on the knot complement M 3 , it is interesting to probe its other protected observables. Here we focus on the S 2 × q S 1 indices of T [M 3 ], and make some preliminary observations as to the nature of the "Poincaré polynomial theories" T poly [M 3 ; r] on the left-hand side of the flow diagram (1.5). The 3d index [37][38][39] of a knot-complement theory, or equivalently a partition function on S 2 × q S 1 , depends on three fugacities q, ξ, τ and two integer monopole numbers n, p : We'll consider "twisted" indices I(ζ, n; τ, p; q) = Tr Hn,p(S 2 ) e iπR q R 2 −J ζ ex τ ep as in [40,41], in which case it's convenient to regroup fugacities into pairs of holomorphic and anti-holomorphic variables , or (x, x) = (q r , 1), whose (logarithmic) residue is the r-th Poincaré polynomial of the colored SU (2) knot homology, ] of a knot-complement theory depends on the ellipsoid deformation b as well as two dimensionless complexified masses m x , m t for U (1) x , U (1) t , which are conveniently grouped into holomorphic and anti-holomorphic parameters Then the S 3 b partition function has poles at m x = ibr, or (x, x) = (q r , 1), with These relations are not altogether surprising, since both I[M 3 ] and Z b [M 3 ] should take the form of a sum of products of holomorphic blocks [41], and our theory T [M 3 ] was engineered so that the x → q r specialization of a specific linear combination of blocks B P would reproduce Poincaré polynomials. Below we will choose a convenient basis of blocks so that B P is one of the B α 's, and manifestly gives the only contribution to the residues (3.24), (3.26). (Nevertheless, in the natural basis of blocks labelled by flat connections at fixed (x, t ≈ −1, q = 1), B P may easily correspond to a sum over multiple flat connections, including the abelian one.) Taking the residue of a pole in an index such as (3.24) has an important physical interpretation, which was discussed in [11] in the context of 4d indices and, closer to our present subject, in [12,13]  Consider, for example, the pole at (x, x) = (1, 1), or (ξ, n) = (1, 0). The pole suggests the presence of an operator O x , of charge +1 under U (1) x , in the zero-th U (1) x monopole sector. The contribution of this operator and its powers to the index is Taking the residue I amounts to giving a vev to O x and decoupling massless excitations around it, thereby Higgsing U (1) x symmetry. One can interpret I as the index of a new superconformal theory, the IR fixed point of a flow triggered by the vev O x . More generally, taking a residue at (x, x) = (q r , 1) or (ξ, n) = (q r 2 , r) gives a spacedependent vev (with nontrivial spin) to an operator in the r-th monopole sector. This not only Higgses the U (1) x symmetry of T [M 3 ] but creates a vortex defect. We therefore expect that the residue of I[M 3 ] at (x, x) = (q r , 1) is the index of a new 3d theory T poly [M 3 , r] in the presence of a (complicated!) line operator.
In the context of 4d theories T [C; G] coming from compactification of the 6d (2, 0) theory on a punctured Riemann surface C, taking the residue at a pole in the index amounted to removing a puncture from C -or more generally replacing the codimension-two defect at the puncture by a dimension-two defect in a finite-dimensional representation of G. Similarly, we expect here that taking a residue replaces the codimension-two defect along a knot K ⊂ M by a dimension-two defect in the (r + 1)-dimensional representation of SU (2). We hope to elucidate this interpretation in future work.
We proceed to examples of (3.24). Our conventions for indices follow [40,41]. Below, all indices depend on fugacities from (3.23) as well as the pair which is used for summations/integrations. We assume |q| < 1, as is physically sensible for the index. Thus, the convergent q-Pochhammer symbols are and theta-functions are Since we do our calculations while maintaining a manifest factorization into holomorphic blocks, results for S 3 b follow immediately from their index analogues, by reinterpreting the meaning of x, t, etc.

Unknot
The index of the unknot theory T [0 1 ] from (3.7) is given equivalently by In the first line, we simply write down the index as defined by the theory -with the massless chiral Φ 3 decoupled in order to remove an otherwise infinite factor. In the second line, we show that this index comes from a fusion norm B 0 1 (x, t; q) 2 id of the holomorphic block (2.19), with (q −1 ) − ∞ removed. Since we are working at |q| < 1, we replace all q-Pochhammer symbols and theta-functions in the definition of the block. In the third line, we explicitly write out what the fusion product means, following [41]. We could take the limit (ξ, n) → (q r 2 , r) in the first line of (3.32); after setting n = r, we would find a pole at ξ → q r 2 whose residue is the Poincare polynomial P r U (t, q). But it is more illustrative to take the equivalent limit (x, x) → (q r , 1) in the factorized expression on the last line. Setting x = 1 produces no divergence. The pole we are looking for comes from (x −1 ) ∞ in the denominator. We get Note how the t dependence completely cancelled out of the problem. If we had taken a more general limit (x, x) → (q r , q r ), we would have found a similar pole, with residue P r U (t; q)P r U ( t; q −1 ). The fact that the t dependence cancels out follows from the simple identity P r =0

Trefoil
For the trefoil, the theory T [3 1 ] of (3.12) leads to an integral formula for the index, where 36) Again, we have chosen to regroup Chern-Simons contributions into ratios of theta-functions, separating out the x and x dependence. The integrand in (3.35) has three pairs of half-lines of zeroes and poles in the σ-plane, coming from the three terms ( ) ∞ /( ) ∞ . They lie at The real, physical contour in (3.35) should lie on or around the unit circle, separating each half-line of zeroes from its corresponding half-line of poles.
We also observe that the integrand of (3.35) vanishes as |σ| → ∞, if we stay away from half-lines of poles. Thus we can attempt to deform the contour outwards, closing it around σ = ∞. We pick up the poles in lines II and III, obtaining an expression of the form The holomorphic blocks B II and B III here correspond to integrals along contours Γ II and Γ III in Figure 3, with substitutions of the form (x) − ∞ → 1/(qx) ∞ to account for |q| < 1. Now, if we send (x, x) → (q r , 1), the leading pole in line I can collide with the leading pole in line III, pinching the integration contour in the σ-plane, and leading to a divergence of the the index. We see this explicitly in the evaluated expression (3.38): while the prefactor I 0 and the blocks ||B II || 2 id are finite in this limit, the blocks ||B III || 2 id have the expected divergence. It comes from the denominator (q m x) ∞ in (3.39b), and occurs only for m = 0. The related factor (q 1−k x) ∞ in the numerator vanishes as x = q r unless k ≤ r. Therefore, we find a residue reproducing the superpolynomial after some straightforward manipulations.

Figure-eight knot
The setup for the figure-eight knot is almost identical to that for the trefoil. Now the index is given by with There are four pairs of half-lines of zeroes and poles in the integrand; three are identical to those in the trefoil integrand above, which we denote I, II, III as in (3.37), and there is one new pair We close the contour around σ = ∞ (where the integrand generically vanishes), picking up the poles in lines II, III, and IV, to give The holomorphic blocks in these expressions correspond to the integration cycles discussed above (2.22) (with the usual translation from |q| > 1 to |q| < 1). Now as (x, x) → (q r , 1), the prefactor I 0 along with ||B II || 2 id and ||B V I || 2 id all have finite limits; while ||B III || 2 id has a pole due 1/(q m x) ∞ at m = 0, and is nonvanishing for k ≤ r. As in the case of the trefoil, the divergence can be attributed to the poles of lines I and III pinching the contour of the integrand (3.41). We then find As above, our analysis will be largely example-driven. In Section 4.1 we examine how the trefoil and figure-eight knot theories of Section 3.2 flow to DGG theories. We verify in Section 4.2 that t → −1 limits induce divergences in S 2 × q S 1 indices, indicative of Higgsing. Then in Section 4.3 we use effective twisted superpotentials on R 2 × S 1 to better understand how vacua corresponding to abelian flat connections decouple.

The DGG theories
We can see an explicit example of the proposed DGG flow by considering the trefoil theory T [3 1 ] of (3.12). If we turn off the real mass for the flavor symmetry U (1) t , then the chiral operator O t = Φ 4 can get a vev, The vev breaks U (1) t , but no other symmetries. Moreover, it induces a complex mass for Φ 1 and Φ 3 due to the superpotential Therefore, taking Λ → ∞, we may decouple fluctuations of Φ 4 and integrate out Φ 1 and Φ 3 , arriving at CS : At this point, we observe that T [3 1 ] has a sector containing a U (1) s gauge theory with a single charged chiral Φ 2 , together with minus half a unit of background Chern-Simons coupling. This sector can be dualized to an ungauged chiral ϕ as in [2, Sec 3.3], a consequence of a basic 3d mirror symmetry [60,61]. Indeed, the dual ungauged chiral is identified with the (anti-)monopole operator CS : The superpotential lets us integrate out Φ 6 and ϕ, leaving behind Here Φ 5 is a fully decoupled free chiral, while T DGG [3 1 ] is a slightly degenerate description of the DGG trefoil theory. Namely, T DGG [3 1 ] here is a "theory" consisting only of a background Chern-Simons coupling at level 3 for the flavor symmetry U (1) x , and some flavor-R contact terms given by the matrix on the RHS of (4.5). A similar "theory" was obtained by DGG methods in [40,Section 4.3], using a degenerate triangulation of the trefoil knot complement into two ideal tetrahedra. It was interpreted as an extreme limit of the true DGG theory T DGG [3 1 ] in marginal parameter space. It is not surprising that we have hit such a limit, since, as discussed at the beginning of Section 3.2, we are ignoring some marginal deformations.
Our T DGG [3 1 ] becomes identical to that in [40,Section 4.3] upon shifting R-charges by minus two units of U (1) x charge. The shift is due to difference of conventions: we initially set x = q r in Poincaré polynomials whereas the equivalent choice for [2,40] would be x = q r+1 .
We can repeat this exercise for the figure-eight knot. The theory T [4 1 ] of (3.17) again has a chiral operator O t = Φ 4 that is charged only under U (1) t , and can get a vev when the real mass corresponding to U (1) t is turned off, (4.7) Then the effective superpotential lets us integrate out Φ 1 and Φ 3 . We flow directly to a theory where Φ 6 is a decoupled chiral and is basically the GLSM description of the CP 1 sigma-model. It is equivalent (after shifting Rcharges by minus two units of U (1) x charge) to the DGG theory obtained from a triangulation of the figure-eight knot complement into two tetrahedra. 9 Again, this triangulation is a little degenerate (as discussed explicitly in [2, Section 4.6]), so (4.10) should be viewed as a limit of the true T DGG [4 1 ], which has the same protected partition functions (index, half-indices, and holomorphic blocks). 9 The equivalence is most directly seen using the polarization discussed in Appendix B and Section 6.3 of [41]. The "degenerate" DGG theory for the figure-eight knot, a.k.a. the CP 1 sigma-model, has three standard duality frames that are analyzed in Section 5.1 of [41], and the most symmetric of these duality frames agrees with (4.10). Another frame matches Section 4.6 of [2].

Indices and residues
The S 2 × q S 1 indices of theories T [M 3 ] help us to further illustrate the breaking of U (1) t by "Higgsing" and the flow to T DGG [M 3 ]. As discussed in Section 3.3, Higgsing corresponds to taking residues in an index. In particular, we expect here to find the indices I DGG [M 3 ] of DGG theories as residues of I[M 3 ] at (t, t) → (−1, −1). Consider, for example, the index I[3 1 ] of the trefoil theory as given by (3.38). Sending t → −1, the prefactor I 0 develops a pole due to the factor 1/(−1/t) ∞ . This factor comes directly from the chiral Φ 4 in T [3 1 ]. (The factor 1/(−1/(qt 3 )) ∞ in I 0 , coming from the chiral Φ 4 , also develops a pole, but it is not relevant for the Higgsing we want to do.) In addition, we see that ||B III || 2 id has a finite limit as (t, t) → (−1, −1), whereas ||B III || 2 id vanishes due to (−q −k t −1 ) ∞ in the numerator. One way to understand this vanishing is to observe that the zeroes in line I of the index integrand perfectly cancel all poles in line II when (t, t) = (−1, −1). Therefore, The resummation in the third line captures the duality between a charged chiral (Φ 2 ) and a free chiral (ϕ = V − ) discussed in Section 4.1. Then the expression q 3n ξ 3n matches the DGG trefoil index of [40], modulo a redefinition of R-charges ξ → q −1 ξ. The infinite prefactor (−q 2 / t 3 ) ∞ /(−1/(qt 3 )) ∞ → (q 2 ) ∞ /(q −1 ) ∞ is the contribution of the decoupled chiral Φ 5 .
When considering the t, t → −1 limit of the figure-eight index I[4 1 ] from (3.44), the prefactor I 0 has the same divergent term (−1/t) −1 ∞ that appeared for the trefoil. Moreover, the contribution ||B II || 2 id to the figure-eight index vanishes, because poles of the index integrand in line II are cancelled by zeroes in line I. Thus, following a short calculation, the figure-eight index takes the form We recognize in this the DGG index of the figure-eight knot, already split into two holomorphic blocks. For proper comparison to [40] or [41], we should again rescale ξ → q −1 ξ, or (x, x) → (q −1 x, q x).

Critical points and missing vacua
We saw in Section 4. We can make this idea much more precise by considering the effective twisted superpotentials that govern theories T [M 3 ] on R 2 × S 1 . For example, for the trefoil, this was given by (3.14): It is important to note that this function on C * (parametrized by the dynamic variable s) has branch cuts coming from integrating out chiral matter that at some points in the s-plane becomes massless. In particular, each term Li 2 (f (s)) has a cut along a half-line starting at the branch point f (s) = 1 and running to zero or infinity. Such cuts and their consequences have been discussed from various perspectives in e.g. [14,[62][63][64]. Often one writes the vacuum or critical-point equations as exp s ∂ W 3 1 /∂s = 1 , (4.14) because in this form they are algebraic in s. However, when analyzing vacua of T [M 3 ] on R 2 × S 1 , one must remember to lift solutions of (4.14) back to the cover of the s-plane defined by W -and to make sure they are actual critical points on some sheets of the cover. Now consider what happens if we send t → −1. The branch points of Li 2 (s) and Li 2 (−1/(st)), located at s = 1 and s = −1/t, collide. (These branch points came directly from the chirals Φ 1 and Φ 3 , which we integrated out of T [3 1 ] in (4.3).) In the process, the half-line cuts originating at these branch points coalesce into a full cut running from s = 0 to s = ∞; this is easy to see from the inversion formula Moreover, one of the solutions s * to (4.14), or rather its lift(s) to the covering of the splane, gets trapped between the colliding branch points and ceases to be a critical point as t → −1. One can see this from the explicit form of the critical-point equations (3.15), which are reduced from quadratic to linear order in s by a cancellation at t = −1. However, to properly interpret this limit, it is helpful to think about the branched cover of the s-plane as we have done.
Physically, each solution of (4.14) is a vacuum of T [M 3 ] on R 2 × S 1 . As t → −1, the vacuum at s * is lost. This is possible precisely because the t → −1 limit is singular. Indeed, we know that t → −1 corresponds to making T [M 3 ] massless, so that the reduction on R 2 × S 1 is no longer fully described by an effective twisted superpotential. The specialized superpotential W (s; x, t = −1) does not describe T [M 3 ] itself at the massless point, but rather the Higgsed T DGG [M ] as found in Section 4.1.
In the case of the trefoil, the vacuum at s * close to t = −1 is labelled (via the 3d-3d correspondence) by the abelian flat connection on M 3 = S 3 \K. Indeed, if we substitute the limiting t → −1 value of s * (namely s * = 1) into the SUSY-parameter-space equation exp x ∂ W /∂x = y, we find corresponding to the abelian factor y − 1 = 0 of the trefoil's classical A-polynomial. Thus we see explicitly that the DGG theory T DGG [3 1 ] loses a vacuum corresponding to the abelian flat connection. We may also perform this analysis at the level of holomorphic blocks. Holomorphic blocks are labelled by (q-deformed) critical points of W -or more precisely by integration cycles Γ α obtained by starting at a critical point of W and approximately following gradient flow with respect to Re 1 log q W . For the trefoil we can choose a basis of integration cycles given by Γ II and Γ III in Figure 3. The precise correspondence with critical points depends on x, t, q. Close to t = −1, however, it is clear that Γ II corresponds to the "abelian" critical point s * . As t → −1, the contour Γ II gets trapped crossing a full line of poles (resolutions of the classical branch cuts described above), and ceases to be a good holomorphic-block integration cycle in the sense of [41]. 10 Most importantly, it no longer flows from any classical critical point. Beautifully, the remaining contour Γ III is isolated away from the point s * where halflines of poles merge. The t → −1 limit of the corresponding block B III (x, t; q) is precisely the holomorphic block of T DGG [3 1 ], labelled by the irreducible flat SL(2, C) connection, and contributing to the index (4.11).
Analogous remarks apply to the figure-eight example. The 3d Higgsing and integrating out of Φ 1 , Φ 3 in T [4 1 ] translates on R 2 × S 1 to branch points of Li 2 (s) and Li 2 (−1/(st)) colliding in (3.19), and trapping a critical point between them. Thus, as t → −1, T [4 1 ] looses one of its three massive vacua on R 2 × S 1 -the one labeled by the abelian connection on

Relation to colored differentials
We expect that the Higgsing procedure found to relate T [M 3 ] to T DGG [M 3 ] in the examples above holds much more generally. We can actually recognize some key signatures of the reduction in a much larger family of examples, which include so-called thin knots. The phenomena described above follow from the structure of colored Poincaré polynomials for these knots. The structure of the Poincaré polynomials is highly constrained by the properties of colored differentials whose existence in S r -colored homologies was postulated in [21,65], as well as by the so-called exponential growth. Using these properties, in [8] colored Poincaré polynomials of many thin knots, including the infinite series of (2, 2p + 1) torus knots and twist knots with 2n + 2 crossings, were uniquely determined.
More precisely, colored differentials enable transitions between homology theories labeled by the r-th and k-th symmetric-power representations S r and S k . The existence of these differentials implies that Poincaré polynomials take the form of a summation (over k = 0, . . . , r), with the summand involving a factor (−aq −1 t; q) k . On the other hand, the exponential growth is the statement that for q = 1 (normalized) colored Poincaré polynomials (superpolynomials) satisfy the relation If the uncolored superpolynomial on the right hand side is a sum of a few terms, its r'th power can be written as a (multiple) summation involving Newton binomials, which for arbitrary q turn out to be replaced by q-binomials [8,9]. This structure can be clearly seen in the example of (2, 2p + 1) torus knots considered in [8,9], whose (normalized) colored superpolynomials take the form P S r T 2,2p+1 (a, q, t) = a pr q −pr 0≤kp≤...≤k 2 ≤k 1 ≤r (1 + aq i−2 t).
Here the last product originating from the structure of differentials, as well as a series of q-binomials originating from the exponential growth, are manifest (in this formula k 0 = r). Poincaré polynomials for infinite families of twist knots derived in [8,9] share analogous features. It becomes clear now that various properties of trefoil and figure-8 knots, discussed earlier, should also be present for other knots, such as thin knots discussed above. For example, as discussed in section 4.2, the divergence at t → −1 in the trefoil and figure-8 indices, I[3 1 ] and I[4 1 ], is a manifestation of a pole due to the factor 1/(−1/t) ∞ . This factor originates from the q-Pochhammer symbol (−aq −1 t; q) k in corresponding Poincaré polynomials (2.4) and (2.16), after setting a = q 2 and rewriting this term in the denominator. As follows from the discussion above, such a factor is present in general for other thin knots (and represents the action of colored differentials), so for such knots an analogous pole at t → −1 should develop. We postulate that the residue at this pole in general reproduces indices I DGG [M ] for theories dual to other (thin) knots.
Similarly, a decoupling of the abelian branch for more general knots is a consequence of the structure of superpolynomials described above. From this perspective, let us recall once more how this works for trefoil and figure-8 knot, just on the level of critical point equations (3.15) and (3.16), or (3.20) and (3.21). If we set t = −1 in (3.15) or (3.20), the ratio 1+st 1−s on the left hand side drops out of the equation (this is a manifestation of the cancelation (4.15) at the level of twisted superpotential ). In this ratio the numerator 1+st has its origin in the (−aq −1 t; q) k term in superpolynomials (2.4) and (2.16), while the denominator 1 − s originates from q-Pochhamer (q; q) k being a part of the q-binomial in those superpolynomials. As explained above, such terms appear universally in superpolynomials for thin knots. Similarly, for t = −1 the equations (3.16) and (3.21) reduce to y = 1 (which represents the abelian branch that drops out when t → −1 is set first) due to a cancellation between the term in their numerator and s−x in denominator. The terms in numerator have the origin in (a(−t) 3 ; q) r from unknot normalization (2.15), possibly combined with another term (aq r (−t) 3 ; q) k representing colored differentials for figure-8 knot (2.16). The term s − x in denominator has its origin in (q; q) r−k ingredient of q-binomial. Analogous terms, responsible for cancellations, are also universally present in superpolynomials for other knots. The analysis is slightly more involved if Poincaré polynomials include multiple summationse.g. for (2, 2p + 1) torus knots (4.18) -however one can check that similar cancellations between "universal" terms decrease the degree of saddle equations and result in the decoupling of the abelian branch.

Boundaries in three dimensions
In this section we discuss the gluing along boundaries of M 3 and the boundary conditions in 3d N = 2 theories T [M 3 ].
In particular, understanding the operations of cutting and gluing M 3 along a Riemann surface C opens a new window into the world of closed 3-manifolds. The basic idea of how such operations should manifest in 3d N = 2 theory T [M 3 ] was already discussed e.g. in [54,56] and will be reviewed below. The details, however, cannot work unless T [M 3 ] accounts for all flat connections on M 3 . This was recently emphasized in [10] where the general method of building T [M 3 ] via gluing was carried out for certain homology spheres.
After constructing 3d N = 2 theories T [M 3 ] for certain homology spheres, we turn our attention to boundary conditions in such theories. Incorporating boundary conditions and domain walls in general 3d N = 2 theories was discussed in [14] and involves the contribution of the 2d index of the theory on the boundary / wall that is a "flavored" generalization of the elliptic genus. For theories of class R that come from 3-manifolds, many such boundary conditions come from 4-manifolds as illustrated in (1.6). In this case, the flavored elliptic genus of a boundary condition / domain wall is equal to the Vafa-Witten partition function of the corresponding 4-manifold [10].

Cutting and gluing along boundaries of M 3
It is believed that a 3-manifold with boundary C gives rise to a boundary condition in 4d N = 2 theory of class S, see Figure 2 in [2] or Figure 6 in [10]. This system can be understood as a result of 6d (2, 0) theory compactified on a 3-manifold with cylindrical end R + × C and to some extent was studied previously. 11 For example, when C = T 2 is a 2-torus (with puncture) the corresponding 4d N = 2 theory is actually N = 4 super-Yang Mills (resp. N = 2 * theory).
A simple class of 3-manifolds bounded by C includes handlebodies, which for a genus-g Riemann surface C is determined by a choice of g pairwise disjoint simple closed curves on C (that are contractible in the handlebody 3-manifold). For example, if C = T 2 , then the corresponding handlebody is a solid torus: It is labeled by a choice (p, q) of the 1-cycle that becomes contractible in M 3 . In the basic case of (p, q) = (0, 1) the Chern-Simons path integral on M 3 defines a state (in the Hilbert space H T 2 ) that is usually denoted |0 , so that we conclude It was proposed in [10] that the corresponding boundary condition in 4d theory T [C] is Nahm pole boundary condition [67,68] that can be described by a system of D3-branes ending on D5-branes 12 More generally, for M 3 ∼ = S 1 × D 2 obtained by filling in the cycle in homology class (p, q) the corresponding boundary condition is defined by a system of D3-branes ending on IIB five-branes of type (p, q). This class of boundary conditions can be easily generalized to other Riemann surfaces C and 3-manifolds with several boundary components. The latter correspond to domain walls in 4d N = 2 theories T [C], see e.g. [2,10,66] for details. For example, each element φ of the mapping class group of C corresponds, on the one hand, to a mapping cylinder M 3 (with two boundary components identified via φ) and, on the other hand, to a duality wall of type φ in the 4d theory T [C]. In the case C = T 2 we have the familiar walls that correspond to the generators φ = S and φ = T of the SL(2, Z) duality group of N = 4 super-Yang-Mills, and the general "solid torus boundary condition" described above can be viewed as the IR limit of a concatenation of S-and T -walls with the basic Nahm pole boundary condition, see [10, pp.20-21] for details. For instance, Clearly, there are still many details to work out, but we have outlined the key elements necessary to glue 3-manifolds along a common boundary and, in particular, to illustrate 11 See e.g. [2,[54][55][56]66] for a sample of earlier work; unfortunately the methods of these papers cannot be used to recover all flat connections for general 3-manifolds, even in the simplest cases of knot complements. 12 Whether we identify the state |0 with D5 or NS5 is a matter of conventions. Here we follow the conventions of [10,14]. A particularly simple and useful operation that involves (re)gluing solid tori a la (5.1)-(5.5) is called surgery. In fact, it is also the most general one in a sense that, according to a theorem of Lickorish and Wallace, every closed oriented 3-manifold can be represented by (integral) surgery along a link K ⊂ S 3 . Since the operation is defined in the same way on any component of the link L it suffices to explain it in the case when K has only one component, i.e. when K is a knot. Then, for a pair of relatively prime integers p, q ∈ Z, the result of q/p Dehn surgery along K is the 3-manifold: where N (K) is the tubular neighborhood of the knot, and S 1 ×D 2 is attached to its boundary by a diffeomorphism φ : S 1 × ∂D 2 → ∂N (K) that takes the meridian µ of the knot to a curve in the homology class The ratio q/p ∈ Q ∪ {∞} is called the surgery coefficient.
In what follows we discuss various aspects of cutting, gluing, and surgery operations. In particular, we shall see how the operations (5.6) and (5.7) manifest at various levels in 3d N = 2 theory T [M 3 ] -at the level of SUSY vacua, at the level of twisted superpotential, and at the level of quantum partition functions -thereby illustrating the important role of abelian flat connections. Needless to say, there are many directions in which one could extend this analysis, e.g. to various classes of 3-manifolds not considered in this paper, as well as more detailed analysis of the ones presented here, to higher rank groups G and to relation with known properties of homological knot invariants.

Compactification on S 1 and branes on the Hitchin moduli space
A useful perspective on our 3d-4d system can be obtained by compactification on S 1 and studying the space of SUSY vacua. Thus, a compactification of 4d N = 2 theory T [C] on a circle yields a 3d N = 4 sigma-model whose target is the hyper-Kähler manifold

Lens space theories and matrix models
In the above discussion we used the solid torus (5.1)-(5.2) as a simple example of a handlebody, in this case bounded by C = T 2 . Likewise, the simplest example of a closed 3-manifold obtained by gluing two solid tori is the Lens space L(p, 1) = 0|ST p S|0 ∼ = S 3 /Z p (5.14) Using the dictionary (5.3) and (5.4), we can identify the corresponding 3d N = 2 theory T [L(p, 1)] as the theory on D3-branes suspended between a NS5-brane and a (p, 1)-fivebrane: T [L(p, 1); G] = SUSY G −p Chern-Simons theory (5.15) Following [10], here we assumed that the gauge group G is of Cartan type A, i.e. G = U (N ) or G = SU (N ). It would be interesting, however, to test the conjecture (5.15) for other groups G. Now, let us discuss this gluing more carefully, first from the viewpoint of flat connections (= SUSY vacua) and then from the viewpoint of partition functions. According to (5.4) and For the case at hand, the result of this operation modifies the space of SUSY parameters from y = 1 to y = x p . Finally, gluing 0|S and T p S|0 in (5.14) means sandwiching N = 4 super-Yang-Mills between the corresponding boundary conditions. In our IR description of the boundary conditions, this makes U (1) x dynamical, so that all critical points of the effective twisted superpotential [10]: i.e. Young diagrams that fit in a rectangle of size N × (p − 1). Note, these are in one-to-one correspondence with integrable representations of su(p) N (equivalently, of u(N ) p ), the fact that plays an important role [70][71][72][73] in the study of Vafa-Witten partition function on ALE spaces bounded by L(p, 1). Next, let us consider the gluing (5.14) at the level of partition functions. The partition function of the theory (5.15) on the ellipsoid S 3 b is given by (see e.g. [74]): where r = rank(G), W is the Weyl group of G, h is the dual Coxeter number of G, Λ + rt is the set of positive roots of G, and ρ is the Weyl vector (half the sum of the positive roots). Furthermore, turning on a FI parameter ζ contributes an extra term e 4πiζTr σ into the integral (5.18). As usual, the S 3 b partition function of T [L(p, 1); G] should have the following structure where q = e = e 2πib 2 and each block B ρ (q) ∼ Z ρ CS (L(p, 1); G) is expected to represent the Chern-Simons partition function computed in the background of a flat connection labeled by ρ. (Recall from our earlier discussion that classical solutions in Chern-Simons theory on M 3 = L(p, 1) are labeled by certain Young diagrams ρ.) Unfortunately, there is no systematic algorithm to define holomorphic blocks in general 3d N = 2 theories and, as a result, the factorization (5. 19) is not known at present for the N = 2 super-Chern-Simons theory (5.15). However, the integral form of the partition function (5.18) does share many key features with the Chern-Simons partition function on the Lens space that will be discussed in the next section (and extended to more general Seifert manifolds). Here, let us just note that log Z S 3 b in (5.18) has the form of a power series in = 2πib 2 that starts with the leading 1 term and terminates at the order O( ). This is indeed the property of Chern-Simons partition function on L(p, 1): according to a famous result of Lawrence and Rozansky [75], higher loop corrections to Z ρ CS (L(p, 1); G) all vanish. Finally, we propose a "lift" of the gluing formula (5.17) to a similar formula at the level of partition functions, cf. (5.6): where the integration measure [dU ] = Z T [C] dx is determined by the 4d N = 2 theory T [C; G] associated with the Riemann surface C = ∂M + 3 = −∂M − 3 . It would be interesting to test this gluing formula in concrete examples, including the Lens spaces and Seifert manifolds discussed here. Note that with the t-variable that keeps track of homological grading, (5.20) basically is a surgery formula for homological knot invariants. Such formulas are indeed known in the context of knot Floer homology and its version for general 3-manifolds, the Heegaard Floer homology.
As explained around (5.5), we can construct closed 3-manifolds by gluing open 3-manifolds along their boundaries. The Chern-Simons partition functions on manifolds with torus boundary depend on a parameter x, which should be integrated out upon gluing. For a particular class of 3-manifolds, the resulting Chern-Simons partition functions can be represented as matrix integrals, much like (5.18), where the integration measure is responsible for integrating out the parameters x. The integrands of such matrix models take the form where V (x) is usually called potential and 2πi = 2πi log q is called the "level". Let us note that in the case of 3-manifolds with boundary, when the parameters x are not integrated out, the same representation of partition functions Z CS ∼ exp( 1 W + . . .) was used to read off the twisted superpotentials of dual N = 2 theories, such as (3.14) or (3.19). One is therefore tempted to postulate, that a matrix model potential V (x) might encode information about the twisted superpotential and field content of the dual N = 2 theory T [M 3 ] associated to a closed 3-manifold M 3 . Let us demonstrate that this is indeed the case.
For non-abelian Chern-Simons theories it is convenient to a write matrix model representation of their partition functions in terms of eigenvalues σ i = log x i . A very well known example is a matrix model representation of the U (N ) Chern-Simons partition function on M 3 = S 3 [30,76], whose measure takes the form of a trigonometric deformation of the Vandermonde determinant, and the potential V (σ) = σ 2 /2 is Gaussian in σ = log x. More generally, the matrix model potential for M 3 = L(p, 1) and G = U (N ) takes the form V (σ) = pσ 2 /2. More involved integral representations of Chern-Simons partition functions on other Lens spaces and Seifert homology spheres can be found in [30,75,76]. Various other matrix integral representations of Chern-Simons or related topological string partition functions, including the refined setting, were constructed in [26,[77][78][79][80][81][82][83].
Let us now consider more seriously the proposal that the potential of a Chern-Simons matrix model determines the dual 3d N = 2 theory T [M 3 ]. For example, as reviewed above, the potential for a theory of the Lens space L(p, 1) takes the form V (σ) = pσ 2 /2. Taking into account a minus sign in (5.21), and using by now familiar 3d-3d dictionary, we might conclude that the dual theory is N = 2 theory at level −p, at least in the abelian case. Due to the universal form of the matrix integral, we might also be tempted to declare that in the nonabelian case the dual theory is U (N ) theory at level −p. This is precisely the dual theory (5.15) which was originally constructed by other means. We also emphasize that the form of the matrix model reflects the structure of the gluing (5.5), namely the fact that the resulting Lens space (5.14) is constructed from two solid tori (unknot complements), glued with a suitable SL(2, Z) twist φ. Indeed, in this case the potential factor (5.21) represents the gluing SL(2, Z) element φ, while the information about two solid tori is encoded in the matrix model measure. This construction is discussed in detail e.g. in [76].

Seifert manifolds and D4-D6 systems
The matrix model potential suggests a dual 3d N = 2 theory T [M 3 ; G] also for other Lens spaces and more general Seifert homology spheres. In this section, we start with a brief review of the most general Seifert fibered 3-manifolds and then discuss how various ways to look at their geometry find application in 3d-3d correspondence. For a nice exposition of Seifert manifolds see e.g. [84]. 14 a) b) D6 Figure 4. a) M-theory on a Seifert fibered 3-manifold M 3 , and b) its reduction to type IIA string theory with D6-branes. Upon reduction on the S 1 fiber the fivebrane system (1.2) turns into a system of D4-branes wrapped on the (orbifold) surface Σ intersecting D6-branes at finitely many points on Σ.
Seifert manifolds were introduced 80 years ago and can be described in a number of equivalent ways: • A Seifert fibered manifold M 3 is a circle bundle over a 2-dimensional orbifold Σ.
• A 3-manifold M 3 is Seifert fibered if and only if it is finitely covered by an S 1 -bundle over a surface.
• Finally, a Seifert manifold M 3 can be constructed by a sequence of surgeries on a trivial circle fibration over a Riemann surface.
Each closed Seifert fibration with n exceptional fibers is classified by the following set of Seifert invariants (also known as the symbol of the Seifert manifold): {b, ( , g); (p 1 , q 1 ), . . . , (p n , q n )} , gcd(p i , q i ) = 1 (5.22) where tells us whether M 3 and Σ are orientable. Since both will be assumed to be orientable, will not play an important role in our discussion. The integer-valued invariant b is (minus) the Euler number of the S 1 -bundle, while the non-negative integer g is the genus of the underlying base orbifold Σ, whose orbifold Euler characteristic is The pair (p j , q j ) of relatively prime integers are the Seifert invariants of the j-th exceptional fiber, locally modeled on the Z p j orbifold: For n = 0, 1, 2 and g = 0, the Seifert fibration produces a Lens space L(p, q) with where cp 1 − d(bp 1 + q 1 ) = 1. For n = 3 it gives a Brieskorn sphere Σ(p 1 , p 2 , p 3 ) for any choice of q 1 , q 2 , q 3 . One can add integers to each of the rational numbers b, q 1 p 1 , . . ., qn pn provided that their sum remains constant. In other words, is an invariant of oriented fibrations. Usually, the symmetries are used to achieve 1 ≤ q i < p i for all i = 1, . . . , n. Another popular choice of (partial) "gauge fixing" is to use the symmetry (5.27) to set b = 0. This choice gives the so-called non-normalized Seifert invariants and clearly is very non-unique [84]. The description of a Seifert manifold M 3 as an S 1 -bundle over a 2-dimensional orbifold Σ is very helpful in understanding the fivebrane system (1.2) and, therefore, the corresponding 3d N = 2 theory (1.1). Indeed, by interpreting the circle fiber of M 3 as the "M-theory circle" we can equivalently describe (1.2) in type IIA string theory. Upon this reduction, the fivebranes supported on R 3 × M 3 become D4-branes with world-volume R 3 × Σ. In addition, singular fibers of the S 1 fibration in general give rise to D6-branes supported at (orbifold) points on Σ and intersecting D4-branes along the R 3 part of their world-volume, as illustrated in Figure 4  Equivalently, a Seifert manifold M 3 can be produced by a sequence of Dehn surgery operations along the fibers of the trivial S 1 bundle over Σ g . Indeed, since the tubular neighborhood of every fiber is bounded by a 2-torus, each surgery operation is specified by the image of the meridian circle or, more precisely, by its homology class where p ∈ Z + and q ∈ Z are coprime integers. The integral surgery (with p = 1) is special and can be represented by a four-dimensional cobordism of attaching a 2-handle. It does not introduce a singular fiber and merely changes the degree of the S 1 bundle by q.
Hence, a Seifert manifold with the symbol (5.22) can be constructed by a sequence of n+1 surgeries on S 1 × Σ g with the surgery coefficients b, q 1 p 1 , . . ., qn pn . In the surgery presentation, the symmetries (5.27) correspond to basic Kirby moves [10] represented by dualities of the 3d N = 2 theory T [M 3 ]. Surgery presentation is especially useful for applications to Chern-Simons theory, quantum group invariants, and their categorification.
For future use, let us note that the fundamental group of M 3 fits into the following exact sequence π 1 (S 1 ) → π 1 (M 3 ) → π 1 (Σ) → 1 (5.29) and where L is a line bundle over Σ associated with the circle bundle M 3 = S(L). In particular, if we want to work with a homology sphere, we need to take g = 0. Then, the resulting space is a plumbing 3-manifold given by the graph in Figure 5 and its (co)homology can be computed using the algorithm described in [10, sec.2.2]. One of the results of this calculation is that A Seifert homology sphere M 3 can be constructed by a surgery on a link in S 3 with n + 1 components, which consists of n parallel mutually unlinked unknots, all linked (with linking number one) to one additional copy of the unknot. The surgery coefficients for n parallel unknots are q i p i , i = 1, . . . n. An integral representation for U (N ) Chern-Simons partition function on a Seifert homology sphere M 3 was found in [30] and it takes the form For the Lens space case n = 1, 2, with p i = 1, the integration measure Dσ in this expression reduces to the standard (trigonometric) Vandermonde determinant k<l (2 sinh σ k −σ l 2 ) 2 , which has straightforward interpretation as a unitary matrix integral; for other cases we get more general integral representation, with modified measure. More precisely, the above integral represents contribution from some particular flat connection, whose choice is specified by the choice of t i and the linear term k lt k σ k in the potential. As discussed after (5.18), such linear terms in the potential and, therefore, the choice of the flat connection ρ corresponds to the choice of the FI term in the partition function of the dual 3d N = 2 theory T [M 3 ]. The full Chern-Simons partition function is given by a sum of such contributions, taking into account all flat connections. Finally, it is important to remember that the coefficient of the Gaussian term in the potential is rescaled and takes form = n j=1 q j p j −1 . (5.33) Again, by applying the standard 3d-3d dictionary, at least for G = U (1), one might conclude that the theory dual to a Seifert homology sphere is 3d N = 2 Chern-Simons theory at a fractional level. This is again consistent with the predictions of [10]. Moreover, it is well known that Chern-Simons theory at fractional level is equivalent to a quiver Chern-Simons theory with integer levels.
To be more specific, focusing on the Lens space M 3 = L(p, q) and G = U (1) let us demonstrate how this data determines the dual quiver theory T [M 3 ; G] and show the equivalence of this quiver theory to a 3d N = 2 abelian Chern-Simons theory at a fractional level. According to [10], at least in the abelian case, the theory dual to L(p, q) Lens space is a U (1) k quiver Chern-Simons theory with interactions between various U (1) gauge fields specified by a quadratic matrix a 1 1 0 0 · · · 1 a 2 1 0 · · · 0 1 a 3 1 · · · 0 0 1 a 4 · · · . . . . . .
where a 1 , . . . , a k arise from the continuous fraction expansion of p/q: Schematically, denoting U (1) Chern-Simons gauge fields by u i , the twisted superpotential of this quiver theory takes the form We can now integrate out u 2 , . . . , u k fields using their equations of motion Solving these equations, starting from the last one and proceeding to the first one, we find We can also use the equations of motion to get rid of all u i (u i−1 + u i+1 ) and u k u k−1 terms in (5.36). Finally, substituting the above result for u 2 , the twisted superpotential takes the form which indeed represents the abelian 3d N = 2 super-Chern-Simons theory at fractional level −p/q. This p/q factor precisely agrees with the rescaling (5.33) of the Gaussian potential in (5.32) for the Lens space X(q/p) = L(p, q). As a special case, let us also note that for a i = 2 we find u i = k+1−i k u 1 and p/q = (k + 1)/k, which corresponds to L(k + 1, k) Lens space. Therefore, by 3d-3d dictionary, the potential rescaled by p/q in (5.32) suggests that the dual 3d theory is N = 2 Chern-Simons at level −p/q, or equivalently the quiver Chern-Simons theory determined by the interaction matrix (5.34). Since matrix integrals result from "abelianization" of non-abelian theories, it is tempting to speculate that similar correspondence holds for non-abelian G as well.
Using this dictionary one could also consider other matrix models representation of Chern-Simons partition functions [26,30,[75][76][77][78][79][80][81][82][83], either for various interesting manifolds or in the refined setting, and predict the form of dual 3d N = 2 theories T [M 3 ; G]. Interestingly, the models derived in loc. cit. have potentials that consist of quadratic and dilogarithmic terms, which indeed are the basic ingredients in modeling the content of dual 3d N = 2 theories. Also, in some cases inequivalent matrix model representations of the same Chern-Simons partition function are known and, hence, might lead to interesting new dualities of 3d N = 2 theories.

Dehn surgery
As a final simple illustration of the necessity of accounting for all flat connections, we return to the basic Dehn surgery operation (5.7). Suppose that the knot K = 3 1 is the trefoil. As we know well from Section 3.2, the A-polynomial 15 for the trefoil, parametrizing M SUSY (T [3 1 ], SU (2)) for the full trefoil-complement theory T [3 1 , SU (2)], is A(x, y) = (y − 1)(y + x 6 ) ⊂ (C * × C * )/Z 2 . (5.39) Here x and y are the C * -valued eigenvalues of longitude and meridian SL(2, C) holonomies on the torus boundary of the knot complement, well defined up to the Weyl-group action (x, y) → (x −1 , y −1 ). We recall that the (y − 1) component of the A-polynomial corresponds to an abelian flat connection on the knot complement, while the (y +x 6 ) component corresponds to an irreducible flat connection. Suppose that we perform Dehn surgery with q/p = ±1 on the trefoil knot complement. The result is a closed 3-manifold -in fact one of the Brieskorn spheres of (5.25) In each case, the moduli space of flat SL(2, C) connections on S 3 p/q (3 1 ), consists of isolated points. It is easy to count them directly from a presentation of the fundamental groups of the Brieskorn spheres, This does not quite match the count of flat connections on the Brieskorn spheres: in each case, there is one extra intersection point in (5.42). In particular, in each case, the intersection point (x, y) = (−1, −1) corresponds to flat connections on the knot complement S 3 \3 1 and the solid surgery torus whose eigenvalues match at the T 2 surgery interface, but whose full holonomies do not. Namely, the flat connection on the solid surgery torus with eigenvalues (−1, −1) is trivial, while the flat connection on the trefoil knot complement with eigenvalues (−1, −1) is parabolic, meaning the full holonomy matrix is −1 1 0 −1 . This is not unexpected, since (x, y) = (−1, −1) lies on the nonabelian branch y + x 6 = 0 of the trefoil's A-polynomial. After subtracting the "false" intersection point from the counts in (5.42), we recover the expected number of flat connections on S 3 +1 (3 1 ) and S 3 −1 (3 1 ). 15 In contrast to the rest of the paper, we take care in this section to write A-polynomials in terms of actual SL(2, C) meridian and longitude eigenvalues rather than their squares. Thus, for the trefoil, the non-abelian A-polynomial is written as y + x 6 rather than y + x 3 . The distinction is important for consistently counting SL(2, C) (as opposed to P SL(2, C), etc.) flat connections resulting from surgery.
Physically, (5.42) is the (naively) expected count of vacua when gluing the trefoil theory to an unknot theory with the appropriate element φ ∈ SL(2, Z) corresponding to the Dehn surgery. The presence of a "false" intersection point (x, y) = (−1, −1) suggests that the corresponding vacuum in the glued theory must be lifted. It would be interesting to uncover the mechanism behind this. The remaining vacua match the count of flat connections on the Brieskorn spheres (i.e. vacua of T [S 3 ±1 (3 1 ), SU (2)]), as they should. Crucially the vacuum corresponding to the intersection point (x, y) = (1, 1) must be included in order for the count to work out; this intersection point sits on the abelian branch (y − 1) of the trefoil A-polynomial, and labels the trivial flat connection on S 3 ±1 . A similar phenomenon occurs when considering simple surgeries on the figure-eight knot complement S 3 \4 1 . For example, the Brieskorn sphere Σ[2, 3, 7] may be constructed from +1 or −1 surgeries on S 3 \4 1 . (The two different surgeries produce opposite orientations on Σ [2,3,7]. , which will never satisfy µ p λ q = I for any p, q.  Much like in Chern-Simons theory on M 3 the presence of non-trivial boundary requires specifying boundary conditions, the same is true in the case of 3d N = 2 theories. One impor-tant novelty, though, is that some boundary conditions are now distinguished if they preserve part of supersymmetry, such as half-BPS boundary conditions that preserve N = (0, 2) supersymmetry on the boundary. These "B-type" boundary conditions have been studied only recently in [14] and then in [86].
In the presence of a boundary (or, more generally, a domain wall) one can define a generalization of the index as a partition function on S 1 × q D with a prescribed B-type boundary condition on the boundary torus S 1 × q S 1 ∼ = T 2 of modulus τ , as illustrated in Figure 2. The resulting half-index I S 1 ×qD is essentially a convolution of the flavored elliptic genus of the 2d N = (0, 2) boundary theory with the index of a 3d N = 2 theory on S 1 × q D. The contribution of (0, 2) boundary degrees of freedom is summarized in Table 1 where, as usual, gauge symmetries result in integrals over the corresponding variables σ i .
The half-index I S 1 ×qD labeled by a particular choice of the boundary condition can be viewed as a UV counterpart of a holomorphic block labeled by a choice of the massive vacuum in the IR. Moreover, since the half-index is invariant under the RG flow, it makes sense to identify some of massive vacua and integration contours in the IR theory with specific boundary conditions in the UV. The latter, in turn, can sometimes be identified with 4manifolds via (1.6), which altogether leads to an interesting correspondence between certain holomorphic blocks and 4-manifolds.
Note, that for theories T [M 3 ; G] labeled by closed 3-manifolds, supersymmetric vacua ρ ∈ M SUSY (T [M 3 ; G]) specify boundary conditions for the Vafa-Witten topological gauge theory on a 4-manifold bounded by M 3 . Therefore, had we missed any of the vacua in constructing T [M 3 ; G] there would be no hope to relate supersymmetric boundary and 4manifolds in (1.6).
For instance, let us consider one of the simplest 3d N = 2 theories, namely the super-Chern-Simons theory with gauge group G = U (N ) that in (5.15) we identified with the Lens space theory. As we mentioned earlier, the holomorphic blocks for this theory are not known. However, their UV counterparts I S 1 ×qD are easy to write down by choosing various B-type boundary conditions constructed in [10,14,86]. Thus, a simple boundary condition involves pN Fermi multiplets on the boundary. According to the rules in Table 1, its flavored elliptic genus can be interpreted as the half-index of 3d N = 2 super-Chern-Simons theory (5.15) with gauge group G = U (N ): is the well known form of the Vafa-Witten partition function on the ALE space (5.45) written in the "discrete basis" [70][71][72][73]. Here, ρ is a Young diagram with at most p − 1 rows and N columns that in the previous section we identified with the choice of flat connection on M 3 = ∂M 4 = L(p, 1).

Conclusions and open questions
In this work we have studied 3d-3d correspondence, which relates (fivebranes compactified on) non-trivial 3-manifolds to the 3d N = 2 theories. To much extent we have focused on examples of theories whose partition functions can be identified with homological knot invariants. We discussed their relation to the theories considered previously by Dimofte-Gaiotto-Gukov, stressed the importance of taking into account all flat connections in the construction of the N = 2 theories, and discussed the role of boundary conditions on both sides of the correspondence. While the approach in the main part of the paper combines the strong points of [1] and [2,24], there is, however, something deeply puzzling between these two lines of development. They both morally describe 3d N = 2 theory associated either to a knot K or a 3-manifold M 3 , but realize the quantum / categorified invariants of K very differently. Indeed, the approach of [1] leads to P r K (q, t, . . .) as a vortex partition function on S 1 × R 2 in a sector with vortex number r, Z vortex (T DGH ) = r z r P r K (q, t, . . .). It is therefore important to understand the relation between these formulas and between the corresponding theories T DGH and T [M 3 ]. At this stage we can suggest some possible explanations, which however require further studies. First, it is very suggestive to compare two partition functions, (6.1) and (6.2), to the fourdimensional Nekrasov partition function and its dual partition function discussed in [87]. The Nekrasov partition function and its dual are indeed related by the Fourier transform, analogous to the one that relates (6.1) and (6.2). In this relation the (original) Nekrasov partition is evaluated on the discrete set of parameters determined by the summation parameters, similar to the form of P r K (q, t, . . .) in (6.2). Moreover, in four-dimensional case one can introduce the dual prepotential, which morally describes the same N = 2 theory, and is related by the Legendre transform to the (original) Seiberg-Witten prepotential. Similarly, one can associate two twisted superpotentials to both sides of (6.1), which are then related by the Legendre transform. Nonetheless, one should be cautious in following this analogy; in particular, the dual partition function was introduced in the non-refined limit 1 = − 2 , and it does not automatically extend to other cases.
As another possibility, one might try to interpret the relation (6.1) as gauging of the global U (1) x symmetry in a theory with the (half-)index B K * (x): while identifying the U (1) z flavor symmetry of T DGH as a topological symmetry for U (1) x . The coefficient of z r in (6.1) is the R 2 × q S 1 half-index of T DGH in an r-vortex sector. In turn, via the logic of Section 3.3, this half-index could be identified with the residue of the index of T [M 3 ] at x = q r -or, equivalently, the specialization of a holomorphic block of T [M 3 ] to x = q r . This provides a possible conceptual explanation of (6.1)-(6.2), whose details must be worked out with some care. This brings us back to the main and final question that remains unanswered: Is there a systematic construction of the 3d N = 2 theory T [M 3 ] that accounts for all classical solutions in G C Chern-Simons theory on M 3 ? Such construction might come from the triangulation data of M 3 , extending the work [2,3], or via representing M 3 in some other way.