BPS Invariants for Seifert Manifolds

We calculate the homological blocks for Seifert manifolds from the exact expression for the $G=SU(N)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mari\~no. For the $G=SU(2)$ case, it is possible to express them in terms of the false theta functions and their derivatives. For $G=SU(N)$, we calculate them as a series expansion and also discuss some properties of the contributions from the abelian flat connections to the Witten-Reshetikhin-Turaev invariants for general $N$. We also provide an expected form of the $S$-matrix for general cases and the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks.


Introduction
The Chern-Simons partition function on knot complement in 3-sphere S 3 with boundary conditions or on S 3 with Wilson loops supported on knots provide knot polynomial invariants, e.g. the Jones or the HOMFLY polynomial with colors or with refinement. The Jones polynomial, which is a polynomial with integer powers and integer coefficients, can be understood as the graded Euler characteristic of Khovanov homology, and this homology provides the categorification of the Jones polynomial. Those integer coefficients can be understood as the dimension of the vector space. Such categorification of knot polynomials have been studied in many literature, but it has not been much studied on closed 3-manifolds. For example, a mathematical definition or construction that categorifies the Chern-Simons (CS) partition function or the Witten-Reshetikhin-Turaev (WRT) invariant on closed 3-manifolds is not available yet. The WRT invariants for closed 3-manifolds have been calculated for a number of 3manifolds including Seifert manifolds. From what is originally expressed, it was not obvious to see whether such WRT invariant can be expressed in terms of q-series with integer powers and integer coefficients, which is the property suitable for categorification.
Meanwhile, in [1], it has been found that the WRT invariant on a Poincaré homology sphere Σ (2,3,5) can be expressed as (linear combination of) false theta functions with modular parameter τ being related to the Chern-Simons level as where K is the quantum corrected Chern-Simons level. More precisely, such expression is actually equal to the WRT invariant upon τ ց 1 K meaning that τ approaches to 1/K from the upper half plane of complex τ -plane. As the Poincaré homology sphere Σ (2,3,5) is an integer homology sphere, in this case it can be seen rather easily that the WRT invariant is expressed as q-series with integer powers and integer coefficients. After [1], a number of examples including the case of rational homology spheres have been calculated in [2][3][4][5][6]. However, the integrality of the WRT invariant was not obvious in those examples on rational homology spheres.
Recently, it has been conjectured in [7,8] that the WRT invariant for closed 3-manifold can be expressed in terms of so called homological block, which is a q-series with integer powers and integer coefficient so that it may admit the categorification. The conjecture for the case of G = U (N ) was also discussed in [8].
Here, we briefly summarize the conjecture of [7,8]. The conjecture states that the partition function of Chern-Simons theory with G = SU (2) can be decomposed into homological blocks, Z b (q), where Z b (q), which is defined on |q| < 1, takes a form of q ∆ b Z[[q]] with rational number ∆ b , K ∈ Z is a quantum-corrected level, and W a denotes the stabilizer subgroup Stab Z 2 (a) for a in Weyl group Z 2 of SU (2). The label a denotes the reducible flat connections or equivalently the abelian flat connections, when G = SU (2). Here, lk(·, ·) denotes the linking form on Tor H 1 (M 3 , Z). More specifically, given two elements a, b ∈ Tor H 1 (M 3 , Z), there is a 2-chain B ′ such that s b = ∂B ′ for some s ∈ Z =0 . Then the linking form is Here, Z a means the contribution from the abelian flat connections to the WRT invariant, which can be obtained from Borel resummation of perturbative expansion with respect to abelian flat connection a. Several examples including Lens space, O(1) → Σ g , 3-manifolds from plumbing graphs, and several rational homology Seifert manifolds were worked out and they supported the conjecture in [7,8].
Interestingly, the homological blocks are labelled by reducible flat connections, and this can be understood in the context of resurgent analysis [9]. According to the resurgent analysis on G = SU (2) case, the exact partition function is expressed in terms of Borel resummation of the perturbative expansion around abelian (or reducible) flat connections, while the contributions from non-abelian (or irreducible) flat connections are encoded in transseries expansions of the Borel resummation of the perturbative expansion with respect to abelian flat connections.
The homological blocks can be understood in the context of the M-theory configuration, where D 2 is a disc, T N is Taub-NUT space, and D 2 ⊂ T N . There are two U (1) symmetries in this system, which are the rotational symmetry U (1) q on D 2 and the U (1) R symmetry. These two symmetries provide two gradings Z × Z in the homological invariants that lead to homological blocks. When M 3 is a Seifert manifold, which is the 3-manifold that is considered in this paper, there is an additional symmetry U (1) β that arises due to the existence of semi-free U (1) action on the Seifert manifold. This will lead another extra grading Z in the homological invariants. Via the 3d-3d correspondence 1  1 Some aspects of the 3d-3d correspondence for Seifert manifolds have been discussed in [10][11][12][13].
where i and j denote the grading for U (1) q and U (1) R . Then the homolgical block, Z a , is given by Z a (q) = Tr Ha q i (−1) j (1.8) and this is a partition function or the half index of T [M 3 , G] on D 2 × q S 1 with a boundary condition a. If M 3 is a Seifert manifold, there is an additional grading due to U (1) β , (1.9) and the homological block is given by Z a (q, t) = Tr Ha q i (−1) j t l (1. 10) where t denotes the fugacity of U (1) β . In this paper, we only consider the case t = 1 of (1.10). The S-transform in (1.2) can be understood from several dualities from the M-theory configuration (1.6), now with S 1 instead of R. The Hilbert space H b encodes the spectrum of massless BPS particles of T [M 3 , G], which are realized as M2 branes ending on M5 branes. If considering G = SU (2) case, at the boundary, M2 branes are wrapped on nontrivial 1-cycle (b ′ , −b ′ ) on M 3 where [b ′ ] = b ∈ TorH 1 (M 3 , Z)/Z 2 and this b is interpreted as charge of the spectrum. Meanwhile, taking the type IIA limit of (1.6) on S 1 of D 2 and then T-dualizing along S 1 of (1.6), the resulting configuration becomes D3-D5 system in the type IIB string theory. Taking the S-duality of type IIB string, it becomes D3-NS5 system and the boundary condition of the system at the infinity of R + on which D3's are supported is provided by the connected component of SL(2, C) flat connections [14]. Meanwhile, since Z SU (2) (M 3 ) is written only in terms of the contributions from abelian flat connections a, Z a , in (1.2), the boundary conditions labelled only by the abelian flat connections a of SU (2) are taken into account. 2 Then, the subscript a of S-matrix S ab corresponds to the connected component of abelian flat connections, Hom(TorH 1 (M 3 , Z), U (1))/Z 2 , while the subscript b of S ab denotes the M2 brane charges in the configuration (1.6). We note that the linking form provides a pairing or isomorphism between b ∈ TorH 1 (M 3 , Z)/Z 2 and a ∈ (TorH 1 (M 3 , Z)) * /Z 2 := Hom(TorH 1 (M 3 , Z), U (1))/Z 2 , TorH 1 (M 3 , Z)/Z 2 lk ∼ = (TorH 1 (M 3 , Z)) * /Z 2 ∼ = π 0 M ab flat (M 3 , SU (2)) .
(1.11) [16]. We see that examples calculated in this paper fit into the expected structure of the WRT invariant of Seifert manifolds in terms of homological blocks which is discussed in section 3.2. The organization is as follows. In section 2, we calculate homological blocks from the integral expression of the G = SU (2) WRT invariant of Seifert manifolds with three singular fibers obtained by Lawrence and Rozansky, which serves as a basic example in this paper. The calculation covers any values of |TorH 1 (M 3 , Z)|. Considering the case that |TorH 1 (M 3 , Z)| is even number, we see that the formula for the S-transform (1.3) can be generalized. From the calculation, we see that the homological block can be expressed in terms of false theta functions that are available in literature. We also discuss resurgent analysis and see that when H = 1 the exact expression of Lawrence and Rozansky can be understood as the exact Borel sum of perturbative expansion of the analytically continued partition function around the trivial flat connections. Then, we move on to the case of higher rank G = SU (N ) in section 3. We discuss some properties of the formula that we obtained and provide some number of examples. From those examples, we provide a general form of the S-matrix for G = SU (N ) and also the structure of the WRT invariant in terms of homological blocks. In Appendix A, we perform similar calculation for the case of four singular fibers with G = SU (2). In this case, the homological block is expressed in terms of the false theta function of a different type that is discussed in [17]. We also calculate homological blocks for Seifert manifolds with more singular fibers and with higher genus base surface when G = SU (2) in Appendix B. In these cases, the homological blocks are expressed in terms of false theta functions discussed in section 2 and Appendix A, and their derivatives with respect to q.
Note added: While preparing the manuscript, we found that there are some partial overlaps with [18] on the G = SU (2) case.

G = SU(2) and three singular fibers
We consider the WRT invariant for G = SU (2) on Seifert manifolds with three singular fibers with genus of base surface being zero, X(P 1 /Q 1 , P 2 /Q 2 , P 3 /Q 3 ), where P j 's are coprime to Q j 's for each j and also P j 's are pairwise coprime. b of the Seifert invariant is set to zero. The order of the torsion of the first homology group |Tor H 1 (M, Z)| is given by where P = P 1 P 2 P 3 . Due to conditions that P j 's and Q j 's are coprime for each j and P j 's are pairwise coprime, P and H are coprime, which is also the case for arbitrary number of singular fibers. We denote quantum corrected level as K = k + 2 and According to [15], contributions of reducible flat connections including the trivial flat connection of G = SU (2) to the WRT invariant on Seifert manifolds described above can be written as and s(P, Q) is the Dedekind sum The integration cycle Γ t is chosen in such a way that for each t the integrand is convergent on both ends of infinity, e.g. when K ∈ Z + and P H is positive, Γ 0 is a line from −(1 + i)∞ to (1 + i)∞ through the origin and Γ t is parallel to Γ 0 and passes through y = −2πi P H t, which is a stationary phase point of the integrand. When P H is negative, the contour is given by clockwise rotation of Γ 0 by π 2 and similarly for Γ t . Also, we note that reducible flat connections in SU (2) case is abelian flat connections, so we use both interchangeably for G = SU (2). Here, t labels abelian flat connections where t = 0 corresponds to the trivial flat connection.
In (2.3) and in the rest of paper, we use the physics normalization (2.7) Also we put additional 1/2 to the expression in [15] to have a same overall coefficient with the result of [16]. We would like to express (2.3) in terms of convergent q-series for the analytically continued theory.

Calculation of the partition function
The last factor in the integral formula (2.3) can be expanded as and ǫ j = ±1. This depends on the choice of P j , but for simplicity of notation we denote it as χ 2P (n). Also, we assumed Re y > 0 and P > 0. We will discuss the case of other possible ranges in section 2.7. Then for 3 j=1 1 Here we didn't move the integration cycle from Γ t to Γ 0 but just changed integration variable from y + 2πi P H t to y for each t. We analytically continue K and take Im K < 0, i.e. |q| < 1. We also take H > 0. Then we choose an integration contour γ as a line parallel to the imaginary axis of y-plane that passes through Re y > 0. Calculating the integral, we obtain the partition function of the analytically continued SU (2) theory for abelian flat connections, We note that though Z ab SU (2) (M 3 ) in (2.13) seemingly contains contributions only from abelian flat connections, it is expected to be the full partition function Z SU (2) (M 3 ) of analytically continued theory, which contains the contributions from nonabelian flat connections as transseries. This will be discussed in section 2.6. So in the following discussion we will denote (2.13) by Z SU (2)

The case H = 1
When H = 1, there is only a trivial flat connection and the partition function is It is possible to decompose χ 2P (n) into another periodic function ψ (l) (n). Then we have where, for simplicity, we denote P ) by R 0 , R 1 , R 2 , and R 3 in (2.17), respectively. Then, when 3 j=1 1 P j < 1, the partition function can be written as It is known that Ψ (l) P (q) is a false theta function, which is the Eichler integral of modular form Ψ (l) 4P of half-integer weight 3/2 [1]. The only coprime P j 's that satisfy 3 j=1 1 P j > 1 is (P 1 , P 2 , P 3 ) = (2, 3, 5) for the case of three singular fibers. Thus, in this case, the partition function is given by The q-series in parenthesis takes a form of q  (2,3,5). As it is an integer homology sphere, there is one homological block from the trivial flat connection, which is (2.20).

The case H ≥ 2
When H ≥ 2, we have rational homology Seifert manifolds and there are contributions from other abelian flat connections in addition to the trivial flat connection. The periodic function ψ when H is odd and when H is even. By introducing the floor and the ceiling function Therefore from (2.17) and (2.25), Then, when 3 j=1 1 P j < 1, the partition function is where we put the contribution from the trivial flat connection (t = 0) separately.
In (2.27), we see that e 2πi t H n in the summation over n can be taken out of summation over n when K ∈ Z. Indeed, consider in (2.27). It is nonzero when n = 2HP m + l and 2HP m ′ − l with m, m ′ ∈ Z ≥0 . Since e 2πi t H (2HP m+l) = e 2πi t H l , the n = 2HP m + l part of (2.28) is given by (2.29) While, for the n = 2HP m − l part of (2.28), we have (2.30) Renaming t to t ′ = H − t, (2.30) can be written as If we want to obtain the usual WRT invariant, the limit q ց e 2πi K with K ∈ Z is taken, so e 2πiK( P H t ′2 −2P t ′ +HP ) becomes e 2πiK P H t ′2 . Then, we see that (2.29) and (2.31) share same e 2πi t H l , so (2.28) can be written as Thus, from (2.27), the WRT invariant is given by In terms of the false theta function Ψ (a) HP (q), when H is odd, (2.33) becomes (2.34) When H is even, Here, we write the expression as sum over distinct Weyl orbits where Weyl reflection is when H is odd, and when H is even.

Properties of the formula
Before providing some examples, we study properties of the formula (2.13) or (2.33). Variable or label t in (2.27) is regarded as t of the diagonal matrix diag (t, −t) ∈ (Z H ) 2 , which gives diag (e 2πi P H t , e −2πi P H t ) of SU (2) holonomy where we note that H and P are coprime. So the Weyl group action on (t, −t) gives (−t, t). The Weyl orbit of (t, −t) corresponds to abelian flat connections where t = 0, in particular, corresponds to the trivial flat connection [15], c.f. section 3.1. Given a t, we see that the summand in (2.34) and (2.35) is invariant under t ↔ −t. We note that t ↔ −t provide complex conjugate of (t, −t) at the level of holonomy. Thus, it can be said that the contributions from the abelian flat connection corresponding to the Weyl orbit of (t, −t) and from the conjugate abelian flat connection corresponding to the Weyl orbit of (−t, t) are same though in the case of SU (2) those two abelian flat connections are equivalent so that contributions from them are obviously same from the beginning.
In addition, the abelian flat connections that are related by the action of the center of SU (2) give the same contribution. The center of SU (2) is given by e 2πi c 2 I 2 , c ∈ Z 2 , which we denote by (c, −c) at the level of (Z 2 ) 2 /Z 2 . As we see below, there are cases that elements in (Z H ) 2 /Z 2 are related by the nontrivial center, (1, −1) ≡ (1, 1) mod (Z 2 ) 2 and it can only be possible when H is even. That is because in order to relate them the center should also be expressed as (m, −m) ∈ (Z H ) 2 , which gives diag (e 2πi P H m , e −2πi P H m ) of SU (2) holonomy, and the nontrivial center is given by is not an integer, so H should be even. When H is even, upon t → t + H 2 , e 2πi l H t + e −2πi l H t get an additional factor e πil . From the assumption that P j 's and Q j 's are coprime for each j and P j 's are coprime to each other, P j should be all odd when H is even, so l's are always even for the case of three singular fibers F = 3. l is also even for other number of singular fibers and higher genus case when H is even considering (B.21) and (B.22) in section B. Hence, e πil = 1, so the abelian flat connections that are related by the center have same e 2πi l H t + e −2πi l H t , so contributions from them are same.
For e 2πiK P H t 2 , upon t → t + H 2 , we have an additional factor e πiK P H 2 when K ∈ Z. Therefore, when H is a multiple of 2 but not of 4, there is an additional factor e πiK for the abelian flat connection (t + H 2 , −t − H 2 ) compared to the case of (t, −t). There is no such factor when H is a multiple of 4. Thus, the contributions from abelian flat connections that are related by the action of the center can have a different coefficient by e πiK .
We denote Weyl orbit of (t, −t) in (Z H ) 2 /Z 2 as W t . When H is even, elements in distinct Weyl orbits, say W t and W t+ H 2 , which are related by the center, give the same contribution to the WRT invariant up to the overall coefficient e πiK , so we group W t and W t+ H when H is multiple of 2 but not of 4, and a = 0, 1, . . . , H 4 when H is multiple of 4. We also denote elements in the Weyl orbit W t byt and a representative of any W t in C b byb.
With the setup above, the S-matrix can be written as where t = (t 1 , −t 1 ) and t ′ = (t ′ 1 , −t ′ 1 ). 3 We often use the notation that lk(a, b) := lk(ã,b). When H is odd, there is only a single Weyl orbit W t in C a , which we denote by W a , as there is no non-trivial center that relates (t, −t)'s. So in that case, we simply have which agrees with the S-matrix provided in [7,8] for odd H. Then, the WRT invariant is given by when H is odd or multiple of 4. Here, b 1 (M 3 ) is the first Betti number, which is 2g when the genus of base surface is g. The linking form lk(a, a) of a is the Chern-Simons invariant for abelian flat connection a, so we call it 1 2 lk(a, a) = CS a . We also note that a factor e 2πiK P H t 2 in (2.34) and (2.35) can be written as e πiKlk(t,t) .
When H is multiple of 2 but not of 4, the WRT invariant can be written as ). Some arrangement of the order of entries can be necessary to put the WRT invariant in the form of (2.43). We also note that Zȧ(q) = Zȧ + H+2 where Zȧ(q) = (I 2 ⊗ S ab )ȧ˙b Z˙b(q) up to arrangement of the order of entries. If we ignore the issue about the center, one can also obtain (I 2 ⊗ S ab )ȧ˙b in (2.43) from (2.41) and (2.39) with some possible arrangement of the order of entries.

Some examples with homological blocks
From the expression (2.34) and (2.35) above, we provide some concrete examples with homological blocks. We omit q ց e 2πi K in the expression from now on. 3 Or this can also be written as where Stab Z 2 (a) is Z2 if a = −a in ZH or is 1 otherwise.

Resurgent analysis
We discuss the resurgent analysis on the expression we obtained. We consider the case H = 1. The overall factor of the analytically continued Chern-Simons partition function or the WRT invariant is proportional to 1/ √ K. Since the partition functions with H = 1 are given by linear combination of Ψ (l) P (q), we consider first the resurgent analysis for 1 so its Borel transform is given by Here, Re y > 0 and also P > 0 are assumed and are used. We note that we have seen (2.97) in (2.8) with (2.17). After taking y → y/P in (2.97) and then multiplying e − K 2πi 1 P y 2 at each side of equality of (2.97), we integrate them over ǫ + iR with ǫ > 0, where we note that the RHS is the integral we evaluated in section 2.1 and it gives Integrating the LHS after changing of variable ξ = 1 2πi 1 P y 2 , we finally have [9] where δ is a small real number and direction of contour is from the origin to infinity. Therefore, the original Ψ As the analytically continued CS partition function on Seifert manifolds with three singular fibers and with H = 1 for abelian flat connections are given by linear combination of Ψ (a) The residues at poles of this integral are the contribution from the irreducible or nonabelian flat connections in the resurgent analysis [9]. That is, by deforming the integral contour for K ∈ Z + , the integral picks the poles and calculation of the residue agrees with the known results on the contributions from non-abelian flat connections. Thus, the Ψ 3) with H = 1 from which we obtained (2.13). So, we obtained a convergent q-series (2.13) from the exact expression (2.3) for reducible flat connections with H = 1, which is basically (2.101) or (2.98) in the resurgent analysis, and the resurgent analysis tells that (2.13) is indeed a full exact partition function. In the context of the exact formula of the WRT invariants in [15], contributions from nonabelian flat connections also appear as the residue part and we can see, for example, that the residues in [15] agree with the result of [9,19] on F = 3 and H = 1. 5 Also when H ≥ 2, residues are interpreted as the contribution from nonabelian flat connections in [15] as well as in the resurgent analysis. We expect that the residues corresponding to nonabelian flat connections attached to abelian flat connections in the resurgent analysis agree with the residues calculated in [15]. In other words, we expect that (2.27) with H ≥ 2 also provide full exact partition function. Similarly, we also expect that the partition functions that we calculate for arbitrary number of singular fibers F , H, and genus g in section A and section B provide a full exact partition function.

Other ranges of P and H and reversed orientation of M 3
So far, we have considered the case that P and H are both positive. We also assumed that Im K < 0 ⇔ |q| < 1. We consider other ranges of their values.
In the expansion (2.8), it was assumed that Re(y/P ) > 0. This is obtained when Re y > 0 and P > 0 and also Re y < 0 and P < 0. The former case with H > 0 has been considered in previous sections with |q| < 1. 5 In [15], though the residue part looks different as it contains additional factor in the denominator, they agree with the result of [9,19] on F = 3 and H = 1. More specifically, when the number of singular fibers is three, we can show that 2P −1 [15] vanishes, while the other part m P − 1 H 3 j=1 sin( mπ P j ) in the residue is nonzero. When H = 1, this agrees with the result of [9,19] on Brieskorn spheres. We found that recently resurgent analysis on arbitrary F with H = 1 was considered in [20].
When H < 0 with Re y > 0 and P > 0, the coefficient of y 2 , which is − K 2πi H P , in the exponential factor in (2.12) needs to be positive so that we can choose a contour that passes through a fixed Re y and extends along the imaginary axis of y-plane for the consistency of the calculation. This requires that Im K > 0 ⇔ |q| > 1. Then the integral gives q n 2 4HP in the expression and |q 4HP is convergent. In sum, if Re y > 0 and P > 0, the calculation is consistent when H > 0 and Im K < 0 ⇔ |q| < 1 or H < 0 and Im K > 0 ⇔ |q| > 1.
We can do similar analysis with Re y < 0 and P < 0. The calculation is consistent if H > 0 and Im K > 0 ⇔ |q| > 1 or H < 0 and Im K < 0 ⇔ |q| < 1.
When Re(y/P ) < 0, the RHS of the expansion (2.8) should be ∞ n=0 χ 2P (n)e n P y . This change only leads to the change of the exponent, e 2πin t H → e −2πin t H in (2.13). But as we saw in (2.34) and (2.35), this doesn't affect the final result. This case Re(y/P ) < 0 is obtained when Re y > 0 and P < 0 and also when Re y < 0 and P > 0. Summarizing the analysis, if Re y > 0 and P < 0, H ≷ 0 and Im K ≷ 0 ⇔ |q| ≷ 1, respectively, provide consistent calculations. Similarly, if Re y < 0 and P > 0, the calculation is consistent when H ≷ 0 and Im K ≶ 0 ⇔ |q| ≶ 1, respectively.
In summary, whatever Re y is, the calculation is consistent when The results stay same in those ranges just with a few differences. For example, when P > 0 and H < 0 so |q| > 1, we obtain 2|H|P (n). Other cases are also similar with P or/and H are replaced with |P | or/and |H| appropriately.
This has an implication on the reverse of orientation of M 3 and sign of K. In above calculation and also below where we have set b of Seifert invariant to be zero, the reverse of orientation of Seifert manifold of M 3 = X(P 1 /Q 1 , · · · , P F /Q F ) is realized by the change of signs of all Q j 's, This also leads to H → −H as H := P F j=1 Q j P j . We see from (2.102) and (2.103) that given a P when H changes its sign, then also the sign of Im K should be changed. This is consistent with the expectation that when the orientation of M 3 is reversed to −M 3 and q is inverted to q −1 it gives the same partition function [8,18]

Higher rank gauge group
The expression of the WRT invariants of Seifert manifolds for the ADE gauge group with arbitrary number of singular fibers F was obtained in [16]. The contributions from the reducible flat connections for gauge group G are given by where r, d, andč g denote the rank, the dimension of group G, and the dual Coxeter number of Lie algebra g of G, respectively. Also |∆ + |, W, Λ r , and Λ w denote, respectively, the number of positive roots, Weyl group, the volume of root and weight lattice. K is quantum corrected CS level K = k +č g . Here Γ r = Γ × · · · × Γ is a multiple contour of Γ in C r where Γ is the contour discussed in previous section. Elements t ∈ Λ r /HΛ r correspond to reducible flat connections where those related by Weyl reflections are regarded as equivalent reducible flat connections. Also, the case t = 0 corresponds to the trivial flat connection. As done in the SU (2) case, we would like to express (3.1) in terms of the q-series with integer coefficients and integer powers. In this paper, we consider the case G = SU (N ).
We leave the cases of other gauge groups as future work.

Calculation of the partition function
The integral and summation part 6 of (3.1) for G = SU (N ) is We consider the case F = 3 with j 1 P j < 1 for simplicity. For larger number of fibers, higher genus, or j 1 P j > 1, we can just use the formula in section B in the following calculations.
Given a pair i and j with i < j and j = N , we have

4)
6 When G = SU (N ), the overall factor to the integral is and for j = N , where we chose 0 < Re β 1 < Re β 2 < · · · < Re β N −1 and P > 0 for convergence. 7 Then, (3.3) is expressed as We do similar calculation as in section 2.1. Expressing the integral in such a way that the contours pass at the origin as in section 2.1, we analytically continue K to complex number with Im K < 0 and take integral contour as γ j , j = 1, . . . , N − 1 that passes through Re β j > 0 and extends parallel to the imaginary axis of β j -plane. Then we obtain the partition function of analytically continued SU (N ) CS theory, Or this can be written as 7 We can choose other ordering of Re βi in the calculations, which gives some of expansions in (3.4) to have −ni,j . But by renaming ni,j's and considering t∈Λr /HΛr we can see that the final expressions are all same.
Contributions from irreducible flat connections would appear as residues of the integral (3.1) [16]. Since all irreducible flat connections are attached to the reducible flat connections in the case of G = SU (2) (when H = 1), we may also expect that similar phenomena happen in the G = SU (N ) case. So we expect that the partition function (3.8) provides a full exact partition function.
Regarding other possible ranges of P , H, Im K, we can do similar analysis as we discussed in section 2.7, and obtain same result also for the case of G = SU (N ). So we will only consider the case P > 0, H > 0, and Im K < 0.

Reducible flat connections
Before moving on to the properties of (3.8), we discuss reducible flat connections for G = SU (N ) on Seifert manifolds.
If the stabilizer subgroup for the holonomy Hol A of the flat connection A or Hom(π 1 (M 3 ), G)/conj., is a center of G, it is called the irreducible flat connection. If not, it is called the reducible flat connection. In the case of G = SU (2), reducible and irreducible flat connections are given by the abelian and nonabelian SU (2) flat connection, respectively. So abelian or nonabelian flat connections were used interchangeably with reducible or irreducible flat connections, respectively, in section 2.
In the case of higher rank, reducible flat connections are not necessarily abelian flat connections but there are reducible flat connections that are nonabelian. For example, when G = SU (3), holonomy of reducible flat connection can be S(U (2) × U (1)).
In order to sort out the type of flat connections that contribute to the partition function in G = SU (N ) case, we briefly review the case of G = SU (2) [15,[21][22][23]. Seifert manifold M 3 is obtained by (P j , Q j ) surgery, j = 1, . . . , F , on the link p j × S 1 in Σ g × S 1 where p j 's are points on Σ g . Let h be the loop that wraps S 1 , x j be the loops around each punctures p j , and c l , d l , l = 1, . . . , g, be the standard generators of π 1 (Σ g ). Then the fundamental group π 1 (M 3 ) of Seifert manifold is generated by h, x j , c l , and d l that satisfies x where h commutes with all x j , c l , and d l , The result is similar for G = U (N ). Instead of above bi( n)'s and ti's, if we choose bi( n) = − i−1 j=1 nj,i + N j=i+1 ni,j and take tN to be an independent variable, we obtain the partition function of analytically continued G = U (N ) CS theory.
Introducing a map φ : π 1 → [0, 1 2 ] such that e 2πiσ 3 φ(h) is in the same conjugacy class of Hol A (h) in SU (2), one can see that holonomies Hol A (h), Hol A (x 1 ), . . . , Hol A (x F ) depend on the choice of φ(b) and some other integers. When φ(b) = 0, 1 2 , Hol A (h) is U (1), and since h commutes with x j 's, Hol A (x j )'s are also U (1). Also, Hol A (c l ) and Hol A (d l ) are U (1). Stabilizer subgroup with respect to them is U (1), so this choice gives reducible flat connection which are abelian. In the notation of (2.3), e 2πiσ 3 φ(h) ∼ = Hol A (h) is equal to e 2πiσ 3 P H t , 1 ≤ t ≤ H − 1 [15]. When φ(b) = 0, 1 2 , Hol A (h) is a center of SU (2), so flat connections can be irreducible. However, in the special case of this choice, one can have the trivial or central flat connections, and one can also find that this particular case gives the contribution from the trivial or central flat connections to the WRT invariant. So φ(b) = 0, 1 2 case can be regarded as trivial and central flat connection in the integral formula (2.3).
So far, we have seen that Hol A (h) determines the type of flat connections in the case of G = SU (2). So we perform similar analysis in the higher rank case. As in the case of G = SU (2), t ∈ Λ r /HΛ r determines Hol A (h) ∼ = e 2πi P H diag (t 1 ,...,t N ) . Therefore, we consider the type of flat connections from given t.

Properties of the formula
We consider some properties of the formula (3.8).

Shift by H
Given a t = (t 1 , t 2 , . . . , t N ) with t N = − N −1 j=1 t j , any shift of t j 's, j = 1, . . . , N − 1, by (multiple of) H doesn't affect e πiK P H N m=1 t 2 m when K ∈ Z and also e − πi H N m=1 tmbm( n) . It is easy to see that such shift only leads to change of N m=1 t 2 m by multiple of 2H. For N m=1 t m b m ( n), we first note that the values that n i,j 's take are all odd or all even, which depends on P j 's, j = 1, . . . , F . We also note that b m ( n) contains N − 1 n i,j 's. If we take , which contains even number of n i,j 's. So whatever N is even or odd, the difference is multiple of 2H. Hence, such shift by H doesn't affect e πiK P  (2) cases. As elements t ∈ Λ r /HΛ r give diag (e 2πi P H t 1 , . . . , e 2πi P H t N ) at the level of holonomy, the center that can relate elements in Λ r /HΛ r take, for example, a form of e 2πi P H c with c := (c, . . . , c where m is an integer such that N c = mH which comes from the condition N i=1 t i = 0. For instance, when N = 2 discussed in the previous section, elements can be related by the nontrivial center when H is even number, and given such an H we can see from N c = mH that c can be 0 or H 2 . Also, when N = 2 and if H is an odd number, only possible c that satisfies N c = mH is zero, which gives identity I 2 . Given a t = (t 1 , t 2 , . . . , t N ), action of the center, for example c, on t doesn't affect . We check whether m N j=N −m+1 b j ( n) is even. As noted earlier, given an F and P j 's, n i,j 's take value in all odd numbers or all even numbers and the total number of n i,j 's in b j ( n) for G = SU (N ) is N − 1. Therefore, when N is odd, N − 1 is even, so b j ( n)'s are always even. Thus, m N j=N −m+1 b j ( n) is even when N is odd. Meanwhile, when N is even, N − 1 is odd. When H is even, from the assumption on P j 's and Q j 's, P j 's should be all odd, which makes all n i,j 's to take values in even number by considering P α = P F j=1 ǫ j P j , (B.21), and (B.22) in section B. Therefore b j ( n) are all even, so m N j=N −m+1 b j ( n) is even. For the case that H is odd, from N c = mH, as the LHS is even but H is odd m should be even. Accordingly, m N j=N −m+1 b j ( n) is even. Therefore, whatever N , H, and F are, elements that are related by the action of the center give same e − πi H N m=1 tmbm( n) . This implies that the elements or the reducible flat connections related by the center give same contribution to the analytically continued CS partition function or the WRT invariant up to e πiK P H N m=1 t 2 m factor. Therefore it is natural to group those related by the center under the same label.
However, such action can change e πiK P H N m=1 t 2 m for some cases. Here, we consider the case with K ∈ Z. After some calculation, possible difference is given by e πiKP (N +m)c . If P (N + m)c is even, then the elements related by such action, which give same contribution to the WRT invariant, have same coefficient e πiK P H N m=1 t 2 m . However, if P (N + m)c is odd, those, which give same contributions to the WRT invariant, have different coefficients by e πiK . When N is odd, in order for P (N + m)c to be odd, both c and P should be odd and m should be even. However, this leads that, from N c = mH, the LHS is odd but the RHS is even, which is a contradiction. Thus, N cannot be odd for P (N + m)c to be odd and extra factor e πiK doesn't appear in the case of odd N . Meanwhile, if N is even, both c and m should be odd to make P (N + m)c to be odd. From N c = mH, we see that H should be even. Therefore, if there are even N , even H, odd m, and odd c that satisfy N c = mH, there is an extra factor e πiK . For example, if N is not divisible by m, from H = N c m , c should be divisible by m. Since c and m are both odd, c/m is also an odd number, so we find that there can be an extra factor e πiK when N is even and H is an odd integer times N . When N = 2, we saw in previous section that the extra factor e πiK appeared when H is multiple of 2 but not multiple of 4, which is consistent with above discussion applied to the SU (2) case. When N is even and H is odd, there is no extra factor e πiK .

Weyl orbit of t and of −t
There are some cases that element t and −t, which are complex conjugate to each other at the level of holonomy, are not in the same Weyl orbit. For example, when N = 3 and H = 4, (2, −1, −1) and (−2, 1, 1) are in different Weyl orbits. We can see that contributions from Weyl orbit of t and from Weyl orbit of −t to the analytically continued CS partition function or the WRT invariant are same. This means that contributions from the reducible flat connection corresponding to Weyl orbit of t and from the conjugate reducible flat connection corresponding to Weyl orbit of −t are same. In

General structure
In order to write an expected general expression for the S-matrix, we introduce some notations. From now on, we take K ∈ Z. We group elements in Λ r /HΛ r by Weyl orbits where elements in Weyl orbit are considered as equivalent reducible flat connections. Among Weyl orbits, there are cases that the orbit contains both t and −t. We denote such Weyl orbit of t in Λ r /HΛ r by W t where t is a label for reducible flat connections. There are also cases that Weyl orbit containing t is distinct from the one containing −t. We denote such orbits as W t and W −t , respectively.
In some cases, it happens that some elements t's are related by the action of the centers. As they give same contributions to the WRT invariant up to overall coefficient e πiK , we group the Weyl orbits by orbits of them under the action of centers. We denote such orbits by C a where a is a label for reducible flat connections in C a . There are cases that W t and W −t are in the same C a . For example, when N = 4 and H = 4, C 0 = {W 0 , W 2 , W 7 , W −7 } where W 0 , W 2 , W 7 , and W −7 are Weyl orbits of (0, 0, 0, 0), (2, −2, 2, −2), (3, −1, −1, −1), and (−3, 1, 1, 1) in Λ 3 /4Λ 3 , respectively. There are also cases that orbits W t and W −t are not related by the action of the center. For instance, when N = 4 and H = 6, W 12 and W 14 are Weyl orbits of (3, −1, −1, −1) and (0, 2, 2, −4) in Λ 3 /6Λ 3 , respectively, and they are related by the action of center, i.e. (−3, 3, 3, −3) in (Z 6 ) 4 . We see that {W 12 , W 14 } = {W −12 , W −14 } but they are complex conjugate to each other at the level of holonomy. As we discussed above, contributions from W t and from W −t to the WRT invariant are same, so we put them in the same class, which we denote by C ±a .
We denote elements in the Weyl orbit W t in Λ r /HΛ r byt. Also, we denote any representative of any of W t in C b or C ±b byb. Then from examples we worked out, we expect that when G = SU (N ) the S-matrix for Seifert manifolds is given by with a N = − N −1 j=1 a j and similarly for b N . Here, C a in the summation can be C a 's or C ±a 's depending on a under consideration. For example, in the calculation below, "C 8 " in the case of N = 4 and H = 6 is C ±8 and W t ∈ C ±8 are W 12 , W 14 , W −12 , and W −14 in Λ 3 /6Λ 3 . We also expect that (3.10) holds for general closed 3-manifold.
In next sections, we provide a number of examples. We omit overall factor (3.13) in the following examples.

The case H ≥ 2
When H ≥ 2, we have sum over t ∈ Λ r /HΛ r in (3.8). We denote simple roots by ..,N are the standard orthonormal basis. We provide some examples for H ≥ 2.
• H = 6 The case H = 6 can be obtained, for example, from (P 1 , P 2 , P 3 ) = (5,7,11 , and Z 9 = Z ′ 15 , the WRT invariant can be written as which satisfies S ab S bc = δ ac . This S-matrix and CS a 's can also be obtained from (3.10) and (3.11) and they agree with results above.

Discussion
We calculated homological blocks for general SU (N ) gauge group and general Seifert manifolds with P j 's being pairwise coprime from the exact expression of Lawrence, Rozansky, and Mariño. We firstly calculated the partition function of the analytically continued SU (N ) Chern-Simons theory or the WRT invariant and obtained the exact expression for Z a 's that are labelled by reducible flat connections. From Z a 's, we extracted homological blocks Z b , which are related to Z a by the S-matrix. We proposed a formula to calculate the S-matrix in general cases and checked that the S-matrix calculated from the formula by using the linking form agrees with the result that was obtained from the calculation of Z a 's and Z b 's. Also, we discussed general expression for the WRT invariant and checked that examples that we discussed fit into the expected form. In addition, we discussed some properties of the SU (N ) WRT invariant. We found a symmetry that the reducible flat connections in the same orbit under the action of the center give same contribution Z a to the WRT invariant up to overall exponential factor e πiK . We also discussed a symmetry that contributions from the reducible flat connection and from the conjugate reducible flat connection are same. In addition, we also saw that the exact expression of Lawrence and Rozansky can be understood in the context of resurgent analysis when H = 1.
There are several interesting directions to consider. In [8], the superconformal index and topologically twisted index can be calculated from homological blocks via and several examples supported it. However, as we saw in some examples, reducible flat connections in the same orbit under the action of center were grouped under a single label. So this should be taken into account when calculating the indices from the homological blocks obtained in this paper. If we consider the multiplicity from the orbit under the action of centers, it is expected that the indices are given by whereas there would be another overall 1/2 factor when considering the case that involves the matrix ((1, 1), (1, 1)). In the calculation discussed above, it was not obvious to calculate Z(q −1 ) on M 3 with |q| < 1, so we didn't calculate (4.3) or (4.4) with t = 1. It would be interesting to calculate the indices for general Seifert manifolds.
We have obtained various homological blocks. When G = SU (2) and the number of singular fibers F is 3 or 4 with genus zero, homological blocks are expressed in terms of false theta functions, which are known in literature. By varying the number of singular fibers or genus, we found that homological blocks are given by false theta functions Φ (l) 2HP (q) and Ψ (l) 2HP (q), and their derivatives. Moreover, considering the higher rank case SU (N ), N ≥ 3, it is expected that the homological blocks in those cases would provide new false theta functions. It would be interesting to study modular properties of homological blocks obtained in this paper including their properties upon q → q −1 [18].
Resurgent analysis was discussed in the case of SU (2) Chern-Simons theory with H = 1 in literature. We considered it in the context of the exact formula in [15] when H = 1. It would be interesting to study resurgent analysis for more general cases.
A mathematical construction or definition for Khovanov-type homology for closed 3manifolds is not available yet. But if it is constructed, from the analogy, it would be expected that there is also a homology theory for closed 3-manifolds analogous to the HOMFLY homology for knots and links. It will be also interesting to study properties of such expected homology theory from homological blocks for G = SU (N ) obtained in this paper.

Acknowledgments
I would like to thank Sergei Gukov for valuable discussions and also helpful remarks on the manuscript. I am also grateful to the Korea Institute for Advanced Study (KIAS) for hospitality at several stages of this work.
Appendix A G = SU(2) and four singular fibers where with the integration cycle Γ t as in section 2. The last factor can be expressed as where R s 's with even (resp. odd) s are P 4 j=1 ǫ j P j with 4 j=1 ǫ j = 1 (resp. −1) up to overall sign where ǫ j = ±1. where M l = −1, 0, 1, · · · such that 2M l P < l < 2(M l + 1)P . By taking derivative with respect to y, it leads to cosh ly (sinh P y) 2 = 1 P where M Rs 's, M Rs = 0, 1, · · · , satisfy 2M Rs P < R s < 2(M Rs + 1)P given R s for each s. As P j 's are coprime in our setup, the maximum value that R s can take is less than 2P . Thus, when F = 4, M Rs is 0, so the terms in the second line vanish. Therefore, we obtain

A.1 The case H = 1
For the integer Seifert homology sphere, the partition function is given by It is known that Φ Therefore, by using (2.25) and (A.12) the partition function can be written as .
As before, when H is odd, the WRT invariant is given by (A.14) When H is even, We note that the structure of the WRT invariant, i.e. the coefficients to the contributions from abelian flat connections including the trivial one are same as in the case of F = 3, (2.34) or (2.35). Therefore the WRT invariant for Seifert manifolds with four singular fibers can also be written similarly as in section 2. We provide one example.
• F = 4 and H = 3 The structure is same as in the case of F = 3. H = 3 can be obtained, for example, from (P 1 , P 2 , P 3 , P 4 ) = (2, 5, 7, 11) and (Q 1 , Q 2 , Q 3 , Q 4 ) = (−1, 2, 2, −2). In this case, the homological blocks are given by with (CS 0 , CS 1 ) = (0, 1 3 ) and S ab = 1 Appendix B G = SU(2), more singular fibers, and the base with higher genus We would like to consider Seifert manifolds with more singular fibers and with higher genus of the base surface. The case of H = 1 with arbitrary number of singular fibers and with genus zero was discussed in [4]. Here we consider general Seifert manifolds with more than 4 singular fibers and also with higher genus case where P j 's are pairwise coprime.
As before, we consider the integral expression for the WRT invariant of Seifert manifolds with F singular fibers and with genus g according to [15,21], with same integration cycle as in section 2.
We will see that calculation for arbitrary genus g can be done easily if calculation for genus zero with arbitrary number of singular fibers is done. So from now on, we proceed calculation with g = 0.
As we saw above, given an H, the structure of the WRT invariant is same for arbitrary number of singular fibers. That is, given an H the S-matrix is same but the homological blocks are different depending on the number of singular fibers, P j 's and Q j 's.

Higher genus
For the case of higher genus, the power of the denominator increases by 2g. Thus, we can use the formula for sinh αy (sinh y) 2m+1 or cosh αy (sinh y) 2m given an F to calculate the homological blocks. Then we obtain the partition function of analytically continued theory or the WRT invariant with the same structure as before but with different homological blocks.
For example, when F = 3 and g = 1, the WRT invariant is given by where R s 's are given by P 3 j=1 ǫ j P j with ǫ = 1 and M s = −1, 0 such that (2M s + 1)P < R s < (2M s + 3)P . Also, since M s = −1, 0, there is no additional term in the expression as in the case of (B.28). Therefore, also when g = 0, given an H we have same S-matrix. The explicit expression of homological blocks for H = 3 with a choice (P 1 , P 2 , P 3 ) = (2, 5, 7) is available in section B.1.