Resurgent Analysis for Some 3-manifold Invariants

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial.


Introduction
Many exact results on the partition function and its perturbative expansions are available in Chern-Simons theory, so it provides various examples for resurgent analysis.A number of interesting aspects of Chern-Simons theory in the context of resurgence have been studied in [1].For example, it was argued that all non-abelian flat connections are attached to the abelian flat connections, i.e. it is possible to recover the information of all non-abelian flat connections from the perturbative expansions around the abelian flat connections on 3-manifolds M 3 via the Stokes phenomena.This was checked for some Brieskorn spheres, which are integer homology spheres.Since they are integer homology spheres, the Witten-Reshetikhin-Turaev (WRT) invariant or the Chern-Simons (CS) partition function on them is given by a homological block introduced in [2,3].
Also, aspects of resurgence in Chern-Simons theory via the modularity of false theta functions have been studied in [4] and the contributions from SL(2, C) flat connections to the SU(2) WRT invariant were calculated for some Seifert rational homology spheres.
Contributions from the abelian flat connections to the partition function for an infinite class of Seifert manifolds with a gauge group G = SU (N ), N ≥ 2, have been calculated in [5].In particular, when G = SU (2), it was noted that the integral expression of the contribution from an abelian flat connection, which is obtained after the analytic continuation of the level K in the expression obtained in [6] with an appropriate choice of the integration contour, is the same as the Borel resummation of the Borel transform of the perturbative expansion around the abelian flat connection under consideration.Similar discussion for the case of a rational level K has also been discussed in [7] but without considering resurgence explicitly.
Meanwhile, a new two-variable invariant for the knot complement in S 3 has been discussed in [8], which is denoted F K (x; q) for a knot K.This is obtained from resurgent analysis on the perturbative expansion around the abelian flat connection, which is trivial, where a thorough resurgent analysis for F K (x; q) has not been discussed.
In this paper, we consider resurgence for the Chern-Simons partition functions on an infinite family of the Seifert manifolds with an integer and a rational level K, and the torus knot complement in S 3 when G C = SL(2, C).In section 2, we first review and study the case of the Seifert integer homology spheres when the level K is taken to be an integer, which serves as a basic example for the rest of the paper.Then we consider the case of Seifert rational homology spheres.As in the case of Seifert integer homology spheres, we see that the contributions from all non-abelian flat connections can be obtained from the contributions from abelian flat connections via the Stokes phenomena.This indicates that the contributions from abelian flat connections with an analytic continuation of the level K can be interpreted as a full partition function of an analytically continued Chern-Simons theory.We also calculate transseries parameters for an example when K is taken to be an integer and discuss the case of a complex K in Appendix A.
In section 3, we do a similar analysis when we take the level K = r s to be a rational number where r and s are coprime.In this case, the structure of the WRT invariant in terms of homological blocks is similar to the case of an integer K but it involves the sum over certain lifts of abeilan flat connections.We also see that the contributions from non-abelian flat connections can be recovered from the contributions from abelian flat connections.The calculation of transseries parameters for an example when K is taken to be a rational number is discussed in Appendix A.
In section 4, we first obtain F K (x; q) for the torus knot complement in S 3 from the integral expression in [6] after analytic continuations of the level K and the color R with an appropriate choice of the contour, which agrees with the Borel resummation of the Borel transform of the perturbative expansion around the abelian flat connection.We see via resurgent analysis that F K (x; q) contains information of contributions from all non-abelian flat connections so it is a full partition function of an analytically continued Chern-Simons theory on S 3 \K, i.e. an analytic continuation of the colored Jones polynomial.In addition, we discuss the case of rational K. Also, we provide brief remarks on the modularity and a limit with a general x.
2 Resurgent analysis for Seifert rational homology spheres with integer level K We consider resurgent analysis of the G = SU (2) WRT invariant on an infinite family of the Seifert rational homology spheres X(P 1 /Q 1 , . . ., P F /Q F ) with F singular fibers where P j and Q j are coprime for each j and P j 's are pairwise coprime, when the level K is an integer.
One of observations in [5] is that after an analytic continuation of the level K and the change of the contour the integral expression of the partition function [6] of Chern-Simons theory on the Seifert manifolds X(P 1 /Q 1 , . . ., P F /Q F ) agrees with the Borel resummation of the exact Borel transform of the perturbative expansion around the abelian flat connections.It was also checked for the Seifert integer homology spheres that contributions from the non-abelian flat connections to the partition function calculated in [6] agree with the residues that are captured when K is taken to be an integer in [1]. 1or discussion on the case of the Seifert rational homology spheres, H = ±|Tor(H 1 (M 3 ), Z)| ≥ 2, we first consider the case of the Seifert integer homology spheres [1,5].We begin with a known exact result of [6] for the WRT invariant of the Seifert manifolds X(P 1 /Q 1 , . . ., P F /Q F ), and see that contributions from all non-abelian flat connections can be recovered from the homological block via the Stokes phenomena.

Resurgent analysis for H = 1
For the Seifert manifolds X(P 1 , Q 1 , . . ., P F , Q F ) the full exact G = SU (2) partition function [6] is given by (2.4) The Gaussian integral parts in (2.2) are the contributions from the abelian flat connections.A Weyl orbit of t corresponds to the abelian flat connection of G = SU (2) whose holonomy for the central element in the fundamental group π 1 (M 3 ) is diag(e 2πi P H t , e −2πi P H t ) where the Weyl action is given by t ↔ −t mod H, and t = 0 corresponds to the trivial flat connection.
The residue parts are the contributions from the non-abelian flat connections.There are two types of residues at the second line of (2.2).The second type of residues in (2.2) arises due to a parallel shift of the integral contour C 0 , which is a line from −(1 + i)∞, the origin, and to (1 + i)∞ in the y-plane, to the contour C t that passes the saddle points, y = −2πi P H t.
We consider the case of F = 3 for concreteness.The case of more singular fibers can be done similarly as in the case of F = 3.When F = 3, the first type of residues in (2.2) is given by The term that contains the factor i HK 3 j=1 1 P j cot πm P j vanishes.
3 j=1 sin πm P j is nonzero when m is coprime to all P j 's.Therefore, if a is coprime to all P j 's, then so is 2P − a.Hence, the term containing the factor 1 H 3 j=1 sin πm P j in (2.5) vanishes because if m = a gives a nonzero contribution, then 2P − a also gives the same value but with an opposite sign.Thus, the first type of residues in (2.2) becomes [5] 2P −1 In particular, when H = 1 and F = 3, (2.2) is simplified to and from (2.6) the residue part is given by (2.8)

Resurgent analysis
We denote a connected component of the SL(2, C) flat connection as and its lift as which is a critical point of an analytically continued Chern-Simons theory.Here, S α ∈ CS(α) + Z denotes the value of a Chern-Simons invariant of α in the universal cover, while CS(α) is a Chern-Simons invariant of α, which is defined in mod 1.A lift of α to α is determined by S α .Taking K = |K|e iθ , a path integral over a Lefschetz thimble for α with a given θ is denoted by It is expected that the partition function of an analytically continued Chern-Simons theory takes a form of where with a chosen lift from a to a [1].
From an analytic continuation of the level K in the Gaussian integral part of (2.2) with a choice of the integration contour γ that is parallel to the imaginary axis of the complex y-plane, the contributions Z t from the abelian flat connections to the partition function of the analytically continued Chern-Simons theory can be obtained [5].These Z t 's that are labelled by abelian flat connections can be further decomposed into the homological blocks with an SL(2, Z) S-transform.We would like to see that contributions from all non-abelian flat connections can be recovered from I t via the Stokes phenomena, which implies that (2.11) indeed can be regarded as a full partition function.
For the integer homology sphere, there is one abelian flat connection, which is trivial, so Z 0 is given by a homological block.When the number of singular fibers is 3, F = 3, the homological block is given by a linear combination with integer coefficients of the false theta function Ψ (a) where In particular, when H = 1, I 0 for the Seifert manifolds X(P 1 /Q 1 , P 2 /Q 2 , P 3 /Q 3 ) was calculated in [5], where ) w 6 g j n 2 X l z 3 u e l a 8 6 i 5 x j + y P n 8 A R C 3 j x g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S E f O X H c p U F r c 5 a T d 5 r n X y M A c Z o M = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L L a C p 5 J 4 0 W P B i 8 e K 9 g P a U D b b S b t 0 s w m 7 G y G E / g Q v H h T x 6 i / y 5 r 9 w 6 g j n 2 X l z 3 u e l a 8 6 i 5 x j + y P n 8 A R C 3 j x g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S E f O X H c p U F r c 5 a T d 5 r n X y M A c Z o M = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L L a C p 5 J 4 0 W P B i 8 e K 9 g P a U D b b S b t 0 s w m 7 G y G E / g Q v H h T x 6 i / y 5 r 9 x 6 9 F + / d + 1 i 2 F r x 8 5 h T + w P v 8 A T r 5 j 1 w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " f 2 k m q / g 3 w 6 g j n 2 X l z 3 u e l a 8 6 i 5 x j + y P n 8 A R C 3 j x g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w 6 g j n 2 X l z 3 u e l a 8 6 i 5 x j + y P n 8 A R C 3 j x g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Q G e 4 N m z 3 q P 3 4 r 3 O W 9 e 8 x c w x / I H 3 9 g O u 1 J E S < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " for P > 0, which satisfies s(−Q, P ) = −s(Q, P ).R j , j = 0, 1, 2, 3, in (2.15) are given by In (2.15), 3 j=1 1 P j > 1 is satisfied only when (P 1 , P 2 , P 3 ) = (2, 3, 5).Also, we see that the Chern-Simons invariant for the abelian flat connection is zero, and we chose a lift such that S 0 = CS(0) = 0.
The integral part of (2.2) with the contour γ that calculates I 0 agrees with the Borel resummation after the change of integration variable.For Ψ (a) P (q), the Borel resummation is given by the average of the Borel sums [1], where π 2 ± δ denotes the integration contours (c.f. the contours in Figure 1a) in the Borel plane, more specifically, where the contour in the Borel plane in Figure 1a becomes the contour γ in Figure 1b.Therefore, we can see that with y = (2πiP ξ) 1/2 , the Borel resummation (2.20) for the Seifert manifolds with three singular fibers is the same as the integral part of (2.7) with the integration contour γ and Im K < 0, j=1 (e y/P j − e −y/P j ) (e y − e −y ) . (2.23) It is more convenient to consider the Borel resummation as in the expression (2.23) rather than as in the standard expression (2.21) in the Borel plane, so we consider the former type of expression in the rest of the paper.
If K ∈ Z + , the contour should be changed so that the integral is convergent and such an integration contour is given by a line that passes from −(1 + i)∞, the origin, and to (1 + i)∞ in the y-plane as in Figure 1c.Due to such a move of the contour, poles at the negative imaginary axis of the y-plane are picked up and their contributions to the residues are given by 2πi (2.24)So m's that are not multiples of P j 's give non-zero contributions in (2.24).For example, when (P 1 , P 2 , P 3 ) = (2, 3, 5), they include m = 1, 11, 19, 29, 31, 41, 49, 59 mod 60 and m = 7, 13, 17, 23, 37, 43, 47, 53 mod 60.Each set of poles corresponds to non-abelian flat connections whose Chern-Simons invariants are − 1 120 and − 49 120 , mod 1, respectively.The infinite sum in (2.24) can be regularized.For example, consider a periodic function C p with a period p, C p (n+p) = C p (n), which has 0 mean value p−1 n=0 C p (n) = 0. Then it is known [10] as w approaches to 0 from the upper half plane, w 0. Here, L(−r, C p ) is given by where B j (x) is the Bernoulli polynomial whose generating function is (2.28) Or simply we may consider a limit y 0 on sinh(P −a)y sinh P y For m = a that is not a multiple of P j 's, the contribution from m = a in (2.24) has an opposite sign of the contribution from m = 2P − a in (2.24).Thus, (2.24) can be regularized to which agrees with (2.8).
Transseries parameters for some examples of H = 1 have been calculated in [1,11], so for the case of H = 1 we don't provide further examples.Instead, we calculate transseries parameters when H ≥ 2 in section 2.2.

A remark on modularity from resurgent analysis
As a simple consistency check, we also discuss a relation between the modularity and the resurgent analysis.It is known that the false theta function Ψ (a) P (q) satisfies a nearly modular property [10,12] Ψ (a) where M ab = 2 P sin πab P and c n is such that P (e −2πiK ) = 1 − a P e − πi 2 1 P Ka 2 , so Ψ (a) The first and the second term in (2.34), which are the non-perturbative and the perturbative part, correspond to the contributions from the non-abelian and the abelian flat connection, respectively.(2.34) can be derived from (2.21) when K is taken to be an integer.
For the Seifert manifolds with H = 1 and F = 3, by using when F = 3 and H = 1, (2.36) agrees with the nonperturbative part of the WRT invariant calculated from (2.34) by using the modularity.

Resurgent analysis for H ≥ 2
When H ≥ 2, we would also like to see that the contributions from the non-abelian flat connections can be recovered from the contributions from the abelian flat connections.In addition, we calculate transseries parameters for an example when K is taken to be an integer.
As discussed in section 2.1, the residue −2πi e mπi/P j − e −mπi/P j (2.38)Therefore, the total residue, i.e. the total contribution from the non-abelian flat connections to the WRT invariant, is (2.39)

Resurgent analysis
The Gaussian integral part of (2.2) for K ∈ Z + can be written as q e 2πi K .
(2.41) when H is odd, and (2.42) when H is even [5].Thus, for an odd H, (2.44) where a = 1, . . ., H−1 2 denotes a Weyl orbit W a , which contains {a, H − a}.When H is even, we have (2.46)where a = 1, . . ., H 2 − 1. Below, we consider the case of an odd H for convenience.The case of even H can be done similarly.
As in the case of H = 1, we can obtain the Borel resummation of the Borel transform of the perturbative expansions around the abelian flat connection t.More explicitly, when t = 0, from (2.13), (2.44) is expressed as (n)q n 2 4HP (2.48) Since the sum over n is nonzero only when n = 2HP l ±c due to the presence of the periodic function ψ (c) 2HP (n), we can have (2.49) (2.50) and the Borel resummation that gives (2.49) is given by This can also be written as 3 j=1 e y/P j − e −y/P j e y − e −y (2.53) since γ is parallel to the imaginary axis of y-plane.We can do similarly for the case t = 0 and we have w 6 g j n 2 X l z 3 u e l a 8 6 i 5 x j + y P n 8 A R C 3 j x g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S E f O X H c p U F r c 5 a T d 5 r n X y M A c Z o M = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L L a C p 5 J 4 0 W P B i 8 e K 9 g P a U D b b S b t 0 s w m 7 G y G E / g Q v H h T x 6 i / y 5 r 9 O / e 7 T 6 g 0 j + W j y R L 0 I z q W P O S M G i s 9 1 L P 6 s F p z G + 4 C Z J 1 4 B a l B g d a w + j U Y x S y N U B o m q N Z 9 z 0 2 M n 1 N l O B M 4 q w x S j Q l l U z r G v q W S R q j 9 f H H w 6 g j n 2 X l z 3 u e l a 8 6 i 5 x j + y P n 8 A R C 3 j x g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S E f O X H c p U F r c 5 a T d 5 r n X y M A c Z o M = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L L a C p 5 J 4 0 W P B i 8 e K 9 g P a U D b b S b t 0 s w m 7 G y G E / g Q v H h T x 6 i / y 5 r 9 O / e 7 T 6 g 0 j + W j y R L 0 I z q W P O S M G i s 9 1 L P 6 s F p z G + 4 C Z J 1 4 B a l B g d a w + j U Y x S y N U B o m q N Z 9 z 0 2 M n 1 N l O B M 4 q w x S j Q l l U z r G v q W S R q j 9 f H H     Therefore, we see that the sum of the integral part of (2.56) and (2.58) is the same as the integral part of (2.2) and the sum of the residues (2.57) and (2.59) agrees with the total residue (2.39).Hence, this indicates that the contributions from the non-abelian flat connections can be recovered from the contributions from the abelian flat connections via the Stokes phenomena also when H ≥ 2.
In addition, we can read off the transseries parameters for an integer K from the residue parts of (2.56) and (2.58).For the examples considered in this paper, which take a form of (2.20), upon θ = 0 the partition function (2.11) can be expressed as where m β a 's are the transseries parameters.We provide an example for transseries parameters for an integer K.We discuss the case of a general K in Appendix A.

Example
We consider an example with H = 5 and (P 1 , P 2 , P 3 ) = (2,3,7).In this case, the WRT invariant is given by where Z a 's, a = 0, 1, 2, are and (2.67) When K ∈ Z + , we change the integration contour as in Figure 2.There are three sets of poles at y = −nπi in (2.62) with (2.68) In this case, it is expected that there are three SL(2, C) non-abelian flat connections, which we denote as α 1 , α 2 , and α 3 , and their Chern-Simons invariants are respectively.This case is rather a special case due to the condition that P j 's are pairwise coprime. .Thus, α 1 and α 2 are real non-abelian flat connections, n α 1 = n α 2 = 1, while α 3 is a complex non-abelian flat connection, n α 3 = 0. We also checked this via the modularity discussed in [4].
From (2.56) and (2.58), we can obtain transseries parameters associated to y = −nπi for each abelian flat connections a when HP (q) and the integrand of the corresponding Borel resummation contains a rational function of sine hyperbolic functions that depend on H.However, at least when Pj's are pairwise coprime, the building blocks Ψ (a) HP (q) are summed up in such a way that Zt contains a rational function of sine hyperbolic functions that doesn't depend on H.
As a side remark, though we consider an example from an infinite family of Seifert manifolds where Pj's are coprime, a similar analysis can be done for the examples that the expression in terms of Ψ (a) HP (q) are known, such as examples discussed in [4], by using (2.22).
The full G = SU (2) WRT invariant for the Seifert manifolds X(P 1 /Q 1 , . . ., P F /Q F ) when the level K = r s is a rational number with coprime r and s [6] 4 can be expressed as We first consider the residue part of (3.1) or the first type of the residue in the second line of (3.2), when F = 3.The case of other singular fibers can also be done similarly.When m is not a multiple of s, there is no pole from (1 − e −2Ky ) −1 if K = r s , but the integrand can have poles of order 1 from (e y − e −y ) −1 of f (y).In this case, the sum of the residues from such poles is zero.Meanwhile, if m is a multiple of s, the integrand has poles of order 2 from (e y − e −y ) −1 (1 − e −2Ky ) −1 for K = r s .The contributions from such poles are given by which can be simplified to The second type of residues in (3.2) is given by 8π Therefore, the total residue is given by the sum of (3.4) and (3.(3.6) 4 More precisely, for the examples discussed in this paper, the condition is that s is coprime to 4KP , which was used in [6] to derive (3.1) via the Galois action on e 2πi 1 4KP .

Resurgent analysis
As in the case of an integer K, by analytically continuing the level K from a rational K and choosing a contour γ in the Gaussian integral part of (3.1), we can see that the WRT invariant can be expressed as up to an overall factor, when H is odd and F = 3 where R j , j = 0, 1, 2, 3, are given in (2.19) [7]. 5 We note that when deriving (3.7) we took t = Hv + u and expressed the sum over t = 0, 1, . . ., Hs − 1 as the sum over u = 0, 1, . . ., H − 1 and v = 0, 1, . . ., s − 1.
The structure of (3.7) takes a form of where Z u 's are given by (2.43) and (2.44).We see that the differences from the case of the integer K are the factor e 2πi r s P H (vH+u) 2 and the summation s−1 v=0 .The structure (3.8) tells that u labels the abelian flat connection as in the case of integer K.In this section, we discuss the case of an odd H and the case of even H can be done similarly.
The structure (3.8) at a rational K is different from (2.11) with (2.12) or its limit when K ∈ Z + in that (3.8) is expressed as a sum over v but there is no such a sum in (2.11).As discussed in [14], a lift for a flat connection is chosen in the analytically continued Chern-Simons partition function as in (2.11).Also, by construction, when K is taken to be an integer, (2.11) with (2.12) or (2.60) becomes the standard path integral of Chern-Simons theory for an integer K.In our discussion, when K ∈ Z + , any value of S a ∈ CS(a) + Z that captures a lift gives e 2πiKCS(a) for an abelian flat connection a, and there is no sum like (3.8).
From a viewpoint of the analytically continued Chern-Simons theory, the case of a rational K can be regarded as another limit of an analytic continuation from an integer K.For a general complex K such that |q| < 1 (or |q| > 1), the construction of an analytically continued Chern-Simons theory doesn't necessarily depend on whether K = r or K = r s for r ∈ C, since they are just one of the value of K with |q| < 1.Thus, it would not be natural to expect that there are two types of expression of the analytically continued Chern-Simons partition function obtained from the analytic continuation from K ∈ Z and from K ∈ Q.Rather, it would be appropriate that for a general value of K ∈ C an analytic continued Chern-Simons partition function would be given by (2.11) with (2.12) and an expression as a sum over lifts 6 as in (3.8) arise at K ∈ Q. (3.7) or (3.8) tell how the sum over lifts of abelian flat connections arise explicitly in the WRT invariant of the Seifert manifolds at K = r s , but we don't have an explanation on why the sum of exponential factors in (3.8) takes such a form precisely.
As done previously, when taking K = r s , we change the contour from γ to C u and C H−u , and we have for the integral part.We can see that (3.9) agrees with the integral part of (3.2).
For the residue part, we obtain 6 A large gauge transformation that shifts the Chern-Simons invariant by the amount of s would be a symmetry.Accordingly, the number of lifts in the sum would be at most s.Meanwhile, considering the value of e and the first line of (3.12) after regularization cancels out the contribution (3.4).Thus, ( can be expressed as

−8π
Hs−1 t=0 and after regularization we see that this agrees with (3.11) when K is a rational number, K = r s .Given a u and a v, we see that there are s+1 2 number of distinct values of e 2πi (3.17) and also (3.11) can be simplified to −4πg(P r; s) As a consistency check, we see that (3.18) also agrees with the non-perturbative part obtained from the modularity.More specifically, from the modularity, the non-perturbative part of Ψ From the regularization, Ψ P (e −2πi r s ) is given by Ψ (b) When F = 3 and H = 1, the WRT invariant is given by Z M 3 (K = r s ) g(P r; s) 3 j=0 Ψ (R j ) P (q) q e 2πi s r , so the non-perturbative part of Z M 3 (K) can be expressed as −g(P r; s) which agrees with the contribution from the non-abelian flat connections obtained via resurgent analysis for Z M 3 (K) g(P r; s) 3 j=0 Ψ (R j ) P (q) q e 2πi s r .
Example: H = 1 We provide an example for H = 1, when K is taken to be a rational number.We discuss an example for H ≥ 2 in Appendix A. For concreteness, we consider the Poincaré sphere Σ(2, 3, 5) where (P 1 , P When K = r s , a natural way to sort the poles would be to consider sets of poles that have the same factor e −2πi r A new q-series invariant with an additional variable x for the knot complement S 3 \K in S 3 was studied in [8].It is denoted as F K (x; q), and we call it the homological block for a knot complement as Z is called the homological block for the closed 3-manifolds.One of conjectures in [8] states that a two variable invariants f K (x; q) = F K (x;q) x 1 2 −x − 1 2 with integer coefficients is obtained from the Borel resummation of the asymptotic expansion of the reduced colored Jones polynomial with a color R around the trivial flat connection discussed in [15], which is where q = e and x = q R .P ] is a Laurent polynomial with P 0 (x) = 1 and ∆ K (x) is the Alexander polynomial.Here the coefficients c m,j are the Vassiliev invariants for a knot K. Therefore, F K (x; q) can be regarded as an analytic continuation of the colored Jones polynomial.In this section, we discuss aspects of the homological block F K (x; q) for a knot complement S 3 \K as a full partition function of an analytically continued Chern-Simons theory by studying the torus knot complement in S 3 via resurgent analysis, which directly indicates that it is indeed an analytic continuation of the colored Jones polynomial.
The integral expression of the G = SU (2) Chern-Simons partition function for a torus knot complement in S 3 was obtained in [6,16]. 7The unnormalized Jones polynomial for (the mirror of) torus knot K P,Q with an irreducible representation of a dimension R is given by where q = e 2πi K with K ∈ Z.The Jones polynomial (4.2) can be expressed as and the contour C is a line that passes from −(1 + i)∞, the origin, and to (1 + i)∞ in the y-plane.The overall factor D is Here what we mean by the unnormalized Jones polynomial is that a normalization for the Jones polynomial of the unknot is chosen as The residue part in (4.3) is given by but we can see that this residue part vanishes. 8Therefore, the Jones polynomial is obtained from the Gaussian integral part of (4.3), (e Kuy − e −Kuy ) (e y/P − e −y/P )(e y/Q − e −y/Q ) e y − e −y .(4.9) Another type of residues arises by moving the contour from C to the stationary phase points, y = ±πiP Qu, of h(u, y) in (4.9).Denoting such contours as C ± , the integral (4.9) can be expressed as  The residue part of (4.11) vanishes if u < 1 P Q .
The integral part of (4.11) corresponds to the contribution from the trivial (abelian) flat connection and the residue part of (4.11) is from non-abelian flat connections of the torus knot complement in S 3 .For a (P, Q) torus knot, there are 1 2 (P −1)(Q−1) non-abelian flat connections [17] and this can also be seen from the number of distinct Chern-Simons invariant − 1 4P Q m 2 mod 1, which can be read off from the factor e −2πiK m 2 4P Q in the residue part of (4.11).

Homological block for torus knot complement in S 3
We would like to discuss that F K (x; q) for the torus knot complement can be obtained from analytic continuations of K and R in the Gaussian integral part of (4.10).For the calculation of the homological block for the torus knot complement, we expand (e y/P −e −y/P )(e y/Q −e −y/Q ) e y −e −y in (4.10) as (e y/P − e −y/P )(e y/Q − e where Here, we took P Q > 0 and Re y > 0. After an analytic continuation of K where we choose Im K < 0 for convergence, we take the integral contour as γ.Then the Gaussian integral part can be written as where the last integral term gives the overall factor, πi 2 K |P Q| 1/2 e − πi 4 sign 1 P Q .Thus, we obtain where x = e 2πiu .(4.17) This agrees with the homological block F K (x, q) of the torus knot in [8] up to an overall factor.
Or we may take the integral part of (4.11) and take an expansion (e y/P − e −y/P )(e y/Q − e when we don't specify the range of y.Then, by substituting (4.18) into the integral (4.11) with analytic continuations of K and R and also with the integration contour γ, we have the same homological block (4.16).

Resurgent analysis
As in the case of the Seifert manifolds, the Borel resummation of the Borel transform of the perturbative expansion of the homological block F K (x; q) 9 is given by γ dy e Given the Borel resummation (4.19), the contributions from all non-abelian flat connections can also be recovered.For K ∈ Z + and also for R = Ku ∈ Z + , we deform the integral contour from γ to C .We note that (4.19) doesn't pick poles along the process due to e Kuy − e −Kuy = e Ry − e −Ry factor in the numerator.Then the (4.19) becomes (4.9) and then by shifting the integral contours C to C + and C − for the part that contains e −Kuy and e Kuy , respectively, the residues in (4.10) are recovered. 10Therefore, contributions from all non-abelian flat connections can be recovered from the homological block F K (x; q).Thus, the homological block F K (x; q) for a torus knot, which is obtained by analytic continuations of K and R in the integral expression (4.10) that gives the Jones polynomial when K is an integer, contains the information of all flat connections so it can be regarded as a full partition function of an analytically continued Chern-Simons theory.This indicates that the homological block F K (x; q) is indeed an analytic continuation of the colored Jones polynomial which is a full G = SU (2) partition function with an integer level K.We expect that this argument hold for general knot complements in 3-manifolds.If this is so, it implies that the surgery operation discussed in [8] is indeed a full topological quantum field theoretical operation with no flat connections left behind. 9While preparing the manuscript, we found that [18] appeared where resurgent analysis for the WRT invariant of Seifert loop was considered while q is analytically continued but the color is fixed as a positive integer, R ∈ Z+, throughout.Our discussion in this section is about resurgent analysis for the homological block FK(x; q) with a general complex R, including the case of the specialization FK(x = q R ; q) with a given R ∈ Z+. 10 We may consider resurgent analysis from 2 γ dy e which agrees with the result of [8] for the trefoil knot when (P, Q) = (2, 3).As discussed above, upon K ∈ Z+ and R = Ku ∈ Z+, we deform γ to C as in Figure 1c.Then, we can see that (4.9) or (4.11) is obtained from (4.20)., −e 13iπK 7 , −e iπK e 11πiK 6 , −e , −e 13iπK 9 , −e , e πiK 6 .
As a remark, if taking x = e 2πi R K = q R in the homological block (4.16)where R is an integer R ∈ Z + and K is still analytically continued, (4.16) can be simplified to

.21)
This agrees with the Jones polynomial (4.2) with an analytically continued q.So the specialization F K (x = q R ; q) with R ∈ Z + of the homological block for torus knot agrees with an analytic continuation of q in the colored Jones polynomial with a given representation R, which we also expect so for a general knot. 11s another remark, in the previous discussion for the Seifert manifolds, we saw that the resurgent analysis provide a nearly modular property for the homological block.For torus knot complement, we have  For some closed 3-manifolds M 3 , the homological block for M 3 is known to have near modularity, which may be understood in the context of the 3d-3d correspondence.From the perspective of the 3d N = 2 theory T [M 3 , SU (2)], the homological block Z(q) of closed 3-manifolds is given by the D 2 × q S 1 partition function of T [M 3 , SU (2)].Since D 2 × q S 1 R + × T 2 , if there is no R + , the partition function would be modular, but due to the bulk part of T [M 3 ], the modularity would be spoiled in such a way that the partition function of T [M 3 ] is nearly modular.As in the case that M 3 is a Seifert manifold, it is expected that (4.22) 12 exhibits a nearly modular property of F K (x; q).It will be interesting to study further the modularity of F K (x; q).

Other roots of unity
We may also consider F K (x; q) at other roots of unity.The unnormalized Jones polynomial for the torus knot can be expressed as [6] 1 4iK |P Q| z where z = e πi 2KP Q .As in the case of the Seifert manifolds, considering the Galois action on z, z is replaced with another primitive 4P QK-th root of unity, z = e sπi 2KP Q where s is coprime to 4P QK.With such an expression, the integral expression can be obtained as in 12 By introducing a periodic function ϕ the case of an integer K and up to an overall factor 13 it is given by J K P,Q (R, q = e 2πi s r ) D where K = r s .We can also see that the residue term in (4.30) vanishes, so (4.30) becomes There is also another type of the residue that arises from moving the contour C to the contour that passes the saddle points, y = ±πiP Qu − 2πiP Qt for each t.After some calculations, (4.31) can be expressed as Dg(P QR, s) 2e where K = r s .
In order to obtain the Chern-Simons partition function at other roots of unity in terms of the homological block, we do a similar calculation as in the case of an integer K, and we have x=q R , R∈Z + . (4.33) We can also see that (4.32) can be recovered from (4.33) via the Stokes phenomena.
Limit q e 2πi K with arbitrary x We may also consider a limit q e 2πi K with R arbitrary in x = e 2πiu = e 2πi R K in resurgence.Since R is not taken to be an integer, there are additional contributions by moving the contour from γ to C 0 as in Figure 1c,  Therefore, the total residue is given by the sum of (4.34) and (4.35), so where the perturbative part is given by the integral part of (4.11), which can be expressed as (4.23) with = 2πi K .We don't have a nicer or closed form expression of (4.36), but we expect that (4.36) provides a near modularity of F K (x; q) for a general x.Also, this would be related to the asymptotic expansion of the Akutsu-Deguchi-Ohtsuki knot invariants, which was recently studied in the context of homological block F K (x; q) [20].
Similarly, we may also consider a limit to other roots of unity, q e 2πi s r .In this case, the limit is the same as (4.36) with K replaced by r s and with an overall factor g(P QR, s).
Weyl orbit with their own lifts, but we see from (A.3) that there is just one exponential factors for the poles m ∈ [− 2P (H − t)/H , − 2P t/H − 1].So in this context, the sum would be over β in (A.7) with (A.8).We note that Z ã itself is not gauge invariant, but if we choose a gauge transformation as a based gauge transformation as discussed in [1], Z ã may be a quantity that can be considered in the middle of the calculation in the context of resurgence.Meanwhile, the total sum of them on the RHS of (A.6) for a given abelian flat connection is a gauge invariant quantity.
If the expressions (A.6) and (A.7) are taken, (A.1) at t = 0 would be expressed as I t + I H−t where for a given pole y = mπi and here β t and β H−t are lifts of β.
With above discussion in mind, we calculate transseries parameter associated to y = −nπi for an example with (P 1 , P 2 , P 3 ) = (2, 3, 7) and H = 5 discussed in section 2.2.We chose lifts such that S t = P H t 2 .For the latter consideration, we would have

f
(y)e Kgt(y) dy − 2πi 2P −1 m=1 Res f (y)e Kg 0 (y) 1 − e −2Ky , y = πim (2Kg 0 (y) 1 − e −2Ky , y = πim − 2πi H = ±|Tor H 1 (M 3 , Z)| = P x l p P 4 9 B 2 x t S M 9 a o 3 F / / z e p m J r o O c y z Q z K N l y U Z Q J Y h I y / 5 s M u U J m x N Q S y h S 3 t x I 2 p o o y Y 9 O p 2 B C 8 1 Z f X S b t R 9 9 y 6 d 9 e o N d 0 i j j K c w T l c g g d X 0 I R b a I E P D E b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A F X b j c E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x l p P 4 9 B 2 x t S M 9 a o 3 F / / z e p m J r o O c y z Q z K N l y U Z Q J Y h I y / 5 s M u U J m x N Q S y h S 3 t x I 2 p o o y Y 9 O p 2 B C 8 1 Z f X S b t R 9 9 y 6 d 9 e o N d 0 i j j K c w T l c g g d X 0 I R b a I E P D E b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A F X b j c E = < / l a t e x i t > ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " x l p P 4 9 B 2 x t S M 9 a o 3 F / / z e p m J r o O c y z Q z K N l y U Z Q J Y h I y / 5 s M u U J m x N Q S y h S 3 t x I 2 p o o y Y 9 O p 2 B C 8 1 Z f X S b t R 9 9 y 6 d 9 e o N d 0 i j j K c w T l c g g d X 0 I R b a I E P D E b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A F X b j c E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x l p P 4 9 B 2 x t S M 9 a o 3 F / / z e p m J r o O c y z Q z K N l y U Z Q J Y h I y / 5 s M u U J m x N Q S y h S 3 t x I 2 p o o y Y 9 O p 2 B C 8 1 Z f X S b t R 9 9 y 6 d 9 e o N d 0 i j j K c w T l c g g d X 0 I R b a I E P D E b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A F X b j c E = < / l a t e x i t > (a) y < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 p h l E a I 8 z 6 x L r b W w m b U E W Z s e l U b A j e 6 s v r p H P V 8 N y G d + / V m m 4 R R x n O 4 B w u w Y N r a M I d t K A N D M b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A J k d j U Q = < / l a t e x i t > y < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 p h l E a I 8 z 6 x L r b W w m b U E W Z s e l U b A j e 6 s v r p H P V 8 N y G d + / V m m 4 R R x n O 4 B w u w Y N r a M I d t K A N D M b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A J k d j U Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " f 2 k m q / g 3 1 f 6 U m A Z j U 2 b u C C D f i c o = " > A A A B 7 3 i c b Z C 7 S w N B E M b n f C b x F b W 0 W U w E q 3 B n o 2 X A x j K C e U B y h L m 9 v W T J 7 t 2 5 u y e E I 7 W 9 j Y U i a f 1 3 7 P x v 3 D w K T f x g 4 c f 3 z b A z E 6 S C a + O 6 3 8 7 G 5 t b 2 z m 6 h W N r b P z g 8 K h + f t H S S K c q a N B G J 6 g S o m e A x a x p u B O u

g j 5 /
M H s o S R M A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w O o n z 2 e m / p X B X E w A N a z X A n j d w D I = " > A A A B 7 3 i c b V A 9 S w N B E J 2 L X z F + R S 1 t F h P B K t z Z a B m w s Y x g P i A 5 w t x m L 1 m y u 3 f u 7 g k h 5 E / Y W C h i 6 9 + x 8 9 + 4 S a 7 Q x A c D j / d m m J k X p Y I b 6 / v f X m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 T J J p y p o 0 E Y n u R G i Y 4 I o 1 L b e C d V L N U E a C t a P x 7 d x v P z F t e K I e 7 C R l o c S h 4 y h L m 9 v W T J 7 t 2 5 u y e E I 7 W 9 j Y U i a f 1 3 7 P x v 3 D w K T f x g 4 c f 3 z b A z E 6 S C a + O 6 3 8 7 G 5 t b 2 z m 6 h W N r b P z g 8 K h + f t H S S K c q a N B G J 6 g S o m e A x a x p u B O u

3 j=1
e y/P j − e −y/P j e y − e −y = ∞ n=0 χ 2P (n)e − n P y r b W w m b U E W Z s e l U b A j e 6 s v r p H P V 8 N y G d + / V m m 4 R R x n O 4 B w u w Y N r a M I d t K A N D M b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A J k d j U Q = < / l a t e x i t > y < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 p h l E a I 8 z 6 x L L i Y r 5 + y l g X c z 2 r b W w m b U E W Z s e l U b A j e 6 s v r p H P V 8 N y G d + / V m m 4 R R x n O 4 B w u w Y N r a M I d t K A N D M b w D K / w 5 g j n x X l 3 P p a t J a e Y O Y U / c D 5 / A J k d j U Q = < / l a t e x i t > C 0 t < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 s 9 A 0 d H 6 F w m o B 2 5 o 8 8 z F f CK P z w Q = " > A A A B 8 3 i c b V D L S g N B E O w 1 P m J 8 R T 2 J l 8 F E 8 B R 2 v e g x k I v H C O Y B y R J m J 7 P J k N n Z Z a Z X C E t + w 4 s H R b z 6 D 3 6 D B 8 G v 0 c n j o I k F D U V V N 9 1 d Q S K F Q d f9 c t Z y 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o a e J U M 9 5 g s Y x 1 O 6 C G S 6 F 4 A w V K 3 k 4 0 p 1 E g e S s Y 1 a Z + 6 5 5 r I 2 J 1 h + y t h A 2 p p g x t T A U b g r f 8 8 i p p X l Y 8 t + L d e q W q C 3 P k 4 R T O 4 A I 8 u I I q 3 E A d G s A g g Q d 4 g m c n d R 6 d F + d 1 3 r r m L G a O 4 Q + c t x 8 W e 5 F W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " F j J h z u X K 2 O e P u F l x y y 9 l q p g Z B q c = "> A A A B 8 3 i c b V A 9 S w N B E J 2 L X z F + R a 3 E Z j E R r M J d G i 0 D a Sw j m A 9 I j r C 3 2 U u W 7 O 0 d u 3 N C O P I 3 b C w U s f X P 2 P l v 3 C R X a O K D g c d 7 M 8 z M C x I p D L r u t 1 P Y 2 t 7 Z 3 S v u l w 4 O j 4 5 P y q

1 F p x 8 5
h z + w P n 8 A U B L j n 4 = < / l a t e x i t > C 0 t < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 s 9 A 0 d H 6 F w m o B 2 5 o 8 8 z F f CK P z w Q = " > A A A B 8 3 i c b V D L S g N B E O w 1 P m J 8 R T 2 J l 8 F E 8 B R 2 v e g x k I v H C O Y B y R J m J 7 P J k N n Z Z a Z X C E t + w 4 s H R b z 6 D 3 6 D B 8 G v 0 c n j o I k F D U V V N 9 1 d Q S K F Q d f 9 c t Zy 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o a e J U M 9 5 g s Y x 1 O 6 C G S 6 F 4 A w V K 3 k 4 0 p 1 E g e S s Y 1 a Z + 6 5 5 r I 2 J 1 h + y t h A 2 p p g x t T A U b g r f 8 8 i p p X l Y 8 t + L d e q W q C 3 P k 4 R T O 4 A I 8 u I I q 3 E A d G s A g g Q d 4 g m c n d R 6 d F + d 1 3 r r m L G a O 4 Q + c t x 8 W e 5 F W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " F j J h z u X K 2 O e P u F l x y y 9 l q p g Z B q c = "> A A A B 8 3 i c b V A 9 S w N B E J 2 L X z F + R a 3 E Z j E R r M J d G i 0 D a Sw j m A 9 I j r C 3 2 U u W 7 O 0 d u 3 N C O P I 3 b C w U s f X P 2 P l v 3 C R X a O K D g c d 7 M 8 z M C x I p D L r u t 1 P Y 2 t 7 Z 3 S v u l w 4 O j 4 5 P y q

= 2 b=0S
to the trivial flat connection, and a = 1, 2 correspond to the abelian flat connections.Z a is expressed in terms of homological block Z b as Z a

2 , 2 number
v = 0, 1, . . ., s − 1, we see that there are s+1 + u) 2 mod 1 given a u where s is odd, which may imply that there are s+1 2 number of lifts.We can also see that(3.11)agrees with the total residue (3.6) above.More specifically, (3.5) can be written as 8π

H ) 2
where s is odd for the examples that we work here.Thus, given a u and a v there is also a sum over lifts of non-abelian flat connections.When H = 1, (3.7) is simplified toZ M 3 (K = r/s) g(P r; s) to an overall factor where g(m; s) denotes the quadratic Gauss sum g(m; s) = s−1 n=0 e 2πimn 2 /s .
such that sin πb P j is nonzero are not divisible by any of P j 's.For such b's in [1, P − 1], n's that give a nonzero result are n = ±b mod 2P .Therefore,(3.22)can be written as −4g(P r; s)

s n 2 4P 2 4P mod 1 .
, i.e. the same − r s n If we denote N as the number of sets of poles for the case of the integer level K = r, for examples that we work where s 4 Resurgent analysis for torus knot complement in S 3

FKK 2πi 1 P Q y 2 , 1 2 1 2
K (x = e 2πiu ; q = e 2πi for K, R ∈ Z + where u = R K .The perturbative part is obtained by taking the Taylor expansion of the rational function of sine hyperbolic function in (4.11) around y = 0 and evaluating the Gaussian integral term by term with the factor e − which is[15] 1 e /2 − e − /2 e 4 (P Q− P e y − x − 1 2 e −y ∆ K P,Q (x e y ) y=0 ,Q (x) = (x − x −1 )(x P Q − x −P Q ) (x P − x −P )(x Q − x −Q ) .(4.24)

1 P Q m 2 (
e πiRm − e −πiRm )(−1) zero when R was set to an integer.Also, by shifting the contour C 0 to the contours C ± that pass y = ±πiP Qu for each integrand containing e ±Kuy , respectively, there is also a contribution, j=1 e y/P j − e −y/P j e y − e −y .(2.54)We note that (2.53) and (2.54) are also obtained from (2.40) with Im K < 0 and with the contour γ.Thus, from (2.11), the analytically continued CS partition function would be expressed as the Borel resummation 2 Z M 3 (K) = e 2πiKS 0 γ dye − K 2πi H P y 23 j=1 e y/P j − e −y/P j e y − e −y + H (H−t))23 j=1 e y/P j − e −y/P j e y − e −y (2.55) when H is odd where CS(t) = P H t 2 mod 1 and S t = CS(t) + Z.When taking K to be an integer, we change the contour from γ to C t that passes y = −2πi P H t as in Figure2.More specifically, the part containing e − K The case of H ≥ 2 Meanwhile, upon the change of the contour from γ to C H−t , the part containing e − K

3
From (2.57) and (2.59), the contributions from the non-abelian flat connections attached to the abelian flat connection a are obtained from the Stokes phenomena.The set of poles that give, for example, the same e 2πiK P H u 2 e − πi 2 K H P (m− 2P u H ) 2 in (3.10) when K is an integer split to the poles that give the same e 2πi r 2 , P 3 ) = (2, 3, 5).When K is an integer, there two sets of poles y = −nπi in (2.23) with 1) m e e 2πiKS t