Resurgence analysis of quantum invariants of Seifert fibered homology spheres

For a Seifert fibered homology sphere X$X$ , we show that the q$q$ ‐series invariant Ẑ0(X;q)$\hat{\operatorname{Z}}_0(X;q)$ , introduced by Gukov–Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(X)$\operatorname{Z}_0(X)$ . We show that for every even k∈N$k \in \mathbb {N}$ there exists a full asymptotic expansion of Ẑ0(X;q)$ \hat{\operatorname{Z}}_0(X;q)$ for q$q$ tending to e2πi/k$e^{2\pi i/k}$ , and in particular that the limit Ẑ0(X;e2πi/k)$\hat{\operatorname{Z}}_0(X;e^{2\pi i/k})$ exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant τk(X)$\tau _k(X)$ . We show that the poles of the Borel transform of Z0(X)$\operatorname{Z}_0(X)$ coincide with the classical complex Chern–Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2,C)$\rm {SL}(2,\mathbb {C})$ ‐connections.

The work [48] by Garoufalidis used these ideas to pose a rich series of conjectures concerning the resurgence properties of quantum invariants, and their connection to complex Chern-Simons theory. More recently, the use of resurgence in connection to the Teichmüller topological quantum field theory (TQFT) of the first author and Kashaev [8], which is the mathematical model of the partition function of complex Chern-Simons theory [52], has been initiated by Garoufalidis, Gu and Marino in [49].
In [55], Gukov, Pei, Putrov and Vafa used arguments from string theory and 3 − 3 correspondence to propose the existence of an invariant of ( , ), ∈ Spin c ( ), which is an integer power series convergent inside the unit discẐ (1. 3) The Gukov-Pei-Putrov-Vafa (GPPV) invariant was argued to be connected via resurgence to Chern-Simons in the work of Gukov, Marino and Putrov [54]. In the case of certain Brieskorn spheres = Σ( 1 , 2 , 3 ) the invariantẐ ( ; ) was conceived as a Borel-Laplace resummation of the large level asymptotic expansion of the Witten-Reshetikhin-Turaev (WRT) invariant (1.1). Subsequently, the contour integral formula for (1.3) for (negatively definite) plumbed 3-manifolds from [55] was proven mathematically to be a topological invariant in [53]. Moreover; the radial limit conjecture [28,53,55] postulates that if 1 ( ) = 0, then the following holds: (see Conjecture 2, where the notation is introduced). The proof of this remarkable conjecture would give an analytic extension of ( ) to the interior of the unit disc. This paper concerns with WRT quantum invariants and their connection to Gukov-Pei-Putrov-Vafa (GPPV) invariants via resurgence and the radial limit conjecture. We now summarize our main results. Let 1 , … , ∈ ℕ be pairwise coprime integers, and let for the rest of this paper denote the oriented Seifert fibered integral homology 3-sphere with ⩾ 3 exceptional fibers: = Σ( 1 , … , ).
Let Z 0 ( ) denote the Ohtsuki series of . This is known by the work [73] to give the series in (1.2) attached to 0 ∈ CS( ). Let  denote the Borel transform (see the Appendix).
We stress that in this paper we work with the mathematical definition ofẐ 0 ( ; ) given in [53]. We now present our results in full detail.

Complex Chern-Simons theory
For a Lie group , let ( ) = ( , ) be the moduli space of flat -connections on . Set = ∏
(1.8) Remark 1. In accordance with the asymptotic expansion conjecture we expect that the sum in (1.6) should only range over the Chern-Simons values of flat SU(2)-connections. This is known to be true for = 3 [59] and in some cases for = 4 [58]. However, we see from (1.8) that the quantum invariants via resurgence determine all the Chern-Simons values of flat SL(2, ℂ)-connections. † A comparison is given at the end of the introduction.

F I G U R E 1
The integration contour = + + −

A resurgence formula for the GPPV invariantL
et Δ ∈ ℚ be given by Equation (4.2). Let denote a complex variable. Consider the GPPV invariant [55] which (up to the pre-factor −Δ ) is given by a power series invariant (1.9) with integer coefficients and radius of convergence equal to unitŷ (1.9) In this paper, we work with the mathematical definition of (1.9) given in [53] for a large class of plumbed 3-manifolds which includes . This definition is recalled below in Definition 1. Set such that for all ∈ ℂ with | | < 1 Let denote the upper half-plane. Let ∈ and set = exp (2 ).

Let
= + + − be the oriented unbounded contour depicted in Figure 1 and let = (−1) 2 (2 ) −1∕2 . We show that the GPPV invariantẐ 0 ( ; ) is a Borel-Laplace resummation (see the Appendix) of the Ohtsuki series Z 0 ( ).  (1.10) In this paper, we use the resurgence formula (1.10) to prove that the GPPV invariantẐ 0 ( ; ) admits a full Poincaré asymptotic expansion when tends to the th root of unity equal to 2 ∕ . We further show that the constant term of this expansion is equal (up to a scaling factor) to the Witten-Reshetikhin-Turaev quantum invariant, and thus we use resurgence to prove the radial limit conjecture for .

1.4
The asymptotic expansion of the GPPV invariantŴ e now present the asymptotic expansion theorem forẐ 0 ( ), which in particular implies the radial limit conjecture for [28,53,55]. This conjecture is recalled in Conjecture 2. Assume that is even. Set = Δ − ∕4 and = (2 ) −1 . For a positive parameter set .

Theorem 4.
For each ∈ CS * ℂ ( ), there exists a unique polynomial (defined in (5.21)) in of degree at most − 3 with coefficients in formal power series without constant termš (1.11) giving a full Poincaré asymptotic expansion for small and fixed even We remark that the existence of a full asymptotic expansion in terms of complex Chern-Simons invariants and polynomials in the level as in (1.12) is a new phenomenon not observed in the literature prior to this work (neither as a conjecture nor as a result). Thus the series (1.11) are new topological invariants of .
In combination with Lemma 14, our Theorem 4 yields a remarkable resummation formula for the (normalized) Witten-Reshetikhin-Turaev quantum invariant in terms of the (normalized) Ohtsuki series Z 0 . Recall that Ω denotes the set of poles of the Borel transform (Z 0 ) of Z 0 . Let be even, let be a small positive parameter and set , = Corollary 5.
. (1.14) Informally, the identity (1.14) can be rewritten as provided the right-hand side is interpreted as the limit on the right-hand side of (1.14).

Comparisons with the literature
We now give a brief comparison with the relevant works from the literature. The existence of an asymptotic expansionZ where ( ) ⊂ ℚ∕ℤ is a finite set which was proven in [73]. In this work, it was also shown that Z 0 is a normalization of the Ohtsuki series. Our contribution in regard to (1.15) is to compute CS ℂ ( ) and to show ( ) ⊂ CS ℂ ( ). In [73], the authors do not address the Borel transform. The -series from Theorem 3 was considered in the study of (Σ( 1 , 2 , 3 )) by Lawrence and Zagier [74], and further explored by Hikami [58]. It is easy to show that for = 3, the series (1.16) have periodic coefficients of mean value equal to zero. These facts, which are not true for ⩾ 3, were used by Lawrence and Zagier to prove that when → , the series Ψ( ) tends to the Witten-Reshetikhin-Turaev invariant. For ⩾ 4 Hikami in [59] considers a differently defined -series. Our Theorem 4 generalizes the result from [74] from = 3 to any number of exceptional fibers ⩾ 3. We remark that to go beyond the case of = 3, we use the resurgence formula (1.10).
The work [54] of Gukov, Marino and Putrov is one of the main inspirations for this paper. In [54], the authors analyze ( ) for some examples with = 3. The identity (1.8) was verified for these examples. For ∈ set ℎ = 2 so that = exp(ℎ). Consider again the integral In [54], identities of the form were discovered. In a sense, the series Ψ( ) was taken as a definition forẐ 0 ( ) for Σ( 1 , 2 , 3 ), and the GPPV formula (Definition 1) was only later introduced in [55]. Prior to this work, it was not proven that the GPPV formula and the Borel-Laplace resummation (1.17) give the same result. In the work [47] of Fuji, Iwaki, Murakami and Terashima, the -series Ψ( ) is also considered for general ⩾ 3, and they prove a radial limit theorem, which is analogous to (1.13). They also prove an identity of the form (1.18). In [47], they do not however work with the definition of the GPPV invariantẐ 0 ( ), although they conjecture that this is equal to Ψ( ). They also consider the case of the WRT invariant of a knot inside and prove a difference equation for Ψ( ).
Our Theorem 3 shows ΔẐ 0 ( ) = I(ℎ) = Ψ( ) for all Seifert fibered integral homology 3-spheres with ⩾ 3 singular fibers, whereẐ 0 ( ) is independently defined via the GPPV formula. We remark that those of our results that overlap with [47] had been presented prior to their submission by the first author in the online seminar [5] and by the second author at a seminar [79] at IST, Austria. The -series Ψ( ) was also conjectured to be a normalization ofẐ 0 ( ) in the second author's thesis [78]. We also remark that our proof of the radial limit formula (1.13) differs from theirs; our stronger Theorem 4 is derived using the resurgence formula forẐ 0 ( ) from Theorem 3, while their proof of their radial limit theorem uses Gaussian reciprocity directly on Ψ( ). We warmly thank them for cordial coordination.

Further perspectives
In a planned sequel to this paper, we give a resurgence analysis of Witten-Reshetikhin-Turaev quantum invariants of hyperbolic surgeries on the figure-eight knot. These manifolds are more complicated than Seifert fibered manifolds, and the resurgence analysis will fully use the Pham-Picard-Lefshetz theory [19,20,35,66,75,[85][86][87] developed for Laplace integrals with holomorphic phase (see the introduction to the second author's thesis [78] for a brief summary of the relevant results), as well as a detailed study of Faddeev's quantum dilogarithm [42,43]. In connection hereto, we mention also the paper [50] on resurgence of Faddeev's quantum dilogarithm and the following work of the first author [4], which concerns resurgence of meromorphic transforms, which is a class of functions that includes Faddeev's quantum dilogarithm. To obtain our results, we prove a conjecture due to the first author and Hansen [6]. A preliminary version of our results in this direction, which assumed the conjecture of [6], featured in the PhD thesis of the second author [78]. The resurgence analysis in this paper was derived from concrete formulae for quantum invariants, obtained combinatorially from a surgery diagram of . It is an important open problem to generalize this work by deriving a similar resurgence analysis for more general and complicated 3-manifolds using either the conformal field theory approach [11,91], or the quantization approach [17,65] to the Witten-Reshetikhin-Turaev TQFT. By a large body of work culminating in the works of the first author and Ueno [10][11][12][13], these approaches are equivalent to the combinatorial construction of .
Let us make a few more remarks in the quantization direction. For a closed oriented surface Σ (possible with labeled points), let V (Σ) be the TQFT Hilbert space. By the works mentioned above, this Hilbert space can be obtained by quantization of  Flat (Σ, SU( )) [17,65] equipped with the Atiyah-Bott-Goldman symplectic form [16,51]. Consider a closed oriented 3-manifold 3 , which contains Σ as an embedded surface. By cutting along Σ, we obtain a 3-manifold cut 3 with boundary  In our joint work [9], we prove the asymptotic expansion conjecture for the mapping tori 3 (with a special colored link) of a generic surface self-diffeomorphism ∶ Σ → Σ (preserving a point ∈ Σ which traces out the link in 3 ) using formula (1.19) and the quantization of moduli spaces approach to V (Σ). This quantization approach to quantum invariants is also considered by other authors in the works of [25][26][27]67]. By building on the work of the first author [2] and Toeplitz operator theory [70], the quantization approach allowed us in [9] to reduce the proof of the asymptotic conjecture to an application of stationary phase approximation applied to oscillatory integrals (1.20) over the moduli space  Σ =  Flat (Σ ⧵ { }, SU( ), 2 ∕ Id) of flat connections on Σ ⧵ { } with holonomomy around conjugate to 2 ∕ Id Such integrals are amenable to resurgence analysis by means of Pham-Picard-Lefshetz theory [19,20,35,66,75,[85][86][87]. Moreover, the phase function̂appearing in (1.20) admits a holomorphic extensions to a suitable complexification of an open neighborhood of the fixed locus  Σ . This is possible connected to Gukov and Witten's theory of brane quantization [56], in which complexification plays a central role, and where Hitchin's moduli space of Higgs bundles [64] plays the role of a complexification of  Flat (Σ, SU( )). The moduli space of Higgs bundles is by non-abelian Hodge theory [31,37,64] isomorphic to  Flat (Σ, SL( , ℂ)), and thus there is an immediate connection to complex Chern-Simons theory. In the future, we hope to use this approach to perform a general resurgence analysis of quantum invariants.

Organization
In Section 2, we prove Theorem 1 in several steps. Theorem 7 gives a decomposition of the moduli space, Corollary 9 computes the Chern-Simons invariants and Theorem 10 proves that components of this moduli space are classified by their Chern-Simons value. In Section 3, we prove Theorem 2. Proposition 12 gives an exact formula for the generating function ofZ ( ), verifying a special case of a conjecture of Garoufalidis [48]. In Section 4 we prove Theorem 3 and in Section 5 we prove Theorem 4. In the Appendix, we present generalities on resurgence.

COMPLEX CHERN-SIMONS THEORY ON
Let be the oriented Seifert fibered homology 3-sphere from the introduction. Choose 1 , … , ∈ ℤ such that ( , ) = 1 and Then has a surgery diagram as depicted in Figure 2. Without loss of generality, we can assume that 2 , … , are odd. The homeomorphism type of is unaltered under a transformation p 1 /q 1 p 2 /q 2 p 3 /q 3 p n /q n F I G U R E 2 Surgery link for ↦ + for any choice of integers 1 , … , such that ( , + ) = 1 and If is odd for > 1, we perform the transformation ↦ + and 1 ↦ 1 − ( − 1) 1 which does not change sum (2.1). Hence, we can assume without loss of generality that 2 , … , are all even. Recall that under our assumptions 1 = ∑
Define the meromorphic function ∈ (ℂ) and g ∈ ℂ[ ] explicitly as follows: In [73], Lawrence and Rozansky show the following results. There exists a finite subset * ( ) ⊂ ℚ * ∕ℤ and non-vanishing polynomials Z ( ) ∈ ℂ[ ], ∈ * ( ) of degree at most − 3 such that for all non-negative integers . Let to be the contour from (−1 − )∞ to (1 + )∞. Observe that ( ) is a steepest descent path for g. Introduce the following notation: Recalling the definition of the normalized quantum invariantZ given in (1.4), Lawrence and Rozansky proved that it can be decomposed into a sum of an integral part Z I and a residue part ZZ ( ) = Z I ( ) + Z ( ). (2.4) This is [73,Section 4.5,eq. 4.8]. We have used the same notation for , g and , whereas constant in their notation is equal to −1 . Thus, if we define and set ( ) = * ( ) ⊔ {0} then we have an asymptotic expansioñ In the work [73], it was observed that Z 0 is in fact a normalization of the Ohtsuki-series [81][82][83]. Let Z ∞ denote the Ohtsuki series (with the normalization used in [73]). Introduce the variable ℎ = − 1, where, as above, = exp(2 ∕ ). In [74,Section 4.6], they show the following identity:

The moduli space and complex Chern-Simons values
We now begin our investigation of ( , SL(2, ℂ)), which closely follows [44]. We have the following presentation of the fundamental group of : Let us first recall a few of Fintushel and Stern's results concerning the moduli space ( , SU(2)) established in [44]. As is an integral homology sphere, the only reducible representation into (2) is the trivial one. For an irreducible representation ∶ 1 ( ) → SU(2) at most − 3 of the ( ) are ± , and if exactly − of the ( ) are equal to ± , then the component of in ( , SU(2)) is of dimension 2( − ) − 6.
By this corollary we can, in particular, conclude that all Chern-Simons values are real and they only depend on ∈ ( 1 , … , ). In Proposition 8, we actually provide an explicit formula.
Before commencing the proof let us introduce the following notation: which should not cause any ambiguities as long as the context shows that we are dealing with a matrix.
Proof. We start with the construction of . Introduce matrices Assume that we can chose 2 , 3 ∈ SL(2, ℂ) such that To see this, observe that ∶= 1 1 = (− ) 1 is central and as 1 is odd whereas is even for ⩾ 2, we also have = , ∀ . The last relation in 1 ( ) is ensured by (2.8). Observe that it will suffice to choose 2 ∈ SL(2, ℂ) such that because this will ensure that there exists some 3 ∈ SL(2, ℂ) with the property that since non-diagonalizable elements of SL(2, ℂ) have trace ±2, given that the unit determinant condition implies that the unique eigenvalue with multiplicity two for such elements must be either 1 or −1 and we have that For (2.9) we used our assumption on 3 . Write We observe that by the conditions on ( 2 , 2 ) and ( 3 , 3 ) we have that , ∉ ℤ.
Thus we have that tr It follows that we must solve Thus it remains to argue + ∉ ℤ and ∉ ℤ. Assume toward a contradiction that + = for some ∈ ℤ. Hence, we would have ( + ) = for some integer , which would imply for ∈ {0, 1}, with = 0 for 1 = 1 and = 0 otherwise. This is a contradiction, as 2 ∏ . We see that ∉ ℤ directly from the conditions on 2 . Thus we can solve (2.10), and hence find the needed , , , , which concludes the proof of the first part of the proposition.
Let us now prove that we can actually choose the functions such that we obtain an SL(2, ℝ)representation. We will denote this new choice of the by ℝ . We set ℝ = for ∈ {1, … 2 − 1, 3 which has the following property: ) .
Introduce the notatioñ= −1 and observe by the above computations that To understand which values, say , this trace can take, we consider in analogy with (2.10) the equation The determinant is = −2 sin( ) sin( ), which is non-vanishing since , ∉ ℤ. Then we have that ( .
From which we see that̃2 if and only if Im( + ) = 0, Re( − ) = 0 or equivalently = whenever ∈ ℝ. But then this implies that Which we can solve when > cos( ± ) by letting and then and when < cos( ± ) then we can take This allows us to complete the construction as follows. First we assume that , ∉ ℤ. For =̃( , , , ) for = 1, 2 we consider the equation which is equivalent to since these are certainly all SL(2, ℂ) matrices. But now, using that we also have that , ∉ ℤ, we can choose bigger than cos( ± ) and cos( ± ) and fix̃as above such that and tr (̃2 (2.13) Thus, we can now conclude that there exists ℝ ∈ SL(2, ℝ) such that Thus if we further set and ℝ = ℝ for ∈ { 2 + 1, … , 3 − 1}, then we find the needed conjugation to obtain an SL(2, ℝ)representation. Let us now consider the remaining cases. Suppose that ∈ but ∉ ℤ. Then the common trace is by (2.12) forced to be cos , so we can solve (2.13) if and only if cos is not contained in the interval spanned by the two values cos( ± ). If this is the case, we proceed with the argument as above. If on the other hand cos is contained in the interval spanned by cos( ± ), then it is well known that we can choose ∈ SU(2) so as to obtain an SU(2)representation. A similar argument of course works in the case where ∈ but ∉ ℤ. If we have , ∈ ℤ, then + ∈ ℤ, but this we have already argued is impossible. Now let ∶ 1 ( ) → SL(2, ℂ) be an arbitrary non-trivial representation. As remarked before any non-trivial representation is irreducible since is an integral homology 3-sphere. Since (ℎ) commutes with the image of , we see that (ℎ) = ± . Hence, the relation = ℎ implies that ( ) = ± , and for = 2, … , we must have ( ) = , since is even. Hence, must be of the form (2.7) for some ′ ∈ ℕ . It only remains to argue that at most − 3 of the ( ) are ± . If not, the relation 1 2 ⋯ = 1 implies that there is 1 < 2 with ( 1 ) ( 2 ) = ± . As 1 and 2 are relatively coprime, this is only possible if ( 1 ) = ± and ( 2 ) = ± . This would imply that ( 1 ( )) ⊂ {±1} = (SU (2)) which contradicts the fact that is irreducible since it was assumed non-trivial.
Then we have that Formula (2.16) was proven for SU(2) connections by Kirk and Klassen and it is stated in [71,Theorem 5.2]. It is proven using the following general result. Let be a closed oriented 3-manifold containing a knot . Let be the complement of a tubular neighborhood of in . With respect to an identification ⧵ ≃ 2 × 1 , choose simple closed curves , on intersecting in a single point such that bounds a disc of the form 2 × {1}. Let ∶ 1 ( ) → SU(2) be a path of representations such that 0 ( ) = 1 ( ) = 1, and for which there exist continuous piecewise differentiable functions ) .
Thinking of 1 , 0 as flat connections on we have Note that formula (2.17) differs from the corresponding formula in [71] by a sign. This discrepancy was already discussed by Freed and Gompf in [46] and is due to a sign convention; see the footnote in [46, p. 98]. The formula (2.17) was also used in the work [6] by the first author and Hansen.
Proof of Proposition 8. Let ⊂ be the th exceptional fiber. Let be the complement of a tubular neighborhood of in . Removing has the effect on 1 of removing the relation = ℎ − , that is, we have a presentation As the meridian and longitude of we can take = ℎ and = − 1 ⋯ −1 ℎ , respectively, These choices of meridian and longitude coincide with the choices made in [44].
Let ∶ 1 ( ) → SL(2, ℂ) be any irreducible representation with its corresponding = ( 1 , … , ) ∈ ( 1 , … , ). Now (ℎ) = exp( ) for some integer ∈ ℤ. Introduce the two quantities, The proof of (2.16) presented here consists analogously with [71, Proof of Theorem 5.2] of two parts. In the first part, we find a path of SL(2, ℂ) connections on connecting to an abelian representation 0 . In fact 0 will be an SU(2) connection on . In the second part, we then find a path from 0 to the trivial representation triv and we then apply Kirk and Klassens formula (2.17). The only difference from the proof in [71] is that we need to explicitly ensure that our paths stay away from parabolic representations. The relevant paths are chosen such that , are mapped to the maximal ℂ * torus of diagonal matrices.

Corollary 9.
We have that Proof. It is clear that CS * ℂ ( ) ⊂ ( 1 , … , ). We must show that for any ∈ ℤ which is not divisible by more than three of the we can find = ( 1 , … , ) ∈ ( 1 , … , ) which solves the congruence equation For ∈ ℤ and ∈ ℕ let [ ] denote the congruence class of in the quotient ring ℤ∕ ℤ. Since is odd for ⩾ 2, it follows that 4 1 , 2 , … , are also pairwise co-prime. Hence, the Chinese remainder theorem applies and the natural ring homomorphism ∶ ℤ → ℤ∕4 1 ℤ ⊕ =2 ℤ∕ ℤ, given by ↦ ([ ] 4 1 , … , [ ] ), descends to an isomorphism of rings It follows that (2.20) is in fact equivalent to the following congruence equations The coprimality conditions ensures that 2 ∏ ≠ is an invertible element in ℤ∕ ℤ and therefore solving the last − 1 of the equations in (2.21) can indeed be done with 0 ⩽ ⩽ ( − 1)∕2. It remains only to consider the first of the equations in (2.21). To this end we first observe that But then we can solve  Proof of Theorem 1. This follows from Theorem 7, Corollary 9 and Theorem 10. □

THE BOREL TRANSFORM AND COMPLEX CHERN-SIMONS
We now provide the proof of Theorem 2. The reader not familiar with the Borel transform  and its relation to the Laplace transform is encouraged to read the Appendix, before reading the proof of Theorem 2. For a measurable function g ∶ ℝ + → ℂ of sufficient decay, we use the notation  ℝ + (g) for the Laplace transform -see Equation (A1).  Therefore we see that if (3.1) holds, then there is somẽ∈  (− 2 ∕4 ) which is divisible by at most − 3 of the . This establishes * ( ) ⊂ CS * ℂ and we get (1.6). Observe that as a corollary we obtain for each ∈ CS * ℂ the formula We now turn to (Z 0 ). The formal series Z 0 is the asymptotic expansion of the Laplace integral Let be the rational function introduced in (1.5) and introduce the multivalued function  0 ( ) given by With this notation, the equation for the Borel transform (1.7) which we want to prove, reads as follows: The function  0 is related to as follows: . Thus We now rewrite Z I ( ) as the Laplace transform of  0 (3.6) The existence of the asymptotic expansion (2.5) Z I ( ) =  ℝ + ( 0 )( ) ∼ →∞ Z 0 ( ) can now be obtained by appealing to the first part of Lemma A1 where we set =  0 . Here we use the existence of the expansion (3.5). Therefore the desired identity (1.7) (Z 0 ) =  0 follows from the second part of Lemma A1 and the convergence of the expansion (3.5). As (− ) = ( ) we note that the factor gives a well-defined meromorphic function. Thus (Z 0 )( ) is a multivalued meromorphic function with a square root singularity at 0 and with singularities for √ 8 ∈  where  is the set of poles of ( ). This set was computed above (see Equation 3.2) and we conclude that the poles of (Z 0 )( ) occur at with ∈ ℤ being divisible by less than or equal to − 3 of the functions . This concludes the proof of (1.8). □ It is of course expected that only a Chern-Simons invariant of a flat SU(2) connection have a non-vanishing polynomial Z ≠ 0, that is, * ( ) = S CS ( * ( , SU(2))).

Resummation of the WRT invariant
We now turn to the resummation of the normalized WRT invariantZ ( ). Recall that for ∈ ℚ∕ℤ we introduced the set We also introduce the residue operator  which for a meromorphic function̂is given by Observe that by definition  ( ) is empty for all but finitely many ∈ ℚ∕ℤ and therefore  is 0 for all but these finitely many .
Corollary 11. The polynomials Z and the quantum invariantZ ( ) are determined by (Z 0 ) as follows: The identity (3.8) of Corollary 11 is reminiscent of the typical resummation process from resurgence [14,38]. The Ohtsuki series is known to determine ( ). The new insight provided by resurgence is that it does so via resummation as stated in Corollary 11.
We now prove Corollary 11.
Proof. It easily follows from (1.7) that .  In the proof of Theorem 2 we obtained the following exact formula: Thus, we see that (3.8) follows from this formula and (3.7). □

Resurgence of the generating function
Let be a simple, simply connected compact Lie group, and let , be the level Reshetikhin-Turaev TQFT constructed from the quantum group ( ), where is the complexification of the Lie algebra of . Letȟ be the dual Coxeter number of , and seť= + ℎ. For a closed oriented 3-manifold (possibly containing a colored framed link) we consider the normalized invariant .

Z ( ; ) ∈ ℂ[[ ]]
given by By the work of Garoufalidis, Z ( ; ) is known to be convergent on the unit disc. Motivated by the paradigm of analytic continuation and resurgence, Garoufalidis posed the following conjecture.
Conjecture 1 [48]. The generating function Z ( ; ) has an analytic continuation to ℂ ⧵ Λ where Λ is a finite set containing zero and the exponentials of the negatives of the complex classical Chern-Simons values.
In other words, the conjecture is that the generating function Z ( ; ) determines the germ at zero of a resurgent function. This conjecture is formally motivated from resurgence of Laplace integrals and the (non-rigorous) path integral formula for the WRT invariant, as explained in [48].
We now specialize to the case of the Seifert fibered homology sphere and = SU(2). Set = + 2 and consider the generating function for the normalized quantum invariantZ ( ) given bỹ For ∈ ℂ consider the polylogarithm (3.11) For = − , ∈ ℕ the polylogarithm is exact and in fact a rational function We introduce the following notation for the exponentials of the negatives of the classical complex Chern-Simons values: ) .
We prove the following proposition.
Proposition 12. The generating functionZ( ; ) is the germ at zero of a holomorphic functioñ Z ∈ (ℂ ⧵ Λ) given by the following formula: .

(3.12)
Proof. From Equation (3.10), it follows that (3.13) The first term can be simplified by interchanging summation and integration and then using the geometric series expansion (3.14) This can be justified by standard complex analysis arguments. To complete the proof, we can consider separately each term in (3.13) corresponding to a complex Chern-Simons value ∈ CS ℂ ( ). We get that ) .

A RESURGENCE FORMULA FOR THE GPPV INVARIANT
We now turn to the -series invariantẐ 0 ( , ). We follow [53]. Let (Γ, ) be an ordered weighted tree, that is, Γ is a tree together with an ordering of its set of vertices and is a map ∶ → ℤ. We recall that two plumbed 3-manifolds and ′ are diffeomorphic if and only their plumbing graphs are related by Neumann moves.
When is a plumbed manifold with weakly negative definite plumbing graph and is not necessarily a homology 3-sphere, the -series invariantẐ ( , ) depend on a label ∈ Spin c ( ). Originally, these labels were thought to be abelian or 'almost abelian' flat connections (see [29]). For a 3-manifold with 1 ( ) = 0, the set of abelian flat connections and Spin c can be identified. As = Σ( 1 , … , ) is an integral homology 3-sphere, we have = 0, and need not go deeper into this discussion. For the sake of completeness however, we recall the GPPV-formula definition as it is stated in terms of Spin c structures. First, we recall how Spin c structures can be described in terms of the adjacency matrix . This is thoroughly explained in [53]. Let be a plumbed 3-manifold with plumbing graph Γ. Let = | |. Let ⃗ ∈ ℤ be the weight vector, that is, = ( ). Let ⃗ ∈ ℤ , be the degree vector, that is, = deg( ). We have isomorphisms Spin c ( ) ≃ (ℤ + ⃗ )∕2 ℤ ≃ (ℤ + ⃗ )∕2 ℤ .
These isomorphisms are compatible with Neumann moves as explained in [53]. We now recall the GPPV formula (4.1).
Definition 1 [55]. Let be a plumbed 3-manifold with weakly definite plumbing graph Γ. Let denote the number of positive eigenvalues of and let denote the signature of .
We recall that the principal value . . is defined such that for every sufficiently small > 0 we have

Proof of Theorem 3
We now consider = Σ( 1 , … , ) in more detail. Choose 1 , … , ∈ ℕ such that for each = 1, … , we have ( , ) = 1, Then = −1∕ < 0 is the Seifert Euler number. Choose a continued fraction expansion of ∕ for each = 1, … , As explained in [53], has a negative definite plumbing graph Γ defined as follows. The graph Γ is star-shaped with arms and central vertex 0 with weight 0 . For each = 1, … , the th arm has vertices. If these are ordered ( ,1 , … , , ) with ,1 being closest to the central vertex 0 , then , have weight − , . This graph is illustrated for = 3 in Figure 3. Before proving Theorem 3, we first give a formula for the rational exponent Δ ∈ ℚ. For each = 1, … , let be the plumbed manifold whose graph Γ is identical to Γ except that on the th arm, we delete the terminal vertex , . Define ℎ ∈ ℕ as

F I G U R E 3 Plumbing graph for in the case = 3
Observe that the total number of vertices of Γ is given by .
Proof. Recall that = exp (2 ) where ∈ . For the sake of notational simplicity, we also introduce the paramter ℎ = 2 so that = exp(ℎ). We start by proving that where Ψ( ) is the series introduced in (1.16) and I(ℎ) is the contour integral introduced in (1.17) (with ℎ = 2 ). Observe that for the purpose of proving (4.3) we can and will assume that because if the identity (4.3) holds true on this half-line, it has to hold on the entire upper half-plane , since both functions are holomorphic in . Set .

F I G U R E 4 The contour Δ
For each ∈ ℤ ⩾ 0 introduce the polynomial This is a Morse function with a unique saddle point at = ℎ 2 √ and we have that Let D( ) be the closed ball centered at the origin with radius > | 0 |. We can deform ℝ slightly to a contour Δ ⊂ { ∈ ℂ ∶ Re( ) < 0} ∪ D( ), which passes through the saddle point and such that the function given by ↦ exp( ( )) has exponential decay along Δ . The orientation of Δ is as depicted in Figure 4. We remark that Figure 4 depicts the situation where 0 ⩾ 0. Recall that if Γ is a steepest descent contour through the unique saddle point of a degree two polynomial ( ) = − 2 + , then we have the following exact formula known as Gaussian integration ) .
As Δ 0 is a small deformation of Δ for each ∈ ℤ ⩾ 0 , we obtain In the second equality of (4.7) we used that √ ∈ { ∈ ℂ ∶ Re < 0} for all ∈ Υ( ), and the contour −Υ(−1) denotes Υ(−1) but oriented in the direction from the origin and toward infinity.
In the third equality of (4.7) we used Equation (4.5) and the identity which follows directly from the definition of . Now introduce the variable = 2 .
This identifies (up to a small deformation) the contour Υ + − Υ − with the contour introduced in Figure 1.
We have In the last equality of (4.8) we used Equation (4.4), which relates  and (Z 0 ). Now recall that = √ 2 , since = 1 and recognize the pre-factor in the last line of (4.8) as where is the scalar introduced in the statement of Theorem 3. By combining Equations (4.6), (4.7) and (4.8), we see that Equation (4.3) holds. WriteẐ 0 ( ; ) =Ẑ 0 ( ). We now show that where Δ ∈ ℚ is the scalar introduced in (4.2). This will establish (1.10) and thereby finish the proof. We start withẐ 0 ( ). By Definition 1 and since in this case = 0, we have that Here it is understood that we have taken the principal value of the integral as explained above.
Recall that for a Laurent series ( ) = ∑ ∈ℤ we have that For our star-shaped plumbing graph Γ, the non-zero contributions to (4.10) comes from ⃗ ∈ 2 ℤ with = 0 for all of the entries corresponding to an internal vertex of an arm, and = ±1 if is a terminal vertex of an arm and then from the central vertex 0 , which we will now consider.
In comparingẐ 0 ( ) with Ψ( ) it is useful to introduce the integer sequence { } ∞ =0 determined by (4.11) The functions can be explicitly evaluated: By the formula for the geometric series and Cauchy multiplication of power series, we see that and therefore one sees that However for the comparison ofẐ 0 ( ) and Ψ( ) given below, we do not need the closed form for , but rather Equation (4.11). Write 0 = and = 0 . We obtain We know that the adjacency matrix is unimodular, and so ℤ  With this notation, the above considerations show that If we apply the symmetry that simultaneously changes the sign of all and , then we obtain The quadratic form was computed for = 3 in [53] in their proof of Proposition 4.8. The size of the matrix −1 is irrevelant to their computation, and their formula can be generalized to our case to give the for-mulaẐ We now compute Ψ( ). For | | < 1 we have It follows that This shows (4.9). □ We obtain the following corollary. Proof. This is a consequence of the integral formula (1.10) from Theorem 3 and Borel-Laplace resummation, which is stated as Theorem A2. □ Let us now recall previous work on the -series Ψ. We start with the case = 3, for which more is known. As already mentioned in the introduction, Lawrence and Zagier have shown in [74] that the quantum invariant ( ) can be recovered as the radial limit of Ψ( ), as tends to exp(2 ∕ ). This was generalized to = 4 by Hikami in [59] but with corrections terms appearing. The series Ψ have interesting arithmetic properties; the coefficients ( ) are periodic functions of period 2 and Ψ is the so-called Eichler integral of a mock modular form with weight 3∕2. As mentioned in the introduction the connection between quantum invariants and number theory was further pursued by Hikami in a number of articles [57][58][59][60][61]63]. For general ⩾ 3 we mention again the work [47] of Fuji, Iwaki, Murakami and Terashima, which was discussed in the introduction.
Let us now discuss what was previously known about the -series invariantẐ 0 ( ). In [53] it was shown that when is a Brieskorn sphere Σ( 1 , 2 , 3 ), (that is, = 3) thenẐ 0 ( ) is a linear combination of so-called false theta functions. The -series invariantẐ 0 was also considered for certain Seifert fibered manifolds (with up to = 4 singular fibers) in the work [30], as well as a proposed analog ofẐ 0 for higher rank gauge group -see also [84] for further developments in this direction. In this paper, we work exclusively with = SU(2)).
In connection with the work [74], Zagier invented the notion of a quantum modular form. This notion was generalized by Bringmann et al. in [22], where they introduce the notion of a higher depth quantum modular form. For any ⩾ 3, it is known, that Ψ is a linear combination of derivatives of quantum modular forms [23,24]. It is interesting to observe that Ψ is obtained from the Borel transform through a resummation process reminiscent of the median resummation of [32]. Moreover, as explained in [28] it is expected that for a general 3-manifold , mock/false modular form duality is related toˆ( ; ), that is, there exists an associated pair of a so-called Mock modular form and a so-called false modular form, and these are related by a ↦ −1 transformation and have the same transseries expression near → 1. This is quite possibly connected to [48,Conjecture 2] (called the symmetry conjecture). Let us also mention the work [36] by Dimofte-Garoufalidis, which connects modularity in quantum topology with complex Chern-Simons theory.

THE ASYMPTOTIC EXPANSION OF THE GPPV INVARIANT
The invention ofẐ was party motivated by an attempt to generalize the following discovery of Lawrence and Zagier. Set = exp(2 ∕ ). For = 3 they proved in [74] the identity (for some ∈ ℚ) ThusẐ ( ; ) can be seen as an analytic extension of ( ) to the interior of the unit disc as illustrated in Figure 6.
F I G U R E 6 Analytic extension of ( )
Recall the decomposition (2.4) of the normalized quantum invariantZ ( ) into an integral part Z I and a residue part Z . In Lemma 14, we prove the existence of an analogous decomposition for Z 0 ( ) into a Laplace integral part  and a residue part where we recall that = 2 . We present in Proposition 15 a standard result in complex analysis [74], which asserts that a -series with periodic coefficients of mean value zero has an asymptotic expansion, as tends to a root of unity. We then show in Proposition 16 that satisfy this hypothesis. Finally, we apply Proposition 15 to prove Theorem 4.
(5.5) From (5.5) we see that is periodic with period , that is, for all ∈ ℤ we have that We will now apply Cauchy's residue theorem to move − across ∕4 ℝ + to + in order to obtain the formula (5.2). Deform ± on the complement of a neighborhood around the origin to two curves ± , which are parallel to ∕4 ℝ + outside this neighborhood of the origin, as indicated in Figure 7. Set We first show that

7)
F I G U R E 7 The contours ± , ± and ± and the subset of poles  drawn rotated, such that ℝ + points straight up and then we show that the right-hand side of (5.7) can be rewritten as a sum of residues. Let > 0 be a positive constant, and let ± be the arc segment of the circle of radius , which connects ± and ± . Because ± is parallel to ℝ + outside a neighborhood of the origin, there exists a real positive constant 0 > 0 such that every ∈ ± is of the form = ± , ( , ) ∈ ℝ + × [ 0 , +∞) and therefore exists a positive real constant > 0 independent of , which gives an upper bound for all ∈ + . It follows that we obtain a uniform estimate By combining the estimates (5.8) and (5.9), and using that the arc length of ± is proportional to , we obtain the estimate ∫ + exp(− 2 ∕ ) ( ) d = ( − 2 2 + 1 ).
Because of this universal bound, it is easy to see that It follows that the right-hand side of (5.11) converges to This also implies that the sum of residues is convergent.
Let us now recall a simple transformation law for residues. Let 0 ∈ ℂ and let ∈  0 (ℂ) be the germ of a meromorphic function with a pole at 0 . Assume 0 ∈ ℂ and that ∈  0 (ℂ) satisfies  Push the contourΓ + to ℝ + . If the integral is invariant under this deformation of the contour, we obtain the desired identity. To see that the integral is invariant under this deformation of the contour, we apply a limiting argument, together with Cauchy's residue formula. To that end, let > 0 be a positive parameter, and let be the arc segment of the circle of radius , which connects to of (Z 0 ), the only difficulty is to show that As remain at least a fixed distance away from the axis of poles of (Z 0 )( ), limit (5.14) follows by (5.13) together with arguments similar to the arguments giving limit (5.10). □

Asymptotic expansions of -series with periodic coefficients
Let ( ) denote the th Bernoulli polynomial, that is, We recall the following result.
Proposition 15 [62,74]. Let ∶ ℤ → ℂ be a periodic function with period and mean value equal to zero Proof. The existence of the analytic extension of the -series of , as well as the explicit evaluation (5.16) are proven in [74].
In [62,74] the following asymptotic expansions are proven where it is understood that = 0 for > . The expansion (5.17) follows formally from differentiating the expansions given in (5.18). This differentiation is valid because Poincaré asymptotic expansions of analytic functions which are valid on suitable sectors can be termwise differentiated.
is an analytic function of in a small tubular neighborhood of (0,1], and from the proof given in [74] it is clear that the asymptotic expansions (5.18) are valid on such a small sector. □ Recall the definition of the meromorphic function given in (2.3). Next we prove that the coefficients of the principal part of at poles are periodic functions with mean value equal to zero. Proof. The periodicity of the functions , = 1, … , − 3 follow directly from the 4 -periodicity of .
We now prove (5.19) assuming is even -which is equivalent to exactly one the being even. Using the definition (2.3) of we obtain This implies that for each = 1, … , − 3 and = 1, … , we have On the other hand, we have for integral g(2 ( + )) = (2 ( + )) 2 It follows that for even we have a pairwise cancellation: This concludes the proof. □

The asymptotic expansion of the GPPV invariant
Recall the definition the functions and g given in (2.3). Let ∈ ℕ. For close to 2 we use the notation of Proposition 16 and write Observe that as is an integral homology sphere, the only Spin c structure is = 0, and therefore the radial limit conjecture reduces (up to a scalar factor) to Equation (1.13). We first focus on the integral part ( ). For every ∈ ℕ * the integral part ( ) extends continuously to = 1∕ and it follows from Equations (2.4), (3.6) and (5.3) that We start with the expansion (5.22). To ease notation we set g(2 ) = .
We have and accordingly √ Note that , ∈ for ∈ (0, 1). The function g has the following transformation property Because of this, and the 2 periodicity of the functions , we can write  This establishes the asymptotic expansion (5.22).
We now turn to the identity (5.23). Set .
We have that and accordingly Recall that the polynomials defined in (5.20) as the coefficients of the Taylor series of g( ) . Therefore it follows from Cauchy's formula for multiplication of power series, the formula for the Taylor expansion of the exponential and the identity (5.25) that the following holds for all , : By comparing with Equation (5.24) and using the facts that has a multiple order zero at multiples of 4 and that we know that the −1 term cancels in Z ( ), we obtain the desired identity + 1 +1 ( 2 ) (5.28) = 0 ( ).

A.1 Resurgent functions and the Borel transform
The theory of resurgence was originally developed by Écalle in [40,41]. See [80] for an introduction to the mathematical theory of resurgence and see [39] for an introduction to the general use of resurgence in quantum field theory. Garoufalidis [48] and Witten [95] were the pioneers of the use of resurgence in quantum Chern-Simons theory.
Definition A1. For a Riemann surface with universal covering space ∶̃→ the group of resurgent functions is ( ) = (̃).
One source of resurgent functions are the Borel transforms of Laplace integrals. We now introduce the Borel transform. Let Γ ∈ (ℂ) be the Gamma function, which for ∈ ℂ with Re( ) > 0 is defined by Γ( ) = ∫  .

A.2 Borel-Laplace resummation
We now discuss in more detail the relation between the Borel transform  and the Laplace transform, which we now introduce. Let ⊂ ℂ be an oriented contour. Let g be a measurable function defined in a neighborhood of . Denote by  (g) the Laplace transform given by for all ∈ ℂ such that the integral is absolutely convergent. Here we think of ∈ ℂ * as a large modulus asymptotics parameter. For any ∈ ℂ * we let the contour ℝ + be oriented in the direction of unless we state otherwise. That the transforms  and  ℝ + are formally inverses of each other should be understood as follows. If ∈ ℂ satisfies Re( ) > −1 and ∈ ℕ then ) .
Let ∈ ℂ with Re( ) > 0 and let ∈ ℕ. We then have that  Proof. The lemma is an elementary consequence of Equations (A2) and (A3). □ The following theorem explains Borel-Laplace resummation. The content of Theorem A2 is standard in resurgence, and a proof can be found in, for example, [90]. One of the goals of Ecalle's theory [40,41] is to consider the case where the formal series( A4) is obtained as a formal solution to some dynamical problem, which can be for instance an ODE or a difference equation (with a singularity at −1 = 0). In such situations, the function  Γ( ) •(̃) will be a holomorphic solution, and resurgence is developed as a tool to analyze the monodromy (known as Stokes phenomena), which occur upon varying the choice of direction in which the Laplace transform is performed. For future studies related to the radial limit conjecture (Conjecture 2), we also mention the work of Marmi and Sauzin [77].

A C K N O W L E D G E M E N T
We warmly thank S. Gukov for valuable discussions on the GPPV invariantẐ ( 3 ; ). The first author was supported in part by the center of excellence grant 'Center for Quantum Geometry of Moduli Spaces' from the Danish National Research Foundation (DNRF95) and by the ERC-Synergy grant 'ReNewQuantum'. The second author received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 754411.

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