Title: Multi-instantons in 2D string theory and median resummation

Abstract: The normalization of multi-instanton effects in two-dimensional string theory involves numerical factors that are hard to determine. I'll show how they can be found using a chiral representation of the dual matrix quantum mechanics.The result is confirmed by a string field theory calculation of annuli amplitudes and nicely agrees with a median resummation of the Gamma function.

Title: Divergence and resurgence

Abstract: In the mathematical phenomenon of resurgence, typically associated with divergent series, the tail of the series – the late terms - can be resummed to generate a series of corrections corresponding to the early terms of the original series or one or more closely related series. The first of two examples concerns the saddle-point approximation to integrals. The divergent tail of the series corresponding to an integration path through one saddle is represented, after resummation, by the terms in the integals through other saddles; this is

‘hyperasymptotics’[1]. The second example concerns the Riemann zeta function at height t on the critical line, in particular the real function Z(t), proportional to ζ(1/2+it). Representing Z(t) by the Dirichlet series gives a divergent series of complex terms. But if the Dirichlet series is truncated by O(t1/2) terms, the tail, approximated by the Poisson sum formula, consists of terms that are the complex conjugates of those in the original Dirichlet series [2-4]. The resulting approximation to Z(t) is real, and equal to the ‘main sum’ in the Riemann-Siegel formula.

[1] Berry, M. V., & Howls, C. J. 1991 Hyperasymptotics for integrals with saddles Proc. Roy. Soc. Lond. A434 657-675.

[2] Berry, M. V. 1986. Riemann's zeta function: a model for quantum chaos? In T. H. Seligman & H. Nishioka (Eds.), Quantum chaos and statistical nuclear physics (Vol. 263, pp. 1-17).

[3] Edwards, H. M. 2001.Riemann's Zeta Function Mineola, New York Dover Publications:

[4] Berry, M. V. 1995 The Riemann-Siegel formula for the zeta function: high orders and remainders Proc.Roy.Soc.Lond. A450 439 - 462.

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Title: A proof of Witten’s asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres

Abstract: D. Sauzin (CNU Beijing / CNRS Paris), joint work with J. E. Andersen, L. Han, Y. Li, W. E. Mistegård and S. Sun

Consider a general Seifert fibered integral homology 3-sphere with r≥3 exceptional fibers. We show that its SU(2) Witten-Reshetikhin-Turaev invariant (WRT) evaluated at any root of unity ζ is (up to an elementary factor) the non-tangential limit of its Gukov-Pei-Putrov-Vafa invariant (GPPV) as q tends to ζ, thereby generalizing a result from Andersen-Mistegård [JLMS 2022]. The quantum modularity results developed by Han-Li-Sauzin-Sun for functions like the GPPV invariant [FAA 2023], based on Écalle's resurgence theory and median summation, then help us to prove Witten’s asymptotic expansion conjecture [CMP 1989] for such a manifold: the asymptotic behaviour of the WRT invariant at exp(2πi/k) as k tends to infinity is given by a sum of contributions, one for each SU(2) Chern-Simons critical values.

Furthermore, we show that, when going to the variable τ defined by q = exp(2πiτ), the GPPV invariant has for each rational α a non-tangential asymptotic expansion that is a resurgent-summable series of τ-α or, equivalently, a resurgent-summable series of q-ζ where ζ = exp(2πiα). The formal series of q-ζ seems to be related to the expansion at ζ of the Habiro invariant [Inv. 2008] (for ζ=1 this is the Ohtsuki series [Inv. 1996]). In the variable τ all these formal series make up a higher depth strong quantum modular form in the sense of D. Zagier.

Title: Recent applications of resurgence in 2d QFT and CFT

Abstract: After a quick overview of recent advances in resurgence in 2d Quantum Field Theories (QFTs), we focus in particular on an application within 2d conformal field theories (CFTs).  We determine the large central charge c asymptotic expansion of Virasoro conformal blocks in four-point functions with external degenerate operators and study its resurgence properties as a function of the conformal cross-ratio z. Starting from the 1/c series of the identity block, we show how a resurgent analysis allows us to reconstruct the full correlator. The so-called forbidden singularities are turning points emanating Stokes lines, and we show how they are non-perturbatively resolved. Implications for gravitational theories in AdS_3 are briefly discussed. Our results are based on new asymptotic expansions for large parameters (a,b,c) of certain hypergeometric functions 2F1(a,b,c;z) which can be useful in general.

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