ReNewQuantum International Conference 2025 - Resurgence in Mathematics and Physics

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: Instantons and Renormalons in the 2-dimensional Sigma Model: The Full Resurgent Transseries

Abstract: We analyze a family of generalized energy densities in the 2-dimensional sigma model. By using the Wiener-Hopf technique to solve the linear thermodynamic Bethe ansatz equations we derive the full analytic trans-series for these observables and give their complete resurgence structure. We demonstrate that the physical value of the energy density is obtained by the median resummation of the perturbative series. The resummation of the trans-series is convergent for all values of the coupling.

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: Painlevé I, topological recursion and resurgence

Abstract: I will review several results and conjectural properties of the topological recursion free energy

associated with the Weierstrass elliptic spectral curve, which is related to Painlevé I.
In particular, I will show that the free energy satisfies the conifold gap condition,
and verify that its large-t expansion coincides with that of the irregular conformal block up to a certain order.
I will also explain how the Gu-Marino resurgence formula can be derived from the integrability of Painlevé I.

Based on joint works with M. Marino (SIGMA 20, 2024), and with N. Iorgov, O. Lisovyy and Y. Zhuravlov (arXiv:2505.16803).

Title: Resurgence and the topological string

Abstract: I will review recent progress regarding the determination of instanton corrections to the topological string partition function on compact Calabi-Yau threefolds. These results yield tools to study how BPS information is encoded in the Borel plane of the topological string. I will discuss this question in the context of one-parameter hypergeometric Calabi-Yau models. The talk is ba-sed on joint work with Jie Gu, Albrecht Klemm and Marcos Marino, and with Simon Douaud.

Title: Painlevé I tau function and irregular conformal blocks

Abstract: I will start by introducing the first Painlevé equation and the associated space of monodromy data. I will explain how the Fourier transform structure of the Painlevé I tau function arises from an extension of the Jimbo-Miwa-Ueno differential. We will then explore the connection between the main building block of this representation - the Painlevé I partition function - and irregular conformal blocks. If time permits, I will also discuss  how these quantities are related to topological recursion and  holomorphic anomaly equation.

Title: A new construction of c=1 Virasoro blocks

Abstract: I will describe a new method for constructing conformal blocks for the Virasoro vertex algebra with central charge c=1 by "nonabelianization", relating them to conformal blocks for the Heisenberg algebra on a branched double cover. The construction is joint work with Qianyu Hao. Special cases give rise to formulas for tau-functions and solutions of integrable systems of PDE, such as Painleve I and its higher analogues. The talk will be reasonably self-contained (in particular I will explain what a conformal block is).

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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Title: Tau-functions, invariants and resurgence: A survey

Abstract: The goal of my talk will be to offer a survey of some older work with I. Coman, P. Longi and and E.Pomoni, proposing a geometric characterization of the topological string partition functions, together with some subsequent developments on the resurgence of the formal series defining such partition functions, in particular the work of Iwaki and Marino. The resulting proposal characterizes the dual partition functions as a canonical section of a holomorphic line bundle on the underlying moduli spaces, such that the transition functions defining this line bundle get identified with the Stokes jumps appearing in this context. The Stokes jumps take an almost universal form related to cluster varieties, fully characterized by numerical data given by certain enumerative invariants. My goal will be to indicate how the different approaches to this proposal appear to complement each other beautifully.