・ 9:30 ー 10:30 Gaetan Borot 1
Title: Introduction to topological recursion
Abstract: I will introduce the basic lexicon and grammar of topological recursion: spectral curves, fundamental bidifferentials, correlators and residue recursion formula, free energies, wave functions, quantum curves, form-cycle duality.
・10:45 ー 11:45 Kohei Iwaki
Title: Conjectures on resurgent structure in topological recursion
Abstract: To a given divergent series, Écalle's theory of resurgent analysis provides a method to investigate the Stokes phenomenon (exponentially small terms behind the power series asymptotics) through the analysis of the singularity of the Borel transform. In this talk, I will briefly review concepts of resurgence theory, and further discuss the relationship between this framework and the topological recursion and BPS invariants. In particular, we will introduce several examples where the resurgence property of the partition function has been mathematically rigorously established (related to hypergeometric functions), and, time permitting, explain conjectural formulas for general spectral curves. This talk is based on joint works with T. Koike, Y.-M. Takei, O. Kidwai and M. Mariño.
・13:15 ー 14:15 Elba Garcia-Failde
Title: Resurgent large genus asymptotics of intersection numbers
Abstract: I will present a new approach to the computation of the large genus asymptotics of intersection numbers, in particular of Witten—Kontsevich, Theta and r-spin intersection numbers. Our technique is based on a resurgent analysis of the generating series of such intersection numbers, and relies on the presence of a quantum curve and the determinant formulae. With this approach, we are able to extend the recent results of Aggarwal with the computation of subleading corrections, solving a conjecture of Guo–Yang and initiating the development of a universal technique to explore large order asymptotics of enumerative problems that appear in the context of topological recursion. Based on joint work with B. Eynard, A. Giacchetto, P. Gregori and D. Lewański.
・14:45 ー 15:45 Alessandro Giacchetto
Title: Topological recursion is Gevrey-2
Abstract: In this talk, I will show that the topological recursion correlators associated with any regular spectral curve with simple ramifications grow at most like (2g)! as the genus g tends to infinity. The proof involves two key steps: (1) showing that topological recursion correlators are bounded by the so-called Painlevé correlators, and (2) bounding these correlators using estimates established by Aggarwal. This result provides a large genus upper bound for many curve counting problems and serves as a preliminary step for a resurgence analysis. This work is based on a joint project with G. Borot and B. Eynard.
・16:00 ー 17:00 Paul Norbury
Title: Weil-Petersson volumes, stability conditions and wall-crossing
Abstract: In this talk, I will describe the Weil-Petersson volumes of the moduli spaces of conical hyperbolic surfaces. The volumes are polynomials, in their cone angles, which generalise Mirzakhani's volume polynomials. The cone angles can be naturally identified with stability conditions defined by Hassett.
The space of stability conditions decomposes into chambers separated by walls and the volumes assign to each chamber a polynomial. The main chamber is assigned Mirzakhani's polynomial which can be related to the polynomial in any chamber via wall-crossing formulae giving a recursive relation.
This talk is based on joint work with Lukas Anagnostou and Scott Mullane.
・ 9:30 ー 10:30 Veronica Fantini
Title: Resurgence and modularity for fermionic spectral traces on local P^2
Abstract: In a joint project with C. Rella, we propose a new paradigm of modular resurgence that focuses on the role of the Stokes constants and the interplay of the their generating functions in form of q-series with the corresponding L-functions. Resurgent series with a modular resurgent structure are conjectured to have specific summability properties as well as to be closely related to quantum modular forms. Remarkably, a pivotal example arises from topological string theory, in the study of the resurgence structure of the first fermionic spectral traces of local P^2. In this talk, after discussing the general paradigm of modular resurgent series, I will focus on the example of local P^2. In particular, I will show that the generating functions of the Stokes constants (both in the weak and strong coupling regimes) are quantum modular forms. This talk is based on arxiv:2404.11550 and arxiv:2404.10695.
・10:45 ー 11:45 Pietro Longhi
Title: q-difference WKB Analysis and q-Painlevé Equations
Abstract: In this talk I will present an approach to the study of second order q-difference equations based on the WKB approximation, and comment on implications for broader correspondences with q-Painlevé equations and wall-crossing in five-dimensional supersymmetric Quantum Field Theory.
・13:15 ー 14:15 Sergey Shadrin
Title: Log TR (logarithmic topological recursion)
Abstract: I'll explain the set up of logarithmic topological recursion (it deals with meromorphic 1-forms dx and dy rather than functions x and y), and show how it interplays with x-y and symplectic duality and provides an efficient computational tool in various applications. It is joint work with Alexandrov, Bychkov, Dunin-Barkowski, and Kazarian.
・14:45 ー 15:45 Kento Osuga
Title: On a recent progress of refined topological recursion
Abstract: The framework of Airy structures due to Kontsevich and Soibelman has helped us understand a deep relation between topological recursion and W-algebras at the self-dual level. However, the story cannot be easily generalised once we consider β-deformation/refinement. In this talk, I will give an overview of what we know about refined topological recursion including previous attempts as well as recent progress. In particular, I will show an intriguing correspondence between deformation condition and quantisation condition in the refined setting. If time permits, I will also discuss applications in enumerative geometry/combinatorics. I will This talk is partly based on joint work with O. Kidwai, and also partly joint with N. Chidambaram and M. Dołęga.
・16:00 ー 17:00 Tom Bridgeland (online)
Title: From Donaldson-Thomas invariants to Painleve equations
Abstract:
・ 9:30 ー 10:30 Nitin Chidambaram
Title: b-Hurwitz numbers from W-algebra representations
Abstract: Hurwitz theory is a classical field of study that asks for a count of branched coverings of a sphere with a prescribed ramification profile. A program, which started almost 15 years ago, to prove that topological recursion governs the structural properties of various types of Hurwitz numbers was recently completed by Bychkov—Dunin-Barkowski—Kazarian—Shadrin.
Chapuy and Dołęga recently introduced a one-parameter deformation of usual Hurwitz theory known as b-Hurwitz theory, that incorporates the counting of certain types of non-orientable branched covers. In this talk, I’ll explain how the generating function of rationally weighted b-Hurwitz numbers can be obtained from a certain Whittaker-type state in a W-algebra representation. Upon setting b=0, b-Hurwitz theory reduces to usual Hurwitz theory, and the connection between TR and Hurwitz numbers can be reproved using this result (see Gaetan’s talk).
This talk is based on joint works with Gaetan Borot, Maciej Dołęga, Kento Osuga and Giacomo Umer.
・10:45 ー 11:45 Bertrand Eynard 1
Title: Differential systems and non-perturbative topological recursion
Abstract:
・ 9:30 ー 10:30 Gaetan Borot 2
Title: Whittaker vectors and topological recursion
Abstract: The AGT correspondence (mathematically established by Maulik-Okounkov in type A and Braverman-Finkelberg-Nakajima in simply laced type) roughly says that a suitable cohomology of instanton moduli spaces (including all degrees) for a Lie group G is a highest-weight module for a W-algebra and identifies the fundamental class with a Whittaker vector. From there the Nekrasov partition function can be accessed by taking the squared-norm of the Whittaker vector. I will show how to use the formalism of Airy structure and analytic continuation to reconstruct this Whittaker vector, first formally in the degree counting parameter, then analytically, using topological recursion. In the "unrefined case" (\epsilon_1 + \epsilon_2 = 0), I will explain the consequences that can be drawn from the result using the existing theory of the topological recursion. The method also apply to the Whittaker vector constructed by Chidambaram-Dolega-Osuga in the context of b-Hurwitz numbers (see Chidambaram's talk). This is based on joint works with Vincent Bouchard, Nitin Chidambaram, Thomas Creutzig, Dmiry Noshchenko, Giacomo Umer.
・10:45 ー 11:45 Taro Kimura
Title: Wall-crossing, R-matrix, and vertex operators
Abstract: We revisit the monodromy/connection problem associated with the (q-deformation of) conformal block, initiated by Kohno, Tsuchiya-Kanie, and Drinfeld, with emphasis on its relation to the associated wall-crossing and the R-matrix structure. We show that the radial ordering of vertex operators plays an essential role in this context, and then demonstrate how it is related to various concepts, e.g., stable envelope, quantum geometric Langlands correspondence, and equivariant DT/PT invariants of Calabi-Yau 3- and 4-fold.
・13:15 ー 14:15 Hiroyuki Fuji
Title: Dynamical triangulations of Riemann surfaces and topological recursions
Abstract: In this talk, we will study non-critical string field theories in terms of dynamical triangulations developed in 1990’s. The dynamical triangulation is known as a physical model of the quantum gravity on the discretized manifold. In works by Ishibashi-Kawai and Ambjørn-Watabiki, a quantum Hamiltonian formulation of the non-critical string field theories is proposed for the two dimensional quantum gravities coupled with minimal matter field based on the continuum limit of the dynamical triangulation of Riemann surfaces. Motivated by recent developments of the quantum studies of the Jackiw-Teitelboim gravity, we discuss the dynamical triangulation for the two dimensional quantum gravities coupled with (2,2m-1) conformal matter field further, and find the topological recursions from the Schwinger-Dyson equation of the non-critical string field theory. This talk is based on a work in progress with Masahide Manabe and Yoshiyuki Watabiki.
・14:45 ー 15:45 Paolo Gregori
Title: Topological Recursion Instantons from Gluing Cylinders
Abstract: In Mathematical Physics, spectral curves with nodal points appear in various contexts, including matrix integrals, Painlevé equations, Weil-Petersson volumes, and many others. Computing non-perturbative contributions (instantons) to partition functions and free energies in such contexts is often a challenging problem which can nevertheless be tackled in the framework of Topological Recursion. In this talk I will show how, using a "gluing prescription" which allows to carefully take into account the presence of nodal points, one can make instanton calculus rigorous and systematic, while also proving the equivalence of two a priori distinct approaches which were proposed in the past in the context of matrix integrals.
・16:00 ー 17:00 Akishi Ikeda
Title: BCOV theory for LG model and topological recursion
Abstract: In this talk, we consider a BCOV theoretic description of LG model with one variable following the work of Costello-Li and Li-Li-Saito.
We start with the same initial data as used in Eynard-Orantin TR and construct correlation functions of BCOV theory.
Finally, we give a few explicit forms of lower degree correlation functions and compare them with w_{g, n} of TR.
This is a joint work with K.Osuga.
・ 9:30 ー 10:30 Olivia Dumitrescu
Title: Lagrangian geometries of the Dolbeault and the de Rham moduli spaces
Abstract: We describe the variation of Hodge structure map on the Deligne moduli space of lambda connections and compare the lagrangian geometries of the two moduli spaces. This comparison leads to the understanding of the geometry of the Deligne moduli space of lambda connections.
・10:45 ー 11:45 Vincent Bouchard
Title: Highest weight vectors and quantum curves
Abstract: Topological recursion can be used to reconstruct the wave-function for a particular quantisation of the spectral curve. What is perhaps a little bit mysterious in this process is that topological recursion selects a particular quantisation of the spectral curve; that is, a particular choice of ordering when going from classical to quantum (and it is often a non-trivial ordering, not the normal ordering). A natural question arises: can topological recursion be used to reconstruct the wave-function for quantum curves corresponding to other choices of ordering? This can sometimes be achieved by varying the integration divisor in the reconstruction process, but this provides only a partial answer. In this work we study this question for the spectral curve building blocks, namely the (r,s) spectral curves (the natural higher degree generalisations of the Airy and Bessel curves). In particular, for the (r,1) spectral curves, we show that we can obtain very large families of quantisations by adding new initial conditions for a few correlators in topological recursion, which corresponds to varying a few of the loop equations. From the point of view of Airy structures, this simply amounts to considering partition functions that are general highest weight vectors for the corresponding W-algebra representations.
This is work in progress with Raphaël Belliard, Reinier Kramer and Tanner Nelson.
・12:00 ー 13:00 Bertrand Eynard 2
Title: Perspectives in topological recursion
Abstract: