Main Lecture 1: Introduction to Resurgence via Wall-crossing Structures (1/4)

• Maxim Kontsevich (IHES)

Room: Marilyn and James Simons Conference Center I'll introduce an alternative approach to the classical Borel-Écalle resummation method of factorially divergent series based on the notion of an analytic wall-crossing structure introduced by Yan Soibelman and myself in arXiv: 2005.10651. Instead of working in the Borel plane, one defines a holomorphic bundle over a small disc directly in the original coordinate, by the gluing of the trivialized bundle on finitely many overlapping sectors by gauge transformations which are convergent series in exponentially small terms. The global geometric object is a bundle over a neighborhood of a wheel of 1-dimensional torus orbits in a higher-dimensional toric variety. I'll illustrate the general theory by several examples, including exponential integrals, a generalization to closed 1-forms, including Stirling formula, as well as the quantum dilogarithm.

Main Lecture 2: Introduction to Resurgence via Wall-crossing Structures (2/4)

• Maxim Kontsevich (IHES)

Room: Marilyn and James Simons Conference Center I'll introduce an alternative approach to the classical Borel-Écalle resummation method of factorially divergent series based on the notion of an analytic wall-crossing structure introduced by Yan Soibelman and myself in arXiv: 2005.10651. Instead of working in the Borel plane, one defines a holomorphic bundle over a small disc directly in the original coordinate, by the gluing of the trivialized bundle on finitely many overlapping sectors by gauge transformations which are convergent series in exponentially small terms. The global geometric object is a bundle over a neighborhood of a wheel of 1-dimensional torus orbits in a higher-dimensional toric variety. I'll illustrate the general theory by several examples, including exponential integrals, a generalization to closed 1-forms, including Stirling formula, as well as the quantum dilogarithm.

Research talk: First Steps in Global Lie Theory: wild Riemann surfaces, their character varieties and topological symplectic structures

• Philip Boalch (IMJ-PRG)

Room: Marilyn and James Simons Conference Center I'll describe some of the story leading up to the construction of the topological symplectic structures (P.B. Oxford thesis 1999, Adv. Math. 2001) and subsequent evolution leading to the general, purely algebraic approach (B. 2002, 2009, 2014, B.-Yamakawa 2015). They generalise the holomorphic version of the symplectic structures of Narasimhan, Atiyah-Bott, Goldman involving the topological fundamental group. Our approach gives a TQFT approach to moduli of meromorphic connections on curves, involving Lie group valued moment maps. The right point of view seems to be to generalise the notion of Riemann surface to the notion of wild Riemann surface, in the spirit of Weil's 1958 Bourbaki talk, and view these symplectic varieties as their character varieties (in the spirit of Weil's 1948 text "Sur les courbes algébriques et les variétés qui s'en déduisent"). The simplest irregular example (involving the wild fundamental group) underlies the Drinfeld-Jimbo quantum group (and deformations of the underlying wild Riemann surface explain the natural G-braid group action of Lusztig, Kirillov-Reshetikhin and Soibelman). Classification of these varieties, as "global analogues of Lie groups", is still at a quite elementary stage, but a rich theory of Dynkin diagrams exists for many examples. If time permits I'll describe how these two-forms fit together with the Bottacin-Markman Poisson structure on the meromorphic Higgs bundle moduli spaces to give the wild nonabelian Hodge hyperkahler manifolds (Biquard-B. 2004). This gives in particular a rich bestiary of new special Lagrangian fibrations. Surprisingly these hyperkahler metrics are often complete even though the corresponding harmonic maps have infinite energy. The simplest examples, certain hyperkahler four-manifolds, are the "spaces of initial conditions" of the Painlevé equations. Painlevé knew his equations were deformations of equations for elliptic functions, and so we can now see this "Painlevé simplification" as a hyperkahler rotation, from meromorphic connections to meromorphic Higgs bundles. Not only does this story encompass many famous classical integrable systems like the Lagrange top (2 poles of order 2), and those studied by Mumford (in Tata lectures on Theta II), but several of these Painlevé integrable systems were used in Seiberg-Witten's 1994 solution of 4d N=2 super Yang-Mills theory for SU(2), and one of the higher rank generalizations, introduced by Garnier in 1919 (the simplified Schlesinger system), underlies the famous Gaudin model. It was solved by Garnier in terms of abelian functions by defining spectral curves, a method rediscovered in the soliton literature in the 1970s (see e.g. Adler-Van Moerbeke 1980, Linearization of Hamiltonian systems, Jacobi varieties and representation theory, p.337, or Verdier's 1980 Séminaire Bourbaki), before being generalised by Hitchin to the case where the base curve has genus >1 and then connected to the harmonic theory.

Research Talk (online): Wall-crossing and Resurgence in Supersymmetric QFT

• Lotte Hollands (Heriot Watt University)

Room: Marilyn and James Simons Conference Center In this lecture I'll try to explain how the topics of wall-crossing and resurgence appear in physics.

Main Lecture 3: Holomorphic Floer Theory and Resurgence (1/3)

• Yan Soibelman (Kansas State Univ. & IHES)

Room: Marilyn and James Simons Conference Center The notion of resurgence ,i.e. the property of a divergent series to have endless analytic continuation of its Borel transform, is due to Jean Écalle. Recently it became clear that the resurgence can be approached via an a priori different notion of analytic wall-crossing structure. The latter concept was defined and studied by Maxim Kontsevich and myself with the original motivation coming from Donaldson-Thomas theory and theory of complex integrable systems (see arXiv: 0811.2435, 1303.3253, 2005.10651). In my lecture course I plan to discuss a source of analytic wall-crossing structures coming from symplectic topology. More precisely it is the Floer theory in the framework of complex symplectic manifolds. When we started to work on it in 2014 we called it ``Holomorphic Floer Theory". Central part of the Holomorphic Floer Theory is played by the generalized Riemann-Hilbert correspondence which relates Fukaya categories (the latter notion is central in the Floer theory of real symplectc manifolds) with the categories of holonomic deformation-quantization modules (the latter is a generalization of the notion of holonomic D-module). I am going to explain in what way our conjectural Riemann-Hilbert correspondence is related to resurgence of perturbative expansions arising in mathematics and mathematical physics. Examples include exponential integrals in finite and infinite dimensions (e.g. partition function of the complexified Chern-Simons theory) and WKB expansions of wave functions associated with quantum spectral curves. In these examples we deal with the simplest non-trivial case of Holomorphic Floer theory related to a pair of complex Lagrangian submanifolds of a complex symplectc manifold.

Main Lecture 4: Holomorphic Floer Theory and Resurgence (2/3)

• Yan Soibelman (Kansas State Univ. & IHES)

Room: Marilyn and James Simons Conference Center The notion of resurgence ,i.e. the property of a divergent series to have endless analytic continuation of its Borel transform, is due to Jean Écalle. Recently it became clear that the resurgence can be approached via an a priori different notion of analytic wall-crossing structure. The latter concept was defined and studied by Maxim Kontsevich and myself with the original motivation coming from Donaldson-Thomas theory and theory of complex integrable systems (see arXiv: 0811.2435, 1303.3253, 2005.10651). In my lecture course I plan to discuss a source of analytic wall-crossing structures coming from symplectic topology. More precisely it is the Floer theory in the framework of complex symplectic manifolds. When we started to work on it in 2014 we called it ``Holomorphic Floer Theory". Central part of the Holomorphic Floer Theory is played by the generalized Riemann-Hilbert correspondence which relates Fukaya categories (the latter notion is central in the Floer theory of real symplectc manifolds) with the categories of holonomic deformation-quantization modules (the latter is a generalization of the notion of holonomic D-module). I am going to explain in what way our conjectural Riemann-Hilbert correspondence is related to resurgence of perturbative expansions arising in mathematics and mathematical physics. Examples include exponential integrals in finite and infinite dimensions (e.g. partition function of the complexified Chern-Simons theory) and WKB expansions of wave functions associated with quantum spectral curves. In these examples we deal with the simplest non-trivial case of Holomorphic Floer theory related to a pair of complex Lagrangian submanifolds of a complex symplectc manifold.

Research Talk: Topological Recursion, Painlevé Equation, Exact WKB and Resurgence

• Kohei Iwaki (The University of Tokyo)

Room: Marilyn and James Simons Conference Center In the first half of the talk, I’ll give a review of the topological recursion and how it is related to WKB, Painlevé equations and BPS structures (I 2019, Eynard--Garcia-Failde-Marchal-Orantin 2019, 2021, I—Kidwai 2020, 2021). In the latter half, I’ll discuss a resurgence property of the topological recursion partition function based on several conjectures which are expected to hold from the view-point of the exact WKB analysis. In particular, I’ll explain how the cluster transformation is related to the non-linear Stokes phenomena for the Painlevé transcendents.

Research Talk: The resurgent structure of topological strings

• Marcos Mariño (University of Geneva)

Room: Marilyn and James Simons Conference Center Topological strings are described in perturbation theory by factorially divergent series, and one can ask what is their resurgent structure, namely the location of their Borel singularities and the corresponding values of alien derivatives. It turns out that an important part of this structure can be obtained by considering trans-series solutions of the holomorphic anomaly equations of BCOV. I describe exact formulae for the relevant trans-series and I propose a general conjecture for the resurgent structure of the topological string, which can be tested in compact and non-compact Calabi-Yau manifolds and also in matrix models. I also compare the resulting structure with similar results on quantum periods. In particular, I point out that Stokes automorphisms in topological string theory involve a more complicated structure than the one captured by the Delabaere-Pham formula.

Introduction to Resurgence via Wall-crossing Structures (3/4)

• Maxim Kontsevich (IHES)

Room: Marilyn and James Simons Conference Center I'll introduce an alternative approach to the classical Borel-Écalle resummation method of factorially divergent series based on the notion of an analytic wall-crossing structure introduced by Yan Soibelman and myself in arXiv: 2005.10651. Instead of working in the Borel plane, one defines a holomorphic bundle over a small disc directly in the original coordinate, by the gluing of the trivialized bundle on finitely many overlapping sectors by gauge transformations which are convergent series in exponentially small terms. The global geometric object is a bundle over a neighborhood of a wheel of 1-dimensional torus orbits in a higher-dimensional toric variety. I'll illustrate the general theory by several examples, including exponential integrals, a generalization to closed 1-forms, including Stirling formula, as well as the quantum dilogarithm.

Main Lecture 6: Holomorphic Floer Theory and Resurgence (3/3)

• Yan Soibelman (Kansas State Univ. & IHES)

Room: Marilyn and James Simons Conference Center The notion of resurgence ,i.e. the property of a divergent series to have endless analytic continuation of its Borel transform, is due to Jean Écalle. Recently it became clear that the resurgence can be approached via an a priori different notion of analytic wall-crossing structure. The latter concept was defined and studied by Maxim Kontsevich and myself with the original motivation coming from Donaldson-Thomas theory and theory of complex integrable systems (see arXiv: 0811.2435, 1303.3253, 2005.10651). In my lecture course I plan to discuss a source of analytic wall-crossing structures coming from symplectic topology. More precisely it is the Floer theory in the framework of complex symplectic manifolds. When we started to work on it in 2014 we called it ``Holomorphic Floer Theory". Central part of the Holomorphic Floer Theory is played by the generalized Riemann-Hilbert correspondence which relates Fukaya categories (the latter notion is central in the Floer theory of real symplectc manifolds) with the categories of holonomic deformation-quantization modules (the latter is a generalization of the notion of holonomic D-module). I am going to explain in what way our conjectural Riemann-Hilbert correspondence is related to resurgence of perturbative expansions arising in mathematics and mathematical physics. Examples include exponential integrals in finite and infinite dimensions (e.g. partition function of the complexified Chern-Simons theory) and WKB expansions of wave functions associated with quantum spectral curves. In these examples we deal with the simplest non-trivial case of Holomorphic Floer theory related to a pair of complex Lagrangian submanifolds of a complex symplectc manifold.

Main Lecture 7: Introduction to Resurgence via Wall-crossing Structures (4/4)

• Maxim Kontsevich (IHES)

Room: Marilyn and James Simons Conference Center I'll introduce an alternative approach to the classical Borel-Écalle resummation method of factorially divergent series based on the notion of an analytic wall-crossing structure introduced by Yan Soibelman and myself in arXiv: 2005.10651. Instead of working in the Borel plane, one defines a holomorphic bundle over a small disc directly in the original coordinate, by the gluing of the trivialized bundle on finitely many overlapping sectors by gauge transformations which are convergent series in exponentially small terms. The global geometric object is a bundle over a neighborhood of a wheel of 1-dimensional torus orbits in a higher-dimensional toric variety. I'll illustrate the general theory by several examples, including exponential integrals, a generalization to closed 1-forms, including Stirling formula, as well as the quantum dilogarithm.

Main Lecture 8: Quantum Chern-Simons Theory, both Real and Complex (1/3)

• Jørgen E. Andersen (SDU)

Room: Marilyn and James Simons Conference Center I shall review the construction of the WRT invariants and treat a few examples in detail. I shall then state the Resurgence conjecture for these invariants. Following this I shall recall our construction of the Teichmüller TQFT joint with Rinat Kashaev and further cover the new formulation of it. I shall recall a few conjectures concerning these TQFT and end with a discussion of explicit examples.

Research Talk: Floer Theory and Exact Results for Non-perturbative Complex Chern-Simons Theory

• Sergei Gukov (Caltech)

Room: Marilyn and James Simons Conference Center Heegaard branes are particular holomorphic Lagrangians in moduli spaces of Higgs bundles. During the past 16 years they played a useful role in a variety of problems in pure mathematics, ranging from the geometric Langlands program to low-dimensional topology. In particular, they played an important role in non-perturbative formulation of complex Chern-Simons theory via quantum groups at generic q and in formulating invariants of 4-manifolds via trisections. Their Floer theory, on the one hand, is related to monodromies (Stokes coefficients) in complex Chern-Simons theory on 3-manifolds and, on the other hand, to the curve count in Calabi-Yau 3-folds. Making these relations explicit and mathematically precise involves a number of interesting details: precise definitions of the moduli spaces (and their compactification) in gauge theory and in curve counting, a similar choice of the mathematical definition for Hom's in the corresponding Fukaya-Seidel category, and the role of Spin-C structures for quantum group invariants at generic q.

Research Talk: Resurgence, Quantum Modularity and Quantum Invariants

• William Mistegård (SDU)

Room: Marilyn and James Simons Conference Center The Witten-Reshetikhin-Turaev (WRT) quantum invariant of three-manifolds is the mathematical realization of the Chern-Simons partition function with compact gauge group. Inspired by the use of resurgence in physics, this invariant has been studied from the viewpoint of resurgence in recent years, and this study has led lead to many important discoveries. The use of resurgence in quantum Chern-Simons field theory was pioneered by Witten and Garoufalidis. This research greatly illuminated the interplay between Chern-Simons theory with compact gauge group and with complexified gauge group and how this interplay manifests itself in resurgence properties of the divergent series arising from pertubation theory. Furthermore, work of Gukov, Putrov and Mariño showed how resurgence connects the WRT quantum invariant to the BPS q-series invariant of Gukov, Pei, Putrov and Vafa (GPPV). The BPS q-series invariant is conjectured to categorify the WRT quantum invariant through the radial limit conjecture of GPPV, and to be a quantum modular form. In this talk, we consider the case where the three-manifold is a Seifert fibered homology sphere. We explain how the WRT quantum invariant determines the complex Chern-Simons invariants through resurgence, and we explain how a resurgence formula for the BPS q-series leads to a proof of the radial limit conjecture as well as a proof of quantum modularity. The former is based on joint work with Andersen, and the proof of quantum modularity is based on joint work in progress with Andersen, Han, Li, Sauzin and Sun.

Research Talk: On resurgent Poisson structures and deformations

• David Sauzin (Observatoire de Paris Meudon)

Room: Marilyn and James Simons Conference Center I will review the definition of the algebra A of simple Z-resurgent series and its alien derivations $\Delta_m$, as given by Jean Ecalle in 1981. In particular, I will recall why one can say that the alien derivations are independent in a strong sense. Then I will explore one consequence of the freeness of the Lie algebra generated by the $\Delta_m$'s under commutators and multiplication by elements of A: since we have so many derivations (although we are dealing with a formal series of *one* variable), one can construct non-trivial Poisson structures on A and, correspondingly, non-commutative deformations of the product of A.

Main Lecture 9: Quantum Chern-Simons Theory, both Real and Complex (2/3)

• Jørgen E. Andersen (SDU)

Room: Marilyn and James Simons Conference Center I shall review the construction of the WRT invariants and treat a few examples in detail. I shall then state the Resurgence conjecture for these invariants. Following this I shall recall our construction of the Teichmüller TQFT joint with Rinat Kashaev and further cover the new formulation of it. I shall recall a few conjectures concerning these TQFT and end with a discussion of explicit examples.

Main Lecture 10: Quantum Chern-Simons Theory, both Real and Complex (3/3)

• Jørgen E. Andersen (SDU)

Room: Marilyn and James Simons Conference Center I shall review the construction of the WRT invariants and treat a few examples in detail. I shall then state the Resurgence conjecture for these invariants. Following this I shall recall our construction of the Teichmüller TQFT joint with Rinat Kashaev and further cover the new formulation of it. I shall recall a few conjectures concerning these TQFT and end with a discussion of explicit examples.

Research Talk: DT Invariants and Holomorphic Curves

• Pierrick Bousseau (University of Georgia)

Room: Marilyn and James Simons Conference Center Kontsevich and Soibelman suggested a correspondence between Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds and holomorphic curves in complex integrable systems. After reviewing this general expectation, I will present a concrete example related to mirror symmetry for the local projective plane (partly joint work with Descombes, Le Floch, Pioline), along with applications in enumerative geometry (partly joint work with Fan, Guo, Wu). I will end by an “explanation” of the general correspondence based on the physics of N=2 4d quantum field theories and holomorphic Floer theory.

Research Talk: q-series, Resurgence and Modularity

• Veronica Fantini (IHES)

Room: Marilyn and James Simons Conference Center In Zagier's paper titled "quantum modular forms", one of the first examples of quantum modular form is related to the q-series $$ \sigma(q)=1+\sum_{n=0}^\infty (-1)^n q^{n+1} (q)_n $$ from Ramanujan's "Lost" Notebook. In this talk, I will discuss the resurgent structure of the formal power series associated with the q-series $\sigma(q)$: it is a simple resurgent structure which conjecturally encodes the modularity properties already studied by Zagier. Furthermore, the same resurgent structure appears when considering formal power series associated to other q-series, such as the Kontsevich--Zagier q-series for trefoil and the q-series coming from the fermionic spectral traces of quantum-mechanical operators related with the quantization of the mirror curve of toric CY 3-folds (recently studied by C. Rella arXiv:2212.10606). Hence we expect to find analougus modularity properties by studying their resurgent structures.

Research Talk (online): Resurgence and quantum modularity

• Campbell Wheeler (MPI Bonn)

Room: Marilyn and James Simons Conference Center Improved analytic properties of solutions to certain q-difference equations have given the ability to compute conjectural formulae for Stokes constants of associated asymptotic series. This was first explored for some invariants associated to simple knots by Garoufalidis-Gu-Mariño. I will explore examples from the perspective of quantum modular forms involving work of the previous authors and additionally work of Kashaev and Zagier.

Discussions Room: Marilyn and James Simons Conference Center