Quantisation of moduli spaces from different perspectives

24 Sept 2023, 19:00 → 29 Sept 2023, 13:00 Europe/Zurich

SRS

Anton Alexeev (Universite de Geneve (CH)), Nezhla Aghaei (unige), Nicolas Orantin

The quantisation of moduli spaces is of fundamental interest for applications both in mathematics and physics. Such a quantisation has applications in diverse fields such as Chern-Simons theory and knot theory, conformal field theories, integrability (Painlev´e theory,…), gauge theory, the geometric Langlands correspondence and furthermore, it can be obtained from different perspectives and methods. The diversity of methods used as well as the different motivations for quantising such moduli spaces makes it an extraordinary ground for developing interactions between different specialities of mathematics and physics. The most recent works on these quantisation procedures seem to indicate that it is now possible to try to unify and compare these various perspectives. This workshop aims at bringing together experts on different ways of quantising these moduli spaces:

• Experts on the quantisation from a conformal field theory and separation of variables point of view;

• Experts on the quantisation using the co-adjoint method, the Poisson structure and the corresponding integrable systems;

• Experts from the geometric quantisation perspective as well as the topological recursion method;

• Experts on the resurgence properties of some of the wave functions obtained in by such a quantisation procedure.

This worksop will not only be the opportunity for these different communities to share their results but also to develop and strengthen a common language as well as understand the motivation of each other for studying such moduli spaces both from a physics and a mathematics point of view. This will allow the researchers to collaborate more easily and hopefully develop a powerful dictionary between their methods and approaches.

Zoom link will be available.

 

Sunday 24 September

19:30 Dinner

Monday 25 September

Lecture: Teschner 1: Quantum analytic Langlands correspondence

The analytic Langlands correspondence can be regarded as a variant of the geometric Langlands correspondence imposing additional conditions of analytic nature.

It predicts a correspondence between opers with real holonomy and eigenfunctions of the quantised Hitchin system. We will discuss a one-parameter deformation of this correspondence called quantum analytic Langlands correspondence. This deformation has natural relations to the quantisation of moduli spaces, the separation of variables, and conformal field theory. A key role is played by the Verlinde line operators. These operators represent a quantum deformation of the grafting operation creating eigenstates of the quantum Hitchin system from a cyclic vector. This is joint work with D. Gaiotto.

10:00 Coffee

Lecture: Teschner 2: Quantum analytic Langlands correspondence

The analytic Langlands correspondence can be regarded as a variant of the geometric Langlands correspondence imposing additional conditions of analytic nature.

It predicts a correspondence between opers with real holonomy and eigenfunctions of the quantised Hitchin system. We will discuss a one-parameter deformation of this correspondence called quantum analytic Langlands correspondence. This deformation has natural relations to the quantisation of moduli spaces, the separation of variables, and conformal field theory. A key role is played by the Verlinde line operators. These operators represent a quantum deformation of the grafting operation creating eigenstates of the quantum Hitchin system from a cyclic vector. This is joint work with D. Gaiotto.

10:30 coffee

12:00 Lunch

16:00 Coffee

Lecture: Schiappa 1: On the Uses and Applications of Resurgence to String Theory

I will briefly review how resurgence is of interest in addressing a number of non-perturbative string theoretic problems, including short introductions to the subject associated to these different applications. I will then focus on alien calculus associated to resonant problems, their genericness in string theory, and how they shed light on a “diagonal take” on non-perturbative large N duality.

Lecture: Schiappa 2: On the Uses and Applications of Resurgence to String Theory

I will briefly review how resurgence is of interest in addressing a number of non-perturbative string theoretic problems, including short introductions to the subject associated to these different applications. I will then focus on alien calculus associated to resonant problems, their genericness in string theory, and how they shed light on a “diagonal take” on non-perturbative large N duality.

19:30 Dinner

Tuesday 26 September

Lecture: Eynard 1 : Quantum curves from topological recursion

Starting from a spectral curve $S$ (a plane curve with some structure), Topological Recursion defines a sequence of differential forms, called $\omega_{g,n}(S)$. From this sequence, one can define a formal series $\ln\psi(\hbar,S,x) = \sum_{g,n} \frac{1}{n!}\hbar^{2g-2+n} \int_\infty^x \dots \int_\infty^x \omega_{g,n}(S)$, called the "wave function" (the TR wave function).
An important claim is that this wave function is annihilated by a differential operator with rational coefficients (and formal series of $\hbar$). In fact to make the claim complete, one has to extend it to transseries. The differential operator is called the "quantum curve" or the "quantization of the spectral curve".

10:00 Coffee

Lecture: Eynard 2 : Quantum curves from topological recursion

Starting from a spectral curve $S$ (a plane curve with some structure), Topological Recursion defines a sequence of differential forms, called $\omega_{g,n}(S)$. From this sequence, one can define a formal series $\ln\psi(\hbar,S,x) = \sum_{g,n} \frac{1}{n!}\hbar^{2g-2+n} \int_\infty^x \dots \int_\infty^x \omega_{g,n}(S)$, called the "wave function" (the TR wave function).
An important claim is that this wave function is annihilated by a differential operator with rational coefficients (and formal series of $\hbar$). In fact to make the claim complete, one has to extend it to transseries. The differential operator is called the "quantum curve" or the "quantization of the spectral curve".

12:00 Lunch

16:00 Coffee

Lecture: Alekseev 1: Combinatorial quantization and BV structures of moduli of flat connections

In these two lectures, we will start by recalling the definition and properies of Goldman brackets and of the Fock-Rosly bivector on the representation space of a fundamental group of an oriented surface. We will then sketch a combinatorioal quantization scheme for modulis paces inspired by the Faddeev-Reshetikhin-Takhtajan (FRT) presentation of quantum groups, and we will point out some links to other quantization schemes.

If time permits, we will also discuss the notion of Turaev cobracket and its relation to Batalin-Vilkovisky (BV) structures on moduli spaces.

These lectures are based on joint works with H. Grosse, F. Naef, J. Pulmann, V. Schomerus and P. Severa.

Lecture: Alekseev 2: Combinatorial quantization and BV structures of moduli of flat connections

In these two lectures, we will start by recalling the definition and properies of Goldman brackets and of the Fock-Rosly bivector on the representation space of a fundamental group of an oriented surface. We will then sketch a combinatorioal quantization scheme for modulis paces inspired by the Faddeev-Reshetikhin-Takhtajan (FRT) presentation of quantum groups, and we will point out some links to other quantization schemes.

If time permits, we will also discuss the notion of Turaev cobracket and its relation to Batalin-Vilkovisky (BV) structures on moduli spaces.

These lectures are based on joint works with H. Grosse, F. Naef, J. Pulmann, V. Schomerus and P. Severa.

19:30 Dinner

Wednesday 27 September

Lecture: Andersen 1

10:00 Coffee

Lecture: Andersen 2

12:00 Lunch

16:00 Coffee

Bridgeland :From the A2 quiver to the Painleve I tau function

Iwaki: Topological recursion, resurgence and BPS structure

I’ll discuss resurgence structures of topological recursion partition function.
I’ll show exactly solvable examples, related to the classical limit of the hypergeometric ODEs,
and describe a relation to BPS structures associated with the Stokes graphs of the ODEs.
I will also consider a family of genus 1 spectral curves which is related to Painlev¥’e I,
and propose conjectures on the resurgence properties and relation to BPS structure.
This talk is based on joint works with O. Kidwai, T. Koike, M. Marino and YM. Takei.

19:30 Dinner

Thursday 28 September

Meinrenken: Symplectic geometry of the Teichmueller space of hyperbolic 0-metrics

A hyperbolic 0-metric on a compact surface with boundary is a
hyperbolic metric on the interior, with a boundary behaviour similar to
that of the Poincare metric on the upper half plane. We show that the
infinite-dimensional Teichmueller space of such metrics has a natural
symplectic structure, and is an example of a Hamiltonian Virasoro space.
Based on joint work with Anton Alekseev.

10:00 Coffee

Rembado :The Knizhnik--Zamolodchikov connection (KZ) from quantum isomonodromic deformations

The KZ equations first arose in 2d conformal field theory, as constraints satisfied by correlation functions in the Wess--Zumino--Novikov--Witten model. The standard mathematical construction of the linear PDEs involves spaces of (co)vacua associated to highest-weight modules for affine Lie algebras, and a different derivation was later given by Reshetikhin and Harnad via quantisation of the Schlesinger Hamiltonians; in turn, the latter control isomonodromic deformations of Fuchsian systems on the Riemann sphere.

In this talk we will aim at a review of part of this story. If time allows we will also present a generalisation involving moduli spaces of (nongeneric) irregular singular connections, as well as `generalised' highest-weight modules for affine Lie algebras.
(This extension is joint work with P. Boalch, J. Douçot, G. Felder, M. Tamiozzo and R. Wentworth.)

12:00 Lunch

16:00 Coffee

Lisovyy: Heun connection matrix from Liouville conformal blocks and Darboux theorem

Conformal blocks of the Liouville CFT are known to have very
simple analytic structure with respect to the positions of degenerate
fields. The corresponding monodromy is « quantum » (operator-valued) as
it involves shifts of internal momenta. In the quasiclassical limit, the
BPZ equation satisfied by the simplest nontrivial example of such
conformal block reduces to Heun equation. I will explain how careful
analysis of the limit allows to solve the Heun connection problem in
terms of quasiclassical Virasoro conformal blocks, generalizing a
conjectural relation between quasiclassical Liouville CFT and Heun
accessory parameter function found by Zamolodchikov in 1986. I will then
discuss how this conjecture can be checked with the help of the
classical Darboux theorem relating the Heun connection matrix to the
large-order asymptotics of the coefficients of the corresponding
Frobenius solutions.

Alim: Resurgence, BPS structures and topological string S-duality

19:30 DInner

Friday 29 September

Open Questions

12:00 Lunch