## Integrable systems, Frobenius manifolds and related topics

Institut de Mathématiques de Bourgogne, Dijon, 6 - 9 July 2022

The aim of this workshop is to bring together experts working on integrable systems, their interplay with Frobenius manifolds and their application to several other fields - including enumerative geometry, cohomological field theory, matrix models, quantum differential equations, topological recursion, vertex algebras, Painlevé equations, ... - to forster collaboration and trigger exchanges.

#### Invited Speakers

• M. Bertola (Montreal)
• X. Blot (Rehovot)
• G. Cotti (Lisbon)
• A. De Sole (Rome)
• M. Feigin (Glasgow)
• E. Ferapontov (Loughborough)
• E. Garcia-Failde (Paris)
• D. Guzzetti (Trieste)
• A. Hone (Kent)
• J. van de Leur (Utrecht)
• P. Lorenzoni (Milan)
• G. Ruzza (Louvain)
• R. Vitolo (Lecce)

#### Participants

• N. Babinet (Dijon)
• O. Brauer (Leeds)
• G. Feitosa de Almeida (Mannheim)
• L. F. Lopez Reyes (Trieste)
• V. Matveev (Dijon)
• G. Moroni (Milano)
• S. Perletti (Milano)
• M. Semenov-Tian-Chansky (Dijon)
• I. Singh (Dijon)
• J. Wright (Glasgow)

#### Schedule

All events will take place at the Salle René Baire on the 4th floor of the building Mirande on the campus, unless otherwise indicated.

The seminars will be streamed on Zoom. Please send an email to guido.carlet@u-bourgogne.fr for the zoom links.

Wed 6 Thu 7 Fri 8 Sat 9
9:00 A. De Sole : Integrability of classical affine $W$-algebras. G. Cotti : Borel-Laplace multi-transform, and integral representations of solutions of qDEs. M. Bertola : Orthogonality on Riemann surfaces, KP tau functions and vector bundles.
(Maison Internationale)
9:45 J. van de Leur : Classical Affine Lie algebras, Conjugacy Classes of the Weyl Group and Hierarchies of KdV Type. P. Lorenzoni : A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of B_n. R. Vitolo : Homogeneous Hamiltonian operators, projective geometry and integrable systems.
(Maison Internationale)
10:30 Break Break Break
(Maison Internationale)
11:00 A. Hone : Continued fractions and hyperelliptic curves. M. Feigin : WDVV and commutativity equations, and their trigonometric solutions. E. Ferapontov : Linearisable Abel equations and the Gurevich-Pitaevskii problem.
(Maison Internationale)
11:45 E. Garcia-Failde : rth roots: better negative than positive. G. Ruzza : Quantum Intermediate Long Wave hierarchy and quasimodular forms.
12:30 Lunch/Registration
(Maison Internationale)
Lunch
(Maison Internationale)
Lunch
(Maison Internationale)
Lunch
(Maison Internationale)
14:30 S. Shadrin : Special integrability for KP-integrable problems in enumerative geometry. L. F. Lopez Reyes : A Criterion for a Semisimple Caustic of a Frobenius Manifolds. X. Blot : The quantum Witten-Kontsevich series.
15:15 Break Break Break
16:00 D. Guzzetti : Some aspects of the theory on non-generic isomonodromy deformations. N. Babinet : Factorizability of the partition function of the supermatrix model.
16:45 P. Rossi : Moduli spaces of residueless meromorphic differentials and the KP hierarchy.
19:00 Social dinner
(Restaurant Gril'Laure)

#### Abstracts

N. Babinet : Factorizability of the partition function of the supermatrix model.

In this talk we will focus on the supermatrix model which can be seen as a specific case of two-matrix models. The former has been introduced long time ago and it has been noticed or suggested that its properties might not drastically differ from ordinary one-matrix model, at least in the planar limit. We want here to present how the partition function of the supermatrix model actually differs from the ordinary one and how its factorized form can lead to further questions in terms of non-planar graphs and perturbative corrections. To illustrate these problems we will make connection with the Fermi gas description of ABJ theory introduced by Marino and Putrov.

M. Bertola : Orthogonality on Riemann surfaces, KP tau functions and vector bundles.

I will introduce different notions of (bi)orthogonality for a pairing associated to a measure on a contour in a Riemann surface and show how they are naturally related to suitable Padé approximation problems thus generalizing the ordinary orthogonal polynomials.

These objects can be framed in the context of a Riemann—Hilbert problem on Riemann surfaces, i.e. a vector bundle of degree $2g$. This formulation is, in fact, of practical applications in at least three contexts:

1. application of steepest descent methods,
2. construction of matrix orthogonal polynomials,
3. constructions of KP/2 Toda tau functions that generalize Krichever’s construction.

X. Blot : The quantum Witten-Kontsevich series.

In 1990, Witten conjectured that the coefficients of the logarithm of a tau function of KdV, later called the Witten-Kontsevich series, are given by intersection numbers of psi classes. In 2016, Buryak, Dubrovin, Guéré and Rossi used the quantum double ramification hierarchies and their quantization to introduce quantum tau functions. In particular, the logarithm of one quantum tau function of quantum KdV is an extension of the Witten-Kontsevich series. We call it the quantum Witten-Kontsevich series. The goal of this talk is to present geometrical interpretations of the coefficients of this series. This include connections with Hurwitz numbers and a conjectural relation with the so-called Chiodo class.

G. Cotti : Borel-Laplace multi-transform, and integral representations of solutions of qDEs.

The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the monodromy of its solutions conjecturally rules also the topology and complex geometry of X. These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena. Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds, the latter being identifiable with the space of isomonodromic deformation parameters of the former. The study of qDE’s represents a challenging active area in both contemporary geometry and mathematical physics. In this talk I will introduce some analytic integral multitransforms of Borel-Laplace type, and I will use them to obtain Mellin-Barnes integral representations of solutions of qDEs.

A. De Sole : Integrability of classical affine $W$-algebras.

All classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in $G_2$, one in $F_4$,and five in $E_8$.

M. Feigin : WDVV and commutativity equations, and their trigonometric solutions.

In the theory of WDVV equations $F_i G^{-1} F_j = F_j G^{-1} F_i$ the constant matrix $G$ is usually assumed to be a linear combination of the matrices $F_i$ of the third order derivatives of the prepotential $F$. I would like to explain that this assumption follows from the equations under a non-degeneracy condition. Then, these equations admit a big class of rational solutions determined by configurations of vectors known as V-systems which include root systems. In the trigonometric settings the situation is much more restrictive, and known examples are given by non-simply-laced root systems and their projections. The talk is based on a joint work with M. Alkadhem.

E. Ferapontov : Linearisable Abel equations and the Gurevich-Pitaevskii problem.

Applying symmetry reduction to a class of $SL(2,R)$-invariant third-order ODEs, we obtain Abel equations whose general solution can be parametrised by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the asymptotic analysis of a simultaneous solution to the KdV equation and the stationary part of its higher non-autonomous symmetry. Based on joint work with Stanislav Opanasenko: S. Opanasenko, E.V. Ferapontov, On a class of linearisable Abel equations, (2022) arXiv:2202.07512.

E. Garcia-Failde : rth roots: better negative than positive.

I will introduce a collection of cohomology classes, which correspond to negative r-th roots of the canonical bundle and form a (generically non semisimple) cohomological field theory (CohFT) without a flat unit, which is the negative analogue of Witten’s r-spin CohFT, but turns out to be geometrically much simpler. The case r=2 was considered by Norbury in 2017: he conjectured that the descendant potential is a tau function of the KdV hierarchy. We consider certain one-parameter deformations of this class that form a semisimple CohFT when the parameter is non-zero, which allows us to make use of the Givental—Teleman reconstruction. We prove that the corresponding intersection numbers can be computed recursively using topological recursion and, equivalently, W-constraints. We also conjecture these W-constraints are equivalent to being the r-BGW tau function of the r-KdV hierarchy, and prove this for r=2 (which establishes Norbury’s conjecture) and r=3. Furthermore, we obtain relations in the tautological ring, which in the special case of r=2 reduce to relations involving only kappa classes that were recently conjectured by Norbury–Kazarian. Based on joint work with N. Chidambaram and A. Giacchetto: https://arxiv.org/pdf/2205.15621.pdf

D. Guzzetti : Some aspects of the theory on non-generic isomonodromy deformations.

I will explain some applications of the theory on non-generic isomonodromy deformations (Cotti, Dubrovin, Guzzetti). I will especially focus on the Laplace-transform version of the theory, which allows an adaptation useful to the the sixth Painleve' equation.

A. Hone : Continued fractions and hyperelliptic curves.

Following van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J -fractions and orthogonal polynomials, we show that in the simplest case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in a particular form by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus 2 satisfy a Somos-8 relation. Moreover, for all g we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system. The connection with the Toda lattice, and also the link to S-fractions via contraction, and a family of maps associated with the Volterra lattice, will briefly be mentioned. Some of this is joint work with John Roberts and Pol Vanhaecke.

J. van de Leur : Classical Affine Lie algebras, Conjugacy Classes of the Weyl Group and Hierarchies of KdV Type.

Kac and Peterson showed that the constructions of the basic representation of the affine Lie algebra of type A, D and E, are in one-to-one correspondence with the conjugacy classes of the corresponding finite Weyl group of the same type. This means that for instance for $E_8$, there are 112 different constructions. Each construction leads to a different hierarchy of differential equations. For instance, for the affine Lie algebra of type $A_1$, there are two hierarchies, viz. the KdV- and the AKNS- hierarchy. In this talk I will describe these constructions for the classical Lie algebras, i.e., the ones of type A, B, C and D, and will discuss the corresponding hierarchies of differential equations. This is based on ongoing work with Victor Kac.

L. F. Lopez Reyes : A Criterion for a Semisimple Caustic of a Frobenius Manifolds.

The tangent space of Frobenius Manifold carries a multiplication. The points where this multiplication is not semisimple is an hypersurface called the Caustic. The tangent space of the caustic is multiplication invariant and under some conditons the restricted multiplication is semisimple. On a semisimple "chamber" the stucture a Frobenius Manifold is determined by its monodromy data. We give a criterion in terms of the monodromy data for the existence of a semisimple caustic "facing" the chamber.

P. Lorenzoni : A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of B_n.

We show that the orbit space of $B_2$ less the image of coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schrodinger equations. Motivated by this example, we study the case of $B_n$ and we show that the defocusing case can be generalized to arbitrary n leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction relies on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin-Saito standard setting. Up to $n=4$ the solutions of WDVV equations we get coincide with those associated with constrained KP equations. The talk is based on a joint work with Alessandro Arsie, Igor Mencattini and Guglielmo Moroni.

P. Rossi : Moduli spaces of residueless meromorphic differentials and the KP hierarchy.

I'll present a recent joint work with A. Buryak and D. Zvonkine, where we study the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues. Our main result is that intersection theory on these spaces is controlled by an integrable system containing the celebrated Kadomtsev-Petviashvili (KP) hierarchy as a reduction to the case of differentials with exactly two zeros and any number of poles. This fact has several deep consequences and in particular it relates the aforementioned moduli spaces with Hurwitz theory, representation theory of sl2(C), integrability and a conjecture of Schmitt and Zvonkine on the r=0 limit of Witten's r-spin classes.

G. Ruzza : Quantum Intermediate Long Wave hierarchy and quasimodular forms.

Exploiting the geometry of Double Ramification cycles in the moduli spaces of curves, Buryak and Rossi provided an effective construction of quantum integrable hierarchies associated with any Cohomological Field Theory. In particular, the Hodge Cohomological Field Theory is associated with the quantum Intermediate Long Wave hierarchy. We show that, in this case, the family of commuting Hamiltonian operators have a quasimodular property: their traces, suitably weighted by a parameter q, are (Fourier expansions of) quasimodular forms of homogeneous weight. I will present the result, its motivations and proof, and related conjectures and open problems. Joint work with J.W. van Ittersum.

S. Shadrin : Special integrability for KP-integrable problems in enumerative geometry.

I want to describe some big picture that roughly looks as follows:

• There is a huge interesting class of problems in enumerative geometry (enumeration of maps / hypermaps / dessins d'enfants / constellations / Hurwitz numbers / colored HOMFLY-PT polynomials of knots / etc. etc.);
• They all are described by hypergeometric KP tau functions;
• Under mild analytic assumptions these KP tau functions are subject to the Chekhov-Eynard-Orantin topological recursion coming from matrix models;
• Which means that these KP tau functions are controlled by some possibly non-conformal Dubrovin-Frobenius manifolds of various finite dimensions;
• In particular, these KP tau functions are restrictions to one set of times of the tau function of the Dubrovin-Zhang hierarchies, for some choices of calibration and genus expansion.
• In particular, the enumerative problems that we started with can be universally resolved via beautiful formulas in terms of intersections on the moduli spaces of curves.

This has to be considered as a general principle that always works, though the details are not always worked out in the literature. Various parts of this story are a joint work with Alexandrov, Borot, Buryak, Bychkov, Carlet, Dunin-Barkowski, Kazarian, Kramer, van de Leur, Lewański, Norbury, Orantin, Popolitov, Posthuma, Sleptsov, Spitz, and Zvonkine.

R. Vitolo : Homogeneous Hamiltonian operators, projective geometry and integrable systems.

First-order homogeneous Hamiltonian operators play a central role in the Hamiltonian formulation of quasilinear systems of PDEs. They have well-known differential-geometric invariance properties which find application in the theory of Frobenius manifolds. In this talk we will show that second and third order homogeneous Hamiltonian operators are invariant under reciprocal transformations of projective type, thus allowing for a projective classification of the operators. Then, we will describe how the above operators generate known and new integrable systems, and discuss the invariance properties of the systems under projective transformations.

#### Practical info

The workshop will take place at the "Institute de Mathématiques de Bourgogne" located in the building Mirande on the university campus.

To reach the campus from the centre of Dijon you can take the tramway T1 (from the train station or from place Darcy) in the direction "Quetigny" and get off at "Université".

To reach Dijon via airplane you can fly to Paris and then take a train from Paris Gare de Lyon to Dijon. Alternatively you can fly to Lyon and the a train to Dijon.

Indico webpage.

#### Registration

To register send an email to guido.carlet@u-bourgogne.fr