# Integrable Systems and Automorphic Forms

** Sirius Mathematics Center, Sochi **

**International Laboratory of Mirror Symmetry and Automorphic Forms,**

**NRU HSE University, Moscow**

**February 24 - February 28, 2020 - ****Sochi, Russia**

**Scientific organizers:**

**Victor Buchstaber** *(Steklov Mathematical Institute, Russian Academy of Sciences)*

**Evgeny Ferapontov*** (Loughborough University, United Kingdom)*

**Valery Gritsenko** *(University of Lille/NRU HSE)*

**Volodya Rubtsov** *(University d'Angers, France/ITEP/IGAP, Trieste)*

**Research program **

Classical modular forms can be viewed as holomorphic functions on the upper half-plane that transform in a certain way under a discrete subgroup of SL(2, R). Multi-variable generalisations include Hilbert, Siegel and Picard modular forms and, more generally, automorphic forms on a lattice of a suitable Lie group. The theory of modular forms is highly non-trivial and full of beauty. Remarkably, these refined mathematical objects occur in a wide range of applications covering nearly all branches of mathematics (and beyond). Thus, they arise in number theory (as generating functions of various number-theoretic sequences), algebra (in the classification of Lorentzian Kac-Moody Lie algebras), algebraic geometry (in the theory of moduli spaces and mirror symmetry), analysis (as eigenfunctions of the Laplace-Beltrami operators), mathematical physics (in the theory of Frobenius manifolds, self-dual Yang-Mills equations, magnetic monopoles and 3D dispersionless integrable systems), theoretical physics (black holes, partition functions, Feynman amplitudes), cryptography and coding theory (as coefficients of weight enumerators), just to mention a few. What makes these transcendental functions particularly important are the (relatively simple!) differential equations they satisfy: it is precisely via these equations that they arise in applications. This supports a point of view that modular forms constitute yet another class of special functions of mathematical physics.

During the conference we are planning to discuss new results in automorphic forms and integrable systems related to Picard and Siegel modular forms, modular forms on orthogonal groups, Jacobi forms and Abelian functions in many variables; Rankin-Cohen brackets in the context of representation theory and deformation quantization.

In addition to approximately 40 scientific reports, three problem sessions will be held at the conference:

- Integrable systems and modular forms
*(moderator - Evgeny Ferapontov);* - Rankin-Cohen modular brackets in the context of representation theory and deformation quantization

*(moderators - Michael Pevzner, Volodya Rubtsov); *

- Jacobi forms, Borcherds automorphic products and Integrable systems
*(moderator - Valery Gritsenko). 1 2 3 4*

**The workshop will take place in Omega Sirius Park Hotel located near Black Sea. **

All questions about the conference should be addressed to the organizers via **valery.gritsenko@univ-lille.fr.POSTER SCHEDULE **

** PARTICIPANTS: **

**Dmitrii Adler** (NRU HSE, Moscow). *Jacobi forms and root systems. *

In 1992 K. Wirthmuller proved an analogue of the Chevalley theorem for weak Jacobi forms associated with irreducible root systems, except $E_8$. However, generators in his proof are not constructed explicitly, but they are important in applications. In my talk, I will discuss the new simple approach to this problem in cases of root systems D_n and F_4. Following this method, it is possible to construct the generators of the corresponding algebras of weak Jacobi forms using only Jacobi theta functions and modular differential operators. Also it will be shown that the main generators of the algebras of weak Jacobi forms for root systems of D_n type satisfy a system of non-linear differential equations. The talk will be based on joint works with V. Gritsenko.

**Alexey Basalaev** (NRU HSE, Moscow). *Theta constants and mirror symmetry for simple-elliptic singularities. *

Gromov-Witten theory of elliptic orbifolds satisfies the certain (quasi)-modularity condition. This fact by itself was a side-product of mirror symmetry isomorhism proved by Milanov and Ruan. It can be later employed to show that genus zero potentials of these orbifolds can be written via the certain quasimodular forms. We will show how the series expansion of these quasimodular forms at the different points can be viewed as the variation of B--side primitive form of a simple-elliptic singularity and present the certain A--side mirror CohFT genus zero potentials obtained in this way.

**Victor Buchstaber** (Steklov Mathematical Institute, Russian Academy of Sciences). * Hyperelliptic sigma functions in mathematical physics. *

We introduce the hyperelliptic sigma functions as the entire functions that under certain initial conditions satisfy a specific system of heat equations in a nonholonomic frame. The focus will be on addition theorems and functional equations for these sigma functions and for the hyperelliptic analogues of Weierstrass elliptic functions. We will show how to derive fundamental equations of mathematical physics from these functional equations. Corresponding periodic and quasiperiodic solutions will be described.

**Elena Bunkova** (Steklov Mathematical Institute, Russian Academy of Sciences, Moscow). *Lie algebras of heat operators. *

We will present the Lie algebras that are generated by the operators corresponding to heat equations in a nonholonomic frame. The latter is used to define hyperelliptic sigma functions. These Lie algebras have been obtained recently in collaboration with V.M. Buchstaber. We will discuss their applications and relations with the theory of integrable systems.

**Benjamin Enriquez** (University of Strasbourg, France). *Universal versions of KZB-type equations and special functions. *

We review the construction of a universal version of the elliptic KZB differential system, which gives rise to an explicit flat connection on the moduli spaces of elliptic curves with marked points. The coefficients of the corresponding monodromy representation given rise to special functions on the Poincaré upper-half plane. We will review the recent progress towards understanding the structure of the space of these functions (Lochak, Matthes, Schneps) (partly joint w. D. Calaque and P. Etingof).

**Mikhail Feigin** (University of Glasgow, United Kingdom). *Trigonometric solutions of WDVV equations. *

Trigonometric solutions of WDVV equations arise within quantum cohomology of resolutions of simple surface singularities (Bryan, Gholampour). In this case they correspond to simply laced root systems. More generally, one can obtain trigonometric solutions for special configurations of vectors with multiplicities, which include arbitrary root systems with Weyl-invariant multiplicities. I am going to explain how one can obtain new solutions by taking restrictions and subsystems for a given trigonometric solution. This extends considerations of Veselov and the speaker for the class of rational solutions of WDVV equations. The talk is based on joint work with M. Alkadhem.

**Evgeny Ferapontov** (Loughborough University, United Kingdom). *Dispersionless integrable systems and modular forms. *

It has been observed that generic cases in the classification of dispersionless integrable systems within various particularly interesting classes are expressed in terms of modular forms. The reason behind this is the Odesskii-Sokolov construction that parametrises broad classes of dispersionless integrable systems by generalised hypergeometric functions (which give rise to modular forms for specific parameter values). I will discuss two manifestations of this phenomenon:

1. Integrable Hirota type equations and Siegel modular forms;

2. Integrable Lagrangians and Picard modular forms. This talk is based on joint work with A. Odesskii and F. Clery.

**Ilya Gayur** (University of Birmingham, United Kingdom). *Dualities for Non-commutative Painlevé-Calogero hamiltonians: Ruijsenaars versus Spectral. *

The extension of the Painlevé-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlevé systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlevé systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. The description of this duality in terms of the spectral curve of non-reduced system will be demonstrated in comparison to the Ruijsenaars duality.

**Vassily Gorbunov** (University of Aberdeen, United Kingdom/ NRU HSE, Moscow). *Electrical Lie algebras and groups and their representations. *

This is the second part of the joint talk with D. Talalaev. The electrical Lie group in the type A was introduced by Lam and Pylyavskyy as a déformation of the unipotent group. Using our approach to electrical networks we construct the natural representation of it. We will indicate how this offers a new approach to studying the electrical varieties which are the déformation of the Lusztig varieties.

**Valery Gritsenko** (University of Lille/NRU HSE). *Modular differential operators and t-deformation of modular forms. *

The definition of t-deformation of modular forms on orthogonal groups belongs to K. Saito. It appeared in the theory of Frobenius varieties. We remark that the Cohen-Kuznetzov-Zagier lifting of modular forms used in the talk of Kalmynin is an example of such t-deformation. In this talk we give an overview of my solution of the Saito problem given in terms of the modular differential operators and its relation to Rankin-Cohen brackets and other constructions. We remark that the Cohen-Kuznetzov-Zagier lifting of modular forms used in the talk of Kalmynin is example of such deformation.

**Nicolas Hemelsoet** (University of Geneva, Switzerland). *A computer algorithm for the BGG resolution. *

The celebrated Borel-Weil-Bott theorem gives the cohomology of homogeneous line bundles on the flag variety G/B. For general vector bundles, one needs to compute equivariant Lie algebra cohomology, or compute explicitly differentials in a spectral sequence. I will present an alternative method due to Lachowska-Qi, which computes this sheaf cohomology, based on the Bernstein-Gelfand-Gelfand resolution of a simple \g-module by Verma modules. With Rik Voorhaar, we implemented this algorithm on a computer, and obtained several applications that I will discuss.

**Tomoyoshi Ibukiyama** (University of Osaka, Japan). *Automorphic differential operators and special functions. *

Holomorphic differential operators on functions on Siegel upper half space which commute under a certain restriction of the domain with the actions of a group by automorphy factors are important arithmetic objects related to critical values of $L$ functions, and also give a new theory of special polynomials and functions of several variables, including usual Gegenbauer polynomials and functions. We give both theoretical and concrete description of such operators and explain associated holonomic systems. (This is partly a joint work with D. Zagier and partly with T. Kuzumaki and H. Ochiai.)

**Alexander Kalmynin** (NRU HSE, Моscow). *Sums of squares, Jacobi forms and differential equations. *

We will discuss a connection between distribution of gaps between consecutive numbers which are sums of two squares and distribution of values of Jacobi-like forms. Different approaches to the proof of main formula and possible ways to generalize these results to gaps in supports of other arithmetical functions will also be presented.

**Boris Kruglikov** (University of Tromso, Norway). *Relative Differential Invariants. *

Modular forms can be approached as sections of line bundles on the moduli spaces of holomorphic curves. Usually these are derived from homogenous bundles wrt actions of simple Lie groups. Relative differential invariants arise in a similar way, but the group may become infinite dimensional. I will explain how absolute rational scalar differential invariants of general Lie pseudogroups can be generated, and what role relative differential invariants play in their computations. Example of second order differential equations will be treated, and if time permits I will show application to integrability.

**Quentin Labriet** (University of Reims, France). *Holographic operators associated with the Rankin-Cohen brackets. *

We will introduce the notion of holographic operators through the example of the Rankin-Cohen brackets, and will present different ways to obtain their explicit expressions.

**Roman Lee** (Budker Institute of Nuclear Physics, Novosibirsk). *Modern methods of multiloop calculations. *

We review the methods of multiloop calculations, with the emphasis on the method based on the differential equations. Those differential equations are in close relations with the Gauss-Manin connection of the parametric integral representation corresponding to a given class of loop integrals, as function of kinematic parameters. We discuss the relation between the properties of these differential equations and the class of functions which appear in the expansion of their solutions in the dimensional regularization parameter.

**Marianne Leitner** (Trinity College Dublin/Dublin Institute for Advanced Studies, Ireland). *Convolutions on the complex torus. *

"Quasi-elliptic" functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated and given by Eisenstein series. A related structure has appeared recently in the computation of Feynman integrals.

**Werner Nahm** (Dublin Institute for Advanced Studies, Ireland). *Automorphic forms and quantum field theory. *

The partition functions of euclidean quantum field theory can be described as those maps from compact manifolds with Riemannian metric which have few generalized derivatives. Their fields are just those generalized derivatives. The conventional derivative with respect to the metric is the energy-momentum tensor. For conformal field theories in two dimensions the dependence on Weyl transformations can be factored out, at the price of introducing an automorphy factor for the action of the mapping class group. In a simple case, this leads to a generalization of the Rogers-Ramanujan functions to arbitrary genus.

**Sergey Natanzon** (NRU HSE/ITEP/IMU). *Formal solution to the h-KP hierarchy. *

We find all formal solutions to the h-dependent KP hierarchy. They are characterized by certain Cauchy-like data that are functions of one variable. The solutions are found in the form of formal series for the tau-function of the hierarchy and for its logarithm. An explicit combinatorial description of the coefficients of the series is provided. The talk is based on join work with A.Zabrodin.

**Valentin Ovsienko** (University of Reims, France). *q-deformed rationals, and q-deformed real numbers. *

A number of q-deformations is known in algebra, geometry and combinatorics. However, q-deformations of numbers are well understood, since Euler and Gauss, only for integers. Many classical sequences of integers have interesting q-analogues, the best known one is the notion of q-binomial coefficients. I will talk of a recent attempt, with Sophie Morier-Genoud, to develop a theory of q-deformed rational and real numbers, based on combinatorial properties of continued fractions. A "q-rational" is a ratio of certain polynomials with positive integer coefficients, I will give a combinatorial interpretation of the coefficients and describe some properties, such as the "total positivity". A "q-real" is understood as a formal power series with integer coefficients, I will give several concrete examples.

**Alexei Pantchichkine** (Institut Fourier, University Grenoble-Alpes, France). *Algebraic differential operators on unitary groups and their applications. *

Algebraic differential operators acting on automorphic forms f on unitary groups U_K(n, n) over an imaginary quadratic field K are described. Applications are given to special L-values L(s, f) attached to f, computed as certain Rankin-Selberg integrals.

**Maxim Pavlov** (Lebedev Physical Institute of Russian Academy of Sciences, Moscow). *Whitham equations and the Chazy equation. *

We show that the Chazy equation appears naturally in the theory of Whitham equations.

**Michael Pevzner** (University of Reims, France). *Symmetry breaking versus holography in Representation Theory. *

We shall present the idea of symmetry breaking transform in the framework of branching rules for infinite dimensional representations of reductive Lie groups and will focus on the dual notion of holographic transform by providing a series of concrete examples.

**Changzheng Qu** (Ningbo University, China). *Solitons and Their Stability of Nonlocal Camassa-Holm-type Equations. *

It is well-known that the Camassa-Holm-type equations possess a number of remarkable properties, including the existence of peaked solitons, wave breaking, and interesting geometric formulations etc. It has been known that the Camassa-Holm-type equations adapt nonlocal extensions. In this talk, we provide a review discussion on the nonlocal Camassa-Holm type equations. We shall investigate structures of peaked solitons, geometric formulations, orbital stability of peaked solitons etc of the nonlocal Camassa-Holm-type equations. Some open questions related to this talk will be addressed

**Hjalmar Rosengren** (Chalmers University of Technology/University of Gothenburg, Sweden). *Correlations of the XYZ spin chain and modular tau functions of Painlevé VI. *

We compute nearest neighbour correlations for the supersymmetric XYZ spin chain. We obtain exact expressions on the finite chain in terms of tau functions of Painlevé VI, which were introduced by Bazhanov and Mangazeev. This is joint work with Christian Hagendorf (Louvain-la-Neuve).

**Emmanuel Royer** (University of Clermont Auvergne/CNRS, France). *Rankin-Cohen deformations of the algebra of Jacobi forms. *

After presentation of the number theoretic background, we shall emphasize the arithmetical and algebraic aspects of the Rankin-Cohen brackets in order to extend them to several natural number-theoretical, including the case of teak Jacobi forms.

**Volodya Rubtsov** (University d'Angers, France/ITEP/IGAP, Trieste). *Fay identity: from Mathematical Physics through Geometry to Number Theory. *

We give a survey of various avatars of the trisecant Fay identity which appear in the context of Integrable systems (as forms of the Associative Yang-Baxter Equation) and as conditions on generating functions for period polynomials of (quasi-)modular forms and group cocycle conditions for some multiparametric modular group.

**Riccardo Salvati Manni** (University of Rome, Italy). *An overview on Kodaira dimension of the moduli space of principally polarized abelian varieties. *

The moduli space of principally polarized complex abelian varieties of dimension n was believed for a long time to be unirational. This was first disproved by Freitag for n = 1 mod 8, n ≥ 17. Hence Tai, Freitag and Mumford proved that it is of general type for all n ≥ 7. On the other hand, this moduli space is unirational for n ≤ 5. Recently M. Dittmann and N. R. Scheithauer and I proved that when $n=6$ the Kodaira dimension is non-negative. I will try to give a general overview on the topic.

**Oleg Sheinman** (Steklov Mathematical Institute, Russian Academy of Sciences, Moscow). *Inverse scattering method in presence of group symmetry. *

The aim of my talk is to show how theta-functions with an additional group symmetry can emerge in the Baker-Akhieser functions and in the solutions of Hitchin systems.

**Nils-Peter Skoruppa** (University of Siegen, Germany). *Invariants of Weill representations of SL(2,Z). *

To each integral lattice (or, if one prefers, to each finite quadratic module) is associated its Weil representation, a representation of SL(2,Z). An explicit description of the invariants of these representations would have a plethora of important and immediate consequences, in particular, for the construction of Jacobi forms on several variables and its applications. Alas, such an explicit description of invariants is not known in general. In this talk we give a short overview of the mentioned topics, and we shall propose a solution of the invariant problem for curtain classes of lattices.

**Vyacheslav Spiridonov** (JINR, Dubna/NRU HSE, Moscow). *Modular quantum dilogarithm, hyperbolic beta-integrals and integrable systems. *

In 1994 Faddeev suggested the modular quantum dilogarithm using the simplest SL(2,Z) group transformation applied to the q-Pochhammer symbol. In the theory of special functions it got the name hyperbolic gamma function. Its generalization based on arbitrary transformation from SL(2,Z) was recently suggested by Dimofte. In the talk I will present the evaluation formula for a general univariate hyperbolic beta-integral, built with the help of this function, and indicate its possible applications to integrable systems. This is a joint work with G.A. Sarkissian.

**Ian Strachan** (University of Glasgow, United Kingdom). *Elliptic trilogarithms and solutions of the WDVV equations. *

Elliptic Solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity are studied. These have an extra SL(2,Z)-symmetry and the solutions are expressed in terms of the elliptic tri-logarithm function. This leads to the notion of an elliptic V-system, generalising the notion of a rational and trigonometric V-system used to define rational and trigonometric solutions of the WDVV equations.

**Dmitry Talalaev ** (MSU, Moscow). *Electrical varieties and discreet integrability. *

The principal subject of the talk is the electrical version of the cluster varieties of the Lusztig type. Our main result is a new vertex realization of the statistical model associated with an electrical network on circular graphs. This result has a wide set of connections with many mathematical fields: cluster algebras, Temperley-Leeb algebras, positive mathematics, and others. In this report I will focus on the vertex form of the statistical mechanical model related to the classical problem of electric networks and the phenomenon of discreet integrability associated with the electrical solution of the Zamolodchikov tetrahedron equation. The resulting integrable system is related to the Painleve equation. The talk is based on the joint work with Vassily Gorbounov.

**Pierre Vanhove** (IPhT, France). *Mirror symmetry and Feynman integrals. *

In this talk we will discuss the algebraic and transcendental features of the computation of multiloop sunset Feynman integrals. Starting from the realization of arbitrary Feynman graph hypersurfaces as (generalized) determinantal varieties, we describe the Calabi-Yau sub-varieties of permutohedral varieties that arise from the multiloop sunset Feynman graphs and some key features of their geometry and moduli. We will explain how the “creative telescoping” algorithm allows to derive the inhomogeneous differential equation when the standard Griffiths-Dwork algorithm fails. We will show how the results can be understood by understanding the geometry and the moduli of the Calabi-Yau varieties of the sunset graph. In particular the how specialization of physical parameters leads to rank jump. We will explain the realization of Calabi-Yau pencils as Landau-Ginzburg models mirror to weak Fano varieties. This is based on work done in collaboration with Charles Doran and Andrey Novoseltsev

**Alexander Veselov** (Loughborough University, United Kingdom). *Automorphic Lie algebras and modular forms. *

I will report about some recent results in the theory of the hyperbolic automorphic Lie algebras following joint work with Vincent Knibbeler and Sara Lombardo.

**Haowu Wang** (MPIM, Bonn, Germany). *Reflective modular forms, Siegel paramodular forms and theta blocks. *

In the talk we will explain how to use reflective modular forms to construct exceptional paramodular forms and theta blocks. We will present three infinite series of paramodular forms of weights 2 and 3 which are simultaneously Borcherds products and additive Jacobi liftings, and an infinite family of anti-symmetric paramodular forms of canonical weight 3.

**Brandon Williams** (University of Darmstadt, Germany). *Higher pullbacks of modular forms on orthogonal groups. *

We apply differential operators to modular forms on orthogonal groups O(2, `) to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form φ are theta lifts of partial development coefficients of φ. For certain lattices of signature (2, 2) and (2, 3), for which there are interpretations as Hilbert-Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

**Federico Zerbini** (IPhT, France). *String amplitudes from genus zero to genus one. *

I will relate formal two-point genus-one string amplitudes to the classical genus-zero amplitudes introduced by Veneziano and Virasoro. Then I will relate open and closed string amplitudes at genus one via Brown's single-valued map. This presentation is based on a joint work with Don Zagier.

**Andrei Zotov** (Steklov Mathematical Institute, Russia/ NRU HSE, Russia). *Kronecker function on supersymmetric elliptic curves and Yang-Baxter equations. *

We describe the Kronecker elliptic function on supersymmetric elliptic curves. In the ordinary case this function is widely used in elliptic integrable systems through constructions of R-matrices, Lax equations and other structures. In supersymmetric case the Kronecker function is also shown to satisfy two main properties – the Fay identity and the heat equation.

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