# Arithmetic, geometry, automorphic forms and physics

### 30.10-03.11.2017

### Research organizers: Valery Gritsenko and Shamit Kachru

The main subject of the workshop is the discussion on the interactions between the subjects mentioned in the title. We are planning to present the recent results and to formulate some questions and problems of joint research interests.

### Timetable

Click the title of a talk to unfold an abstract

##### MONDAY, October 30 (room 306, Usacheva, 6)

12:15–13:15 Alexandr Belavin (Landau Institute for Theoretical Physics)

A new approach to computation of the Special geometry on the Moduli space of Calabi-Yau manifolds

The requirement the Space-time supersymmetry in the String theory is equivalent to the geometrical condition of the compactification 6 of 10 dimensions on Calabi-Yau (CY) threefold. The massless sector of the Superstring theory after the compactification is physically the most important. The properties of the Effective Lagrangian of the model, which describe this sector, are defined in terms of the so-called Special Kahler geometry on CY moduli space. I'd like to present a new approach to computing the Special Kahler geometry based on the relation to the Landau-Ginzburg superpotential of the model with a Frobenius manifold structure. I'll demonstrate this approach for computing the Kahler metric on the 101-dimentional complex structure moduli space of the Quintic threefold.

13:30–14:30 Vyacheslav Nikulin (IM RAN/University of Liverpool)

New class of Lorentzian Kac-Moody algebras 1

In our talks I, II with Valery A. Gritsenko we describe a new large class of Lorentzian Kac-Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac-Moody algebras of restricted arithmetic type which are graded by S. For most of them we construct Lorentzian Kac-Moody algebras which give their automorphic corrections: they are graded by S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings. See arXiv:1602.08359 (and a forthcoming paper in Proceedings of the London Math. Society) for some details.

14:30–15h15 Coffee break

15:15–16:15 Vasily Golyshev (IITP RAN/NRU HSE)

Eichler integrals and dimensions of intermediate Jacobians

16:20–17:20 Nils Sheithauer (TU Darmstadt)

Construction and classification of vertex algebras

17:20–18:00 Coffee break

18:00–19:00 Alejandra Castro (University of Amsterdam)

Siegel Modular Forms and Black Hole Entropy

In the language of statistical physics, an extremal black hole is a zero temperature system with a huge amount of residual entropy. Understanding which class of counting formulas can account for a large degeneracy will undoubtedly unveil interesting properties of quantum gravity. In this talk I will discuss the application of Siegel modular forms to black hole entropy counting. The role of the Igusa cusp form in the D1D5P system is well-known in string theory, and its transformation properties are what allows precision microstate counting in this case. We implement this counting for other Siegel modular and paramodular forms, and we show that they could serve as candidates for other types of black holes.

##### TUESDAY, October 31

(On Tuesday, the conference takes place in the Steklov Mathematical Institute of RAN, ul. Gubkina, 8)

12:00–14:00 Arnav Tripathy (Harvard University)

Special cycles and BPS jumping loci

I'll sketch an attempt to bring the theory of special cycles, a deep part of number theory, into the domain of supersymmetric string compactifications. I'll describe a construction based on jumping loci for BPS state counts -- a separate phenomenon from the better-known wall-crossing! -- and explain in what cases these jumping loci generalize some parts of the theory of special cycles. Finally, I'll conclude with a host of physical and mathematical conjectures raised by this line of investigation.

13:00–14:00 Natalie Paquette (California Institute of Technology)

Moonshine in Spacetime: Genus Zero, Algebras, and String Compactifications

New examples of moonshine — relationships between finite groups and special classes of (mock) modular forms— have been proliferating in recent years, starting with the discovery of a Mathieu group moonshine apparently connected to conformal field theory (CFT) on the K3 surface. While many aspects of these new moonshines remain mysterious, in this talk we will stress the power of spacetime string theory---as opposed to worldsheet string theory or CFT—to shine light on some of moonshine's outstanding mysteries.

We will exhibit this by focusing instead on a comparatively well understood, classical example of moonshine: Monstrous moonshine. In spite of the explicit construction of a Monster module (a chiral CFT) and the subsequent proof of Monstrous moonshine by Borcherds, the genus zero property of Monstrous moonshine has lacked a physical context and thereby remained mysterious. We give a conceptual, physical explanation of the genus zero property of Monstrous moonshine for the first time, using properties of a heterotic string compactification. A key role is played by automorphic, i.e. duality-invariant, counting functions of spacetime BPS states which are denominator formulae for certain Generalized Kac-Moody algebras, including the Monster Lie algebra.

14:00–15:00 Lunch

15:00–16:00 John Duncan (Emory University, Atlanta)

Moonshine and Arithmetic

We will explain some recently discovered connections between sporadic simple groups and the arithmetic of modular abelian varieties.

16:00–17:00 Emanuel Scheidegger (University of Freiburg)

Periods and quasiperiods of modular forms and the mirror quintic at the conifold

We review the theory of periods of modular forms and extend it to quasiperiods. General motivic conjectures predict a relation between periods and quasiperiods of certain weight 4 Hecke eigenforms associated to hypergeometric one-parameter families of Calabi-Yau threefolds. We verify this prediction and discuss some of its implications on D-brane masses at the conifold on the corresponding mirrors.

##### WEDNESDAY, November 1 (Room 306, Usacheva 6)

13:30–14:30 Vyacheslav Spiridonov (JINR/NRU HSE)

Elliptic hypergeometric functions associated with the lens space

14:35–15:35 Aleksey Litvinov (LITP/NRU HSE)

Liouville reflection operator and integrable systems in conformal field theory

15:35–16:15 Coffee break

16:15–17:15 Hee Cheol Kim (Harvard University)

5d Duality domain walls and 4d E-string theories with flux

17:30–19:00 (**room 110, a joint session with the seminar “Mathematical Physics” **)

Vasily Gorbunov (University of Aberdeen)

Schubert Calculus Quantum Integrable Systems and correlation functions

We describe a particular quantum group which reproduces the equivariant quantum Schubert Calculus for the Grassmanian varieties. This system is a particular limit of the Yangian. The statistical models defined by the Yangian have very interesting number theoretical invariants-the correlation functions. We will discuss how these can descent to the Schubert calculus.

##### THURSDAY, November 2 (Room 306, Usacheva 6)

14:15–15:15 Sergey Galkin (NRU HSE)

Polar dg-algebra, and another view on Calabi-Yau geography

15:20–16:20 Gregory Sankaran (University of Bath)

Hyperkahler manifolds and degenerations

Degenerations of K3 surfaces were studied in the 1970s by Kulikov, in one of the first examples of minimal model theory. He divided them into three types. For irreducible hyperkahler manifolds a similar division takes place. I will describe some recent work of many people on Type II degenerations and then focus on the problems of studying Type III degenerations, where much less is known.

16:20–17:00 Coffee break

17:00–18:20 Johanna Knapp (TU Wien)

Gauged linear sigma model and Calabi-Yau geometry

##### FRIDAY, November 3

14:15–15:15 (**room 110**) Valery Gritsenko (University Lille 1/IUF/NRU HSE)

New class of Lorentzian Kac-Moody algebras 2

In our talks I, II with Vyacheslav Nikulin we describe a new large class of Lorentzian Kac-Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac-Moody algebras of restricted arithmetic type which are graded by S. For most of them we construct Lorentzian Kac-Moody algebras which give their automorphic corrections: they are graded by S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings. See arXiv:1602.08359 (and a forthcoming paper in Proceedings of the London Math. Society) for some details.

15:30–16:30 (**room 306**) John Alexander Cruz Morales (Universidad Nacional de Colombia)

On Stokes matrices for Frobenius manifolds

In this talk we will discuss how to compute the Stokes matrices for some semisimple Frobenius manifolds by using the so-called monodromy identity. In addition, we want to discuss the case when we get integral matrices and their relations with mirror symmetry. This is part of an ongoing project with Maxim Smirnov which extends previous work with Marius van der Put for the case of quantum cohomology of projective and weighted projective spaces to other Frobenius manifolds not necessarily of quantum cohomology type.

16:30–17:00 Coffee break

17:00–18:00 (**room 306, Colloquium ILMS and LAG**)

Grigory Mikhalkin (University of Geneva)

Examples of tropical-Lagrangian correspondences

18:00–18:30 Coffee break

18:30–19:30 Continuation of Mikhalkin’s talk

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