# PARTITION FUNCTIONS AND AUTOMORPHIC FORMS

28 January 2018 to 2 February 2018
BLTP JINR
Europe/Moscow timezone

## Scientific Programme

LECTURE COURSES
/about four one hour lectures each /

Hermann Nicolai (AEI, Golm, Germany) 2 lectures
ON K(E10) AND ITS FERMIONIC REPRESENTATIONS

In these lectures I will review basic properties of K(E10), the `maximal compact' subalgebra of the maximally extended hyperbolic Kac-Moody algebra E10, and its applications to the fermionic sector of maximal supergravity and M theory.
Slides.pdf

John Duncan (Emory University, USA) 4 lectures
MOONSHINE IN NUMBER THEORY AND GEOMETRY

Moonshine arose in the 1970s as a collection of coincidences connecting modular functions to the monster simple group, which was newly discovered at that time. The effort to elucidate these connections led to new algebraic structures (e.g. vertex algebras and generalized Kac--Moody algebras) which have since found applications in representation theory, number theory, geometry and string theory. In this century the theory has been further enriched, with the discovery of connections between K3 surfaces and certain distinguished groups in the early part of this decade, and connections between sporadic simple groups and the arithmetic of modular abelian varieties in recent months. In these lectures we will review monstrous moonshine, explain the number theoretic foundations of umbral moonshine, and describe recent results which reveal a role for sporadic groups in arithmetic geometry. We will also review vertex algebra, and the known applications of that theory to these topics. There are many interesting open questions in this area.

Slides.pdf

Seok Kim (CTP, Seoul National University, Korea) 4 lectures
SUPERCONFORMAL INDICES AND INSTANTON PARTITION FUNCTIONS

We discuss recent advances in the instanton partition functions of quantum field theories with 8 SUSY in spacetime dimensions d=4,5,6 (compactified to 4 dimensions for d=5,6). Apart from their original use (originally explored by Nekrasov) to describe the Seiberg-Witten theory, these partition functions played crucial roles recently to better understand aspects of strongly-coupled conformal field theories in d=4,5,6, often via various curved space partition functions (including "superconformal indices" or other partition functions). We shall also review some of these developments.
Slides: Lecture 1.pdf   Lecture 2.pdf
Lecture 3.pdf  Lecture 4.pdf

Yasuhiko Yamada (Kobe University, Japan) 4 lectures
THEORY AND APPLICATIONS OF THE ELLIPTIC PAINLEVE EQUATION

I will give an introduction on the theory of discrete Painleve equations focusing mainly on the elliptic difference case. The elliptic Painleve equation is the master case in the Sakai's scheme and has the largest affine Weyl group symmetry (E8). I will present the geometric picture behind the system, its Lax form and the tau-functions. The current status on some recent developments will also be reviewed.

Pierre Vanhove (Saclay, France)   4 lectures
FEYNMAN INTEGRALS AND MODULAR FORMS

Scattering amplitudes are central tools for analyzing fundamental interactions in particle physics. They are the bridge between the theoretical models and the experimental data. It is therefore of extreme theoretical and experimental importance to be able to evaluate scattering amplitudes for numerous physical processes. Despite being used for more than 60 years, the mathematical nature of scattering amplitudes is still poorly understood. Scattering amplitudes are expressed in terms of Feynman integrals. Feynman integrals have rich mathematical structures: algebraic geometry, motivic cohomology and modular forms. These mathematical structures provide powerful ways of evaluating analytically and numerically physical processes.

In this series of lecture we will present the recent progresses in the evaluation of Feynman integral. We will review the modern unitarity method and fundamental properties amplitudes in quantum field theory have to satisfy. We will then explain how Feynman integral can be understood as motivic periods. In the context of the special family of sunset integral, we will discuss the appearance of modular forms and some surprising connection with mirror symmetry.
Slides.pdf

Vyacheslav Nikulin (MIAN|University of Liverpool) 1 lecture
LORENTZIAN KAC-MOODY ALGEBRAS, I

Valery Gritsenko (University of Lille 1|IUF|NRU HSE) 1 lecture
LORENTZIAN KAC-MOODY ALGEBRAS, II

In our talks I, II 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 (see Proceedings of the London Math. Society) for some details. The part I is based on the theory of hyperbolic lattices. The part II is based on the theory of automorphic forms and Borcherds products.

From a non-formal point of view, in these two talks we show that some hyperbolic Kac-Moody algebras have a so-called automorphic correction, i.e. one can construct a generalised hyperbolic Kac-Moody algebra with the same system of real  simple roots but with an automorphic form as the Kac-Weyl-Borcherds  denominator functions. This means that the new lorentzian algebra has a huge group of hidden symmetries. The class of such automorphic  algebras is finite. We fully classified the class of all such algebras in all hyperbolic ranks with the maximal hyperbolic 2-Weyl groups and a Weyl vector with a negative square.

Slides Nikulin.pdf

Du Pei (Caltech) 1 lecture
BPS SPECTRA AND INVARIANTS  FOR THREE- AND FOUR-MANIFOLDS

Much like harmonics of musical instruments, spectra of quantum systems contain wealth of useful information. In this lecture, I will discuss how to obtain new invariants of three- and four-manifolds from BPS spectra of quantum field theories. We will see, in concrete examples, how several well-known invariants of three- and four-manifolds can be realized as partition functions of supersymmetric gauge theories and computed using localization technique.

Vyacheslav Spiridonov (BLTP JINR and NRU HSE)  1 lecture
INTRODUCTION TO THE THEORY OF ELLIPTIC HYPERGEOMETRIC INTEGRALS

Superconformal indices of 4d supersymmetric field theories are described by elliptic hypergeometric integrals. I will describe the most important mathematical facts about this class of special functions and present some methods of proving identities for them supporting the Seiberg duality conjectures.
Slides.pdf

TALKS BY PARTICIPANTS

THE RING OF WEAK JACOBI FORMS FOR D8 ROOT SYSTEM

In my talk I will consider weak Jacobi forms associated with lattices D8. The main result states that the space of these forms is a free algebra with 9 generators over the ring of modular forms.
Slides.pdf

P. Agarwal  (Seul National University)
N=1 LAGRANGIANS FOR GENERALIZED ARGYRES-DUGLAS THEORIES

We  find  N=1  Lagrangian  gauge theories  that  flow  to  generalized Argyres-Douglas theories  with  N=2 supersymmetry.  We  find  that certain  SU-quiver gauge theories  flow  to generalized Argyres-Douglas theories  of  type  Ak-1, Amk-1  and (Im,km,S).  We also find quiver  gauge  theories  of $SO/Sp$  gauge  groups  flowing  to the (A2m-1,D2mk+1), (A2m,D2m(k-1)+k) and Dm(2k+2)m(2k+2)[m]  theories.

Slides.pdf

L. Astrakhantsev (MIPT)
MOVING MIRRORS IN 2 DIMENSIONAL QFT

In  my  talk cases  of  ideal  and  nonideal  moving mirrors in 2 dimensional Minkowski space-time will  be considered. I will pay attention to Hamiltonian and total momentum of the system. Also I will consider corrections to the Keldysh propagator in case of nonideal mirror. In  my  talk  I  will consider  weak  Jacobi  forms  associated  with lattices D8 . The  main result states that the space of these forms
is a free algebra with 9 generators over the ring of modular forms.

Slides.pdf

M. Bershtein (Landau Inst.|Skoltech |NRU HSE)
DIFFERENCE PAINLEVE EQUATIONS FRON 5d GAUGE THEORIES

In the talk I review two relations between difference Painleve equations and supersymmetric 5d gauge theories. First, these equations can be constructed as deautonomization of cluster integrable systems.
Second, them can be solved using Nekrasov partition functions of the 5d gauge theories.

Slides.pdf

A. Bolshov (MIPT)
GRASSMANIANS AND FORM FACTORS

The talk is about a representation of form factors in  N=4 SYM as integrals over Grassmannian  manifolds. A procedure for obtaining such a description will be described and illustrated by several examples.
Slides.pdf

L. Bork (ITEP, CFAP VNIIA)
Integrands for form factors in N=4 SYM and ambitwistor strings.

We are going to discuss how to reconstruct loop integrands for form factors of different operator types in N=4 SYM from integrands of  on-shell amplitudes via "gluing operation". This "gluing operation" was conjectured in context of construction of vertex operators for form factors in ambitwistor string models.

A. Chattopadhyaya (Indian Institute of Science)
Dyon degeneracies from Mathieu moonshine symmetry

We study Siegel modular forms associated with the theta lift of twisted elliptic genera of $K3$ orbifolded with $g'$ corresponding to the conjugacy classes of the Mathieu group M24 . For this purpose we re-derive the explicit expressions for all the twisted elliptic genera for all the classes which belong to M23 ⊂ M24. We show that the Siegel modular forms satisfy the required properties for them to be generating functions of $1/4$ BPS dyons of type II string theories compactified on $K3\times T^2$ and
orbifolded by $g'$ which acts as a $\mathbb{Z}_N$ automorphism on $K3$ together with a $1/N$ shift on a circle of $T^2$. In particular the inverse of these Siegel modular forms admit a Fourier expansion with integer coefficients together with the right sign as predicted from black hole physics. This observation is
in accordance with the conjecture by Sen and extends it to the partition function for dyons for all the $7$ CHL compactifications. We construct Siegel modular forms corresponding to twisted elliptic genera whose twining character coincides with with the class $2B$ and $3B$ of $M_{24}$ and show that they also satisfy similar properties. Apart from the orbifolds corresponding to the $7$ CHL compactifications, the rest of the constructions are purely formal.
Slides.pdf

A. Chekmenev (LPI)
Conformal Lagrangians from the (formal) near boundary analysis of AdS gauge fields

A  simple  generating  procedure allowing  to  obtain  Lagrangians  of  conformal massless and massive fields of mixed-symmetry type in explicit form is presented. The approach originates from  the analysis  of  equations  on  leading  boundary value  of  AdS  field  in  the case  of odd-dimensional AdS space.
Slides.pdf

A. Chowdhury (ITP, TU Wien)

A  few  years  ago  a connection  between  the  elliptic  genus  of  the $K3$ manifold  and  the largest Mathieu group $M_{24}$ was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic  genera  and  sporadic groups.  While  the  Calabi-Yau 3-fold  case  is rather  uninteresting,  the elliptic genera of certain Calabi-Yau d-folds for $d>3$ have  expansions  that  could  potentially arise  from  underlying  sporadic symmetry groups. We explore  such  potential connections  by  calculating twined  elliptic  genera  for  a large  number  of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces,
for a toroidal orbifold and two Gepner models. [arXiv:1711.09698]

Slides.pdf

I. Gahramanov
Supersymmetric indices, elliptic hypergeometric functions and integrability

In this talk, we will review connection of integrable statistical models to supersymmetric indices and elliptic hypergeometric functions. The supersymmetric index is one of the efficient tools in the study of supersymmetric gauge theories providing the most rigorous mathematical check of supersymmetric dualities. The supersymmetric index of a four-dimensional supersymmetric gauge theory is expressed in terms of elliptic hypergeometric integrals. Such integrals are the new class of special functions and are of interest both in mathematics and in physics. Another intriguing physical interpretation of these integral identities stems from the role they play as the Yang-Baxter equation for certain integrable two-dimensional statistical models. As a result, one can construct a correspondence between supersymmetric gauge theories and integrable models such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the supersymmetric index and the Yang-Baxter equation expresses the index identity for dual pairs.

Slides.pdf

A. Lobashev (MSU)
Kontsevich integral in topological models of gravity

We consider the procedure of Kontsevich integral calculation in topological Poincare gauge gravitational theory. The Lagrangian of this theory is the sum  of  two parts - one is& the Chern-Simons term for local Lorentz connection and other is the Chern-Simons term for tetrads field. The Kontsevich  integral  is  not  defined  for non-compact  gauge  groups.  We consider the local morphism between Poincare gauge gravitational theory and gauge gravitational theory of SO(4) gauge gravitational theory and  reduce the  result  of calculation  by means  Wigner-Ihnonue algorithm. We  discuss  the application  of  obtained  results  to  construction of  knot invariants  and  some connections with theory of theta functions.

Slides.pdf

R. Matias

Abstract: In this talk we explore the Hecke Algebra of the group $\Gamma_0(2)+$ in order to define a new form of replication which turns out to respect the power map structure of 2.B, where B is the Baby Monster Sporadic Group, in the same way that usual replication respects the power map structure in the Monster Group.

K. Sun (SISSA, Trieste)
Blowup equations for refined topological strings

Gottsche-Nakajima-Yoshioka K-theoretic  blowup  equations  characterize  the Nekrasov partition function of five dimensional N=1 supersymmetric gauge theories compactified on a circle, which  via geometric  engineering  correspond to the refined topological string theory on SU(N) geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined  topological  string,
and the  crucial  ingredients  are  the  r  fields introduced  in  our  previous paper.  These  blowup  equations  are  in  fact the functional  equations  for  the partition  function  and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other than the refined  topological vertex  in  the  A-model and  the  refined holomorphic  anomaly equations  in  the B-model.  We study  the  modular  properties  of  the  blowup equations and  provide a  procedure  to determine all the vanishing and unity r fields from the polynomial part of refined topological string at large  radius  point.  We  also  find  that
certain  form  of blowup  equations  exist at generic  loci  of  the moduli space. This talk is based on arXiv:1711.09884

Slides.pdf