Workshop on Modular forms around string theory (2013)

Rankin-Selberg methods for closed string amplitudes.

After integrating over location of vertex operators and supermoduli, scattering amplitudes in closed string theories at genus $h \leq 3$ are expressed as an integral of a Siegel modular form on the fundamental domain of Siegel’s upper half plane. I will describe techniques to compute such modular integrals explicitly, by representing the integrand as a Poincar´e series and applying the unfolding trick. The focus will be mainly on genus one, but some results on higher genus will be presented.

Based in part on work in collaboration with C. Angelantonj and I. Florakis.

Rose, Simon

Fields Institute

Towards a reduced mirror symmetry for the quartic K3.

Mirror symmetry in terms of Yukawa couplings for a K3 is relatively trivial, due to the triviality of its Gromov-Witten invariants. Using reduced invariants, however, we can still tease out a lot of enumerative details of these surfaces. As these reduced invariants satisfy the same relations that ordinary invariants do, this raises the natural question: Is there a reduced B-model theory?

In this talk we wil go over our current work on this project, which is joint with Helge Ruddat.

Ruan, Yongbin

Michigan University

(I) and (II): Mirror symmetry and modular forms.

Traditionally, we use mirror symmetry to map a difficult problem (A-model) to an easier problem (B-model). Recently, there is a great deal of activities in mathematics to understand the modularity properties of Gromov-Witten theory, a phenomenon suggested by BCOV almost twenty years ago. Mirror symmetry is again used in a crucial way. However, the new usage of mirror does not map a difficult problem to easy problem. Instead, we make both side of mirror symmetry to work together in a deep way. I will explain this interesting phenomenon in the talk. This is a two-parts talk. In the first part, we will give an overview of entire story. In the second part, we will focus on the appearance of quasi-modularity.

Wan, Daqing

UC Irvine

(I) Rational points on a singular CY hypersurface.

The study of higher moments of Kloosterman sums naturally leads to a singular CY hypersur-face. In this talk, we explain how to estimate the number of rational points on the singular CY hypersurface via results on the Kloosterman sheaf.

(II) Mirror symmetry for the slope zeta function.

The slope zeta function is the slope part of the zeta functions of a variety over a finite field.

It is an arithmetic object. We expect that the slope zeta function satisfies the expected arithmetic mirror symmetry property for a mirror pair of sufficiently large families of CY hypersurfaces. We shall explain some evidence for this conjecture.

Whitcher, Ursula

Wisconsin-Eau Claire

Mirror quartics, discrete symmetries, and the congruent Zeta function.

We use Greene-Plesser-Roan and Berglund-Huebsch-Krawitz mirror symmetry to describe the structure of the congruent zeta function for a set of pencils of quartic K3 surfaces which admit discrete group symmetries.

Yui, Noriko

Queen’s University

Automorphy of Calabi-Yau threefolds of Borcea-Voisin type.

Calabi-Yau threefold of Borcea-Voisin type are constructed as the quotients of products of ellptic curves and K3 surfaces by non-symplectic involutions. Resolving singularities, one obtains smooth Calabi-Yau threefolds. We are interested in the modularity (automorphy) of the Galois representations associated to these Calabi–Yau threefolds. We establish the automorphy of some Calabi-Yau threefolds of Borcea-Voisin type.

This is a joint work with Y. Goto and R. Livn'e.

Zagier, Don

MPIM Bonn and College de France

(I) Quasimodular forms and holomorphic anomaly equation

Quasimodular forms are a special class of holomorphic functions that are nearly modular and become modular after the addition of a suitable non-holomorphic correction term. They are thus similar to, but much simpler than, mock modular forms. They occur in mirror symmetry in several ways, one of these being the so-called ”holomorphic anomaly equation” which describes a sequence of quasimodular forms with the non-modularity of each form being defined inductively in terms of its predecessors. We will describe how this works and how one can understand the structure of the HAE in terms of deformations of power series solutions to linear differential equations and ”bimodular forms”, which are yet another type of nearly modular object.

This is joint work with Jan Stienstra.

(II) Some number theory coming from string amplitude calculations Calculations of amplitudes in string theory lead in a natural way to multiple zeta values at the ”tree” (genus 0) level and to interesting modular functions at the ”1-loop” (genus 1) level. The talk will discuss various calculations related to this that seem to have interesting arithmetic aspects, including certain very specific rational linear combinations of multiple zeta values that rather mysteriously occur in both the tree and 1-loop level calculations, and also some proven and conjectural identities for special values of Kronecker-Eisenstein type lattice sums.

Zhou, Jie

Harvard University