SCIENTIFIC PROGRAMS AND ACTIVITIES

Modular forms associated to K3 surfaces endowed with lattice polarizations of high Picard rank

I will discuss several cases of lattice polarizations of high Picard rank on a K3 su rface. A classification for these objects will be presented in terms of quartic normal forms. Modular forms of appropriate group appear as coefficients of the normal forms.

Families of lattice-polarized K3 surfaces with monodromy

We extend the notion of a lattice-polarized K3 surface to families, study the action of monodromy on the Neron-Severi group of the general fiber, and use this to ”undo” the Kummer and Shioda-Inose structures in families.This technique sheds important light on the 14 families of Calabi-Yau threefolds with $h_{2,1}$ = 1 studied by Doran Morgan.
This is joint work with Andrew Harder, Andrey Novoseltsev, and Alan Thompson.

Extremal bundles on Calabi-Yau manifolds

Motivated by the goal to better understand the implications of stability conditions on numerical invariants, we study explicit constructions of (heterotic string) vector bundles on Calabi-Yau 3 folds. This includes both the monad construction and spectral cover bundles over elliptically fibered CY threefolds. We compare our results with the DRY (Douglas-Reinbacher-Yau) conjecture about generalized Bogomolov-Yau inequalities.
This is a joint work with Y.H. He and S.T. Yau.

Fano threefolds and mirror duality

We discuss recent joint work with Coates, Corti, Galkin and Kasprzyk on a mirror link between Fano threefolds and a class of threefolds obtained by generalizing certain modular threefolds.

Mirror symmetry of determinantal quintics

I describe mirror symmetry of determinantal quintics defined by generic 5 × 5 matrices with entries linear in coordinates of $P^4$. A generic determinantal quintic is singular at 50 nodes, and has a small resolution which is a Calabi-Yau threefold of $h^{1,1}$ = 2 and $h^{2,1}$ = 52. I will consider the mirror family of this quintic by the orbifold construction starting from a special family of the determinantal quintic. It turns out that the singularities of the special family are similar to the Barth-Nieto quintic, although there are some complications in our case. After making a crepant resolution, we obtain the mirror family, namely we find that the orbifold group Gorg is trivial in this case. I will also describe Calabi-Yau manifolds related to determinantal quintics which admit free $Z_2$ quotients.
This is based on the collaborations with Hiromichi Takagi.

Kelly, Tyler
University of Pennsylvania

Berglund–Hübsch–Krawitz Mirrors via Shioda Maps

We will introduce the Shioda map into the Berglund-Hübsch-Krawitz mirror duality proven by Chiodo and Ruan. In particular, we will find a new proof of birationality of BHK mirrors to certain orbifold quotients of hypersurfaces of weighted-projective n-space. We hope to talk about work-in-progress about generalized Shioda maps, BHK mirrors and Picard-Fuchs equations.

Another product formula for a Borcherds form

In his celebrated 1998 Inventiones paper, Borcherds constructed meromorphic automorphic forms $\Psi(F)$ for arithmetic subgroups associated to even integral lattices M of signature (n, 2). The input to his construction is a vector valued weakly holomorphic modular form F of weight 1-n/2, and the resulting Borcherds form has an explicit divisor on the arithmetic quotient X = $\Gamma_M \ D$. Most remarkably, in the neighborhood of each cusp (= rational point boundary component), there is a beautiful product formula for $\Psi(F)$, reminiscent of the classical product formula for the Dedekind eta-function. In this lecture, we will describe an analogous product formula for $\Psi(F)$ in the neighborhood of each 1-dimensional rational boundary component. This formula, which, like that of Borcherds, is obtained through the calculation of a regularized theta integral, reveals the behavior of $\Psi(F)$ on a (partial) smooth compactification of X.

Malmendier, Andreas
Colby College

Lecture Notes

Heterotic/F-theory duality and lattice polarized K3 surfaces.

The heterotic string compactified on $T^2$ has a large discrete symmetry group SO(2, 18;Z), which acts on the scalars in the theory in a natural way; there have been a number of attempts to construct models in which these scalars are allowed to vary by using SO(2, 18;Z)-invariant functions. In our new work (which is joint work with David Morrison), we give a more complete construction of these models in the special cases in which either there are no Wilson lines–and SO(2, 2;Z) symmetry– or there is a single Wilson line–and SO(2, 3;Z) symmetry. In those cases, the modular forms can be analyzed in detail and there turns out to be a precise theory of K3 surfaces with prescribed singularities which corresponds to the structure of the modular forms. This allows us to construct interesting examples of smooth Calabi–Yau threefolds as elliptic fibrations over Hirzebruch surfaces from pencils of irreducible genus-two curves.

(I) Quantum black holes, wall crossing, and mock modular forms.

In the quantum theory of black holes in superstring theory, the physical problem of counting the number of quarter-BPS dyonic states of a given charge has led to the study of Fourier coefficients of certain meromorphic Siegel modular forms and to the question of the modular nature of the corresponding generating functions. These Fourier coefficients have a wall-crossing behavior which seems to destroy modularity. In this talk I shall explain that these generating functions belong to a class of functions called mock modular forms. I shall then discuss some interesting examples that arise from this construction.
This is based on joint work with Atish Dabholkar and Don Zagier.

I shall discuss a conjecture of Eguchi, Ooguri and Tachikawa from 2010 that relates the elliptic genus of K3 surfaces and representations of M24, the largest Mathieu group. The generating function of these representations is a mock theta function of weight one-half. After discussing some properties of this function, I shall present a particular appearance of this function in string theory that suggests a construction of a non-trivial infinite-dimensional M24-module.
This is based on joint work with Jeff Harvey.

Pioline, Boris
University of Jussieu

Lecture Notes

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.

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.

(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.

(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.

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.

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.

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.

(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.

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

Lecture Notes

Gromov-Witten invariants and modular forms.

In this talk we shall solve the topological string amplitudes in termsof quasi modular forms for some noncompact CY 3-folds.
After a quick review of the polynomial recursion technique which is used to solve the BCOV holomorphic anomaly equations, we will construct the special polynomial ring which has a nice grading and show that topological string amplitudes are polynomials of these generators. For the cases in which the moduli space of complex structures could be identified with a modular curve, this ring is exactly the differential ring of quasi modular forms constructed out of periods. Moreover, the Fricke involution serves as a duality relating the amplitudes at the large complex structure limit and the conifold point. Combing the polynomial recursion technique and the duality, we will then be able to express the topological string amplitudes in terms of quasi modular forms. For other cases, the special polynomial ring gives a generalization of the ring of quasi modular forms without knowing much about the arithmetic properties of the moduli space.