Schedule for: 16w5009 - Modular Forms in String Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature $(n-2,2)$. Their functions, which depend on two pairs of time like vectors, are obtained by `completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. We show that their completed modular series arises as integrals of the 2-form valued theta functions, defined in old joint work of the author and John Millson, over a surface S determined by the pairs of time like vectors. This gives an alternative construction of such series and a conceptual basis for their modularity.

A partial theta function is a function associated to a positive cone of a positive definite rational lattice. Such objects appear in representation theory, knot theory and partitions. These functions are not modular but have modular-like transformation properties and these can be used to determine their asymptotic behavior. My motivation for studying them is that they appear as characters of certain logarithmic conformal field theories and that I expect that asymptotic dimensions coincide with Hopf links in the braided tensor category of the conformal field theory. I will illustrate this picture in the examples of Kostant false theta functions of ADE-type root lattices and corresponding W-algebras.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Holomorphic indefinite theta series are approximately the sum over the intersection of a lattice and a closed cone in the associated real quadratic space. It is necessary for convergence that this cone is non-negative. Polyhedral cones are cones which correspond to hyperbolic polyhedra in the projectivisation of the real quadratic space. They can be used to approximate all other cones, and on the other hand can be built up from tetrahedral cones. Zwegers's thesis contains the case of a 1-tetrahedron in the projectivisation of a quadratic space of signature $(n,1)$. In summer, the case of positive, rectangular cones that are positive in signature $(n,2)$ was treated.

Vector-valued modular forms have recently been studied for applications to conformal field theory. In this talk, given an n-dimensional representation of the metaplectic group (e.g. the Weil representation of a finite quadratic module) we study the module of vector-valued modular forms for this representation, using methods from algebraic geometry. We prove that this module is free of rank n over the ring of level one modular forms, and we discuss the problem of finding the weights of a generating set. For Weil representations of cyclic quadratic modules of order 2p, p a prime, we show how the generating weights can be expressed in terms of class numbers of quadratic imaginary fields, and compute the distribution of the weights as p goes to infinity. This is joint work with Cameron Franc (U. Sask.) and Gene Kopp (U. Michigan).

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

It is known that birational geometry of Calabi-Yau manifolds naturally appears when we describe mirror symmetry. In this talk, I will show some interesting examples which have birational automorphisms of infinite order, and identify them with monodromy transformations in their mirror families. I will also discuss some applications to the boundary conditions fixing the so-called holomorphic ambiguities in BCOV holomorphic anomaly equation. This talk is based on collaborations with Hiromich Takagi.

I will explain how local mirror symmetry allows us to use A-model (enumerative/intersection theoretic) computations to evaluate Bloch-Beilinson regulators for families of elliptic curves. This will then be applied in two directions, both inspired by physics: to evaluate the sunset Feynman integral (in the case of unequal masses); and to explore possible implications of a recent and far-reaching conjecture of Marino on quantum curves. This talk is based on joint work with S. Bloch, C. Doran and P. Vanhove.

Given a spectral curve, the Eynard-Orantin topological recursion constructs an infinite sequence of meromorphic differentials. Those can be assembled into a wavefunction, which is believed to be the WKB asymptotic solution of a differential equation that is a quantization of the original spectral curve. This connection may have implications for many areas of enumerative geometry. In a recent paper we proved that this statement is true for a large class of genus zero spectral curves, that includes most (if not all) cases previously studied in the literature. I would also like to report on recent progress for genus one spectral curves. We consider the family of elliptic spectral curves in Weierstrass normal form, and show explicitly that the perturbative wavefunction constructed from the topological recursion is not the right object to consider; one must include non-perturbative corrections. However, we can prove, to order five in hbar, that the non-perturbative wavefunction is the WKB solution of a quantization of the spectral curve, albeit a non-trivial one. We obtain along the way interesting identities for cycle integrals of elliptic functions. This is joint work with N. Chidambaram, B. Eynard and T. Dauphinee.

I will outline the recent proof of Norton's Generalized Moonshine conjecture following the Borcherds-Hoehn program.

(Joint with Carlo Verschoor.) The curve in question was introduced as a Riemann surface by R. Fricke in 1899 and as an algebraic curve by A.M. Macbeath in 1965. It is an example of a Shimura curve; moreover, it is the unique Hurwitz curve of genus 7. Only quite recently models of it over the rational numbers were found. We describe how to compute the number of rational points on reductions mod p of these models.

Shimura curves are generalisations of classical modular curves. However, because of the lack of cusps on Shimura curves, there have been very few explicit methods for Shimura curves. In this talk, we will introduce two realisations of modular forms on Shimura curves, one in terms of solutions of Schwarzian differential equations and the other in terms of Borcherds forms, and discuss their applications, including special values of hypergeometric functions, equations of (hyperelliptic) Shimura curves, and explicit description of quaternionic loci in Siegel's modular threefold.

Via deformation theory, Calabi-Yau varieties in positive characteristic are associated with one-dimensional formal groups. Such formal groups are determined by the height. In this talk, we consider some Calabi-Yau threefolds arising in mirror symmetry, and calculate the height of their formal groups and related quantities.

The speaker will discuss recent work on Manin's theory of zeta polynomials for modular forms. He will describe recent results which confirm Manin's speculation that there is such a theory which arises from periods of newforms. More precisely, for each even weight $k>2$ newform $f$ the speaker will describe a canonical polynomial $Z_f (s)$ which satisfies a functional equation of the form $Z_f (s) = Z_f (1 - s)$, and also satisfies the Riemann Hypothesis: if $Z_f (\rho) = 0$, then $Re(\rho) = 1/2$. This zeta function is arithmetic in nature in that it encodes the mo- ments of the critical values of $L(f, s)$. This work builds on earlier results of many people on period polynomials of modular forms. This is joint work with Seokho Jin, Wenjun Ma, Larry Rolen, Kannan Soundararajan, and Florian Sprung.

In my talk I will examine instances of modularity of (rigid) Calabi--Yau manifolds whose periods are expressed in terms of hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form can be often read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of $p$ and from Weil's bounds. I will also discuss some hypergeometric features that relate the non-truncated sums---the periods, to the critical $L$-values of the modular forms.

The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this talk I will give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, etc. The orbifold singularity (rather than the cusp-like singularity) on the base of the family is responsible for this extra solution. The extra solution plays an important role in the so-called Calabi-Yau/Landau-Ginzburg correspondence which matches different enumerative theories. The same discussion also applies to slightly more general cases, e.g. Calabi-Yau hypersurfaces in projective spaces.

For a hypergeometric abelian variety, we can express the number of it rational points over finite fields in terms of the so-called hypergeometric functions over finite fields, which gives us the information on the corresponding local zeta function. Based on this relationship, we will talk about some interesting results on certain hypergeometric abelain varieties such as generalized Legendre curves and their higher dimensional analogues.

Hypergeometric functions of order 4 appear as hemisphere partition functions of the A-model of the quintic threefold and as periods of the B-model of the mirror quintic. We discuss their analytic continuation to the singular or conifold point and its conjectured relation to the L-series of the quintic at the conifold point determined by Schoen.

The most standard mathematical reformulations of conformal field theory are vertex operator algebras (Wightman axioms) and conformal nets (Haag-Kastler axioms). A conformal field theory lives in the representations of such structures. Representation theory is most elegantly formulated in the language of categories, and the relevant categories for both vertex operator algebras and conformal nets (in the best-understood=semisimple case) are called modular tensor categories. These are the categories which give rise to knot and link invariants for 3-manifolds. One of the deepest open questions in the theory, with the most ramifications, is Reconstruction: which modular tensor categories arise from vertex operator algebras or conformal nets? This is the analogue here of Tannaka-Krein duality: if you look like the category of representations of a compact group, then you ARE the category of representations of a compact group. In my talk I'll explain why this is such an important question, and review the little that is currently understood about it. Then I'll explain the fundamental role which modular forms play in attacking this problem.

We give an explicit description for the relation between generic Kummer surfaces with principal polarization and associated to a (1, 2)-polarized abelian surfaces from the point of view of 1) the geometry of quartic surfaces in $P^3$ using even-eights, 2) Jacobian elliptic K3 surfaces of Picard-rank 17 over $P^1$ using Nikulin involutions, 3) Siegel modular forms using two-isogeny. This is joint work with Adrian Clingher.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.