Showing papers in "Communications in Number Theory and Physics in 2017"

Modular graph functions

Abstract: We consider properties of modular graph functions, which are non-holo- morphic modular functions associated with the Feynman graphs for a conformal scalar field theory on a two-dimensional torus. Such functions arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We demonstrate that these functions are sums, with rational coefficients, of special values of single-valued elliptic multiple polylogarithms, which will be introduced in this paper. This insight suggests the many interrelations between these modular graph functions (a few of which were motivated in an earlier paper) may be obtained as a consequence of identities involving elliptic polylogarithms.

Notes on Motivic Periods

Abstract: The second part of a set of notes based on lectures given at the IHES in 2015 on Feynman amplitudes and motivic periods.

The Galois coaction on $\phi^4$ periods

Abstract: We report on calculations of Feynman periods of primitive log-divergent $\phi^4$ graphs up to eleven loops. The structure of $\phi^4$ periods is described by a series of conjectures. In particular, we discuss the possibility that $\phi^4$ periods are a comodule under the Galois coaction. Finally, we compare the results with the periods of primitive log-divergent non-$\phi^4$ graphs up to eight loops and find remarkable differences to $\phi^4$ periods. Explicit results for all periods we could compute are provided in ancillary files.

On asymptotics and resurgent structures of enumerative Gromov–Witten invariants

Abstract: Making use of large-order techniques in asymptotics and resurgent analysis, this work addresses the growth of enumerative Gromov–Witten invariants—in their dependence upon genus and degree of the embedded curve—for several different threefold Calabi–Yau toric-varieties. In particular, while the leading asymptotics of these invariants at large genus or at large degree is exponential, at combined large genus and degree it turns out to be factorial. This factorial growth has a resurgent nature, originating via mirror symmetry from the resurgent-transseries description of the B-model free energy. This implies the existence of nonperturbative sectors controlling the asymptotics of the Gromov–Witten invariants, which could themselves have an enumerative-geometry interpretation. The examples addressed include: the resolved conifold; the local surfaces local P2 and local P1 × P1; the local curves and Hurwitz theory; and the compact quintic. All examples suggest very rich interplays between resurgent asymptotics and enumerative problems in algebraic geometry.

On the Rankin-Selberg method for higher genus string amplitudes

Abstract: Closed string amplitudes at genus h ≤ 3 are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin-Selberg method, which consists of inserting an Eisenstein series Eh(s) in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of s. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier’s extension of the Rankin-Selberg method at genus one, we develop the Rankin-Selberg method for Siegel modular functions of degree 2 and 3 with polynomial growth near the cusps. In particular, we show that the renormalized modular integral of the Siegel-Narain partition function of an even self-dual lattice of signature (d, d) is proportional to a residue of the Langlands-Eisenstein series attached to the h-th antisymmetric tensor representation of the T-duality group O(d, d,Z). E-mail: ioannis.florakis@cern.ch boris.pioline@cern.ch ar X iv :1 60 2. 00 30 8v 2 [ he pth ] 1 2 Fe b 20 16

Hemisphere Partition Function and Analytic Continuation to the Conifold Point

Abstract: We show that the hemisphere partition function for certain U(1) gauged linear sigma models (GLSMs) with D-branes is related to a particular set of Mellin-Barnes integrals which can be used for analytic continuation to the singular point in the K\"ahler moduli space of an $h^{1,1}=1$ Calabi-Yau (CY) projective hypersurface. We directly compute the analytic continuation of the full quantum corrected central charge of a basis of geometric D-branes from the large volume to the singular point. In the mirror language this amounts to compute the analytic continuation of a basis of periods on the mirror CY to the conifold point. However, all calculations are done in the GLSM and we do not have to refer to the mirror CY. We apply our methods explicitly to the cubic, quartic and quintic CY hypersurfaces.

Calabi-Yau modular forms in limit: Elliptic Fibrations

Abstract: We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

Equivariant K3 Invariants

Abstract: In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformai field theory. We consider the invariants calculated by Yau-Zaslow (capturing the Euler characters of the moduli spaces of D2-branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz-Klemm-Vafa (KKV), and Katz-Klemm-Pandharipande (KKP). We observe that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently been used to establish mock modular moonshine for a variety of sporadic simple groups that are subgroups of Conway's group. This observation leads us to conjectural descriptions of equivariant versions of the KKV and KKP invariants. A K3 sigma model is specified by a choice of 4-plane in the K3 D-brane charge lattice. Symmetries of K3 sigma models are naturally identified with 4-plane preserving subgroups of the Conway group, according to the work of Gaberdiel-Hohenegger-Volpato, and one may consider corresponding equivariant refined K3 Gopakumar-Vafa invariants. The same symmetries naturally arise in the auxiliary CFT state space, affording a suggestive alternative view of the same computation. We comment on a lift of this story to the generating function of elliptic genera of symmetric products of K3 surfaces, and the connection to work of Oberdieck-Pandharipande on curve counts for the product of a K3 surface with an elliptic curve.

On cubic Hodge integrals and random matrices

Abstract: A conjectural relationship between the GUE partition function with even couplings and certain special cubic Hodge integrals over the moduli spaces of stable algebraic curves is under consideration.

Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

Abstract: We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors-Keating, and Smilansky, we formulate an analog of the Hardy-Littlewood prime ...

I-functions of Calabi-Yau 3-folds in Grassmannians

Abstract: We study $I$-functions of Calabi--Yau 3-folds with Picard number one which are zero loci of general sections of direct sums of globally generated irreducible homogeneous vector bundles on Grassmannians.

Hecke-type formulas for families of unified Witten-Reshetikhin-Turaev invariants

Abstract: Every closed orientable 3-manifold can be constructed by surgery on a link in S. In the case of surgery along a torus knot, one obtains a Seifert fibered manifold. In this paper we consider three families of such manifolds and study their unified WittenReshetikhin-Turaev (WRT) invariants. Thanks to recent computation of the coefficients in the cyclotomic expansion of the colored Jones polynomial for (2, 2t+ 1)-torus knots, these WRT invariants can be neatly expressed as q-hypergeometric series which converge inside the unit disk. Using the Rosso-Jones formula and some rather non-standard techniques for Bailey pairs, we find Hecke-type formulas for these invariants. We also comment on their mock and quantum modularity.

Ramanujan identities and quasi-modularity in Gromov–Witten theory

Abstract: We prove that the ancestor Gromov–Witten correlation functions of one-dimensional compact Calabi–Yau orbifolds are quasi-modular forms. This includes the pillowcase orbifold which can not yet be handled by using Milanov–Ruan’s B-model technique. We first show that genus zero modularity is obtained from the phenomenon that the system of WDVV equations is essentially equivalent to the set of Ramanujan identities satisfied by the generators of the ring of quasi-modular forms for a certain modular group associated to the orbifold curve. Higher genus modularity then follows by using tautological relations.

Stringy Chern classes of singular toric varieties and their applications

Abstract: Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional reflexive polytopes to the number 12 in dimension d>3.

Tropical count of curves on abelian varieties

Abstract: We investigate the problem of counting tropical genus g curves in g-dimensional tropical abelian varieties. For g = 2, 3, we prove that the tropical count matches the count provided by Gottsche, Bryan-Leung, and Lange-Sernesi in the complex setting.

Properties of the Extended Graph Permanent

Abstract: Previously, the graph permanent was introduced as a single-valued invariant for graphs $G$ with $|E(G)| = k(|V(G)|-1)$ for some $k \in \mathbb{Z}_{>0}$. Herein, we construct the extended graph permanent, an infinite sequence for all graphs. We prove that, like the graph permanent, the extended graph permanent is invariant under the graph operations that are known to preserve the period. Further, the original construction and extension arise from permanents of matrices, but we construct a novel graph polynomial such that the sequence can be generated from the point count of this polynomial, as a residue over prime-order finite fields.