String Duality and Enumeration of Curves by Jacobi Forms

Abstract

For a Calabi-Yau threefold admitting both a $K3$ fibration and an elliptic fibration (with some extra conditions) we discuss candidate asymptotic expressions of the genus 0 and 1 Gromov-Witten potentials in the limit (possibly corresponding to the perturbative regime of a heterotic string) where the area of the base of the $K3$ fibration is very large. The expressions are constructed by lifting procedures using nearly holomorphic Weyl-invariant Jacobi forms. The method we use is similar to the one introduced by Borcherds for the constructions of automorphic forms on type IV domains as infinite products and employs in an essential way the elliptic polylogarithms of Beilinson and Levin. In particular, if we take a further limit where the base of the elliptic fibration decompactifies, the Gromov-Witten potentials are expressed simply by these elliptic polylogarithms. The theta correspondence considered by Harvey and Moore which they used to extract the expression for the perturbative prepotential is closely related to the Eisenstein-Kronecker double series and hence the real versions of elliptic polylogarithms introduced by Zagier.

35 pages, submitted to the Proceedings of the Taniguchi Symposium 1997 on "Integrable Systems and Algebraic Geometry", Kobe/Kyoto

References in corpus (17)

• Compactifications of F-Theory on Calabi--Yau Threefolds -- II
• Geometric Engineering of Quantum Field Theories
• Geometric Singularities and Enhanced Gauge Symmetries
• Exact Results for N=2 Compactifications of Heterotic Strings
• Mirror Manifolds And Topological Field Theory
• Mirror Symmetry for Two Parameter Models -- II
• K3 Surfaces and String Duality
• Topological Field Theory and Rational Curves
• BPS States of Exceptional Non-Critical Strings
• BPS States, String Duality, and Nodal Curves on K3
• On the Ubiquity of K3 Fibrations in String Duality
• The SO(32) Heterotic String on a K3 Surface
• New Higgs Transitions between Dual N=2 String Models
• Perturbative gravitational couplings and Siegel modular forms in $D=4,N=2$ string models
• Making enumerative predictions by means of mirror symmetry
• Counting rational curves on K3 surfaces
• Gromov-Witten Invariants via Algebraic Geometry

Cited by in corpus (8)

• Tensionless Strings and the Weak Gravity Conjecture
• A Stringy Test of the Scalar Weak Gravity Conjecture
• Counting higher genus curves in a Calabi-Yau manifold
• Weyl groups and Elliptic Solutions of the WDVV equations
• The Ruled Vertex and D-E Degenerations
• Jacobi Forms of Higher Index and Paramodular Groups in N=2, D=4 Compactifications of String Theory
• Cosmic String, Harvey-Moore Conjecture and Family Seiberg-Witten Theory
• State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture