Integrality theorems in the theory of topological strings

TL;DR

This paper simplifies the derivation of instanton numbers and mirror maps in topological string theory using p-adic cohomology, and proves integrality theorems for these quantities and holomorphic disk counts.

Contribution

It provides a new, simplified proof of integrality theorems in topological strings using Frobenius maps on p-adic cohomology, and extends this to holomorphic disk counts.

Findings

Proof of integrality of instanton numbers and mirror maps

Verification of integrality of holomorphic disk counts

Expression of these quantities in terms of Frobenius map on p-adic cohomology

Abstract

We give a simplified derivation of the expression of instanton numbers and of mirror map in terms of Frobenius map on p-adic cohomology and use this expression to prove integrality theorems. Modifying this proof we verify that the Aganagic-Vafa formulas for the number of holomorphic disks can be expressed in terms of Frobenius map on p-adic relative cohomology; this expression permits us to prove integrality of this number.