Integrality of Open Instantons Numbers

TL;DR

This paper proves the integrality of open instanton numbers for specific Calabi-Yau threefolds by analyzing binomial coefficient congruences and deriving factorial expressions modulo prime powers.

Contribution

It establishes the integrality of open instanton numbers in new examples and generalizes classical number theory theorems related to binomial coefficients and factorials.

Findings

Confirmed integrality of open instanton numbers for the resolved conifold and degenerate P^1 x P^1.

Derived new congruences for binomial coefficients modulo prime powers.

Extended classical theorems of Wolstenholme and Wilson to broader contexts.

Abstract

We prove the integrality of the open instanton numbers in two examples of counting holomorphic disks on local Calabi-Yau threefolds: the resolved conifold and the degenerate $ \P \times \P $. Given the B-model superpotential, we extract by hand all Gromow-Witten invariants in the expansion of the A-model superpotential. The proof of their integrality relies on enticing congruences of binomial coefficients modulo powers of a prime. We also derive an expression for the factorial $(p^k-1)!$ modulo powers of the prime $p$. We generalise two theorems of elementary number theory, by Wolstenholme and by Wilson.