Integrality of Open Instantons Numbers

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.

13 pages

References in corpus (2)

• Knot Invariants and Topological Strings
• Mirror Symmetry, D-Branes and Counting Holomorphic Discs

Cited by in corpus (2)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories