Integrality of Open Instantons Numbers

Details Integrality of Open Instantons Numbers Integrality of Open Instantons Numbers

2003-05-07

arxiv.org/abs/hep-th/0305057v1

J.Geom.Phys. 52 (2004) 284-297

Physics

High Energy Physics

High Energy Physics - Theory

13 pages

Scientific paper

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.

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