Kontsevich, Maxim; Schwarz, Albert; Vologodsky, Vadim Integrality of instanton numbers and \(p\)-adic B-model. (English) Zbl 1247.14058 Phys. Lett., B 637, No. 1-2, 97-101 (2006). Summary: We study integrality of instanton numbers (genus zero Gopakumar-Vafa invariants) for quintic and other Calabi-Yau manifolds. We start with the analysis of the case when the moduli space of complex structures is one-dimensional; later we show that our methods can be used to prove integrality in general case. We give an expression of instanton numbers in terms of Frobenius map on \(p\)-adic cohomology; the proof of integrality is based on this expression. Cited in 17 Documents MSC: 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14J33 Mirror symmetry (algebro-geometric aspects) 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 11Z05 Miscellaneous applications of number theory 81T45 Topological field theories in quantum mechanics 14F20 Étale and other Grothendieck topologies and (co)homologies Keywords:genus zero Gopakumar-Vafa invariants; Calabi-Yau manifolds × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Candelas, P.; de la Ossa, X.; Green, P.; Parkes, L., Nucl. Phys. B, 359, 21 (1991) · Zbl 1098.32506 [2] Givental, A. [3] Gopakumar, R.; Vafa, C. [4] V. Vologodsky, On the canonical coordinates, in preparation; V. Vologodsky, On the canonical coordinates, in preparation · Zbl 1235.18011 [5] M. Kontsevich, A. Schwarz, V. Vologodsky, On integrality of instanton numbers, in preparation; M. Kontsevich, A. Schwarz, V. Vologodsky, On integrality of instanton numbers, in preparation · Zbl 1247.14058 [6] Morisson, D. [7] Cox, D.; Katz, S., Mirror Symmetry and Algebraic Geometry (1999), American Mathematical Society: American Mathematical Society Providence · Zbl 0951.14026 [8] Candelas, P.; de la Ossa, X.; Rodriguez-Villegas, F. [9] Wan, D. [10] Mazur, B., Ann. Math., 98, 58 (1973) · Zbl 0261.14005 [11] Steenbrink, J., Invent. Math., 31 (1976) · Zbl 0303.14002 [12] G. Faltings, Crystalline cohomology and \(p\); G. Faltings, Crystalline cohomology and \(p\) · Zbl 0805.14008 [13] Lian, B.; Yau, S. T. [14] A. Schwarz, I. Shapiro, Supergeometry and arithmetic geometry, in preparation; A. Schwarz, I. Shapiro, Supergeometry and arithmetic geometry, in preparation · Zbl 1215.14016 [15] Stienstra, J. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.