Frobenius transformation, mirror map and instanton numbers. (English) Zbl 1246.14053
Summary: We show that one can express Frobenius transformation on middle-dimensional \(p\)-adic cohomology of Calabi-Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on \(p\)-adic cohomology. We discuss a \(p\)-adic interpretation of the conjecture about integrality of Gopakumar-Vafa invariants.
MSC:
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14F25 | Classical real and complex (co)homology in algebraic geometry |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 81T45 | Topological field theories in quantum mechanics |
References:
| [1] | Kontsevich, M.; Schwarz, A.; Vologodsky, V., Phys. Lett. B, 637, 97 (2006) · Zbl 1247.14058 |
| [2] | Vologodsky, V. |
| [3] | Schwarz, A.; Shapiro, I. |
| [4] | Cox, D.; Katz, S., Mirror Symmetry and Algebraic Geometry (1999), Amer. Math. Soc. · Zbl 0951.14026 |
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| [8] | Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Commun. Math. Phys., 165, 211 (1994) |
| [9] | Witten, E. |
| [10] | Schwarz, A.; Tang, X. |
| [11] | Katz, S. |
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