Arithmetic properties of mirror map and quantum coupling. (English) Zbl 0867.14017

The mirror map and the quantum coupling are known to have many interesting number theoretic properties. For example, the Fourier coefficients of the mirror map are integral in all known cases. In this paper, the authors prove that some K3 mirror maps are algebraic over \(\mathbb{Q} (J)\) by solving the Schwarzian differential equation in terms of the \(J\)-function, which leads to a uniform proof that these mirror maps have integral Fourier coefficients. The authors also give another proof that those mirror maps are integral by using the fact that such maps are genus zero functions as Riemann mappings, and conjecture a connection between K3 mirror maps and the Thompson series. Finally, they discuss \(\text{mod } p\) congruences for the cases of threefolds and the quintics.


14J28 \(K3\) surfaces and Enriques surfaces
14M30 Supervarieties
14J30 \(3\)-folds
Full Text: DOI arXiv


[1] Candelas, P., De la Ossa, X., Green, P., Parkes, L.: Nucl. Phys.B359, 21 (1991) · Zbl 1098.32506
[2] Essays on Mirror Manifolds, S.-T. Yau, (ed.) Hong Kong: International Press, 1992
[3] Batyrev, V., van Straten, D.: Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties. Preprint 1993 · Zbl 0843.14016
[4] Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces. HUTMP-93/0801, LMU-TPW-93-22 (hep-th/9308122), Commun. Math. Phys., to appear · Zbl 0814.53056
[5] Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces. Preprint HUTMP-94-02. hep-th 9406055 · Zbl 0920.32020
[6] Clemens, H.: Proc. ICM, Berkeley (1986) 634
[7] Katz, S.: Math. Z.191, 293 (1986) · Zbl 0563.14020
[8] Klemm, A., Lian, B. H., Roan, S. S., Yau, S.-T.: A Note on ODEs from Mirror Symmetry. To appear in the Conference Proceedings in honor of I. Gel’fand
[9] Klemm, A., Lian, B. H., Roan, S. S., Yau, S.-T.: Differential Equations from Mirror Symmetry. In preparation · Zbl 0853.14019
[10] Batyrev, V.: Dual Polyhedra and the Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties. Univ. Essen Preprint (1992), to appear in J. Alg. Geom. · Zbl 0829.14023
[11] Roan, S.-S.: Intern. J. Math.1, 211–232 (1990) · Zbl 0793.14031
[12] Greene, B., Plesser, R.: Nucl. Phys. B338, 15–37 (1990)
[13] Morrison, D. R.: In Essays on mirror manifolds, S.-T. Yau (ed) Singapore: International Press, 1992
[14] Verlinde, E., Warner, N.: Phys. Lett.269B, 96 (1991)
[15] Klemm, A., Theisen, S., Schmidt, M.: Int. J. Mod. Phys.A7, 6215 (1992) · Zbl 0954.81543
[16] Bateman, H.: Higher Transcendental Functions. New York: McGraw Hill, 1953 · Zbl 0051.34703
[17] Conway, J. H., Norton, S. P.: Monstrous Moonshine. Bull. London Math. Soc.11, 308–339 (1979) · Zbl 0424.20010
[18] Birch, B. J., Kuyk, W. ed: Modular Functions of One Variable IV. Proc. Intern. Summer School, Antwerp, 1972, Berlin, Heidelberg, New York: Springer
[19] Kluit, P. G.: On the Normalizer of{\(\Gamma\)} 0(n). In: Modular Functions on One Variable V. Bonn: Springer 1976, pp. 239–246
[20] Frenkel, I., Lepowsky J., Meurman, A.: Vertex Operator algebras and the monster. Boston: Academic Press, 1988 · Zbl 0674.17001
[21] Borcherds, R.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math.109, 405–444 (1992) · Zbl 0799.17014
[22] Mukai, S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math.94, 183–221 (1988) · Zbl 0705.14045
[23] Strominger, A.: Commun. Math. Phys.133, 163 (1990); Candelas P., della Ossa X.: Nucl. Phys.B355, 455 (1991) · Zbl 0716.53068
[24] Tian, G.: In Mathematical Aspects of String Theory. S.-T. Yau (ed) Singapore: World Scientific 1987
[25] Aspinwall, P., Morrison, D.: Commun. Math. Phys.151, 45 (1993) · Zbl 0776.53043
[26] Witten, E.: Commun. Math. Phys.118, 411 (1988) · Zbl 0674.58047
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