Mirror symmetry is in fact a duality involving families of varieties. Dependence on the moduli is defined using the mirror map, which can therefore be described using corresponding Gauss-Manin connection and corresponding Picard-Fuchs equations.
S. Hosono, Albrecht Klemm, Stefan Theisen, Shing-Tung Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301-350 [doi:10.1007/BF02100589, arXiv:hep-th/9308122]
Bong H. Lian, Shing-Tung Yau, Arithmetic properties of mirror map and quantum coupling, Commun. Math. Phys. 176 (1996) 163–191 doi
B. Lian, S-T. Yau, Mirror symmetry, rational curves on algebraic manifolds and hypergeometric series, 163–183 in: XIth International Congress of Mathematical Physics (Paris 1994), Daniel Iagolnitzer (eds.), Intern. Press 1995
Bong H. Lian, Shing-Tung Yau, Mirror maps, modular relations and hypergeometric series I, arXiv:hep-th/9507151
Bong H. Lian, Shing-Tung Yau, Mirror maps, modular relations and hypergeometric series II, arXiv:hep-th Nuclear Physics B - Proc. Suppl. 46:1–3 (1996) 248–262 doi
Number theoretic examples are studied in
A criterium when a mirror map is in fact a elliptic modular function is in
A special case of mirror maps is discussed in video
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