Mathematics

The coefficients of Fourier expansions of quasi-modular forms appear naturally in enumerative geometry of elliptic curves, from the classical Hurwitz numbers, to modern Gromov- Witten invariants. The Taylor expansions of quasi-modular forms at complex multiplication points, despite its beauty, has been studied much less. In this talk, we will briefly review two enumerative theories related to the simple elliptic polynomials: the Gromov-Witten invariants and the FJRW invariants. They are related by the Landau-Ginzburg/Calabi-Yau correspondence in a much more general setting in physics/mathematics literature. Use tautological relations, we are able to prove such a correspondence via Cayley transformations of quasi-modular forms, and realize these FJRW invariants as Taylor coefficients of certain quasi-modular forms.