Applications of CFT determinant formulas in number theory
D. Levin, H. -G. Shin, A. Zuevsky
TL;DR
This work extends determinant-based identities from genus-one CFT to genus two by exploiting torus correlation structures and their deformed elliptic functions. It derives generalized Garvan-type formulas for powers of the modular discriminant $\Delta(\tau)$ in both genus-one and genus-two settings, expressing them as determinants of matrices built from deformed Weierstrass/Eisenstein data. Central to the approach are Fay's trisecant identity, the bosonization of fermionic partition functions, and genus-two self-sewing constructions, which yield explicit determinant representations involving $S^{(2)}$ kernels, Theta constants, and deformed Eisenstein series. The results provide new, structurally explicit links between CFT determinant formulas and classical modular forms, with potential implications for number theory and the theory of vertex operator algebras on higher-genus surfaces.
Abstract
In this note we show how to use the determinant representations for correlation functions in CFT to derive new determinant formulas for powers of the modular discriminant expressed via deformed elliptic functions with parameters. In particular, we obtain counterparts of Garvan's formulas for the modular discriminant corresponding to the genus two Riemann surface case.