Determinant representations for Garvan formulas

Abstract

In this note, we demonstrate how determinant representations for correlation functions in conformal field theory can be used to derive explicit determinant formulas for powers of the classical $\eta$-function, expressed via deformed elliptic functions with parameters. At genus one, comparing the fermionic and bosonic evaluations of a twisted torus partition function, together with an elliptic Frobenius/Fay determinant identity, gives for every $n\ge 1$ a determinant formula for $\eta (\tau {)}^{3n}$ built from a twisted Szegő kernel; at $n\equiv 0,1(\mathrm{mod}8)$ this gives determinant identities for powers of the modular discriminant $\Delta ={\eta }^{24}$ of the same shape as the classical formulas of Garvan and Milne, although those classical determinants are not recovered. At genus two, an analogous comparison for a torus formed by self-sewing, combined with a finite/infinite determinant factorization of the genus two Szegő kernel, gives a determinant formula for $\eta (\tau {)}^{3{\kappa }^2}$, $-1/2<\kappa <1/2$, in terms of genus two data; identifying the genus two modular form to which this specializes is left open.