Spectral determinants of flat metrics on the bielliptic genus-two locus

Abstract

We obtain a closed explicit formula for the spectral determinant of flat conical metrics on the bielliptic genus-two locus. The metrics are generated by holomorphic one-forms with two simple zeros. For a symmetric reference metric, an exact Klein-four spectral identity reduces the spectral determinant to scalar determinants on spheres and tori; the singular anomaly formula then yields the general case. The resulting formula involves an elementary binary sextic in the coefficients of the one-form and two explicit hypergeometric areas of four-cone metrics on a quotient sphere. We apply this formula to the separating, one-node nonseparating, and simultaneous two-node degenerations of the curve and obtain complete asymptotics of the spectral determinant in all cases. The one-node degeneration has two distinct metric limits, according to whether the limiting one-form is holomorphic or meromorphic. Comparison of the cylindrical cases with the Bismut--Bost asymptotics determines the corresponding constants explicitly; in the separating case it also evaluates the relative determinant appearing in the M\"uller--M\"uller formula. As a by-product, the separating asymptotics evaluate the multiplicative constants left undetermined in earlier general conical and variational determinant formulas.