The genus-two heterotic string cosmological constant

Abstract

After a re-examination of the different expressions proposed up to now for the cosmological constant of the heterotic string at genus two, we argue that none of them is fully satisfactory or useful for further calculations. Starting from Knizhnik's formula, written in terms of SL(2,C)-invariant coordinates, and applying the Alvarez-Osorio technique to restore modular invariance (which is connected with the Seiberg-Witten recipe to sum over spin structure contributions), we construct an alternative expression whose four main characteristics are: (1) its modular dependence is explicit and manageable, modular transformations can be performed in the obvious way and it is fully and explicitly modular invariant: no total derivatives are invoked at all; (2) it vanishes only after summation over all even spin characteristics, and before integration over moduli space; (3) its behavior in the limit in which a non-trivial homology cycle degenerates is the same obe finds in the genus-one case; (4) it factorizes explicitly in the expected way.