The Mumford form and the Polyakov measure in string theory (Q1083494)

The authors consider the moduli space \(M_ g\) of algebraic curves and obtain the Mumford form and the Polyakov measure on \(M_ g\) (for genus \(g\geq 2)\) by formulae written in terms of the complex geometry of the surface itself, instead of its spectral invariants. They compute the Polyakov measure by explicitly describing the Mumford form, a priori defined only by the implicit global conditions. They thus obtain a shorter formula than the one obtained by \textit{Yu. I. Manin} [JETP Lett. (USA) 43, No.4, 204-206 (1986); translation from Pis'ma Zh. Eksp. Teor. Fis. (USSR), 43, No.4, 161-163 (1986)]. They also obtain a new proof of a theorem of Mumford relative to sheaves associated with a family of smooth projective curves [\textit{D. Mumford}, Enseign. Math., II. Ser. 23, 39-110 (1977; Zbl 0363.14003)]. The authors also discuss the possibility of computing the Polyakov measure for the superstring in the critical dimension \(10\) by an intermediary Mumford-Berezin form on a suitable superalgebraic moduli space.