1209.6049 (Marco Matone)

Modular Invariant Regularization of String Determinants and the Serre
GAGA principle    [PDF]

Marco Matone

We show that the ratios of determinants of Laplacians which appear in string theories admit a modular invariant regularization. The key point is the modular invariance of the Bergmann reproducing kernel. This is achieved by normalizing the b-c and beta-gamma systems of weight n by the determinant of the n-fold Hadamard product of the Bergmann kernel. As a consequence, such ratios of determinants, which correspond to the path integral on the world-sheet metric, together with space-time coordinates, b-c and/or beta-gamma systems, become volume forms on the moduli space of genus g curves M_g. The construction leads to introduce a non-chiral analog of the Mumford forms without integrating over the fibers of the determinant line bundles \lambda_n, avoiding in this way a Weyl anomaly. It turns out that the building blocks are the recently introduced vector-valued Teichmueller modular forms. These naturally appear in the determinant of the Hadamard product of the Bergman kernel, that, roughly speaking, maps sections of the modulo square of the Hodge bundle |\lambda_1|^2 to sections of |\lambda_n|^2 and absorbs the point dependence of the path integral zero modes insertion, without introducing any anomaly.

View original: http://arxiv.org/abs/1209.6049