Modular invariant regularization of string determinants and the Serre GAGA principle

Abstract

Since any string theory involves a path integration on the world-sheet metric, their partition functions are volume forms on the moduli space of genus $g$ Riemann surfaces ${\mathcal{M}}_{g}$, or on its super analog. It is well known that modular invariance fixes strong constraints that in some cases appear only at higher genus. Here we classify all the Weyl and modular invariant partition functions given by the path integral on the world-sheet metric, together with space-time coordinates, $b\text{\ensuremath{-}}c$ and/or $\ensuremath{\beta}\text{\ensuremath{-}}\ensuremath{\gamma}$ systems, that correspond to volume forms on ${\mathcal{M}}_{g}$. This was a long standing question, advocated by Belavin and Knizhnik, inspired by the Serre GAGA principle and based on the properties of the Mumford forms. The key observation is that the Bergman reproducing kernel provides a Weyl and modular invariant way to remove the point dependence that appears in the above string determinants, a property that should have its superanalog based on the super Bergman reproducing kernel. This is strictly related to the properties of the propagator associated to the space-time coordinates. Such partition functions $\mathcal{Z}[\mathcal{J}]$ have well-defined asymptotic behavior and can be considered as a basis to represent a wide class of string theories. In particular, since noncritical bosonic string partition functions ${\mathcal{Z}}_{D}$ are volume forms on ${\mathcal{M}}_{g}$, we suggest that there is a mapping, based on bosonization and degeneration techniques, from the Liouville sector to first order systems that may identify ${\mathcal{Z}}_{D}$ as a subclass of the $\mathcal{Z}[\mathcal{J}]$. The appearance of $b\text{\ensuremath{-}}c$ and $\ensuremath{\beta}\text{\ensuremath{-}}\ensuremath{\gamma}$ systems of any conformal weight shows that such theories are related to $W$ algebras. The fact that in a large $N$ 't Hooft-like limit two-dimensional ${W}_{N}$ minimal models conformal field theories are related to higher spin gravitational theories on ${\mathrm{AdS}}_{3}$, suggests that the string partition functions introduced here may lead to a formulation of higher spin theories in a string context.