Deformation Theory of Holomorphic Vector Bundles, Extended Conformal Symmetry and Extensions of 2D Gravity

TL;DR

This paper develops a geometric framework for two-dimensional field theories with extended conformal symmetry using deformation theory of holomorphic and Hermitian vector bundles, revealing new insights into anomalies and their splitting.

Contribution

It introduces a novel geometric formulation of extended conformal symmetry in 2D field theories based on deformation theory of vector bundles, including a detailed analysis of anomalies.

Findings

Extended Weyl anomaly can be split into automorphism and Weyl contributions.

The Donaldson action generalizes the Liouville action for vector bundles.

The induced central charge can be explicitly computed.

Abstract

Developing on the ideas of R. Stora and coworkers, a formulation of two dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric background consists of a vector bundle $E$ over a closed surface $\Sigma$ endowed with a holomorphic structure and a Hermitian structure subordinated to it. The symmetry group is the semidirect product of the automorphism group ${\rm Aut}(E)$ of $E$ and the extended Weyl group ${\rm Weyl}(E)$ of $E$ and acts on the holomorphic and Hermitian structures. The extended Weyl anomaly can be shifted into an automorphism chirally split anomaly by adding to the action a local counterterm, as in ordinary conformal field theory. The dependence on the scale of the metric on the fiber of $E$ is encoded in the Donaldson action, a vector bundle generalization of the Liouville action. The Weyl and automorphism anomaly split into two contributions corresponding respectively to the determinant and projectivization of $E$. The determinant part induces an effective ordinary Weyl or diffeomorphism anomaly and the induced central charge can be computed.