The role of the Beltrami parametrization of complex structures in 2-d Free Conformal Field Theory

TL;DR

This paper reviews how the Beltrami parametrization of complex structures on Riemann surfaces enhances the understanding of 2D conformal field theory, connecting complex geometry, index theorems, and holomorphic factorization.

Contribution

It demonstrates the application of Beltrami parametrization to relate complex structures with locality, index theorems, and determinant line bundles in 2D conformal field theory.

Findings

Beltrami parametrization aligns with locality in field theory.

Application of local index theorem to families of elliptic operators.

Connection between determinant line bundles and holomorphic factorization.

Abstract

This talk gives a review on how complex geometry and a Lagrangian formulation of 2-d conformal field theory are deeply related. In particular, how the use of the Beltrami parametrization of complex structures on a compact Riemann surface fits perfectly with the celebrated locality principle of field theory, the latter requiring the use infinite dimensional spaces. It also allows a direct application of the local index theorem for families of elliptic operators due to J.-M. Bismut, H. Gillet and C. Soul\'{e}. The link between determinant line bundles equipped with the Quillen\'s metric and the so-called holomorphic factorization property will be addressed in the case of free spin $j$ b-c systems or more generally of free fields with values sections of a holomorphic vector bundles over a compact Riemann surface.