Normal Coordinates in Kahler Manifolds and the Background Field Method

TL;DR

This paper introduces Kahler normal coordinates (KNC) as a holomorphic alternative to Riemann normal coordinates for Kahler manifolds, enabling a covariant background field method that preserves complex structure in supersymmetric sigma models.

Contribution

It proves that KNC transform holomorphically and extends RNC to Kahler manifolds, facilitating covariant calculations in supersymmetric theories.

Findings

KNC are holomorphic tangent vectors under coordinate transformations.

KNC expansion maintains complex structure covariance.

Provides a new framework for supersymmetric nonlinear sigma models.

Abstract

Riemann normal coordinates (RNC) are unsuitable for \kahler manifolds since they are not holomorphic. Instead, \kahler normal coordinates (KNC) can be defined as holomorphic coordinates. We prove that KNC transform as a holomorphic tangent vector under holomorphic coordinate transformations, and therefore that they are natural extensions of RNC to the case of \kahler manifolds. The KNC expansion provides a manifestly covariant background field method preserving the complex structure in supersymmetric nonlinear sigma models.