Hermitian Geometry and Complex Space-Time

TL;DR

This paper explores a unique Hermitian geometric action in four complex dimensions, linking complex geometry with low-energy effective theories involving metrics and antisymmetric tensors, emphasizing gauge invariance and boundary conditions.

Contribution

It identifies a unique Chern-Simons type action for Hermitian manifolds that reduces to known low-energy physics in the real limit.

Findings

The action is uniquely determined by the linearized Einstein and antisymmetric tensor coupling.

The resulting equations of motion require boundary conditions on the Hermitian metric.

The antisymmetric tensor field exhibits gauge invariance from diffeomorphisms in complex directions.

Abstract

We consider a complex Hermitian manifold of complex dimensions four with a Hermitian metric and a Chern connection. It is shown that the action that determines the dynamics of the metric is unique, provided that the linearized Einstein action coupled to an antisymmetric tensor is obtained, in the limit when the imaginary coordinates vanish. The unique action is of the Chern-Simons type when expressed in terms of the K\"{a}hler form. The antisymmetric tensor field has gauge transformations coming from diffeomorphism invariance in the complex directions. The equations of motion must be supplemented by boundary conditions imposed on the Hermitian metric to give, in the limit of vanishing imaginary coordinates, the low-energy effective action for a curved metric coupled to an antisymmetric tensor.