# Topological field theories of 2- and 3-forms in six dimensions

## TL;DR

This paper explores topological field theories involving 2- and 3-forms in six dimensions, showing their invariance under various potential terms and connecting them to 3D gravity through dimensional reduction.

## Contribution

It introduces a class of topological 6D theories with different potential terms, relating them to Hitchin theory and demonstrating their reduction to 3D gravity.

## Key findings

- Adding potential terms preserves topological nature.
- The $C ilde{C}$ theory relates to complex and para-complex manifolds.
- Dimensional reduction yields 3D gravity.

## Abstract

We consider several diffeomorphism invariant field theories of 2- and 3-forms in six dimensions. They all share the same kinetic term $BdC$, but differ in the potential term that is added. The theory $BdC$ with no potential term is topological - it describes no propagating degrees of freedom. We show that the theory continues to remain topological when either the $BBB$ or $C\hat{C}$ potential term is added. The latter theory can be viewed as a background independent version of the 6-dimensional Hitchin theory, for its critical points are complex or para-complex 6-manifolds, but unlike in Hitchin's construction, one does not need to choose of a background cohomology class to define the theory. We also show that the dimensional reduction of the $C\hat{C}$ theory to three dimensions, when reducing on S3, gives 3D gravity.