Five-dimensional fermionic Chern-Simons theory

TL;DR

This paper explores a novel five-dimensional fermionic Chern-Simons theory derived from twisted 5d super Yang-Mills, revealing its topological nature and connections to 6d theories and knot invariants.

Contribution

It introduces a new 5d fermionic Chern-Simons theory, computes its partition function, and discusses its relation to 6d theories and applications to generalized knots.

Findings

Partition function is a topological invariant involving Ray-Singer torsion.

Mismatch between 5d partition function and 6d index due to dimensional reduction.

Application to generalized knots with 2d sheets in 5d.

Abstract

We study 5d fermionic CS theory with a fermionic 2-form gauge potential. This theory can be obtained from 5d MSYM theory by performing the maximal topological twist. We put the theory on a five-manifold and compute the partition function. We find that it is a topological quantity, which involves the Ray-Singer torsion of the five-manifold. For abelian gauge group we consider the uplift to the 6d theory and find a mismatch between the 5d partition function and the 6d index, due to the nontrivial dimensional reduction of a selfdual two-form gauge field on a circle. We also discuss an application of the 5d theory to generalized knots made of 2d sheets embedded in 5d.