Tunneling solutions in topological field theory on R x S^3 x I

TL;DR

This paper investigates solutions in a topologically twisted five-dimensional supersymmetric Yang-Mills theory on a manifold with a three-sphere, analyzing how solutions evolve as the interval length varies, and explicitly constructing interpolating instanton solutions.

Contribution

It provides a detailed analysis of tunneling solutions connecting static solutions in a twisted 5D Yang-Mills theory on R x S^3 x I, including explicit power series solutions.

Findings

Solutions exist for large interval I, disappear for short I

Interpolating instanton solutions connect static solutions

Explicit power series expressions for solutions are derived

Abstract

We consider a topologically twisted version, recently introduced by Witten, of five-dimensional maximally supersymmetric Yang-Mills theory on a five-manifold of the form M_5 =R x W_3 x I. If the length of the interval I is sufficiently large, the supersymmetric localization equations admit pairs of static solutions (with the factor R interpreted as Euclidean time). However, these solutions disappear for a sufficiently short I, so by the topological invariance of the theory, they must be connected by an interpolating dynamic instanton solution. We study this for the case that W_3 is a three-sphere S^3 with the standard metric by making a spherically symmetric Ansatz for all fields. The solution is given as a power series in a parameter related to the length of I, and we give explicit expressions for the first non-trivial terms.