Partition Functions for Maxwell Theory on the Five-torus and for the Fivebrane on S1XT5

TL;DR

This paper compares the partition functions of 5D abelian gauge theory on a five-torus and a fivebrane on S1 times T5, revealing that they match only in the small circle limit, indicating the 6d theory as an ultraviolet completion.

Contribution

It provides a detailed calculation of partition functions for both theories and clarifies their relationship, especially the conditions under which they are equivalent.

Findings

Partition functions match in the small circle limit.

6d tensor theory acts as an ultraviolet completion.

Kaluza-Klein modes are removed in the limit.

Abstract

We compute the partition function of five-dimensional abelian gauge theory on a five-torus T5 with a general flat metric using the Dirac method of quantizing with constraints. We compare this with the partition function of a single fivebrane compactified on S1 times T5, which is obtained from the six-torus calculation of Dolan and Nappi. The radius R1 of the circle S1 is set to the dimensionful gauge coupling constant g^2= 4\pi^2 R1. We find the two partition functions are equal only in the limit where R1 is small relative to T5, a limit which removes the Kaluza-Klein modes from the 6d sum. This suggests the 6d N=(2,0) tensor theory on a circle is an ultraviolet completion of the 5d gauge theory, rather than an exact quantum equivalence.