Electric-magnetic Duality of Abelian Gauge Theory on the Four-torus, from the Fivebrane on T2 x T4, via their Partition Functions

TL;DR

This paper explores the electric-magnetic duality of abelian gauge theory on a four-torus, demonstrating its origin from six-dimensional tensor theory symmetries via partition function analysis, revealing deep connections between 4d and 6d theories.

Contribution

It explicitly links 4d S-duality of abelian gauge theory to 6d tensor theory symmetries through partition function computations on T4 and T2 x T4.

Findings

SL(4,Z) and SL(2,Z) symmetries in 4d gauge theory partition function

Origin of 4d S-duality traced to 6d tensor theory symmetries

Partition function factorization in small T2 limit

Abstract

We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T4 with flat metric using Dirac quantization. In addition to an SL(4, Z) symmetry, it possesses SL(2,Z) symmetry that is electromagnetic S-duality. We show explicitly how this SL(2, Z) S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2,0) tensor theory, by computing the partition function of a single fivebrane compactified on T2 x T4, which has SL(2,Z) x SL(4,Z) symmetry. If we identify the couplings of the abelian gauge theory \tau = {\theta\over 2\pi} + i{4\pi\over e^2} with the complex modulus of the T2 torus, \tau = \beta^2 + i {R_1\over R_2}, then in the small T2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4d gauge theory. In this way the SL(2,Z) symmetry of the 6d tensor partition function is identified with the S-duality symmetry of the 4d gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the SL(2,Z) acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.