Electric-magnetic duality of Abelian gauge theory on the four-torus, from the fivebrane on Tn 2 × Tn 4, via their partition functions

摘要

We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T 4 with flat metric using Dirac quantization. In addition to an ( mathrm{S}mathrm{L}left(4,;mathcal{Z}right) ) symmetry, it possesses ( mathrm{S}mathrm{L}left(2,;mathcal{Z}right) ) symmetry that is electromagnetic S-duality. We show explicitly how this ( mathrm{S}mathrm{L}left(2,;mathcal{Z}right) ) 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 T 2 times T 4, which has ( mathrm{S}mathrm{L}left(2,;mathcal{Z}right)times mathrm{S}mathrm{L}left(4,;mathcal{Z}right) ) symmetry. If we identify the couplings of the abelian gauge theory ( tau =frac{theta }{2pi }+ifrac{4pi }{e^2} ) with the complex modulus of the T 2 torus ( tau ={beta}^2+ifrac{R_1}{R_2} ), then in the small T 2 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 ( mathrm{S}mathrm{L}left(2,;mathcal{Z}right) ) 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 ( mathrm{S}mathrm{L}left(2,;mathcal{Z}right) ) acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization