Global Aspects of Electric-Magnetic Duality

Abstract

We show that the partition function of free Maxwell theory on a generic Euclidean four-manifold transforms in a non-trivial way under electric-magnetic duality. The classical part of the partition sum can be mapped onto the genus-one partition function of a 2d toroidal model, without the oscillator contributions. This map relates electric-magnetic duality to modular invariance of the toroidal model and, conversely, the $O(d,d',\Z)$ duality to the invariance of Maxwell theory under the 4d mapping class group. These dualities and the relation between toroidal models and Maxwell theory can be understood by regarding both theories as dimensional reductions of a self-dual 2-form theory in six dimensions. Generalizations to more $U(1)$-gauge fields and reductions from higher dimensions are also discussed. We find indications that the Abelian gauge theories related to 4d string theories with $N=4$ space-time supersymmetry are exactly duality invariant.