# Abelian Duality

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm9916.html
**Parent**: https://www.maths.tcd.ie/research/papers/

**Abelian Duality**

We show that on three--dimensional Riemannian manifolds without boundaries
and with trivial first real de Rham cohomology group (and in no other
dimensions) scalar field theory and Maxwell theory are equivalent: the
ratio of the partition functions is given by the Ray--Singer torsion of
the manifold. At the level of interaction with external currents, the
equivalence persists provided there is a fixed relation between the
charges and the currents.