Abelian Duality

TL;DR

This paper demonstrates an equivalence between scalar field theory and Maxwell theory on certain three-dimensional manifolds, linking their partition functions through Ray-Singer torsion, and extends this to interactions with external currents.

Contribution

It establishes a novel duality between scalar and Maxwell theories on specific 3D manifolds, connecting their partition functions via topological invariants.

Findings

Partition functions are related by Ray-Singer torsion.

Equivalence holds for manifolds with trivial first de Rham cohomology.

Duality extends to external current interactions with fixed charge-current relations.

Abstract

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. On the level of interaction with external currents, the equivalence persists provided there is a fixed relation between the charges and the currents.