Spin and abelian electromagnetic duality on four-manifolds

TL;DR

This paper explores the electromagnetic duality of abelian gauge theories on four-manifolds, revealing how partition functions transform under modular groups and linking topological properties to duality symmetries.

Contribution

It demonstrates the full modular group symmetry of partition functions on four-manifolds, including cases with fractional fluxes and multiple partition functions, using lattice and theta function constructions.

Findings

Partition functions are invariant under the full modular group.

Modular transformations depend on topological invariants like Euler number and signature.

Multiple partition functions are related through modular group actions.

Abstract

We investigate the electromagnetic duality properties of an abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be "modular weights" which are linear combinations of the Euler number and Hirzebruch signature of the four-manifold. But sometimes the partition function transforms only under a subgroup of the modular group (the Hecke subgroup). In this case it is impossible to define real spinor wave functions on the four-manifold. But complex spinors are possible provided the background magnetic fluxes are appropriately fractional rather that integral. This gives rise to a second partition function which enables the full modular group to be realised by permuting the two partition functions, together with a third. Thus the full modular group is realised in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest.