2000/03/17 by M. A. G. Álvarez, David I. Olive, D. Olive +1 · 1 citation
Physics and Astronomy · #Black Holes and Theoretical Physics #Physics of Superconductivity and Magnetism #Topological Materials and Phenomena #hep-th
paper · pdf · doi:10.1007/s002200000354
published as Commun.Math.Phys.217:331-356,2001 · 29 pages, Plain TeX
arxiv created 2000/03/17 · openalex publication_date 2001/03/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.