The Dirac Quantisation Condition for Fluxes on Four-Manifolds

Abstract

A systematic treatment is given of the Dirac quantisation condition for electromagnetic fluxes through two-cycles on a four-manifold space-time which can be very complicated topologically, provided only that it is connected, compact, oriented and smooth. This is sufficient for the quantised Maxwell theory on it to satisfy electromagnetic duality properties. The results depend upon whether the complex wave function needed for the argument is scalar or spinorial in nature. An essential step is the derivation of a "quantum Stokes' theorem" for the integral of the gauge potential around a closed loop on the manifold. This can only be done for an exponentiated version of the line integral (the "Wilson loop") and the result again depends on the nature of the complex wave functions, through the appearance of what is known as a Stiefel-Whitney cohomology class in the spinor case. A nice picture emerges providing a physical interpretation, in terms of quantised fluxes and wave-functions, of mathematical concepts such as spin structures, spin C structures, the Stiefel-Whitney class and Wu's formula. Relations appear between these, electromagnetic duality and the Atiyah-Singer index theorem. Possible generalisation to higher dimensions of space-time in the presence of branes is mentioned.

Deligne-Beilinson Cohomology and Abelian Link Invariants

“…The applications of the Deligne-Beilinson (DB) cohomolgy [7,8,9,10,11] -and of its various equivalent versions such as the Cheeger-Simons Differential Characters [12,13] or Sparks [14] in quantum physics has been discussed by various authors [15,16,17,18,19,21,20,22,23]. For instance, geometric quantization is based on classes of U (1)-bundles with connections, which are exactly DB classes of degree one (see Section 8.3 of [24]); and the Aharanov-Bohm effect also admits a natural description in terms of DB cohomology.…”

Charges and Fluxes in Maxwell Theory on Compact Manifolds with Boundary

“…Then there is a possibility of fractional quantisation conditions when the wave function is of a spinor nature (involving half-integers instead of integers). The precise rule is easy to state when m = 4 [Alvarez and Olive 2000].…”

“…Taking the latter into account requires that the schematic term (1.1) in the action be unambiguous when suitably exponentiated. This constrains the values of the magnetic fluxes to satisfy a generalisation of Dirac's celebrated quantisation when compared with any of the electric charges [Dirac 1931, Wu and Yang 1975, Alvarez and Olive 2000.…”

Non-linear electrodynamics in curved backgrounds

“…On the other hand, we can require a rather high symmetry such as 26) where the diagonal matrix denotes the one in which only the i-th diagonal entry is nonvanishing. Then of course we can derive a much restrictive condition…”