Charges and Fluxes in Maxwell Theory on Compact Manifolds with Boundary

Abstract

We investigate the charges and fluxes that can occur in higher-order Abelian gauge theories defined on compact space-time manifolds with boundary. The boundary is necessary to supply a destination to the electric lines of force emanating from brane sources, thus allowing non-zero net electric charges, but it also introduces new types of electric and magnetic flux. The resulting structure of currents, charges, and fluxes is studied and expressed in the language of relative homology and de Rham cohomology and the corresponding abelian groups. These can be organised in terms of a pair of exact sequences related by the Poincaré-Lefschetz isomorphism and by a weaker flip symmetry exchanging the ends of the sequences. It is shown how all this structure is brought into play by the imposition of the appropriately generalised Maxwell's equations. The requirement that these equations be integrable restricts the world-volume of a permitted brane (assumed closed) to be homologous to a cycle on the boundary of space-time. All electric charges and magnetic fluxes are quantised and satisfy the Dirac quantisation condition. But through some boundary cycles there may be unquantised electric fluxes associated with quantised magnetic fluxes and so dyonic in nature.* Presently at NORDITA, Blegdamsvej 17, DK2100 Copenhagen K q (B) = {classes of H q (B) extending to coclosed forms on M}.(10.9)

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.…”

Topological characterization of higher-dimensional charged Taub–NUT instantons

“…where c 1 = F/2π and F is the field strength of the potential (19). This is the only nonzero Chern number for U (1) configurations which are then completely classified by c 1 -see for example the standard reference by Eguchi, Gilkey and Hanson [15]. We write this number as a u(1) Chern-Simons invariant on the boundary as Taub instantons of the same dimension share a common asymptote…”

Flux Quantization in Dilatonic Taub–Nut Dyons

“…Since the U(1) structure can be restated in terms of spacetime cohomology, a similar approach can be performed for field strengths of higher rank. This is the main idea of our method and for which we follow the concepts presented in [4]. Therein, the distinction is stressed between charges and fluxes as they are classified by distinct homological structures of spacetime.…”

“…But starting from a four dimensional low energy string solution does not allow for a quantization following from a compactification scenario, as mentioned above. Nevertheless, topological methods have been formulated to study Abelian fields over general spacetimes [4]. Motivated by understanding topological quantum numbers in gravitation, in this work we study the field fluxes of the bosonic part of a truncation four dimensional low energy heterotic string theory and the homology of the spacetime on which they gravitate.…”

“…The distinction between charges and fluxes is emphasized, as they relate to cycles of different homological type. By using the magnetic flux conditions of extended objects [10], a Dirac quantization condition [11] is proved for branes of electric type of arbitrary order, living in spacetimes of any dimension [4]. This Dirac condition relates the charge created by an electric current and the magnetic flux created by the field.…”