p-brane Dyons, θ-terms and Dimensional Reduction

We present two novel derivations of the recently established (−) factor in the charge quantization condition for p-brane dyon sources in spacetime dimension D=2p+2. The first requires consistency of the condition under the charge shifts produced by (generalized) θ-terms. The second traces the sign difference between adjoining dimensions to compactification effects. [email protected]; [email protected]; [email protected] It was recently established [1] that the generalized dyon quantization condition, for (p–1)-brane dyons coupled to Abelian p-forms in spacetime dimension D=2p+2, involves a p-dependent sign: eḡ + (−)ēg = 2πnh̄, n ∈ Z . (1) The − sign is of course that familiar in D=4 electrodynamics [2, 3, 4]. This sign dependence was actually anticipated [5] through analysis of chiral sources coupled to chiral 2p-forms. It was particularly stressed in [6], where it was related to supergravity duality groups in higher dimensions [7]. Another approach is based on dyon-dyon scattering; the relation between D=10 and D=4 is also discussed there [8]. Our aim is to illuminate this phenomenon through two new arguments. The first is based on the study of the shift in the “electric” charges induced by a (p-generalized) θ-term. The other uses dimensional reduction, or rather enhancement, to relate the conditions (1) in adjoining dimensions. For concreteness, we shall work primarily with D=4 one-forms and D=6 two-forms to illustrate the generic situations. 1. θ-terms. a) In D=4 it is well known that adding a θ-term, (θ/2)Fμν F μν (* always represents dualization) to the Lagrangian has the effect of shifting the electric charge of an (e, g) dyon according to [9] e′ = e− 2gθ . (2) A remarkable feature of this shift is its compatibility with the usual Dirac quantization condition for electric and magnetic charges. Namely, if one simultaneously shifts all dyon electric charges according to (2) starting from values (ea, ga) that obey (1), then the charges 1 (e′a, ga) also do, because e′agb − e ′ bga = (ea − gaθ)gb − (eb − gbθ)ga = eagb − ebga . (3) The − sign is crucial in this result. Indeed, it is the answer to the converse question: what sign in the quantization condition (1) leaves it invariant under the shift (2)? b) In D=6, there is no θ-term for a single 2-form since FABC F vanishes identically. However, a θ-term is possible with two 2-forms A, i = 1, 2. The sources here are strings characterized by four strengths (“charges”) (e a , g (i) a ), the respective electric (magnetic) charges of string a coupled to A. We use a uniform convention for the signs of the couplings (e a , g (i) a ) to the 2-forms: the electric couplings enter with the same sign in the minimal coupling term 2