ar X iv : h ep - t h / 99 05 06 9 v 3 1 9 O ct 1 99 9 A note on Seiberg - Witten central charge

The central charge for the Seiberg-Witten low-energy effective Action is computed using Noether supercharges. A reliable method to construct supersymmetric Noether currents is presented. PACS numbers: 11.30.Pb , 11.30.-j , 12.60.Jv , 11.10.Ef The electromagnetic duality of Seiberg-Witten [1] relies heavily on the BPS mass-formula M = |Z| where Z = (nea+ nmaD) (1) is the topological central charge of the N=2 supersymmetry. In Ref. [2] the classical Z was computed and it was conjectured that if the fields belong to the small representation of the N=2 supersymmetry they should saturate the BPS mass-formula, giving M = |Z|, even at the quantum level. But so far the only direct evidence of this is a BPS-type computation [3] of the minimum of the Hamiltonian, neglecting fermionic fields. As stressed by Seiberg and Witten, another way to find the modification to the classical BPS formula is to compute the central charge Z from the low-energy U(1) effective Action because only the massless (and neutral) degrees of freedom contribute to Z. A way to compute the central charge Z is from the commutation relations of the supersymmetric Noether charges. The purpose of this paper is to present the result of such a computation for the low-energy U(1) part of the effective Action. The computation is by no means trivial but at the end the formula (1) is exactly what is obtained. Apart from the result itself, it is illuminating to see precisely how the centre remains formally insensitive to the quantum corrections. As for any space-time symmetry, the Noether procedure for supersymmetry needs to be handled with care because of the variation δL = ∂μV μ of the Lagrangian density and we present below a reliable way to write supercurrents taking this term carefully into account. We also show how to commute the charges, paying due attention to the way they are expressed in terms of fields and momenta. In this letter we shall give only a brief account of the computation with special emphasis on the application of the Noether technique to Seiberg-Witten centre, leaving to later work a detailed discussion [6]. One of the most important results of the detailed discussion is that the general form of a supersymmetric Noether current is Duality means (