Conserved Charges Sample Clauses

Conserved Charges. To proceed we note that we need to use a slightly different definition of the conserved charge. In particular the problem with the standard definition given in section 7.1 is that the integral over all space includes an integral over x5. However one can simply change integration variable from x5 to x− so that the integral is over all the coordinates. The resulting conserved charge is therefore constant not for dynamical reasons but because we have integrated over all the coordinates upon which the fields depend. On the other hand we can consider Pµ = g2 ∫ d4x Tµ+ , Q = g2 ∫ d4x J− , (7.28) where we have included a factor of g2 to ensure that they have the canonical dimensions. Since D+ = 0, Pµ and Q are conserved in the sense that ∂−Pµ = ∂−Q = 0. Note that this assumes that the fields vanish sufficiently quickly at infinity so that the boundary terms in the integrals can be discarded. In particular conservation of Q requires that D−XI and [XI , XJ ] → 0 as xi → ∞. Therefore, in this chapter, in order to obtain conserved charges that can be used to define the quantum theory we assume that [vI , vJ ] = [vI , w] = 0 . (7.29) 2 XxX That is we require that the scalar fields and gauge field are in a vacuum configuration at infinity. More explicitly these expressions are (in the case of vanishing fermions): P− =Tr d x 2g2 Fi−F − + ∫ 4 1 i g2 ∫— + 8g2 ij P = 1 Tr d4x εijklF Fkl ∫ i 2g2 ij P = 1 Tr d4x F F−j , (7.32) Q =Tr ∫ d4x Fi− ΓiΓ−ψ 1 F ΓijΓ+Γ−ψ + g2DiXJ ΓJ ΓiΓ−ψ . (7.33) Note that P+ = —4π2g−2k, where k is the instanton number. Thus the P+ eigenvalues are discrete. Physically we interpret this as arising because the x+ direction is restricted to lie on a circle with radius R = g2/4π2. We can further decompose Q = Q+ + Q− where Γ−+Q± = ±Q±. In this case the superalgebra becomes + ij+ {Q−α, Q−β} = — 2P−(Γ−C−1)αβ + ZI (Γ−ΓI C−1)αβ + ZIJ (ΓijΓ−ΓIJ C−1)αβ , (7.34) {Q+α, Q+β} = — 2P+(Γ+C−1)αβ , (7.35) {Q−α, Q+β} = — 2Pi(ΓiC−1)αβ + ZI(ΓiΓI C−1)αβ , (7.36) where C = Γ0 is the charge conjugation matrix and the central charges are I Z+ = — 2Tr ∫ d4x F−iDiXI , (7.37) i ZI = — Tr ∫ d4x GijDjXI , (7.38) IJ ij+ = — g2Tr ∫ d4x D[iXI Dj]XJ . (7.39) Note that although there are 16 supersymmetry charges only 8 of them (Q−) have a nontrivial rela- tion with P−. This is a well-known feature of light-cone gauge (c.f. the Xxxxx-Xxxxxxx superstring). Furthermore any state with a nonvanishing P+ must break the Q+ supersymmetries. We also see that Gij only appears through its contri...