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We will introduce now a BRST complex in a geometrical way. As in the preceding section, we introduce auxiliary fields πi with the meaning of a basis of differential forms for the fibre. 51). 49), we get: dXˆ (dxi h) = u ij xj + (Kij − uθij (XP ))xj . 59) Remembering that φ is equivalent to the equivariant curvature of P , the BRST operator for the fibre is naturally given by: Qρi = πi , Qπi = (u ij + φij )ρj . F. Labastida, M. Mari˜no Following [10] we introduce a “localizing” and a “projecting” gauge fermion: 1 Ψloc = −ρi (ixi + πi ), 4 Ψproj = i λ, ν .

We consider the action of the structural group G on P × V given by (p, v)g = (pg, g −1 v), and we form the quotient E = (P × V )/G. Notice that P × V can be considered as a principal bundle over E. 39) where λ(t, p) is an endomorphism of V . We also assume that this flow commutes with the G-action on P × V : P (φP t p)g = φt (pg), λ(t, pg) = g −1 λ(t, p)g. 40) Because of the above condition, a vector field action on E is induced in the natural way, and the one-parameter flow φP t gives in turn a vector field action on M = P/G in the way considered in Sect.

Let θ and K be respectively the connection and curvature of P . Assume now, as in is a constant matrix commuting with all the Sect. 3, that L(XP )θ = 0, and that A ∈ g. Then D = 0. We want to construct an equivariantly closed differential form on P × V with respect to the vector field action Xˆ = (XP , XV ), where XV is associated to the flow λ(t). First of all we define an equivariant curvature on P × V : KX = K + u( − θ(XP )). 44) Notice that − θ(XP ) is a tensorial matrix of the adjoint type, and if P (A1 , · · · , Ar ) is an invariant symmetric polynomial for the adjoint action of g, then we can go through the arguments of Sect.

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