Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1111W k,0 = φkk! , W k,1 = ψW k−1,0 ,W k,2 = FW k−1,0 − 1 2 ψ ∧ ψW k−2,0, (6.176)W k,3 = F ∧ ψW k−2,0 − 1 3! ψ ∧ ψ ∧ ψW k−3,0,W k,4 = 1 2 F ∧ F W k−2,0 − 1 2 F ∧ ψ ∧ ψW k−3,0 − 1 4! ψ ∧ ψ ∧ ψ ∧ ψW k−4,0.These operators are clearly independent of the metric g of X. Although theyare not BRST invariant except for W k,0 , they obey the following equations[Zhang et. al. (1995)]dW k,0 = −[Q, W k,1 ], dW k,1 = [Q, W k,2 ],dW k,2 = −[Q, W k,3 ], dW k,3 = [Q, W k,4 ], dW k,4 = 0,which allow us to construct BRST invariant operators from the the W ’sin the following way: Let X i , i = 1, 2, 3, X 4 = X, be compact manifoldswithout boundary embedded in X. We assume that these submanifolds arehomologically nontrivial. Define∫Ô k,0 = W k,0 , Ô k,i = W k,i , (i = 1, 2, 3, 4). (6.177)X iAs we have already pointed out, Ô k,0 is BRST invariant. It follows fromthe descendent equations that[Q, Ôk,i] =∫[Q, W k,i ] =X i∫dW k,i−1 = 0.X iTherefore the operators Ô indeed have the properties (6.173) and (6.174).Also, for the boundary ∂K of an i + 1D manifold K embedded in X, wehave∫ ∫∫W k,i = dW k,i = [Q, W k,i+1 ],∂KKKis BRST trivial. The correlation function of ∫ ∂K W k,i with any BRSTinvariant operator is identically zero. This in particular shows that the Ô’sonly depend on the homological classes of the submanifolds X i .