Ivancevic_Applied-Diff-Geom

1132 Applied Differential Geometry: A Modern IntroductionObservablesWe can now discuss the observables. For this purpose, let us define [Baulieuand Grossman (1988)]A = A + c, F = F + ψ − φ, K = db + ξ,where we have introduced differential form notations, but their meaningswould be obvious. A and c are considered as a (1, 0) and (0, 1) part of1–form on (Y, M). Similarly, F, ψ and φ are (2, 0), (1, 1) and (0, 2) part ofthe 2–form F, and db and ξ are (1, 0) and (0, 1) part of the 1–form K. ThusA defines a connection 1–form on (Y, M) and F is a curvature 2–form.Note that the exterior derivative d maps any (p 1 , p 2 )−form X p of totaldegree p = p 1 + p 2 to (p 1 + 1, p 2 )−form, but δ B maps any (p 1 , p 2 )−form to(p 1 , p 2 + 1)−form. Also note that X p X q = (−1) pq X q X p . Then the actionof δ B is(d + δ B )A = F, (d + δ B )b = K. (6.216)F and K also satisfy the following Bianchi identities in Abelian theory:(d + δ B )F = 0, (d + δ B )K = 0. (6.217)Equations (6.216) and (6.217) mean anti–commuting property between theBRST variation δ B and the exterior differential d, i.e., {δ B , d} = 0.The BRST transformation rule in geometric sector can be easily readfrom (6.213), i.e., δ B A, δ B ψ, δ B c and δ B φ. (6.217) implies(d + δ B )F n = 0, (6.218)and expanding the above expression by ghost number and form degree, weget the following (i, 2n − i)−form W n,i ,W n,0 = φnn! , W n,1 = φn−1(n − 1)! ψ,W n,2 =W n,3 =φn−22(n − 2)! ψ ∧ ψ − φn−1(n − 1)! F,φn−36(n − 3)! ψ ∧ ψ ∧ ψ − φn−2F ∧ ψ, (6.219)(n − 2)!where 0 = δ B W n,0 , dW n,0 = δ B W n,1 , (6.220)dW n,1 = δ B W n,2 , dW n,2 = δ B W n,3 , dW n,3 = 0.