New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

3 Lectures on Quantization of Gauge Systems 153∫ ( if(x, c)expM (n|2k)h∑a)g(x, c)dxdc = h n−2k2 (2π) n 21√ | det(B(a))|det(L(a)) exp⎛⎝1 + ∑ Ɣ̸=Ø( ih f (a) + iπ 4 sign(B(a)) )⎞(ih) −χ(Ɣ) (−1) c(D(Ɣ)) F a (D(Ɣ))⎠ , (28)| Aut(Ɣ)|where all ingredients are the same as in (27) except that the summation is takenover the graphs with oriented fermionic edges and with weights from Fig. 3.10. Anexample is given in Fig. 3.11.3.6 Finite-Dimensional Faddeev–Popov Quantizationand the BRST DifferentialIn this section we will study the integral (13) when a Lie group G acts on X faithfully(with no stabilizers) and the function f is invariant with respect to this action.3.6.1 Faddeev–Popov TrickLet X be a manifold with the action of a Lie group G. We assume here that theaction is free, i.e., that the stabilizer of every point in X is trivial. Assume also thatX/G is a manifold. (Note that what is really important is the assumption that X/Gis smooth near orbits where f is critical.) In this casedim(X/G) = dim(X) − dim(G).Assume that the manifold X has a G-invariant volume form and that X is compact.It is clear that such restrictions are too strong, but we will see in the next sectionhow they can be relaxed to reasonable assumptions.Let f (x) be a G-invariant real analytic function. The goal of this section is toprepare the setup for the description of the asymptotic expansion of the integral∫I h =X(exp i f (x) )dx (29)has the sum of Feynman diagrams, just as it was done in section for functions on Xwith simple critical points.Since the function f is G-invariant, its critical points are not simple, except whena critical point is a fixed point of the G-action, but since we assume faithfulness,there are no such points.