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

176 N. ReshetikhinD 2 A = A = d ∗ A d A + d A d ∗ A ,where A is Hodge Laplace operator. The operator D A effectively appears in thequadratic part being restricted to odd forms. This operator will be denoted byD − A : { 1 → 1 ⊕ 3 3 → 1 .Now the question is whether the operator D − Ais invertible on the surface ofthe constraint. In other words, whether the Lorentz gauge is really a cross sectionthrough gauge orbits.3.9.2.1 The PropagatorFirst, assume that the complex ( i (M, g), d A ) is acyclic, i.e., H i (M, g) ={0} forall i = 0, 1, 2, 3 (by Poincaré duality H i ≃ H 3−i , so it is enough to assume thevanishing of H 0 and H 1 ). In this case, the representation of π 1 (M) in G defined byholonomies of a flat connection A is irreducible (implied by H 0 ={0}) and isolated(implied by H 1 ={0}).Since the spaces H i can be naturally identified with harmonic forms and thereforewith zero eigenspaces of Laplace operators, in this case all Laplace operatorsare invertible and so is D A . Denote by G A the inverse to A , i.e., the Green’s function,thenP A = ( D − A) −1 = D−AG A = G A D − A .Thus, in this case the quadratic part is non-degenerate and we can write contributionsfrom Feynman diagrams as multiple integrals of the kernel of the integraloperator ( D − ) −1.A The analysis of the contributions of Feynman diagrams to thepartition function was studied in this case by Axelrod and Singer in [8, 9], and byKontsevich [42].Another important special case arises when the flat connection is reducible butstill isolated. For example, a trivial connection for rational homology spheres [13,14, 16] has such property. In this case, we still have H 1 = H 2 ={0} and the Lorentzgauge for α together with the exactness of ∗ (2) is still equivalent to the Lorentzgauge for , i.e., d ∗ = 0. However, now there are harmonic forms in 0 (M) and 3 (M) corresponding to the fundamental class of M and because of this, D − Ais notinvertible on the space of all forms.Nevertheless, in this case (and in a more general case when H 1 ̸= {0}) onecan construct an operator which is “almost inverse” to D − A. Such an operator isdetermined by the chain homotopy K : i → i−1 and the Hodge decompositionof . For details about such operator P see [8, 9, 13, 14, 16] and Sect. 3.3.2 of [17].An important example of a rational homology sphere is S 3 itself. In this case, theinverse to D − for trivial a connection can be constructed explicitly by puncturing of