Ivancevic_Applied-Diff-Geom

1100 Applied Differential Geometry: A Modern IntroductionHence, observables can be computed either in the weak coupling limit, g →0, or in the strong coupling limit, g → ∞.So far we have presented a rather general definition of TQFT and madea series of elementary remarks. Now we will analyze some aspects of itsstructure. We begin pointing out that given a theory in which (6.154) holdsone can build correlators which correspond to topological invariants (in thesense that they are invariant under deformations of the metric g µν ) justby considering the operators of the theory which are invariant under thesymmetry. We will call these operators observables. In virtue of (6.156),if one of these operators can be written as a symmetry transformation ofanother operator, its presence in a correlation function will make it vanish.Thus we may identify operators satisfying (6.154) which differ by an operatorwhich corresponds to a symmetry transformation of another operator.Let us denote the set of the resulting classes by {Φ}. By restricting theanalysis to the appropriate set of operators, one has that in fact,δ 2 = 0. (6.160)Property (6.160) has consequences on the features of TQFT. First, thesymmetry must be odd which implies the presence in the theory of commutingand anticommuting fields. For example, the tensor G µν in (6.154) mustbe anticommuting. This is the first appearance of an odd non–spinorialfield in TQFT. Those kinds of objects are standard features of cohomologicalTQFTs. Second, if we denote by Q the operator which implements thissymmetry, the observables of the theory can be described as the cohomologyclasses of Q:{Φ} = Ker QIm Q , Q2 = 0. (6.161)Equation (6.154) means that in addition to the Poincaré group thetheory possesses a symmetry generated by an odd version of the Poincarégroup. The corresponding odd generators are constructed out of the tensorG µν in much the same way as the ordinary Poincaré generators are builtout of T µν . For example, if P µ represents the ordinary momentum operator,there exists a corresponding odd one G µ such thatP µ = {Q, G µ }. (6.162)Now, let us discuss the structure of the Hilbert space of the theory invirtue of the symmetries that we have just described. The states of thisspace must correspond to representations of the algebra generated by the