Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1099where T µν is the SEM–tensor of the theory, i.e.,T µν (φ i ) =δδg µν S[φ i]. (6.155)The fact that δ in (6.154) is a symmetry of the theory implies thatthe transformations δφ i of the fields are such that both δA[φ i ] = 0 andδO αi (φ i ) = 0. Conditions (6.154) lead, at least formally, to the followingrelation for VEVs:∫δδg µν 〈O α 1O α2 · · · O αp 〉 = − [Dφ i ]O α1 (φ i )O α2 (φ i ) · · · O αp (φ i )T µν e −S[φ i ]∫= − [Dφ i ]δ ( O α1 (φ i )O α2 (φ i ) · · · O αp (φ i )G µν exp (−S[φ i ]) ) = 0, (6.156)which implies that the quantum field theory can be regarded as topological.This second type of TQFTs are called of Witten type. One of its mainrepresentatives is the theory related to Donaldson invariants, which is atwisted version of N = 2 supersymmetric Yang–Mills gauge theory. Itis important to remark that the symmetry δ must be a scalar symmetry,i.e., that its symmetry parameter must be a scalar. The reason is that,being a global symmetry, this parameter must be covariantly constant andfor arbitrary manifolds this property, if it is satisfied at all, implies strongrestrictions unless the parameter is a scalar.Most of the TQFTs of cohomological type satisfy the relation:S[φ i ] = δΛ(φ i ), (6.157)for some functional Λ(φ i ). This has far–reaching consequences, for it meansthat the topological observables of the theory, in particular the partitionfunction, (path integral) itself are independent of the value of the couplingconstant. Indeed, let us consider for example the VEV:∫〈O α1 O α2 · · · O αp 〉 =[Dφ i ]O α1 (φ i )O α2 (φ i ) · · · O αp (φ i ) e − 1g 2 S[φ i ] . (6.158)Under a change in the coupling constant, 1/g 2 → 1/g 2 −∆, one has (assumingthat the observables do not depend on the coupling), up to first–orderin ∆:∫+ ∆ [Dφ i ]δ〈O α1 O α2 · · · O αp 〉 −→ 〈O α1 O α2 · · · O αp 〉[O α1 (φ i )O α2 (φ i ) · · · O αp (φ i )Λ(φ i ) exp(− 1 )]g 2 S[φ i]= 〈O α1 O α2 · · · O αp 〉. (6.159)