Consider a action functional S(gμν,ψ) with spacetime metric gμν and fermionic matter fields ψ, which is invariant under diffeomorphisms/ coordinate transformations. However, in the partition function
Z=∫D[gμν]D[ψ]μeiS
the path integration measure μ is NOT diffeomorphism invariant and therefore, the covariant divergence of the energy-momentum tensor Tμν=2δS√gδgμν does not vanish.
Now I assume the following model: There are only distributions of the metric field allowed which correspond to triangulations of spacetime. That means the quantum gravitational theory should have an isomorphism of categories:
c:Riem→Tri.
Here, Riem is the category of Riemannian manifolds and Tri the category of discrete manifolds that arise from triangulation of Riemannian manifolds. Hence, μ must be an indicator function that has the value 1 if a distribution of metric fields gμν(x) can be mapped functorially to category Tri and has value 0 if this is not possible. Mathematically, the map c must be a forgetful functor that forgets some kind of metric field distributions.
Question: Is that a plausible theory with diffeomorphism invariant action, but diffeomorphism non-invariant measure (a theory of gravitational anomaly)? Can on this way be constructed a quantum gravity theory (with category theoretic assumptions), e.g. causal triangulation theory?