It is believed that LQG can unify all presently known interactions by implementing their common symmetry group, the four-dimensional diffeomorphism group
, which is almost completely broken in perturbative approaches.
But there is a globally defined symmetry group, which is an invariance group of GR, namely the diffeomorphism group
, diff, which, from the local point of view, is the active version of local coordinate transformations.
The main moduli spaces in which we are interested are moduli spaces of manifolds, Or equivalently classifying spaces of diffeomorphism groups
The monograph explores Fredholm properties and index theory, uniform structures of Clifford bundles, ILH diffeomorphism groups
, Lie groups of Fourier integral operators, Teichmuller theory, functional algebraic topology, and relative zeta functions.