Beyond its importance in mathematical physics, it appears also as a geodesic equation on diffeomorphism group
of a circle with right invariant [L.sup.2] metric (see  and references therein).
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.
Physics extensively uses "infinite" groups: the gauge groups in Particle Physics; the diffeomorphism group
, the Bondi-Metzner-Sachs group, and so on in general relativity; the Kac-Moody and loop groups in string theories; and so on are all infinite dimensional.
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.