which is about a Hilbert space of differential forms with appropriately defined inner product replacing an exterior algebra (with exterior derivatives) over differential forms.
(RoC) xi, "The problem of what happens to classical general relativity at the extreme short-distance Planck scale of 10*exp(-33) cm is clearly one of the most pressing in all of physics. It seems abundantly clear that profound modifications of existing theoretical structures will be mandatory by the time one reaches that distance scale. There exists several serious responses to this challenge. These include effective field theory, string theory, loop quantum gravity, thermo-gravity, holography, and emergent gravity. .....
The characteristic of an affine geometry is the fact, that only parallel distances can be measured against each other, i.e. other kinds of infinitesimal small "actions" between not parallel "objects" are not considered in this kind of "continuum". Vectors are the mathematical model of such translations (resp. parallel displacements) and the underlying (affine) geometry is mathematically described by the group properties of vectors (WeH). An affine geometry with space dimension n is the “same” as its related (n-1)-dimensional projective group. The “enrichment” of the today's n-dimensional space-time affine geometry (manifolds and affine connexions, (ScE), (WeH1) and quotes § 18 below) goes along with the concept of exterior derivatives to allow “measurements” and the definition of appropriate metrics resp. to link to the Riemannian metric and the concept of curvature.
The Sobolev H(1/2) space on the circle plays a key role of universal period mapping universal Teichmüller parameter space for all Riemann surfaces via quantum calculus:
Biswas I., Nag S., Jacobians of Riemann Surfaces and the Sobolev Space H(1 2) on the Circle
Nag S., Sullivan D., Teichmüller theory and the universal period mapping via quantum calculus and the H(1 2) space on the circle.
We propose a quantum gravity model