Hopf algebroids in quantum differential geometry

While cohomology theories of various kinds are known on algebras, here we explore the much harder problem of what is the ‘homotopy’ of an algebra as a geometric object? For example, when is an algebra ‘simply connected’?

The project will make sense of this notion using a constructive approach to noncommutative differential geometry in which the possibly noncommutative algebra A is extended to a graded algebra of ‘differential forms’.

The Experienced Researcher will first develop and study a recent proposal of a Hopf algebroid D_A of ‘differential operators’ associated to this data, the existence of which is implied by the More-Eilenberg theorem applied to the category of bimodules on A equipped with flat bimodule connections.

In the classical case of functions on a smooth manifold, this would be a version of the path groupoid and Morita equivalent to π_1.

He will then relate it to a proposed new construction of a universal (co)measuring bialgebra adapted to the differential graded case as a generalised ‘diffeomorphism group’ and to a proposed new notion of differential ‘character variety’ defined by each Hopf algebra H as the moduli of flat connections up to equivalence on quantum principal bundles over A with fibre H.

Classically, the holonomy associated to a flat connection identifies this as maps from π_1 to the fibre group modulo conjugation.

Using these ingredients, the further aim will be to arrive at a quantum differential geometric picture of the Turaev-Viro invariant of 3-manifolds and generalise it to a suitable class of differential algebras A.

The project will also study an analogue of D_A in Connes’ spectral triple approach to noncommutative geometry based on an axiomatic ‘Dirac operator’, explore generalisations at the level of 2-categories and Hopf monads and look for applications to algebraic models of quantum gravity, where both diffeomorphism invariance and ‘loops’ are expected to play a fundamental role.