New ideas from Higher Spin Gravity for old problems in Deformation Quantization and Conformal Geometry
The project will try to address longstanding problems in two areas of mathematics and theoretical high energy physics: (A) Deformation Quantization; (B) Conformal Geometry; (C) quantum gravity models via Higher Spin Gravity.
(A) Deformation quantization is a well-established area of mathematics where one of the main problem is to construct a deformation of the algebra of classical observables into the quantum one. For the case of Poisson manifolds, the solution was found by Kontsevich in the form of a much more general statement: the Formality theorem. However, in applications Poisson manifolds have some discrete symmetries and it turns out that the algebra of observables invariant or extended with such symmetries can have new deformations not captured by the Formality theorem. This is the problem of deformation quantization of Poisson orbifolds. Kontsevich Formality can be reinterpreted as the quantization of a simple topological field theory, the Poisson sigma model. One part of the project is to solve the problem of deformation quantization of Poisson orbifolds via Poisson sigma models on orbifolds.
(B) Conformal geometry is a relatively old branch of mathematics, yet it is still full of unsolved problems that are easy to formulate and hard to solve. One of the most basic ones is the problem of exhausting conformal invariants: in particular an explicit basis for obtaining Type-B invariants is not known beyond few lower dimensions. The project offers a new and constructive approach to the problem, which is related to part (C).
(C) Constructing viable models of quantum gravity has been an important open problem for decades. Higher spin gravities, which are extensions of gravity with higher spin particles, constitute one of the approaches to the problem. It is notoriously difficult to construct such theories, which can be attributed to the complexity of the quantum gravity problem. There is a handful of higher spin gravities known at present, one of them being conformal higher spin gravity, which is closely related to conformal geometry. One of the ideas of the project is to develop efficient ways of constructing new conformal higher spin gravities, which would automatically lead to conformal invariants and invariant operators, and would significantly extend the list of known higher spin gravities. The techniques used to attack this problem are also inspired from deformation quantization, and hence tightly linked to (A).