Noncommutative Algebraic Geometry and its Applications to Physics

Topic: the workshop is about the interaction and/or unification of various flavors of noncommutative geometry (such as algebraic, categorical, differential-geometric), in relation to their applications in theoretical physics, notably quantum field theory and string theory.

Description and aim

There are many incarnations of the notion of noncommutative geometry.

After the work of A. Connes, noncommutative geometry emerged as a new and powerful evolution of modern differential geometry. Among other things, it allows for a natural description of singular spaces like leaf spaces of foliations and its applications range in such disparate fields as the standard model of particle physics and the geometric interpretation of formulas of number theory.

There is another variant, known as “projective noncommutative geometry”, that appeared in the mid 80’s, owing to work of M. Artin, J. Tate and M. van den Bergh. Roughly speaking, it deals with noncommutative analogues of algebraic varieties. It is based on the possibility to somehow twist the usual homogeneous coordinate ring of a projective variety to obtain noncommutative analogues of various geometric objects of projective geometry. These noncommutative analogues are usually described in terms of an abelian and/or triangulated category, possibly endowed with additional structure. The link with ordinary algebraic geometry stems from the possibility of reconstructing, under some conditions, a scheme its derived category of quasi-coherent sheaves.

There are other ways to describe noncommutativity in the realm of algebraic geometry (and associated to such names as A. Rosenberg, M. Kontsevich, V. Ginzburg, ...). All these approaches have an interesting and intriguing common point, that could be called a “noncommutative hamiltonian formalism” (Kontsevich necklace brackets, Le Bruyn and Crawley-Boevey’s noncommutative symplectic geometry, double Poisson structures of M.Van den Bergh, et cetera), and which includes, among other things, noncommutative Poisson geometry, Calabi-Yau algebras, preprojective algebras, quiver representations.

We believe that now is exactly the time where all these geometries have ripened to a stage where they are well-developed enough to start interacting — whence the timeliness of the workshop.

Much of this research is inspired and motivated by questions of modern theoretical physics and, reciprocally, some important mathematical discoveries have generated new directions in modern quantum field theory. Thus Kontsevich’s results, like deformation quantization of Poisson phase spaces and formality, inspired a number of noncommutative quantum field theory models (Seiberg-Witten, A. Schwarz, M. Douglas, D. Gross, N. Nekrasov and others). Moreover, it appears that the “integrable sector” in these examples has deep relations with the above-mentioned noncommutative algebro-geometric constructions and has produced results such as noncommutative instantons, twisted Hilbert schemes of points on complex plane, or Calogero-Moser spaces.

The aim of the workshop to provide a unified view of all these different aspects of noncommutative geometry, and help the “physics oriented” young researchers to understand the possible interactions and respective advantages of different approaches vis-a-vis the challenges and the demands of modern quantum field theory. The Workshop will have an interdisciplinary nature, not only because of the participation of scientists both from the physics and mathematics communities, but also for the diversity of topics that will be touched (algebra, algebraic geometry, differential geometry, category theory, functional analysis in mathematics, quantum mechanics, quantum field theory, string theory, integrable systems in physics).

This topic lies in the intersection of some very “hot” areas of research in present-day pure mathematics and mathematical physics, such as noncommutative geometry, the theory of geometric invariants, enumerative geometry, string theory, integrable systems. We believe that for this reason this workshop will attract considerable interest from the mathematics and physics communities.