nLab prophk2015

The quantization is one of the leading problems in physics. Introducing constraints, or factoring the symmetries (reduction) in geometry of phase space leads to generalizations of Poisson structures like Dirac structures, and from the point of view of quantization the situation is singular and needs cohomological resolutions. New geometric objects which appear in such constructions like Courant algebroids and higher analogues like $L_\infty$-algebroids (and almost algebroids of Strobl et al.) will be in focus of that research. More systematically cohomological obstructions are seen in a higher categorical framework called derived geometry. Most notably, work of Pantev, Toen et. al. natural notions of derived versions of symplectic structures (motivated by field theory) are systematically studied. We are interested to investigate concrete resolutions appearing in quantization from this point of view.

Another topic is to understand appearance of cyclic cohomology in the case of some simple cases of singularities like orbifold singularities. A result of Hinich and independently Skoda is that inertia orbifold (or homotopically equivalent loop orbifold, that is loop space of the orbifold) which plays the central role in this picture has category of equivariant sheaves which is Drinfeld center of the category of sheaves on the original manifold. The orbifold cohomology is, as shown by Baranovsky in the simplest cases (we propose to further extend his result, say for the case with a gerbe) related to cyclic homology. HKR theorem relates de Rham cohomology to Hochschild but in smooth case only, singular case requires cyclic homology again. Toen and Vezzosi have shown further that derived loop spaces (in algebraic context) are related to cyclic homology again (similar result with formal loop space is by Kapranov and Vasserot). We want to study similar correspondences in the setting of differential geometry with singularities.

Quantum phase spaces are recently studied by Skoda, Meljanac and Stojic as carrying Hopf algebroid structures; they also enable to use Hopf algebroid generalization of Drinfeld twist in quantization problems. Recently Szabo, Schupp, Jur\v co and others explored nonabelian cocycles on spaces (e.g. coming from gerbes) to study Drinfeld twist operators leading not only to twisted theories but also to nonassociative geometry. We are interested in considering this problem from more general point of view where the phase is Hopf algebroid/like structure.

A.Kotov, T.Strobl, Generalizing geometry - algebroids and sigma models, arxiv/1004.0632, in Handbook on Pseudo-Riemannian Geometry

T. Pantev, B. Toen, M. Vaquie, G. Vezzosi, Shifted symplectic structures, arxiv/1111.3209, to appear Publ. IHES

B. Toën, G. Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry, arxiv/0804.1274, Proc. of the 2007 Abel Symposium

V. Baranovsky, Orbifold cohomology as periodic cyclic homology, math.AG/0206256

G. E. Barnes, A. Schenkel, R. J. Szabo, Working with nonassociative geometry and field theory, arXiv:1601.07353