2011-10-07

Yes, Yes, These Are All Quantizations!

Vasily Dolgushev, Temple University

Deformation quantization is a procedure which assigns a formal deformation of the associative algebra of functions on a variety to a Poisson structure on this variety. Such a procedure can be obtained from Kontsevich's formality quasi-isomorphism and, it is known that, homotopy inequivalent formality quasi-isomorphisms give ``different'' quantization procedures. I will propose a framework in which all homotopy classes of formality quasi-isomorphisms can be described. More precisely, I will show that homotopy classes of ``stable'' formality quasi-isomorphisms form a torsor for the group $\exp(H^0(\fGC))$, where $\fGC$ denotes the full graph complex. This result may be interpreted as a description of all quantization procedures.