Abstracts Category : Other

Deformation Quantization over a Z-graded base

by Elif Altinay-Ozaslan

Mathematics We investigate the problem how to describe the equivalence classes of formal deformations of a symplectic manifold M in the case when we have several deformation parameters ve1, ve2, dots, veg of non-positive degrees. We define formal deformations of M over the base ring bC[[ve, ve1, dots, veg]] as Maurer-Cartan elements of the differential graded Lie algebra (ve, ve1, dots, veg) sPD^ullet(M)[[ve, ve1, dots, veg]] where sPD^ullet(M) denotes the algebra of polydifferential operators on M. The interesting feature of such deformations is that, if at least one formal parameter carries a non-zero degree, then the resulting Maurer-Cartan element corresponds to a bC[[ve, ve1, dots, veg]]-multilinear A_-structure on the graded vector space cO(M)[[ve, ve1, dots, veg]] with the zero differential, where cO(M) is the algebra of smooth complex-valued functions M. This dissertation focuses on formal deformations of cO(M) with the base ring bC[[ve, ve1, dots, veg]] such that corresponding MC elements satisfy these two conditions: The Kodaira-Spencer class of is ve al and satisfies the equation vertve=0 =0. The main result of this study gives us a bijection between the set of isomorphism classes of such deformations and the set of all degree 2 vectors of the graded vector space , igoplusq 0 , (ve, ve1, dots, veg) , Hq(M, bC)[[ve, ve1, dots, veg]] where H^ullet(M, bC) is the singular cohomology of M with coefficients in bC and every vector of Hq(M, bC) carries degree q. Temple University ThesesAdvisors/Committee Members: Dolgushev, Vasily;, Letzter, E. S. (Edward S.), Lorenz, Martin, Stienon, Mathieu;.