On the trace density in deformation quantization

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

“…2 Following terminology from [31], we say that N a i (14 ) and * N ij (15) are L-dual if L and H are related by Legendre transform (5), or (6). In the following constructions, we shall consider that to Legendre transform there are associated the diffeomorphisms…”

“…[5,6,35] to define two classes of canonical operators which are necessary to quantize the Hamilton-Fedosov spaces and related subspaces on cotangent bundles defined by lifts of Einstein metrics. We shall address precisely the question how the geometry of cotangent bundles and related deformation quantization change under symplectic transforms and elaborate a formalism which preserves the form of Hamilton-Jacobi equations both on classical and quantum level.…”

“…There were proposed different sophisticate constructions with formal and partial solutions for quantum gravity and field interactions theories. We cite here the BRST quantization methods for non-Abelian and open gauge algebras [1,2,3,4], deformation quantization [5,6,7,8], quantization of general Lagrange structures and, in general, BRST quantization without Lagrangians and Hamiltonians [9,10], W -geometry and Moyal deformations of gravity via strings and branes [11,12,13] and quantum loops and spin networks [14,15,16].…”

On the index theorem for symplectic orbifolds

“…Our hope was that combining the methods of Kawasaki [13] and Vergne [15] for the index theorem for elliptic operators on orbifolds on the one hand, with the methods of [9] theory of transversally elliptic operators [1] However, we decided to give such a formula as a conjecture in this paper, since it sheds some new light on the whole concept of deformation quantization. There are many facts which support it, for instance, a formula for contributions of fixed point manifolds to the G-index [12], or a direct calculation of the first three terms. Moreover, for a virtual bundle with compact support over an orbifold cotangent bundle our index formula coincides with Kawasaki's topological index.…”

“…In this section we propose a conjecture for the index formula prompted by the Kawasaki index theorem [13], the index theorem for deformation quantization [9] and the G -index formula [12]. For the time being we have a proof only in very particular cases, cf.…”