We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space.
Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.
We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kahler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.
In this paper, we will provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)). In particul
ar, we recall the differential geometric construction of Hitchins projectively flat connection in the bundle over Teichmuller space obtained by push-forward of the determinant line bundle over the moduli space of rank n, fixed determinant, semi-stable bundles fibering over Teichmuller space. We recall the relation between the Hitchin connection and Toeplitz operators which was first used by the first named author to prove the asymptotic faithfulness of the SU(n) quantum representations of the mapping class groups. We further review the construction of the formal Hitchin connection, and we discuss its relation to the full asymptotic expansion of the curve operators of Topological Quantum Field Theory. We then go on to identifying the first terms in the formal parallel transport of the Hitchin connection explicitly. This allows us to identify the first terms in the resulting star product on functions on the moduli space. This is seen to agree with the first term in the star product on holonomy functions on these moduli spaces defined by Andersen, Mattes and Reshetikhin.
We give a differential geometric construction of a connection in the bundle of quantum Hilbert spaces arising from half-form corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid, family of Kahler structures,
all of which give vanishing first Dolbeault cohomology groups. In [And1] Andersen gave an explicit construction of Hitchins connection in the non-corrected case using additional assumptions. Under the same assumptions we also give an explicit solution in terms of Ricci potentials. Morover we show that if these are carefully chosen the construction coincides with the construction of Andersen in the non-corrected case.
