8 research outputs found
The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory
We give a direct calculation of the curvature of the Hitchin connection, in
geometric quantization on a symplectic manifold, using only differential
geometric techniques. In particular, we establish that the curvature acts as a
first-order operator on the quantum spaces. Projective flatness follows if the
K\"ahler structures do not admit holomorphic vector fields. Following Witten,
we define a complex variant of the Hitchin connection on the bundle of
prequantum spaces. The curvature is essentially unchanged, so projective
flatness holds in the same cases. Finally, the results are applied to quantum
Chern-Simons theory, both for compact and complex gauge groups
A Universal Formula for Deformation Quantization on K\"ahler Manifolds
We give an explicit local formula for any formal deformation quantization,
with separation of variables, on a K\"ahler manifold. The formula is given in
terms of differential operators, parametrized by acyclic combinatorial graphs.Comment: 20 pages, 8 figure