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.
Let $X$ be a compact connected Riemann surface of genus at least two. Let $M_H(r,d)$ denote the moduli space of semistable Higgs bundles on $X$ of rank $r$ and degree $d$. We prove that the compact complex Bohr-Sommerfeld Lagrangians of $M_H(r,d)$ ar
e precisely the irreducible components of the nilpotent cone in $M_H(r,d)$. This generalizes to Higgs $G$-bundles and also to the parabolic Higgs bundles.
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 o
n the quantum spaces. Projective flatness follows if the Kahler 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.
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.
