Assume that the surface ∑ is equipped with the structure of a smooth algebraic curve over C. We denote by H0(M∑,L⊗k) the space of sections of L⊗k on M∑, where M∑ is the moduli space of semi-stable rank N bundles with trivial
determinant over ∑ , and L is the determinant line bundle on M∑. It is known that H0(M∑,L⊗k) is isomorphic to V(∑) of a TQFT(V,Z) derived from the quantum group Uq(slN) at a (k+N)-th root of unity. In this sense, H0(M∑,L⊗k) gives a geometric construction of such a V(∑).
How can we find a geometric way to associate a vector in H0(M∑,L⊗k) to a 3 manifold M with δM=∑.
In physics one can obtain such vector by applying infinite dimensional analogue of geometric invariant theory and sympletic quotients of Chern-Simons integral. We would like to make mathematical sense in that argument.
This post imported from StackExchange MathOverflow at 2017-02-02 22:59 (UTC), posted by SE-user Soutrik