Abelian Chern-Simons theory
Summary:
I describe how to quantize Chern-Simons theory for an abelian gauge group. In particular, I describe the Hilbert space which Chern-Simons assigns to a surface. I follow the geometric quantization procedure, where the Hilbert space is the space of holomorphic sections over the moduli space of line bundles. I describe this space explicitly using theta functions. Next, I describe how the quantization depends on the choice of complex structure on our surface. I describe the projective connection on the bundle of Hilbert spaces, and its relationship with the heat equation.
Presented at:
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Some sources:
Atiyah, [Geometry of Knots and physics], chapter 2
Mihaela Manoliu, Abelian Chern-Simons theory. Here the author describes the vector in the Hilbert space associated to a 3 manifold with boundary.