Abstract
We use radial quantization to compute Chern-Simons partition functions on handlebodies of arbitrary genus. The partition function is given by a particular transition amplitude between two states which are defined on the Riemann surfaces that define the (singular) foliation of the handlebody. The final state is a coherent state while on the initial state the holonomy operator has zero eigenvalue. The latter choice encodes the constraint that the gauge fields must be regular everywhere inside the handlebody. By requiring that the only singularities of the gauge field inside the handlebody must be compatible with Wilson loop insertions, we find that the Wilson loop shifts the holonomy of the initial state. Together with an appropriate choice of normalization, this procedure selects a unique state in the Hilbert space obtained from a Kähler quantization of the theory on the constant-radius Riemann surfaces. Radial quantization allows us to find the partition functions of Abelian Chern-Simons theories for handlebodies of arbitrary genus. For non-Abelian compact gauge groups, we show that our method reproduces the known partition function at genus one.
Highlights
The reason why one may think that a Chern-Simons theory may be exactly soluble on handlebodies is that these spaces are almost factorized as the topological product [0, R]×Σ
The partition function is given by a particular transition amplitude between two states which are defined on the Riemann surfaces that define the foliation of the handlebody
By requiring that the only singularities of the gauge field inside the handlebody must be compatible with Wilson loop insertions, we find that the Wilson loop shifts the holonomy of the initial state
Summary
The first is a path integral, in which on the final surface ΣR we impose a holomorphic boundary condition that fixes the antiholomorphic part Azdzof the gauge connection A, while on the initial surface Σ0 we fix the component of A along the contractible cycles. The third is a wave function in a coherent state basis, obtained by integrating over the gauge orbit a seed wave function which is an eigenstate of the holonomy operator along the contractible cycles. These quantities will be compared to the Chern-Simons partition functions that are identified with the wave functions obtained by a holomorphic quantization on the Riemann surface Σ [15,16,17]. For genus one on the flat metric, coincides with the Dedekind eta function:
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