Spin Chern-Simons and Spin TQFTs

TL;DR

This paper explores the quantization of classical spin Chern-Simons theory for SO_3, relating it to known spin TQFTs and analyzing its properties through geometric quantization and partition function formalism.

Contribution

It applies geometric quantization to classical spin Chern-Simons theory for SO_3 and connects the resulting quantum theory to established spin TQFTs.

Findings

Quantum SO_3 spin Chern-Simons corresponds to Blanchet-Masbaum spin TQFT.

The theory relates to standard SU_2 Chern-Simons and known TQFTs.

Partition function properties are analyzed in the Lagrangian framework.

Abstract

In a previous paper we constructed classical spin Chern-Simons for any compact Lie group $G$: a gauge theory whose action depends on the spin structure of the 3-manifold. Here we apply geometric quantization to the classical Hamiltonian theory and investigate the formal properties of the partition function in the Lagrangian theory, all in the case $G = SO_3$. We find that the quantum theory for $SO_3$ spin Chern-Simons corresponds to the spin TQFT constructed by Blanchet and Masbaum in the same way that the quantum theory for standard $SU_2$ Chern-Simons corresponds to the TQFT constructed by Reshetikhen and Turaev or the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel.