Spin Chern-Simons and Spin TQFTs

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.