Spin TQFTs and Chern-Simons gauge theory
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In the first half we construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-PatodiSinger η -invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed spin 3-manifolds and compact spin 3-manifolds with boundary where the action is properly thought of as a unitary element of a Pfaffian line of twisted Dirac operators. We then investigate the properties of the Hamiltonian field theory over 3-manifolds of the form R × Y , where Y is a closed spin 2-manifold. From the action we derive a unitary line bundle with connection over the moduli stack of flat gauge fields on Y . In the second half we conjecture the quantization of our classical gauge theory is a topological quantum field theory, or TQFT. To investigate its properties we apply the procedure of geometric quantization to our field theory when the gauge group is SO3 . We compare our results to the knot-theoretic spin TQFT constructed by Blanchet and Masbaum and find evidence that the two theories are isomorphic.