Classical Chern-Simons on manifolds with spin structure

We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed, spun 3-manifolds and compact, spun 3-manifolds with boundary where the action is interpreted 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 x Y), where Y is a closed, spun 2-manifold. From the action we derive a unitary line bundle with connection over the moduli stack of flat gauge fields on Y.