TOPOLOGICAL UNITARITY IDENTITIES IN CHERN-SIMONS THEORIES

Starting from the generating functional of the theory of relativistic spinors in 2+1 dimensions interacting through the pure Chern-Simons ga uge field, the S matrix is constructed and seen to be formally the sam e as that of spinor quantum electrodynamics in 2+1 dimensions with Fey nman diagrams having external photon lines excluded, and with the prop agator of the topological Chern-Simons photon substituted for the Maxw ell photon propagator. It is shown that the absence of real topologica l photons in the complete set of vector states of the total Hilbert sp ace leads in a given order of perturbation theory to topological unita rity identities that demand the vanishing of the gauge-invariant sum o f the imaginary parts of the Feynman diagrams with a given number of i nternal on-shell free toplogical photon lines. It is also shown that t hese identities can be derived outside the framework of perturbation t heory. The identities are verified explicitly for the scattering of a fermion-antifermion pair in one-loop order.