The Topological Unitarity Identities in Chern-Simons Theories

TL;DR

This paper derives and verifies topological unitarity identities in 2+1 dimensional Chern-Simons theories, showing that certain sums of diagrammatic imaginary parts vanish, ensuring unitarity without real topological photons.

Contribution

It introduces topological unitarity identities in Chern-Simons theories and demonstrates their validity both perturbatively and non-perturbatively.

Findings

Identifies identities that ensure unitarity in Chern-Simons theories.

Verifies identities explicitly at one-loop order for fermion-antifermion scattering.

Shows identities can be derived outside perturbation theory.

Abstract

Starting from the generating functional of the theory of relativistic spinors in 2+1 dimensions interacting through the pure Chern-Simons gauge field, the S-matrix is constructed and seen to be formally the same as that of spinor quantum electrodynamics in 2+1 dimensions with Feynman diagrams having external photon lines excluded, and with the propagator of the topological Chern-Simons photon substituted for the Maxwell photon propagator. It is shown that the absence of real topological photons in the complete set of vector states of the total Hilbert space leads in a given order of perturbation theory to topological unitarity identities that demand the vanishing of the gauge-invariant sum of the imaginary parts of Feynman diagrams with a given number of internal on-shell free topological photon lines. It is also shown, that these identities can be derived outside the framework of perturbation theory. The identities are verified explicitly for the scattering of a fermion-antifermion pair in one-loop order.