Unitarity, Crossing Symmetry and Duality of the S-matrix in large N Chern-Simons theories with fundamental matter

TL;DR

This work provides a comprehensive analysis of 2→2 scattering in large $N$ Chern-Simons theories with fundamental bosonic or fermionic matter, deriving all-orders S-matrices in several exchange channels and revealing a nontrivial forward-scattering piece along with a modified crossing framework. The boson–fermion duality, encoded by level-rank transpositions, maps S-matrices between the theories (with exchange-channel transposition) and accounts for the anyonic phase in the singlet channel, while a conjectured singlet-channel S-matrix preserves unitarity and non-relativistic limits. A detailed, gauge-dependent calculation in the scalar sector is complemented by a Landau-gauge one-loop check in the fermionic sector, and a careful discussion of the identity-channel forwards, double-analytic-continuation, and potential Schrodinger-based derivations is provided. The findings suggest that modified analyticity and crossing may be general features of matter CS theories, with potential implications for ABJ/ABJM and topological quantum computation contexts. Overall, the paper advances exact S-matrix structure in CS–matter theories and deepens understanding of Bose-Fermi duality in a three-dimensional, gauge-theoretic setting.

Abstract

We present explicit computations and conjectures for $2 \to 2$ scattering matrices in large $N$ {\it $U(N)$} Chern-Simons theories coupled to fundamental bosonic or fermionic matter to all orders in the 't Hooft coupling expansion. The bosonic and fermionic S-matrices map to each other under the recently conjectured Bose-Fermi duality after a level-rank transposition. The S-matrices presented in this paper may be regarded as relativistic generalization of Aharonov-Bohm scattering. They have unusual structural features: they include a non analytic piece localized on forward scattering, and obey modified crossing symmetry rules. We conjecture that these unusual features are properties of S-matrices in all Chern-Simons matter theories. The S-matrix in one of the exchange channels in our paper has an anyonic character; the parameter map of the conjectured Bose-Fermi duality may be derived by equating the anyonic phase in the bosonic and fermionic theories.

Figures (20)

• Figure 1: Propagator of bosonic particles.
• Figure 2: Propagator of fermionic particles.
• Figure 3: This diagram would contain a diagrammatic representation of the exact amplitude $V$ as a sum over ladders, where the 'rungs' in the ladder are the triple line propagators.
• Figure 4: A diagrammatic representation of the effective single particle exchange four point amplitude for bosons. This amplitude is give by the sum of the tree level exchange of a gluon, dressed tree level exchanges of the gluon and the point interaction controlled by the parameter $b_4$
• Figure 5: A diagrammatic depiction of the integral equation obeyed by offshell four point scattering amplitudes. The blob here represents the all orders scattering amplitude while the triple line represents the effective single particle exchange four point interaction between quanta. Here, and in every Feynman diagram in this paper, all momenta flow in the direction of the arrows of the propagators.