Flows, Fixed Points and Duality in Chern-Simons-matter theories

TL;DR

This paper investigates dualities in 3d Chern-Simons-matter theories, analyzing fixed points, beta functions, and phase structures, and constructs RG flows linking supersymmetric and non-supersymmetric theories, supporting the duality conjecture at large N.

Contribution

It computes the beta function for the sextic coupling in large N Chern-Simons-matter theories and identifies multiple fixed points, extending duality relations to RG flows from supersymmetric theories.

Findings

Beta function is cubic in x_6 at order 1/N.

Three fixed points for x_6 at each non-zero 't Hooft coupling.

Dual pairs of RG flows connect supersymmetric and non-supersymmetric theories.

Abstract

It has been conjectured that 3d fermions minimally coupled to Chern-Simons gauge fields are dual to 3d critical scalars, also minimally coupled to Chern-Simons gauge fields. The large $N$ arguments for this duality can formally be used to show that Chern-Simons-gauged {\it critical} (Gross-Neveu) fermions are also dual to gauged `{\it regular}' scalars at every order in a $1/N$ expansion, provided both theories are well-defined (when one fine-tunes the two relevant parameters of each of these theories to zero). In the strict large $N$ limit these `quasi-bosonic' theories appear as fixed lines parameterized by $x_6$, the coefficient of a sextic term in the potential. While $x_6$ is an exactly marginal deformation at leading order in large $N$, it develops a non-trivial $\beta$ function at first subleading order in $1/N$. We demonstrate that the beta function is a cubic polynomial in $x_6$ at this order in $1/N$, and compute the coefficients of the cubic and quadratic terms as a function of the 't Hooft coupling. We conjecture that flows governed by this leading large $N$ beta function have three fixed points for $x_6$ at every non-zero value of the 't Hooft coupling, implying the existence of three distinct regular bosonic and three distinct dual critical fermionic conformal fixed points, at every value of the 't Hooft coupling. We analyze the phase structure of these fixed point theories at zero temperature. We also construct dual pairs of large $N$ fine-tuned renormalization group flows from supersymmetric ${\cal N}=2$ Chern-Simons-matter theories, such that one of the flows ends up in the IR at a regular boson theory while its dual partner flows to a critical fermion theory. This construction suggests that the duality between these theories persists at finite $N$, at least when $N$ is large.