Chern-Simons Theory with Vector Fermion Matter

TL;DR

We solve a simple yet nontrivial 3D fixed-line of conformal field theories: U(N) Chern-Simons theory at level k coupled to a fundamental massless fermion, in the large N limit with fixed $\lambda=N/k$. Using a Schwinger-Dyson equation in light-cone gauge and a dimensional-reduction regularization, we obtain an exact planar free energy at finite temperature $F(T,\lambda)$, showing the fixed line exists for $|\lambda|<1$ and that the thermal mass scale $\tilde{c}$ satisfies $\tilde{c}=2\lambda\ln(2\cosh(\tilde{c}/2))$. The operator spectrum comprises a scalar $J^{(0)}$ and an infinite tower of higher-spin currents $J^{(s)}$ with dimensions $\Delta=s+1$ that remain unrenormalized at leading order in $1/N$, while their divergences encode anomalous dimensions at subleading order; an integral equation is presented to determine all correlators of these currents at leading order. Three-point functions are computed up to two loops, revealing parity-odd structures at one loop and parity-even contributions at two loops consistent with a bulk parity-violating higher-spin dual, while a bilocal $W_{\infty}$-algebra framework provides a formal large-$N$ solution path. The results point toward a holographic dual in terms of a parity-violating Vasiliev theory in $AdS_4$ (with ABJ-related connections) and raise intriguing questions about bulk interpretation, phase structure near $|\lambda|\to 1$, and extensions to other CS-matter systems.

Abstract

We study three dimensional conformal field theories described by U(N) Chern-Simons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger-Dyson equation in lightcone gauge, we compute the exact planar free energy of the theory at finite temperature on R^2 as a function of the 't Hooft coupling lambda=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |lambda|=1; the conformal theory does not exist for |lambda|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three point functions up to two loops. We also discuss a lightcone Hamiltonian formulation of this theory where a W-infinity algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.

Figures (1)

• Figure 1: The free energy on $S^1\times \mathbb{R}^2$ as a function of $\lambda$.