Eur. Phys. J. C (2012) 72: 2112
https://doi.org/10.1140/epjc/s10052-012-2112-0

Regular Article - Theoretical Physics

Chern–Simons theory with vector fermion matter

¹ Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
² Dept. of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Rd, Mumbai, 400005, India
³ International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Homi Bhabha Rd, Mumbai, 400005, India
⁴ TIFR Centre Building, Indian Institute of Science, Bangalore, 560012, India
⁵ Center for the Fundamental Laws of Nature, Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

^* e-mail: sgiombi@pitp.ca

Received: 29 June 2012
Published online: 25 August 2012

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 light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ℝ² as a function of the ’t Hooft coupling λ=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |λ|=1; the conformal theory does not exist for |λ|>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 light-cone Hamiltonian formulation of this theory where a W _∞ 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.