Chern-Simons Theory with Vector Fermion Matter (1110.4386v2)

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.

• The paper computes the exact planar free energy in Chern-Simons vector fermion theories via large-N analysis, revealing its vanishing at |λ|=1.
• It demonstrates a non-renormalized spectrum featuring one scalar and an infinite tower of higher spin currents, deepening insights into conformal dynamics.
• Detailed computations of three-point functions uncover distinct parity-even and parity-odd contributions, supporting conjectured higher-spin AdS/CFT dualities.

Insights into Chern-Simons Theory with Vector Fermion Matter

The paper presents an in-depth analysis of three-dimensional conformal field theories described by $U(N)$ Chern-Simons theory at level $k$ coupled to massless fermions in the fundamental representation. The authors obtain several exact results by leveraging the large $N$ limit and solving the Schwinger-Dyson equation in lightcone gauge. This work not only revisits conformal field theories and their intriguing connection with higher spin gauge theories but also provides strong numerical results in terms of free energy at different coupling regimes and explores the implications for conjectured dualities with three-dimensional higher spin gauge theories.

Summary of Key Results

1. Exact Computation of Free Energy and Planar Limit Analysis:
   • Through the large N expansion, the authors compute the exact planar free energy of Chern-Simons vector fermion theories at finite temperature. They report a remarkable result: the free energy vanishes at $|\lambda| = 1$, indicating that the conformal theory does not exist beyond this threshold.
   • The paper further provides insights into the universality of the $1/N$ corrections for anomalous dimensions of operators, applicable to higher spin currents. These results rely on various techniques, including dimensional reduction regularization and analysis in lightcone gauge.
2. Operator Spectrum and Anomalous Dimensions:
   • The authors find that the operator spectrum of the conformal theory consists of one scalar ($\bar{\psi}\psi$) and an infinite number of higher spin currents, with spins $s=1,2,...$ , none of which develop anomalous dimensions at leading order in $N$.
   • The non-renormalization of scaling dimensions for these currents enriches our understanding of the infrared behavior of the theory and its proposed conjectural higher-spin dual.
3. Three Point Functions and Correlation Structures:
   • This paper carefully computes three-point functions at two loops for higher-spin currents, highlighting both parity-even and odd structures. The computations reveal a parity-odd structure that appears at one-loop order, and a parity-even structure at two-loop order, corresponding to contributions from free scalars and free fermions from a conformal field theory perspective.
4. Implications for AdS/CFT Correspondence:
   • The paper comments on the holographic implications, speculating that the dual higher spin theory in $AdS_4$ is governed by a function dependent on 't Hooft coupling, exemplified by functions in the Vasiliev higher spin theory. Specific attention is devoted to the duality between the massive fermion theory and generalized Vasiliev theory, suggesting an elaborate connection via parity-violating interactions.

Future Prospects

• The paper leaves open several provocative questions regarding the exact dual form of the bulk theory, potential symmetry breaking, and realizations in string theory for the highest $\mathcal{N}$ supersymmetric case.
• It also invites further exploration of the dual scenario in both weak and strong coupling limits for various values of the 't Hooft coupling $\lambda$ and at finite physical quantities like volume and temperature.
• The framework provided is ripe for extension to theories with either fundamental bosons or with applied supersymmetry, as well as to explore finite volume or finite chemical potential configurations in these contexts.

The outcomes of this research emphasize the rich interplay between topological aspects uniquely present in three-dimensional Chern-Simons theories, scaling properties of conformal field theories with vector matter, and the symmetry constraints imposed by higher-spin gauge symmetries.