Anomalous Dimensions in Non-Supersymmetric Bifundamental Chern-Simons Theories

TL;DR

The paper investigates non-supersymmetric bifundamental Chern-Simons theories in three dimensions by developing an $M/N$ expansion with $M\ll N$ to access strong coupling and computes the anomalous dimension of the scalar operator $\bar{\psi}\psi$ to first order in $M/N$ and to all orders in the 't Hooft coupling $\lambda_M$ while setting $\lambda_N=0$. Using Euclidean Chern-Simons theory in light-cone gauge with a hard cutoff, the authors sum scalar self-energy, gauge self-energy, and vertex/ladder diagrams to obtain a closed form for the anomalous dimension: $\delta = \frac{128 \lambda^2 (\pi^2 \lambda^2-128)}{3(\pi^2 \lambda^2+64)^2} \frac{M}{N}$. In the strong-coupling limit $\lambda_M\to\infty$, $\delta$ remains finite, approaching $\delta\to \frac{128}{3\pi^2}\frac{M}{N}$, which has implications for possible holographic duals and the IR behavior of related theories. The work also outlines a path to all-orders results in $\lambda_N$ and discusses higher-spin operators, dualities, and the M-theory limit, offering a framework to test holographic ideas in non-supersymmetric settings.

Abstract

Non-abelian Chern-Simons theories coupled to fermions are known to provide an interesting class of non-supersymmetric conformal fixed points \cite{Giombi:2011kc}. These theories, particularly those based on bifundamental matter, are important because they may provide simple non-supersymmetric examples of the AdS/CFT correspondence. For instance, it seems natural to conjecture that $O(N)_{-k}\times O(N)_k$ Chern-Simons theory coupled to Majorana fermions transforming in a bi-vector representation may be dual to pure Einstein gravity with a small negative cosmological constant in the "M-theory" limit where $k=1$ and $N$ is large. While it is extremely difficult to directly study such bifundamental theories when $k=1$ or even at strong 't Hooft coupling $λ=\frac{N}{k}$, it is possible to calculate physical quantities to all orders in $λ$ in a $U(M)_{k_M} \times U(N)_{k_N}$ theory, in the limit $M \ll N$, in an $M/N$ expansion. To illustrate this, we calculate the anomalous dimension of the primary operator $\barψψ$, to first order in $M/N$, to all orders in $λ_M=\frac{N}{k_M}$, but with $λ_N=\frac{N}{k_N}=0$. We also comment on possible bosonization dualities for bifundamental Chern-Simons theories.

Figures (5)

• Figure 1: Correction to the scalar vertex due to the $O(M/N)$ fermion self-energy
• Figure 2: Another correction to the vertex.
• Figure 3: Additional contribution to the two-point function.
• Figure 4: The anomalous dimension $\delta/(\frac{M}{N})$ of $\text{tr~} \bar{\psi}\psi$, from equation \ref{['result']} as a function of $\lambda$. It reaches a minimum of $\delta=-\frac{128}{9\pi^2} \frac{M}{N}$ at $\lambda \pi= 4 \sqrt{2}$. The limiting value as $\lambda \rightarrow \infty$ is $\delta \rightarrow \frac{128}{3\pi^2} \frac{M}{N}$.
• Figure 5: The two point function of the scalar primary to all orders in $\lambda_N$.