Chern-Simons theory coupled to bifundamental scalars

TL;DR

The paper analyzes a three-dimensional Chern-Simons theory with gauge group $SU(N)_k \times SU(M)_{-k}$ coupled to a bifundamental scalar in the large-$N$ limit, introducing the parameters $\lambda = N/k$ and $\xi = M/N$ to organize the theory. It establishes two-loop lines of fixed points for all $\xi$, connects the small-$\xi$ limit to CS theories with fundamental matter (dual to parity-violating Vasiliev in AdS$_4$), and discusses a possible bulk dual at strong coupling. On a spatial torus, it shows the absence of exact light states and quantifies a finite gap that scales as $\Delta \sim \xi\sqrt{\lambda\pi}$ for small $\xi$, using a diagrammatic perturbation theory organized by $\xi$ and large-$N$ counting. The results imply that bifundamental matter can regulate the light-state degeneracies characteristic of fundamental CS-matter theories and offer insight into the interplay between CS dynamics, holography, and topology (higher genus). The work also points to future directions, including all-loop proofs of fixed lines and a deeper understanding of the bulk duals across the parameter space $\xi$ and $\lambda$.

Abstract

We study the three-dimensional theory of two Chern-Simons gauge fields coupled to a scalar field in the bifundamental representation of the $SU(N)_k \times SU(M)_{-k}$ gauge group. At small but fixed $M \ll N$, this system approaches the theory of a Chern-Simons field coupled to fundamental matter, conjectured to be dual to a parity-violating version of Vasiliev's higher-spin gauge theory in $AdS_4$. At finite $M/N$ and large 't Hooft coupling this theory (or its SUSY version) is expected to be dual to an Einstein-like gravity. We show at two loops that this theory possesses a line of fixed points at any value of $M/N$. We also prove that turning on a finite but small $M/N$ gaps out the light states that Chern-Simons theory coupled to fundamental matter develops when placed on a torus. We also comment on the higher genus case.

Figures (5)

• Figure 1: The Feynman diagrams that contribute to the flowing of six-point couplings at two loops. This representation is schematic. Each six-point coupling in diagrams (A1), (A4), and (A8) must be treated as a single-, double-, or triple-trace coupling in turn. Similarly, each gauge boson must be treated as either an $A_\mu$ or a $B_\mu$ boson.
• Figure 2: A schematic depiction of diagrams contributing to the field strength renormalization to two loops.
• Figure 3: Two sets of fixed points parametrized by the 't Hooft coupling $\lambda$ in the weak coupling regime at $\xi = 1/2$. The diagram on the left corresponds to choosing the plus sign in eqs. \ref{['lambda1']}--\ref{['lambda3']}.
• Figure 5: Converting to double-line notation. Dashed lines correspond to indices running from $1$ to $N$, and full lines correspond to indices running from $1$ to $M$.
• Figure 6: The three diagrams that contribute to $E(\mathcal{C}_2)$ up to second order in perturbation theory, shown in single-line notation (above) and double-line notation (below). The first diagram is the zero-vertex (unperturbed) energy, and the other two are the only two-vertex diagrams that contribute in the 't Hooft limit.