Exceptional Chern-Simons-Matter Dualities

TL;DR

This work extends the landscape of 3d Chern-Simons dualities by leveraging conformal embeddings of exceptional affine Kac-Moody algebras to derive new dualities among TQFTs with exceptional gauge groups, thus generalizing level-rank dualities beyond classical groups. The authors catalog explicit exceptional dualities, track gravitational Chern-Simons counterterms, and map line content across dual presentations, establishing a framework to obtain boson-boson Chern-Simons-matter dualities. They propose numerous Chern-Simons-matter dualities, including $(E_N)_1+N_f\Phi_{\mathbf{fund}} \leftrightarrow SU(9+N_f-N)_{-1}+N_f\Phi'_{\mathbf{fund}}-16\mathrm{CS}_{\mathrm{grav}}$, and analyze Higgsing chains among $E_N$, $F_4$, $G_2$, and related groups. Finally, they explore phase diagrams for time-reversal invariant $G_{2}$ gauge theories with adjoint or fundamental fermions, demonstrating consistency via anomaly matching and gravitational counterterms and illustrating rich quantum phases connected by dual descriptions.

Abstract

We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is $(E_{N})_{1}\leftrightarrow SU(9-N)_{-1}$. Others such as $SO(3)_{8}\leftrightarrow PSU(3)_{-6},$ are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant $G_{2}$ gauge theory coupled to either an adjoint fermion, or two fundamental fermions.

Figures (3)

• Figure 1: A conjectural phase diagram of $(G_2)_0$ coupled to an adjoint Majorana fermion as a function of the fermion mass $m$. When $m=0$ the theory is time-reversal invariant, and has $\mathcal{N}=1$ supersymmetry. At large $|m|$ the IR is described by a TQFT visible semiclassically. As $|m|$ is reduced there is a transition to a quantum phase. This transition is weakly coupled in dual variables with a $U(2)$ gauge field and a fundamental fermion. At $m=0$ there is also massless fermion $\bar{\chi}$ which is the Goldstino arising from the expected spontaneous breaking of supersymmetry Witten:1999ds. In each phase, we also indicate the coefficient of the gravitational Chern-Simons term which can be used as a consistency check on the phase diagram (see section \ref{['g2sec']}).
• Figure 2: A conjectural phase diagram of $(G_2)_0$ gauge theory with 2 fermions in the fundamental representation where $m_1=m_2$. In the center is a quantum phase where the $U(1)$ global symmetry is spontaneously broken.
• Figure 3: A conjectural phase diagram for the $(G_2)_0$ gauge theory with two real fermions $2\psi_{\mathbf{7}}$ in the fundamental representation of $G_2$. The diagonal line where $m_1=m_2$ is identical to Figure \ref{['fig:G21']}. As explained there, on the solid line in the middle, the theory flows to $\frac{SU(2)_2\times S^1}{\mathbb{Z}_2}$, and at the black and white stars the theory is dual to $U(2)_{\pm\frac{5}{2},\pm\frac{1}{2}}+\Psi_{\mathbf{fund}}$. On the solid and dashed doubled lines we expect that the theory flows to a gapless theory dual to $(G_2)_{\pm \frac{1}{2}}+\psi_{\mathbf{7}}$, and intersects with the $m_1=m_2$ line at the black and white stars.