(0,2) Trialities

TL;DR

This work analyzes non-abelian $2$d $\mathcal N=(0,2)$ gauge theories arising from the 4-manifold/2d correspondence and uncovers a triality: an IR equivalence among three $2$d SQCD-like frames connected by a cyclic permutation of flavor blocks. The authors formulate the triality for a class of $U(N_c)$ theories, derive a local rank transformation $N_i' = \sum_{j\in\mathcal X_i} N_j - N_i$, and validate the duality via anomaly matching, central charges, and the equivariant (elliptic) index, with R-charges fixed by c-extremization. This framework extends to general quivers, where local triality moves generate large networks of dual theories; positivity conditions on ranks delineate SUSY-preserving regions, while many generic quivers dynamically break SUSY. The results deepen the 4-manifold/2d dynamics connection, reveal rich IR structures and dualities, and point to brane realizations and integrable-system connections as promising avenues for further exploration.

Abstract

Motivated by the connection between 4-manifolds and 2d N=(0,2) theories, we study the dynamics of a fairly large class of 2d N=(0,2) gauge theories. We see that physics of such theories is very rich, much as the physics of 4d N=1 theories. We discover a new type of duality that is very reminiscent of the 4d Seiberg duality. Surprisingly, the new 2d duality is an operation of order three: it is IR equivalence of three different theories and, as such, is actually a triality. We also consider quiver theories and study their triality webs. Given a quiver graph, we find that supersymmetry is dynamically broken unless the ranks of the gauge groups and flavor groups satisfy stringent inequalities. In fact, for most of the graphs these inequalities have no solutions. This supports the folklore theorem that generic 2d N=(0,2) theories break supersymmetry dynamically.

Figures (13)

• Figure 1: The $(0,2)$ SQCD. We use oriented solid arrows to label chiral fields with their representations, while unoriented dotted lines represent Fermi multiplets. The $\Omega$ multiplet in the $\hbox{det}$ representation is shown with a wavy line.
• Figure 2: The 2d ${\mathcal{N}}=(0,2)$ triality. In this and the following figures the $\Omega$ multiplets are suppressed.
• Figure 3: The space of UV SQCDs. The triangular slice is the projective space that labels the theories up to a simultaneous rescaling of all $N_i$. Each edge of this slice has size $\tfrac{2}{\sqrt 3}$ and the center of mass coordinates $\nu_i$ are the distances of a given interior point from the three edges.
• Figure 4: The triangle labeling the SQCD. The area of the inscribed circle is equal to $\frac{\pi}{3} c_R$.
• Figure 5: The green triangle $ABC$ is the space of theories preserving supersymmetry, while the points in red correspond to SUSY breaking theories. The triality acts as a $\frac{2\pi}{3}$ rotation. The triangle $AHC$ shaded in dark green is the fundamental domain under the action of the triality. The points on the edges correspond to degeneration of the $N_i$ triangle in figure \ref{['Ntriangle']}. The small triangle at $F$ denotes a typical degeneration. The corresponding theories are dual to free fermion theories. The $N_i$ triangle degenerates even further at the vertices of $ABC$. See the small triangle at $C$ for an example. The corresponding theories are empty in the infra-red except for two Fermi multiplets. The theories on the segments $AE$, $BF$ and $CG$ are expected to have exactly marginal deformations. The point $H$ is invariant under the triality. It correspond to the theory with $N_1=N_2=N_3$.