Breaking bad theories of class $\mathcal S$

Abstract

We study weakly-coupled descriptions/channel decompositions of the 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $\mathfrak{su}(N)$, from the perspective of the 3d $\mathcal{N}=4$ mirror duals of their circle compactifications. This is a delicate problem when the channel decomposition produces pathological, or bad, 4d configurations that correspond to spheres with non-maximal punctures. The star-shaped quivers, describing the 3d mirrors associated with such bad 4d configurations, are bad 3d $\mathcal{N}=4$ theories. Leveraging recent results regarding 3d bad theories, we identify a new and interesting family of bad theories, which we coin \textit{broken} theories, that naturally arise in this context. Using these broken theories, we develop a systematic and analytic method that determines the generically non-Lagrangian matter sectors and the weakly-coupled gauge groups in such channel decompositions. We understand these weakly-coupled descriptions as emerging dynamically via Higgs mechanisms triggered by operators acquiring vacuum expectation values.

Figures (42)

• Figure 1: Sketch of the analysis performed on the channel decomposition of the theory of class ${\cal S}$ of type $\mathfrak{su}(4)$ associated to a sphere with two maximal and two minimal punctures. We depict the channel decomposition on line two, wherein the theory $S_1$ is bad. Running the electric algorithm on this produces the intermediate step, depicted on line three. This distribution then acts across the gauging to produce the final line. The result represents the effect of the monopole VEV, acquired in $S_1$, that partially Higgses the $U(4)$ gauge group along with the theory $S_2$. Note that we highlight twisted hypermultiplets, with respect to the hypers in the initial star-shaped theory, in blue instead of black, and depict the gauging of an emergent Coulomb branch symmetry by hook-arrows.
• Figure 2: Pictorial representation of gluing together two maximal punctures. This can also be seen as decomposing the four-punctured sphere into two pairs-of-pants.
• Figure 3: Three-dimensional mirror dual to the circle compactification of ${\cal T}\left[\mathfrak{su}(N),{\cal C}_{g,s};[\rho_1],[\rho_2],\dots,[\rho_s]\right]$. Here the dotted line connected to a $\rho_i$ indicates the $T_{\rho_i}[SU(N)]$ tail.
• Figure 4: HB and CB fusion to Identity. Notice that the gluing prescription involves coupling the adjoint chiral $\phi$ of the $U(N)$ gauge nodes to the HB/CB moment maps ${\cal H}_i/{\cal C}_i$ of the first and the second quiver tails. We use $_{\vec{X}}\mathbb{I}_{\vec{Y}}$ as the shorthand for the distribution on the r.h.s. of \ref{['eq:tsun_delta']}. Meanwhile, on the r.h.s. of the figure, we depict the explicit Lagrangian depiction of the two gluings. The top right depicts the linear quiver obtained by diagonal gauging of two $T[U(N)]$ theories, whereas, on the bottom right, we depict the gauging of the emergent symmetries on the Coulomb branches of two $T[U(N)]$ theories using hook arrows.
• Figure 5: Parallel between the channel decomposition in 3d and in 4d. The procedure at the level of the 3d $S^3_b$ partition function is reported in \ref{['eq:chaneldecomp']}. We recall that dashed lines connecting two nodes labelled by $N$ represent $T[U(N)]$ theories, while those that connect the label $\rho_i$ represent $T_{\rho_i}[U(N)].$