We show that the $\mathcal{N}=(1,0)$ superconformal theory on a single M5 brane on the ALE space of type $G=A_n, D_n, E_n$, when compactified on $T^2$, becomes a class S theory of type $G$ on a sphere with two full punctures and a simple puncture. We study this relation from various viewpoints. Along the way, we develop a new method to study the 4d SCFT arising from the $T^2$ compactification of a class of 6d $\mathcal{N}=(1,0)$ theories we call very Higgsable.
We study the mirror symmetry of abelian 3d $\mathcal{N}=2$ theories with mixed Chern-Simons levels by turning them into $\mathcal{T}_{A,N}$ theories that are defined as $N$ copies of $U(1)-[1]$ theory coupled together by mixed Chern-Simons levels $k_{ij}$. We find that $\mathcal{T}_{A,N}$ theories have many mirror dual theories with different mixed CS levels and FI parameters. As an example, we analyze $U(1)_k+ N_C \,\mathbf{C} + N_{AC} \, \mathbf{AC}$ theories by transforming these theories into certain $\mathcal{T}_{A,N}$ theories and find many equivalent effective Chern-Simons levels. Final…
Even though for generic $\mathcal{N}=1$ theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $\mathcal{N}=1$ SCFTs, these of Class $\mathcal{S}_k$ for which it is possible to define Coulomb and Higgs branches precisely as for the $\mathcal{N}=2$ theories of Class $\mathcal{S}$ from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we…
We propose dualities of $\mathcal{N} = (0,2)$ supersymmetric boundary conditions for 3d $\mathcal{N} = 2$ gauge theories with orthogonal and symplectic gauge groups. We show that the boundary 't Hooft anomalies and half-indices perfectly match for each pair of the proposed dual boundary conditions.
We compute superconformal indices for $\mathcal{N} = 3$ $\hat{ADE}$ Chern-Simons quiver gauge theories with a product gauge group $\prod_i U(N)_i$, using the method of supersymmetric localization. We also perform a large $N$ analysis of the index. This index includes contribution from non zero magnetic flux sector. The fact that these theories have a weakly coupled UV completion in terms of $\mathcal{N}= 3$ supersymmetric Chern-Simons Yang-Mills theories enables us to apply the localization technique. Such theories have dual M-theory description on $\mathrm{AdS}_4\times M_7$, where $M_7$ is a …
It is expected on general grounds that the moduli space of 4d $\mathcal{N}=3$ theories is of the form $\mathbb{C}^{3r}/Γ$, with $r$ the rank and $Γ$ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d $\mathcal{N}=3$ theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d $\mathcal{N}=3$ theories proposed in \cite{Garcia-Etxebarria:2016erx}, constructed via non-geometric quotients…
We describe the topological $A$ and $B$ twists of 3d $\mathcal{N}=4$ theories of hypermultiplets gauged by $\mathcal{N}=4$ vector multiplets as certain deformations of the holomorphic-topological ($HT$) twist of those theories, utilizing the twisted superfields of Aganagic-Costello-Vafa-McNamara describing $HT$-twisted 3d $\mathcal{N}=2$ theories. We rederive many known results from this perspective, including state spaces on Riemann surfaces, deformations induced by flavor symmetries, the boundary VOAs of Costello-Gaiotto, and the category of line operators as proposed by Costello-Dimofte-Gai…
For a special family of 4d $\mathcal{N}=2$ superconformal quiver theories, the worldsheet dual corresponding to the free theory is identified. This result is obtained from the recently proposed worldsheet dual of free $\mathcal{N}=4$ SYM by applying a $\mathbb{Z}_k$ orbifold. In support of our proposal we show that the spectrum of both the untwisted as well as the twisted sectors coincides between the two descriptions.
Mirror symmetry of 3d $\mathcal{N}=4$ gauge theories and supersymmetric indices
我们计算超对称指数来检验三维$\mathcal{N}=4$规范理论的镜像对称性和四维$\mathcal{N}=4$ Super Yang-Mills理论中半BPS富集边界条件和界面的对偶性。在IIB型弦论的S-对偶作用下，我们发现指数的匹配是Gaiotto和Witten提出的三维界面的各种对偶性的有力证据.
The $\mathcal{N}_3=3\to \mathcal{N}_3=4$ enhancement of Super Chern-Simons theories in $D=3$, Calabi HyperKähler metrics and M2-branes on the $\mathcal{C}(\mathrm{N^{0,1,0}})$ conifold
This paper studies supersymmetric ground states of 3d $\mathcal{N}=4$ supersymmetric gauge theories on a Riemann surface of genus $g$. There are two distinct spaces of supersymmetric ground states arising from the $A$ and $B$ type twists on the Riemann surface, which lead to effective supersymmetric quantum mechanics with four supercharges and supermultiplets of type $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ respectively. We compute the space of supersymmetric ground states in each case, graded by flavour and R-symmetries and in different chambers for real mass and FI parameters, for a large…
In this paper we begin mapping out the space of rank-2 $\mathcal{N}=2$ superconformal field theories (SCFTs) in four dimensions. This represents an ideal set of theories which can be potentially classified using purely quantum field-theoretic tools, thus providing a precious case study to probe the completeness of the current understanding of SCFTs, primarily derived from string theory constructions. Here, we collect and systematize a large amount of field theoretic data characterizing each theory. We also provide a detailed description of each case and determine the theories' Coulomb, Higgs a…
We apply the technique of sequential deconfinement to the four dimensional $\mathcal{N}\!=\!1$ $Usp(2N)$ gauge theory with an antisymmetric field and $2F$ fundamentals. The fully deconfined frame is a length-$N$ quiver. We use this deconfined frame to prove the known self-duality of $Usp(2N)$ with an antisymmetric field and $8$ fundamentals. Along the way we encounter a subtlety: in certain quivers with degenerate holomorphic operators, a naive application of Seiberg duality rules leads to an incorrect superpotential or chiral ring. We also consider the reduction to $3d$ $\mathcal{N}\!=\!2$ th…
We derive general BPS boundary conditions in two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories. We analyze the solutions of these boundary conditions, and in particular those that allow the bulk fields to have poles at the boundary. We also present the brane configurations for the half- and quarter-BPS boundary conditions of the $\mathcal{N}=(2,2)$ supersymmetric gauge theories in terms of branes in Type IIA string theory. We find that both A-type and B-type brane configurations are lifted to M-theory as a system of M2-branes ending on an M5-brane wrapped on a product of a holo…
We compute the partition functions of $\mathcal{N} = 1$ gauge theories on $S^2 \times \mathbb{R}^2_\varepsilon$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of $\mathbb{R}^2_\varepsilon$. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the $\mathcal{N} = 1$ partition functions on the $Ω$-background, and show that the Nekrasov partition functions can be recovered from these building blocks.
We initiate the geometric engineering of 2d $\mathcal{N}=(0,1)$ gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds. Spin(7) orientifolds explicitly realize the perspective on 2d $\mathcal{N}=(0,1)$ theories as real slices of $\mathcal{N}=(0,2)$ ones. Remarkably, this projection is geometrically realized as Joyce's construction of Spin(7) manifolds v…
Any four-dimensional $\mathcal{N} \geqslant 2$ superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra \cite{Beem:2013sza}. The goal of these lectures is to provide an introduction to the subject, covering the construction of the chiral algebras, the consequences for four-dimensional physics, as well as a brief summary of recent progress. This is the writeup of the lectures given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.
