We consider compactifications of rank $Q$ E-string theory on a genus zero surface with no punctures but with flux for various subgroups of the $text{E}_8times text{SU}(2)$ global symmetry group of the six dimensional theory. We first construct a simp
le Wess-Zumino model in four dimensions corresponding to the compactification on a sphere with one puncture and a particular value of flux, the cap model. Using this theory and theories corresponding to two punctured spheres with flux, one can obtain a large number of models corresponding to spheres with a variety of fluxes. These models exhibit interesting IR enhancements of global symmetry as well as duality properties. As an example we will show that constructing sphere models associated to specific fluxes related by an action of the Weyl group of $text{E}_8$ leads to the S-confinement duality of the $text{USp}(2Q)$ gauge theory with six fundamentals and a traceless antisymmetric field. Finally, we show that the theories we discuss possess an $text{SU}(2)_{text{ISO}}$ symmetry in four dimensions that can be naturally identified with the isometry of the two-sphere. We give evidence in favor of this identification by computing the `t Hooft anomalies of the $text{SU}(2)_{text{ISO}}$ in 4d and comparing them with the predicted anomalies from 6d.
In principle, there is no obstacle to gapping fermions preserving any global symmetry that does not suffer a t Hooft anomaly. In practice, preserving a symmetry that is realised on fermions in a chiral manner necessitates some dynamics beyond simple
quadratic mass terms. We show how this can be achieved using familiar results about supersymmetric gauge theories and, in particular, the phenomenon of confinement without chiral symmetry breaking. We present simple models that gap fermions while preserving a symmetry group under which they transform in chiral representations. For example, we show how to gap a collection of 4d fermions that carry the quantum numbers of one generation of the Standard Model, but without breaking electroweak symmetry. We further show how to gap fermions in groups of 16 while preserving certain discrete symmetries that exhibit a mod 16 anomaly.
We show that the $4d$ ${cal N}=1$ $SU(3)$ $N_f=6$ SQCD is the model obtained when compactifying the rank one E-string theory on a three punctured sphere (a trinion) with a particular value of flux. The $SU(6)times SU(6)times U(1)$ global symmetry of
the theory, when decomposed into the $SU(2)^3times U(1)^3times SU(6)$ subgroup, corresponds to the three $SU(2)$ symmetries associated to the three punctures and the $U(1)^3 times SU(6)$ subgroup of the $E_8$ symmetry of the E-string theory. All the puncture symmetries are manifest in the UV and thus we can construct ordinary Lagrangians flowing in the IR to any compactification of the E-string theory. We generalize this claim and argue that the ${cal N}=1$ $SU(N+2)$ SQCD in the middle of the conformal window, $N_f=2N+4$, is the theory obtained by compactifying the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter SCFT on a sphere with two maximal $SU(N+1)$ punctures, one minimal $SU(2)$ puncture, and with a particular value of flux. The $SU(2N+4)times SU(2N+4)times U(1)$ symmetry of the UV Lagrangian decomposes into $SU(N+1)^2times SU(2)$ puncture symmetries and the $U(1)^3times SU(2N+4)$ subgroup of the $SO(12+4N)$ symmetry group of the $6d$ SCFT. The models constructed from the trinions exhibit a variety of interesting strong coupling effects. For example, one of the dualities arising geometrically from different pair-of-pants decompositions of a four punctured sphere is an $SU(N+2)$ generalization of the Intriligator-Pouliot duality of $SU(2)$ SQCD with $N_f=4$, which is a degenerate, $N=0$, instance of our discussion.
We classify ${cal N}=1$ gauge theories with simple gauge groups in four dimensions which possess a conformal manifold passing through weak coupling. A very rich variety of models is found once one allows for arbitrary representations under the gauge
group. For each such model we detail the dimension of the conformal manifold, the conformal anomalies, and the global symmetry preserved on a generic locus of the manifold. We also identify, at least some, sub-loci of the conformal manifolds preserving more symmetry than the generic locus. Several examples of applications of the classification are discussed. In particular we consider a conformal triality such that one of the triality frames is a $USp(6)$ gauge theory with six fields in the two index traceless antisymmetric representation. We discuss an IR dual of a conformal $Spin(5)$ gauge theory with two chiral superfields in the vector representation and one in the fourteen dimensional representation. Finally, an extension of the conformal manifold of ${cal N}=2$ class ${cal S}$ theories by conformally gauging symmetries corresponding to maximal punctures with the addition of two adjoint chiral superfields is commented upon.
We suggest three new ${cal N}=1$ conformal dual pairs. First, we argue that the ${cal N}=2$ $E_6$ Minahan-Nemeschansky (MN) theory with a $USp(4)$ subgroup of the $E_6$ global symmetry conformally gauged with an ${cal N}=1$ vector multiplet and certa
in additional chiral multiplet matter resides at some cusp of the conformal manifold of an $SU(2)^5$ quiver gauge theory. Second, we argue that the ${cal N}=2$ $E_7$ MN theory with an $SU(2)$ subgroup of the $E_7$ global symmetry conformally gauged with an ${cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of a conformal ${cal N}=1$ $USp(4)$ gauge theory. Finally, we claim that the ${cal N}=2$ $E_8$ MN theory with a $USp(4)$ subgroup of the $E_8$ global symmetry conformally gauged with an ${cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of an ${cal N}=1$ $Spin(7)$ conformal gauge theory. We argue for the dualities using a variety of non-perturbative techniques including anomaly and index computations. The dualities can be viewed as ${cal N}=1$ analogues of ${cal N}=2$ Argyres-Seiberg/Argyres-Wittig duals of the $E_n$ MN models. We also briefly comment on an ${cal N}=1$ version of the Schur limit of the superconformal index.
We study the compactification of the 6d ${cal N}=(2,0)$ SCFT on the product of a Riemann surface with flux and a circle. On the one hand, this can be understood by first reducing on the Riemann surface, giving rise to 4d ${cal N}=1$ and ${cal N}=2$ c
lass ${cal S}$ theories, which we then reduce on $S^1$ to get 3d ${cal N}=2$ and ${cal N}=4$ class ${cal S}$ theories. On the other hand, we may first compactify on $S^1$ to get the 5d ${cal N}=2$ Yang-Mills theory. By studying its reduction on a Riemann surface, we obtain a mirror dual description of 3d class ${cal S}$ theories, generalizing the star-shaped quiver theories of Benini, Tachikawa, and Xie. We comment on some global properties of the gauge group in these reductions, and test the dualities by computing various supersymmetric partition functions.
We consider compactifications of $6d$ minimal $(D_{N+3},D_{N+3})$ type conformal matter SCFTs on a generic Riemann surface. We derive the theories corresponding to three punctured spheres (trinions) with three maximal punctures, from which one can co
nstruct models corresponding to generic surfaces. The trinion models are simple quiver theories with $mathcal{N}=1$ $SU(2)$ gauge nodes. One of the three puncture non abelian symmetries is emergent in the IR. The derivation of the trinions proceeds by analyzing RG flows between conformal matter SCFTs with different values of $N$ and relations between their subsequent reductions to $4d$. In particular, using the flows we first derive trinions with two maximal and one minimal punctures, and then we argue that collections of $N$ minimal punctures can be interpreted as a maximal one. This suggestion is checked by matching the properties of the $4d$ models such as `t Hooft anomalies, symmetries, and the structure of the conformal manifold to the expectations from $6d$. We then use the understanding that collections of minimal punctures might be equivalent to maximal ones to construct trinions with three maximal punctures, and then $4d$ theories corresponding to arbitrary surfaces, for $6d$ models described by two $M5$ branes probing a $mathbb{Z}_k$ singularity. This entails the introduction of a novel type of maximal puncture. Again, the suggestion is checked by matching anomalies, symmetries and the conformal manifold to expectations from six dimensions. These constructions thus give us a detailed understanding of compactifications of two sequences of six dimensional SCFTs to four dimensions.
SCFTs in six dimensions are interrelated by networks of RG flows. Compactifying such models on a Riemann surface with flux for the $6d$ global symmetry, one can obtain a wide variety of theories in four dimensions. These four dimensional models are a
lso related by a network of RG flows. In this paper we study some examples of four dimensional flows relating theories that can be obtained from six dimensions starting with different SCFTs connected by $6d$ RG flows. We compile a dictionary between different orders of such flows, $6dto 6dto 4d$ and $6dto 4dto 4d$, in the particular case when the six dimensional models are the ones residing on M5 branes probing different $A$-type singularities. The flows we study are triggered by vacuum expectation values (vevs) to certain operators charged under the six dimensional symmetry. We find that for generic choices of parameters the different orders of flows, $6dto 6dto 4d$ and $6dto 4dto 4d$, involve compactifications on different Riemann surfaces with the difference being in the number of punctures the surface has.
Recently a very interesting three-dimensional $mathcal{N}=2$ supersymmetric theory with $SU(3)$ global symmetry was discussed by several authors. We denote this model by $T_x$. This was conjectured to have two dual descriptions, one with explicit sup
ersymmetry and emergent flavor symmetry and the other with explicit flavor symmetry and emergent supersymmetry. We discuss a third description of the model which has both flavor symmetry and supersymmetry manifest. We then investigate models which can be constructed by using $T_x$ as a building block gauging the global symmetry and paying special attention to the global structure of the gauge group. We conjecture several cases of $mathcal{N}=2$ mirror dualities involving such constructions with the dual being either a simple $mathcal{N}=2$ Wess-Zumino model or a discrete gauging thereof.
We study 4d N=1 supersymmetric theories of class S_k, obtained from flux compactifications on a Riemann surface of 6d (1,0) conformal theories describing the low energy physics on a stack of M5 branes probing a Z_k singularity. We conjecture that the
protected spectrum of class S_k theories contains a freely generated ring, generalizing the Coulomb branch of the N=2 theories. We derive this by examining a limit of the supersymmetric index of 4d N=1 class S_k theories. The limit generalizes the Coulomb limit of N=2 theories, which coincides with the case of k=1 for a particular choice of flux. We conjecture a general simple formula for the index in the aforementioned limit.
