We discuss compactifications of rank QQ E-string theory on a torus with fluxes for abelian subgroups of the E_8E8 global symmetry of the 6d6d SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as E[USp(2Q)]E[USp(2Q)]. This model has a variety of non trivial properties. In particular the global symmetry is USp(2Q)\times USp(2Q)\times U(1)^2USp(2Q)×USp(2Q)×U(1)2 with one of the two USp(2Q)USp(2Q) symmetries emerging in the IR as an enhancement of an SU(2)^QSU(2)Q symmetry of the UV Lagrangian. The E[USp(2Q)]E[USp(2Q)] model after dimensional reduction to 3d3d and a subsequent Coulomb branch flow is closely related to the familiar 3d3dT[SU(Q)]T[SU(Q)] theory, the model residing on an S-duality domain wall of 4d4d\mathcal{N}=4𝒩=4SU(Q)SU(Q) SYM. Gluing the E[USp(2Q)]E[USp(2Q)] models by gauging the USp(2Q)USp(2Q) symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of 4d4d theories with emergent IR symmetries. We support our claims by various checks involving computations of anomalies and supersymmetric partition functions. Many of the needed identities satisfied by the supersymmetric indices follow directly from recent mathematical results obtained by E. Rains.

We construct a family of 4d$$ \mathcal{N} $$ N = 1 theories that we call $$ {E}_{\rho}^{\sigma } $$ E ρ σ [USp(2N)] which exhibit a novel type of 4d IR duality very reminiscent of the mirror duality enjoyed by the 3d$$ \mathcal{N} $$ N = 4 $$ {T}_{\rho}^{\sigma } $$ T ρ σ [SU(N)] theories. We obtain the $$ {E}_{\rho}^{\sigma } $$ E ρ σ [USp(2N)] theories from the recently introduced E[USp(2N )] theory, by following the RG flow initiated by vevs labelled by partitions ρ and σ for two operators transforming in the antisymmetric representations of the USp(2N) × USp(2N) IR symmetries of the E[USp(2N)] theory. These vevs are the 4d uplift of the ones we turn on for the moment maps of T[SU(N)] to trigger the flow to $$ {T}_{\rho}^{\sigma } $$ T ρ σ [SU(N)]. Indeed the E[USp(2N)] theory, upon dimensional reduction and suitable real mass deformations, reduces to the T[SU(N)] theory. In order to study the RG flows triggered by the vevs we develop a new strategy based on the duality webs of the T[SU(N)] and E[USp(2N)] theories.

We investigate the relation between 3d N = 2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S 2 × S 1 partition functions of some known 3d Seiberg-like dualities reduce, in a suitable 2d limit, to known basic duality identities for DF correlators. These identities are applied in a variety of contexts in CFT, as for example in the derivation of the DOZZ 3-point function. Reversing the logic, we can try to guess new 3d IR dualities which reduce to more intricate duality relations for the DF correlators. For example, we show that a recently proposed duality relating the U(N) theory with one flavor and one adjoint to a WZ model can be regarded as the 3d ancestor of the evaluation formula for the DF integral representation of the 3-point correlator. We are also able to interpret the analytic continuation in the number of screening charges, which is performed on the CFT side to reconstruct the DOZZ 3-point function, as the geometric transition relating the 3d U(N) theory to the 5d T 2 theory.

Argyres-Douglas (AD) theories constitute an infinite class of superconformal field theories in four dimensions with a number of interesting properties. We study several new aspects of AD theories engineered in A-type class $$ \mathcal{S} $$ S with one irregular puncture of Type I or Type II and also a regular puncture. These include conformal manifolds, structures of the Higgs branch, as well as the three dimensional gauge theories coming from the reduction on a circle. We find that the latter admits a description in terms of a linear quiver with unitary and special unitary gauge groups, along with a number of twisted hypermultiplets. The origin of these twisted hypermultiplets is explained from the four dimensional perspective. We also propose the three dimensional mirror theories for such linear quivers. These provide explicit descriptions of the magnetic quivers of all the AD theories in question in terms of quiver diagrams with unitary gauge groups, together with a collection of free hypermultiplets. A number of quiver gauge theories presented in this paper are new and have not been studied elsewhere in the literature.