Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

TL;DR

This paper explores supersymmetric surface defects in 6d conformal matter theories, linking them to generalizations of the van Diejen model and providing new insights into 4d dualities and Lagrangian descriptions.

Contribution

It introduces new $A_N$, $C_N$, and $(A_1)^N$ generalizations of the van Diejen model related to 6d conformal matter compactifications, with explicit operator constructions and duality checks.

Findings

Derived properties of generalized van Diejen operators from 4d dualities.

Explicitly computed operators for the $A_N$ case.

Provided Lagrangian descriptions for certain 6d SCFT compactifications.

Abstract

We consider supersymmetric surface defects in compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. For the case of $N=1$ such defects are introduced into the supersymmetric index computations by an action of the $BC_1\,(\sim A_1\sim C_1)$ van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras $A_{N=1}$, $C_{N=1}$, and $(A_1)^{N=1}$. The indices of these $4d$ theories give rise to three different Kernel functions for the $BC_1$ van Diejen model. We then consider the generalizations with $N>1$. The operators introducing defects into the index computations are certain $A_{N}$, $C_N$, and $(A_1)^{N}$ generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the $A_N$ case, and derive various properties these operators have to satisfy as a consequence of $4d$ dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal $(D_5,D_5)$ conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.