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

We consider supersymmetric surface defects in compactifications of the 6d minimal (DN+3, DN+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 BC1 (similar to A(1) similar to 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 BC1 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 (DN+3, DN+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.

Automated summary

In this study, supersymmetric surface defects in compactifications on punctured Riemann surfaces of the 6d minimal (DN+3, DN+3) conformal matter theories are considered. The introduction of these defects into supersymmetric index computations through the action of the BC1 van Diejen model is investigated using three field theoretic descriptions of the four dimensional models. The generalizations with N > 1 are also explored, revealing the direct relationship to different effective gauge theory descriptions obtained by compactifying the theories on a circle to five dimensions.