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

Abstract

We consider supersymmetric surface defects in compactifications of the 6d6d minimal (D_{N+3},D_{N+3})(DN+3,DN+3) conformal matter theories on a punctured Riemann surface. For the case of N=1N=1 such defects are introduced into the supersymmetric index computations by an action of the BC_1\,(\sim A_1\sim C_1)BC1(∼A1∼C1) 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}AN=1, C_{N=1}CN=1, and (A_1)^{N=1}(A1)N=1. The indices of these 4d4d theories give rise to three different Kernel functions for the BC_1BC1 van Diejen model. We then consider the generalizations with N>1N>1. The operators introducing defects into the index computations are certain A_{N}AN, C_NCN, and (A_1)^{N}(A1)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})(DN+3,DN+3) conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the A_NAN case, and derive various properties these operators have to satisfy as a consequence of 4d4d 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)(D5,D5) conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.