E-string compactifications and the van Diejen model

We study supersymmetric field theories in four dimensions, obtained from compactifications of the 6d E-string theory on Riemann surfaces, in presence of surface defects.

We show that the index of a theory with defect can be obtained by acting with a difference operator on the index of the theory without defect, and show that this difference operator is, up to a constant shift, the van Diejen difference operator.