Surface Defects in the $\mathcal{N}=2$ Lens Space Index and Elliptic Integrability

We determine the surface-defect difference operators of the 4d $\mathcal{N}=2$ lens space index for class $\mathcal{S}$ theories of type $A_{N-1}$ on $({\mathbb{S}}^3\mathrm{/}{\mathbb{Z}}_r)\times {\mathbb{S}}^1$. For generic fugacities $(p,q,t)$ and arbitrary rank $N$, orbifold order $r$ and discrete holonomies, their coefficients are finite products of theta functions with nomes $p^r$ and $q^r$. After an explicit gauge transformation, Gaussian in the holonomies, each operator factorizes into two elliptic Ruijsenaars–Schneider-type operators with step $pq$, one with nome $p^r$ and one with nome $q^r$, acting on variables glued along a lattice. Using this factorization and a Wronski-type relation, we prove that all the operators commute. We also establish a sufficient criterion for self-adjointness with respect to the bilinear vector multiplet pairing. It applies to the basic mixed operator and, in suitable parameter chambers, to the pure $p$- and $q$-type operators; for general mixed operators the contour prescription remains open. These results support the expectations from S-duality.