Surface Defects in the 𝒩=2 Lens Space Index and Elliptic Integrability

Abstract

We determine the surface-defect difference operators of the 4d 𝒩=2 lens space index for class 𝒮 theories of type A_N−1 on (𝕊³/ℤ_r)×𝕊¹. 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.

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BibTeX RIS

BibTeX

@misc{yamazaki2026surface,
  title = {{Surface Defects in the $\mathcalN=2$ Lens Space Index and Elliptic Integrability}},
  author = {Masahito Yamazaki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30849},
  eprint = {2610.30849},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30849},
}

RIS

TY  - UNPB
TI  - Surface Defects in the 𝒩=2 Lens Space Index and Elliptic Integrability
AU  - Masahito Yamazaki
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30849
UR  - https://arxiv.org/abs/2610.30849
AB  - We determine the surface-defect difference operators of the 4d 𝒩=2 lens space index for class 𝒮 theories of type A_N−1 on (𝕊^3/ℤ_r)×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.
ER  -

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