Surface defects on lens space and transfer matrices

Abstract

We study surface defects in four-dimensional 𝒩=2 theories of class 𝒮 and 𝒩=1 theories of class 𝒮_k placed on L(r,1)× S¹. The lens space index depends on the fugacities p and q, and the effect of the defects on it is captured by difference operators acting on the fugacities and holonomies of flavor symmetries. For theories of type A₁ we determine in closed form the difference operators for the basic defects, labeled by p^r, q^r and pq, and show that for odd r there exist additional elementary defects, whose operators shift the holonomy by one half modulo r. All the operators commute, and the elementary operators generate the others. For even r the same holds once the holonomy is extended to a twisted sector. We show that the elementary operators are traces of L-operators whose entries generate a Sklyanin algebra, with nome q^r and shift pq. The lens space enters only through the parameters of the algebra and through its representation, which is realized on a doubled holonomy label. The p- and q-type algebras form a lens space version of the elliptic modular double. For class 𝒮_k theories we propose, and test in several ways, that the defect operators are transfer matrices of inhomogeneous spin chains built from these L-operators. Throughout, we use the TQFT structure of the lens space index, which follows from S-duality, both to fix and to check the operators.

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BibTeX RIS

BibTeX

@misc{maruyoshi2026surface,
  title = {{Surface defects on lens space and transfer matrices}},
  author = {Kazunobu Maruyoshi and Toshihiro Ota and Masahito Yamazaki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28165},
  eprint = {2610.28165},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28165},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Surface defects on lens space and transfer matrices
AU  - Kazunobu Maruyoshi
AU  - Toshihiro Ota
AU  - Masahito Yamazaki
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28165
UR  - https://arxiv.org/abs/2610.28165
AB  - We study surface defects in four-dimensional 𝒩=2 theories of class 𝒮 and 𝒩=1 theories of class 𝒮_k placed on L(r,1)× S^1. The lens space index depends on the fugacities p and q, and the effect of the defects on it is captured by difference operators acting on the fugacities and holonomies of flavor symmetries. For theories of type A_1 we determine in closed form the difference operators for the basic defects, labeled by p^r, q^r and pq, and show that for odd r there exist additional elementary defects, whose operators shift the holonomy by one half modulo r. All the operators commute, and the elementary operators generate the others. For even r the same holds once the holonomy is extended to a twisted sector. We show that the elementary operators are traces of L-operators whose entries generate a Sklyanin algebra, with nome q^r and shift pq. The lens space enters only through the parameters of the algebra and through its representation, which is realized on a doubled holonomy label. The p- and q-type algebras form a lens space version of the elliptic modular double. For class 𝒮_k theories we propose, and test in several ways, that the defect operators are transfer matrices of inhomogeneous spin chains built from these L-operators. Throughout, we use the TQFT structure of the lens space index, which follows from S-duality, both to fix and to check the operators.
ER  -

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