Rank-one 4d 𝒩=3 SCFTs: Schur index, VOA modules, and modularity

Abstract

We study the representation theory of the vertex operator algebras (VOAs) associated with rank-one 4d 𝒩 = 3 superconformal field theories. For the ℤ₃ S-fold theory, whose VOA 𝒲_ℤ₃ has central charge c_2d = −15, we use the 𝒩 = 1 Lagrangian description to obtain the unflavored Schur index in terms of Dedekind eta functions, while Wilson-loop indices yield the unflavored non-vacuum characters. These characters all solve a modular linear differential equation (MLDE) whose solution space also contains a logarithmic character. Combining flavored MLDEs from null states with Zhu's associative algebra and a free-field realization, we study four highest-weight modules of 𝒲_ℤ₃ and their flavored characters in closed form. A parallel analysis applies to the 𝒩 = 3 theories obtained by gauging a discrete ℤ_n flavor subgroup of 𝒩 = 4 U(1) and SU(2) super-Yang--Mills, for which we also obtain closed-form Schur indices and a new free-field realization of the VOA of the ℤ₄ quotient.

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BibTeX RIS

BibTeX

@misc{guo2026rankone,
  title = {{Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity}},
  author = {Zhaoting Guo and Satoshi Nawata and Yiwen Pan and Qituan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10685},
  eprint = {2609.10685},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10685},
}

RIS

TY  - UNPB
TI  - Rank-one 4d 𝒩=3 SCFTs: Schur index, VOA modules, and modularity
AU  - Zhaoting Guo
AU  - Satoshi Nawata
AU  - Yiwen Pan
AU  - Qituan Zhang
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10685
UR  - https://arxiv.org/abs/2609.10685
AB  - We study the representation theory of the vertex operator algebras (VOAs) associated with rank-one 4d 𝒩 = 3 superconformal field theories. For the ℤ_3 S-fold theory, whose VOA 𝒲_ℤ_3 has central charge c_2d = −15, we use the 𝒩 = 1 Lagrangian description to obtain the unflavored Schur index in terms of Dedekind eta functions, while Wilson-loop indices yield the unflavored non-vacuum characters. These characters all solve a modular linear differential equation (MLDE) whose solution space also contains a logarithmic character. Combining flavored MLDEs from null states with Zhu's associative algebra and a free-field realization, we study four highest-weight modules of 𝒲_ℤ_3 and their flavored characters in closed form. A parallel analysis applies to the 𝒩 = 3 theories obtained by gauging a discrete ℤ_n flavor subgroup of 𝒩 = 4 U(1) and SU(2) super-Yang--Mills, for which we also obtain closed-form Schur indices and a new free-field realization of the VOA of the ℤ_4 quotient.
ER  -

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