Spherical DAHA as an algebra of framed BPS states

Abstract

Line operators of 4d 𝒩=2 theories of class 𝒮 of type A_N−1 in a half Omega background provide a physical model for SL_N skein algebras. In this paper we extend this correspondence to GL_N skein algebras and prove that the GL₂ skein algebra of the punctured torus is isomorphic to the GL₂ spherical double affine Hecke algebra SḦ^q,t₂. The construction is based on q-nonabelianization for the SU(2) 𝒩=2^* gauge theory, which realizes the skein algebra as a quantum torus algebra associated with the Seiberg-Witten curve. Different regions of the Coulomb branch yield distinct presentations of SḦ^q,t₂. These range from the Macdonald q-difference module in Fenchel-Nielsen (weak coupling) charts to cluster-type realizations in Fock-Goncharov (strong coupling) charts. Transitions between these descriptions are governed by the vanilla BPS spectrum of the 𝒩=2^* theory via framed wall-crossing, which provides a unified physical framework for several representations of spherical DAHA.

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BibTeX RIS

BibTeX

@misc{gupta2026spherical,
  title = {{Spherical DAHA as an algebra of framed BPS states}},
  author = {Kunal Gupta and Pietro Longhi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07911},
  eprint = {2609.07911},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07911},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Spherical DAHA as an algebra of framed BPS states
AU  - Kunal Gupta
AU  - Pietro Longhi
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07911
UR  - https://arxiv.org/abs/2609.07911
AB  - Line operators of 4d 𝒩=2 theories of class 𝒮 of type A_N−1 in a half Omega background provide a physical model for SL_N skein algebras. In this paper we extend this correspondence to GL_N skein algebras and prove that the GL_2 skein algebra of the punctured torus is isomorphic to the GL_2 spherical double affine Hecke algebra SḦ^q,t_2. The construction is based on q-nonabelianization for the SU(2) 𝒩=2^* gauge theory, which realizes the skein algebra as a quantum torus algebra associated with the Seiberg-Witten curve. Different regions of the Coulomb branch yield distinct presentations of SḦ^q,t_2. These range from the Macdonald q-difference module in Fenchel-Nielsen (weak coupling) charts to cluster-type realizations in Fock-Goncharov (strong coupling) charts. Transitions between these descriptions are governed by the vanilla BPS spectrum of the 𝒩=2^* theory via framed wall-crossing, which provides a unified physical framework for several representations of spherical DAHA.
ER  -

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