Generalizing Verlinde Formulas to Higgs Bundles via Quasi-Hopf Algebras — E8 Intelligence Research

Abstract

FINDING: The Verlinde formula, central to fusion categories and modular tensor categories, is being actively generalized to Higgs bundle moduli spaces and probed via weak quasi-Hopf algebras, with classification efforts for small fusion categories. | MATH: Verlinde formula: N_ij^k = ∑_{λ ∈ 𝒫_k} (S_iλ S_jλ S^*_kλ)/S_0λ, where S is the modular S-matrix; categorical dimension dim(𝒞) = ∑_i |θ_i|² d_i² (for modular categories, with twist θ_i, quantum dimension d_i); root system data enters via affine Weyl alcove 𝒫_k = { λ : ⟨ λ + ρ, θ^∨ ⟩ ≤ k } for level k, with ρ Weyl vector, θ^∨ coroot of highest root. | CONNECTION: Direct link to affine Weyl alcoves — the summation domain is a simplex whose geometry encodes crystallographic root systems (e.g., A_n, D_n, E₆, E₇, E₈). Quantum dimensions d_i are o

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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BibTeX

@misc{caldin2026generalizing,
  title = {{Generalizing Verlinde Formulas to Higgs Bundles via Quasi-Hopf Algebras — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23635163},
  url = {https://zenodo.org/records/23635163},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Generalizing Verlinde Formulas to Higgs Bundles via Quasi-Hopf Algebras — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/25
PB  - Zenodo
DO  - 10.5281/zenodo.23635163
UR  - https://zenodo.org/records/23635163
AB  - FINDING: The Verlinde formula, central to fusion categories and modular tensor categories, is being actively generalized to Higgs bundle moduli spaces and probed via weak quasi-Hopf algebras, with classification efforts for small fusion categories. | MATH: Verlinde formula: N_ij^k = ∑_λ ∈ 𝒫_k (S_iλ S_jλ S^*_kλ)/S_0λ, where S is the modular S-matrix; categorical dimension dim(𝒞) = ∑_i |θ_i|^2 d_i^2 (for modular categories, with twist θ_i, quantum dimension d_i); root system data enters via affine Weyl alcove 𝒫_k = { λ : ⟨ λ + ρ, θ^∨ ⟩ ≤ k } for level k, with ρ Weyl vector, θ^∨ coroot of highest root. | CONNECTION: Direct link to affine Weyl alcoves — the summation domain is a simplex whose geometry encodes crystallographic root systems (e.g., A_n, D_n, E_6, E_7, E_8). Quantum dimensions d_i are o  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -

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