A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras

Abstract

The Melvin--Morton--Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras. We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra g and a dominant integral weight λ, we use the embedding V_dλ↪ V_λ^{⊗ d} to analyse the state sum. Under the specialization q=e^h/d, we show that, as d→∞, only the identity terms and those proportional to E_α⊗ F_α contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the sl₂ case. Writing t_α=e^−2(λ,α)h, the limit is therefore given by the product of Δ_K(t_α)⁻¹ over the positive roots α.

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BibTeX RIS

BibTeX

@misc{nagasaka2026semiclassical,
  title = {{A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras}},
  author = {Atsuhide Nagasaka},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10269},
  eprint = {2610.10269},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10269},
}

RIS

TY  - UNPB
TI  - A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras
AU  - Atsuhide Nagasaka
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10269
UR  - https://arxiv.org/abs/2610.10269
AB  - The Melvin--Morton--Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras. We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra g and a dominant integral weight λ, we use the embedding V_dλ↪ V_λ^⊗ d to analyse the state sum. Under the specialization q=e^h/d, we show that, as d→∞, only the identity terms and those proportional to E_α⊗ F_α contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the sl_2 case. Writing t_α=e^−2(λ,α)h, the limit is therefore given by the product of Δ_K(t_α)^−1 over the positive roots α.
ER  -

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