Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911779964059648 |
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| author | Kanade, Shashank |
| author_facet | Kanade, Shashank |
| contents | We study coloured invariants of torus knots $T(p,p')$ (where $p,p'$ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when $p,p'\geq h^\vee$, we use the representation theory of the corresponding principal affine $\mathcal{W}$ algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain $\mathcal{W}$ algebra modules (up to some factors). This result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15230 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities Kanade, Shashank Quantum Algebra Mathematical Physics Geometric Topology Representation Theory 17B10, 17B69, 57K14 We study coloured invariants of torus knots $T(p,p')$ (where $p,p'$ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when $p,p'\geq h^\vee$, we use the representation theory of the corresponding principal affine $\mathcal{W}$ algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain $\mathcal{W}$ algebra modules (up to some factors). This result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples. |
| title | Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities |
| topic | Quantum Algebra Mathematical Physics Geometric Topology Representation Theory 17B10, 17B69, 57K14 |
| url | https://arxiv.org/abs/2401.15230 |