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arXiv · 2401.15230

Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities

Shashank Kanade
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Abstract

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.

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Shashank Kanade. 2024-01-26. Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities. https://arxiv.org/abs/2401.15230

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