Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2024
- DOI:
- 10.48550/arXiv.2401.15230
- arXiv:
- arXiv:2401.15230
- Bibcode:
- 2024arXiv240115230K
- Keywords:
-
- Mathematics - Quantum Algebra;
- Mathematical Physics;
- Mathematics - Geometric Topology;
- Mathematics - Representation Theory;
- 17B10;
- 17B69;
- 57K14