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 simplylaced, 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 Liealgebraic conjecture on asymptotic weight multiplicities which we verify in some examples.
 Publication:

arXiv eprints
 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