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

Shashank Kanade

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.

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

Abstract

We study coloured invariants of torus knots (where are coprime positive integers). When the colouring Lie algebra is simply-laced, and when , we use the representation theory of the corresponding principal affine 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 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.
Paper Structure (24 sections, 12 theorems, 132 equations)

This paper contains 24 sections, 12 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on affine Lie algebras
  3. Triangular decomposition
  4. Bases and pairings
  5. Bilinear form
  6. Roots, weights, etc.
  7. Weyl group
  8. Admissible weights
  9. Simply-laced case
  10. Principal $\mathscr{W}$ algebras
  11. Coloured invariants of torus knots
  12. The Rosso--Jones formula
  13. Minimum exponent
  14. Limits
  15. Relative asymptotic weight multiplicities
  16. ...and 9 more sections

Key Result

Theorem 1

The functor $H^0_-: \dot{\mathcal{O}}_k\rightarrow \mathcal{O}(\mathscr{W}^k)$ satisfies the following properties.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • ...and 22 more