Shading A-polynomials via huge representations of $U_q(\mathfrak{su}_N)$

TL;DR

This paper introduces a new method for constructing classical A-polynomials for Lie groups $SU(N)$ using CG chords and large representation limits, extending previous $SU(2)$ frameworks.

Contribution

The authors develop a formalism based on CG chords and large representation limits to generalize A-polynomials to arbitrary $SU(N)$ and representations, connecting to quantum groups and knot contact homology.

Findings

Constructed classical A-polynomials for knots 3_1, 4_1, 5_1 in $\mathfrak{su}_3$.

Extended the CG chord formalism to arbitrary $\mathfrak{su}_N$.

Proposed techniques for deriving quantum A-polynomials for general Lie (super)algebras.

Abstract

Classical A-polynomials $A(\ell,m)$ define constraints on coordinates $\ell$ and $m$ in $SL(2,\mathbb{C})$ (a complexification of $SU(2)$) character varieties associated to knot complements $S^3\setminus K$. Quantum A-polynomials $\hat A(\hat \ell,\hat m)$ are difference operators annihilating Jones polynomials believed to represent wave functions of 3d Chern-Simons theory with gauge group $SU(2)$ on a toroidal pipe surrounding the knot $K$ strand -- a boundary of the knot complements $S^3\setminus K$. We suggest a construction of classical shaded A-polynomials $A_a(\ell_b,m_c)$ associated to Lie groups $SU(N)$. We exploit a formalism of Clebsh-Gordan (CG) chords, where indices $a$, $b$, $c$ run over $1,\ldots,N-1$. CG chords have a natural interpretation in terms of 2d CFTs of WZW type, or, alternatively, in terms of quantum group $U_q(\mathfrak{su}_N)$. In the case of $\mathfrak{su}_2$ CG chords could be associated to Reeb chords in a knot contact homology (KCH) framework. KCH suggests its own analogue of A-polynomials known as augmentation polynomials allowed to have extra spurious roots in principle. Yet the CG chord formalism could be easily extended to arbitrary $\mathfrak{su}_N$ allowing us to generalize the construction of A(ugmentation)-polynomials to arbitrary $\mathfrak{su}_N$ and arbitrary representation as well. Primarily we aim at classical A-polynomials by considering a double scaling limit when $q=e^{\hbar}$, $\hbar\to 0$ and the representations are huge, in particular, highest weight vector components $w_i\to \infty$ so that $\hbar w_i\sim m_i$ remain finite. Still we expect the presented techniques would be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras $\mathfrak{g}$. Also we discuss explicit examples of A-polynomials for knots $3_1$, $4_1$ and $5_1$ for $\mathfrak{g}=\mathfrak{su}_3$.