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

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 complement. We suggest a construction of classical shaded A-polynomials $A_a({\ell }_b,m_c)$ associated to Lie groups $SU(N)$, assuming that color $N$ adds some extra "depth" and "shade" to the classical notion of A-polynomials. We exploit a formalism of Clebsch-Gordan (CG) chords, where indices $a$, $b$, $c$ run over $1,\dots ,N-1$. CG chords have a natural interpretation in terms of 2d CFTs of WZW type, or 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. The CG chord formalism can 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 representations as well. We primarily aim at classical A-polynomials by considering a double scaling limit in which $q=e^{\hslash }$, $\hslash \to 0$ and the representations are huge: highest weight vector components $w_i\to \infty$ while combinations $\hslash w_i\sim m_i$ remain finite. We expect the presented techniques will be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras $\mathfrak{g}$. We discuss explicit examples of A-polynomials for knots $3_1$, $4_1$ and $5_1$ for $\mathfrak{g}={\mathfrak{su}}_3$.

arXiv comment36 pages, 8 figures, v2: minor corrections