Refined Chern-Simons theory and Hilbert schemes of points on the plane (Q2875946)

\textit{E. Witten} [Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)] interpreted the Jones invariants of links in \(S^3\) (such as the Jones polynomial) in terms of his topological quantum field theory using Chern-Simons theory (see \textit{K. Marathe} [in: The mathematics of knots. Theory and application. Banagl, Markus (ed.) et al., Berlin: Springer. Contributions in Mathematical and Computational Sciences 1, 199--256 (2011; Zbl 1221.57023)] for a recent survey). This theory depends on a rank \(N\) and level \(k\), the quantum Hilbert space being identified with level \(k\) highest weight representations of the Lie algebra corresponding to \(\text{SU}(N)\) and acted on by the group \(\text{SL}(2,\mathbb Z)\) via matrices \(S\) and \(T\). Recently \textit{M. Aganagic} and \textit{Sh. Shakirov} proposed a refinement of the \(\text{SU}(N)\) Chern-Simons theory for links in 3-manifolds having \(S^1\) symmetry [String-Math 2011, Proc. Symp. Pure Math. 85, Amer. Math. Soc., Providence RI 2012, 3--31 (2012)] in which the matrices \(S\) and \(T\) are replaced by matrices used by \textit{I. Cherednik} [Invent. Math. 122, No. 1, 119--145 (1995; Zbl 0854.22021)] and \textit{A. A. Kirillov, jun.} [J. Am. Math. Soc. 9, No. 4, 1135--1169 (1996; Zbl 0861.05065)]. In the refined theory, the Hilbert space is identified with the MacDonald polynomials of type \(\text{SU}(N)\) with parameters \(q,t\) satsifying \(q^k t^N = 1\).NEWLINENEWLINEIn the paper under review, the author computes the limit of the matrix \(S\) as \(N \to \infty\) for the refined theory. Starting with the explicit form of \(S\) given by \textit{M. Aganagic} and \textit{Sh. Shakirov} [``Knot homology from refined Chern-Simons theory'', Preprint, 2011, \url{arXiv:1105.5117}], he replaces \(t^N\) by a variable \(u\) to obtain a stable version expressed in terms of the modified MacDonald polynomials of \textit{A. Garsia} et al. [Sémin. Lothar. Comb. 42, B42m, 45 p. (1999; Zbl 0920.05071)]. To compute the kernel function, he starts with the Cherednik-MacDonald-Mehta identity in the form used by Garsia, Haiman and Tesler [loc. cit.] and using the relation between MacDonald polynomials and Hilbert schemes of \(n\) points in \(X=\mathbb C^2\) due to \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001), Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)] he expresses two of the terms as power series in \(u\): the coefficient of \(u^n\) in each case is an equivariant Euler characteristics of certain sheaves on the Hilbert scheme \(X^{[n]}\) for the group action of \(\mathbb C^* \times \mathbb C^*\) - in one case the structure sheaf \({\mathcal O}_{X^{[n]}}\) and in the other case the tensor product of arbitrary Schur functors \(s_{\lambda}\) and \(s_{\mu}\) applied to a universal sheaf.NEWLINENEWLINEFor the entire collection see [Zbl 1285.00037].