Instanton slices and their superpolynomials

Abstract

The key theorem is a connection between motivic superpolynomials of plane curve singularities in any ranks with superpolynomials of the corresponding instanton slices, Nekrasov-type instanton sums with conductors. In this case, instanton slices are related to compactified Jacobians, but they form a much wider class and, generally, have no connection to plane curve singularities. This development is expected to impact theory of affine Springer fibers (at least, in type $A$), and related fields. For instance, we obtain a motivic interpretation (counting $\mathbb{F}_q$-points of some stacks) of superpolynomials for hyperbolic knots K12n242, K12n725, among many others. New formulas for instanton sums in any ranks are obtained using that they are inductive limits of superpolynomials of proper families of plane curve singularities. The main conjecture states that instanton superpolynomials for arbitrary Young diagrams (almost all are not from plane curve singularities) can be transformed to the corresponding DAHA superpolynomials if and only if the latter are positive or (equivalently) the former are superdual. Our DAHA superpolynomials are parallel to Galashin-Lam EHA ones, but we used the nonsymmetric approach. The connection with reduced Khovanov-Rozansky polynomials of the corresponding Coxeter knots is expected. Among applications, formulas for Nekrasov's instanton sums with an impressive connection to nonsymmetric Macdonald polynomials, topological invariance of unibranch motivic superpolynomials for valuation semigroups with 3-4 generators, some mixed-characteristic conjecture, and an upgrade of Weak Riemann Hypothesis.