Zeta-polynomials, superpolynomials, DAHA and plane curve singularities
Ivan Cherednik
TL;DR
The work proposes a unifying framework linking modular form periods, DAHA-invariants, and motivic zeta–polynomials for plane curve singularities, conjecturing that motivic, DAHA, and L-function constructions coincide in relevant regimes and satisfy RH-type constraints in certain sectors. It develops a comprehensive DAHA toolkit (including polynomial representations, Verlinde algebras, and Galois action) and applies it to knot invariants, torus links, and iterated structures, while introducing motivic polynomials from compactified Jacobians and affine Springer fibers. By formulating zeta-like constructions for singularities and establishing connections to physics (Lee–Yang, LG models, Witten index, S-duality), the paper proposes a broad, conceptually unified picture in which zeta-polynomials, superpolynomials, and L-functions reflect common topological and arithmetic data. The work also lays groundwork for future links to surface singularities, higher-dimensional invariants, and a common algebraic–geometric interpretation of RH-type phenomena across number theory, topology, and mathematical physics.
Abstract
We begin with modular form periods, a focal point of several Yuri Manin's works. The similarity is discussed between the corresponding zeta-polynomials and superpolynomials of algebraic links, closely related to Khovanov-Rozansky polynomials. We focus on DAHA superpolynomials and motivic ones, defined via compactified Jacobians of plane curve singularities and their counterparts in arbitrary ranks; the non-unibranch construction is new. They conjecturally coincide with the corresponding generalizations of L-functions and satisfy the Riemann Hypothesis in some sectors of the parameters. Presumably, the motivic ones an be interpreted as certain partition functions of Landau-Ginzburg models associated with plane curve singularities; RH for them is remarkably similar to the Lee-Yang circle theorem for Ising models. A q,t-deformation of the Witten index is obtained as an application. General perspectives of the motivic theory of isolated curve and surface singularities are discussed, including possible implications in number theory. Also, we introduce super-analogs of $ρ_{ab}$-invariants and discuss super-deformations of the Riemann's zeta. Among other topics: Verlinde algebras, and the topological vertex.