Catalan combinatorics versus nonabelian Hodge theory

Abstract: The Oblomkov–Rasmussen–Shende conjecture relates the homologies of the Hilbert schemes of a plane curve singularity to the triply-graded Khovanov–Rozansky (i.e., HOMFLYPT) homology of its link, via an identity in variables a, q, t. Two major cases are known: (1) the t = -1 limit, settled a decade ago by Maulik; (2) the lowest-a-degree, q = 1 limit of the "torus link" case, settled jointly by Elias–Hogancamp, Mellit, and Gorsky–Mazin, using (q, t)-Catalan combinatorics as an essential bridge. An unpublished research statement of Shende speculated that the ORS conjecture could be proved in a third, totally different way, via a wild analogue of the P = W phenomenon in nonabelian Hodge theory. He and his coauthors carried out most of this approach for the "torus-knot" subcase of case (1). We extend their work, and also refine it enough to handle the (more difficult) torus-knot subcase of case (2). The key is our new geometric model for Khovanov–Rozansky homology, which realizes the t variable as cohomological degree. If there is time, we will explain how this flavor of nonabelian Hodge theory is related to the noncrossing-nonnesting dichotomy in Catalan combinatorics.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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