Parabolic refined invariants and Macdonald polynomials

Abstract

A string theoretic derivation is given for the conjecture of Hausel, Letellier, and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with G. Pan. Haiman's geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.

77 pages

References in corpus (5)

• The Refined Topological Vertex
• M-theory and a Topological String Duality
• The refined BPS index from stable pair invariants
• Topological Strings, Instantons and Asymptotic Forms of Gopakumar-Vafa Invariants
• Hilbert schemes as moduli of Higgs bundles and local systems

Cited by in corpus (16)

• Singular Geometry and Higgs Bundles in String Theory
• Refined large N duality for knots
• Motivic classes of moduli of Higgs bundles and moduli of bundles with connections
• Seiberg-Witten for $Spin(n)$ with Spinors
• A Nekrasov-Okounkov formula for Macdonald polynomials
• The quiver at the bottom of the twisted nilpotent cone on ${\mathbb{P}}^1$
• BPS states, torus links and wild character varieties
• Moduli spaces of framed flags of sheaves on the projective plane
• Topological strings, quiver varieties and Rogers-Ramanujan identities
• Meromorphic Higgs bundles And Related Geometries
• Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve
• The geometry of double nested Hilbert schemes of points on curves
• Flags of sheaves, quivers and symmetric polynomials
• Equality of the wobbly and shaky loci
• Twisted spectral correspondence and torus knots
• Holomorphic pre-symplectic form on the nested Hilbert scheme $\mbox{Hilb}^{3,4}(\mathbb{C}^2)$