Parabolic Refined Invariants and Macdonald Polynomials
Copyright policy )Parabolic Refined Invariants and Macdonald Polynomials
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
- University of Illinois at Urbana Champaign United States
- National Taiwan University of Arts Taiwan
- Stanford University United States
- University of Pennsylvania United States
- University of Maryland University College United States
High Energy Physics - Theory, FOS: Physical sciences, Calabi-Yau manifolds (algebro-geometric aspects), HLRV conjecture, refined BPS expansion, Mathematics - Algebraic Geometry, Mirror symmetry (algebro-geometric aspects), refined stable pair invariants, FOS: Mathematics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic Geometry (math.AG), MacDonald polynomials, geometric engineering, Symmetric functions and generalizations, Combinatorial aspects of partitions of integers, parabolic Higgs bundles, local orbifold curves, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, parabolic P=W conjecture, Relationships between surfaces, higher-dimensional varieties, and physics, High Energy Physics - Theory (hep-th), Combinatorial aspects of representation theory, refined Gopakumar-Vafa invariants
High Energy Physics - Theory, FOS: Physical sciences, Calabi-Yau manifolds (algebro-geometric aspects), HLRV conjecture, refined BPS expansion, Mathematics - Algebraic Geometry, Mirror symmetry (algebro-geometric aspects), refined stable pair invariants, FOS: Mathematics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic Geometry (math.AG), MacDonald polynomials, geometric engineering, Symmetric functions and generalizations, Combinatorial aspects of partitions of integers, parabolic Higgs bundles, local orbifold curves, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, parabolic P=W conjecture, Relationships between surfaces, higher-dimensional varieties, and physics, High Energy Physics - Theory (hep-th), Combinatorial aspects of representation theory, refined Gopakumar-Vafa invariants
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