3d mirror symmetry and HOMFLY-PT homology

Abstract: Since the original physical prediction of triply-graded HOMFLY-PT link homology by Gukov-Schwarz-Vafa, and its mathematical definition by Khovanov-Rozansky, many other (conjectural) constructions of HOMFLY-PT link homology have appeared --- with different algebraic and geometric origins, and manifesting different properties. One recent proposal of Oblomkov-Rozansky (closely related to work of Gorsky-Neguț-Rasmussen) associated to a link L a coherent sheaf E_L on a Hilbert scheme, whose cohomology reproduces HOMFLY-PT homology. Another proposal, by Gorsky-Oblomkov-Rasmussen-Shende, computes HOMFLY-PT homology of algebraic knots via Borel-Moore homology of affine Springer fibers. I will explain how the first (Hilbert scheme) construction is realized in the "B" twist of a 3d supersymmetric gauge theory, and then carefully apply 3d mirror symmetry to discover a variant of the second (Springer fiber) construction. I will also indicate how both 3d gauge theory setups are related to the original work of Gukov-Schwarz-Vafa based using M-theory on the resolved conifold. (Preprint soon to appear, with N. Garner, J. Hilburn, A. Oblomkov, and L. Rozansky).

mathematical physicsalgebraic geometrycategory theoryrepresentation theory

Audience: researchers in the topic

UMass Amherst Representation theory seminar