3D TQFT and HOMFLYPT homology

Abstract

In this note we propose a 3D TQFT such that its Hilbert space on $S^2$, which intersects with defect surfaces along a (possibly self-intersecting) curve $C$ is the HOMFLYPT homology of the link whose diagram is $C$. Previously this homology was interpreted as the space of sections of a special 2-periodic complex of coherent sheaf on $Hilb_n(\mathbb{C}^2)$. TQFT perspective provides a natural explanation for this interpretation, since the category $D^{per}Coh(Hilb_n(\mathbb{C}^2))$ is the Drinfeld center of the two-category of assigned to a point by our TQFT.Comment: 22 pages, 5 figures, comments are welcom