We define new algebras, local bimodules and bimodule maps in the spirit
of Ozsváth–Szabó’s bordered knot Floer homology. We equip them
with the structure of 2-representations of the categorified negative half
of
, 1-morphisms
of such and 2-morphisms, respectively, and show that they categorify representations
of
and maps between them. Unlike with Ozsváth–Szabó’s algebras, the algebras
considered here can be built from a higher tensor product operation recently
introduced by Rouquier and the author.
Our bimodules are all motivated by holomorphic disk counts in Heegaard
diagrams; for positive and negative crossings, the bimodules can also be expressed as
mapping cones involving a singular-crossing bimodule and the identity bimodule. In
fact, they arise from an action of the monoidal category of Soergel bimodules via
Rouquier complexes in the usual way; the first time (to the author’s knowledge) such
an expression has been obtained for braiding bimodules in Heegaard Floer
homology.
Furthermore, the singular crossing bimodule naturally factors into two
bimodules for trivalent vertices; such bimodules have not appeared in
previous bordered-Floer approaches to knot Floer homology. The action
of the Soergel category comes from an action of categorified quantum
on the 2-representation
2-category of
in
line with the ideas of skew Howe duality, where the trivalent vertex bimodules are associated to
1-morphisms
in
categorified quantum
.
