# Hochschild cohomology of (generalizations) of Khovanov's arc algebra

Backgroud: In his seminal paper A functor-valued invariant of tangles, Khovanov (among many other things) introduced the arc algebra $$H^{n}$$ and several functors between $$H^{n}$$ and $$H^{m}$$ related to tangles. In particular, Khovanov homology of links could be recovered from these functors/bimodules.

Stroppel defined a generalization of Khovanov's arc algebra in Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology by considering arcs with "semi-infinite" ends and the resulting algebra is denoted by $$\mathcal{K}^n$$. Brundan and Stroppel generalized these algebras further in Highest weight categories arising from Khovanov's diagram algebra I: cellularity, such as $$K_{m}^{n}$$.

As these papers suggest, the algebras $$K_{m}^{n}$$ are important for geometric representation theory. Some more recent work shows the connection between such algebras and annular Khovanov homology.

Question: Do we have some concrete computations of Hochschild cohomologies of the algebras $$H^n$$ and $$K_{m}^n$$? Certainly we can compute everything out by writing some computer programs, so the question is really about patterns/geometric intepretations of the Hochschild cohomologies in these cases.

• Is it possible to generalize the technique of Diogo-Lisi and compute symplectic cohomologies instead? Jan 11 '20 at 4:59
• That's possibly true, but it is know that the total rank of Hochschild cohomology of the ordinary arc algebra grows exponentially w.r.t. $n$. To compute the differentials, one needs to compute GW invariants inside some Hilbert scheme, and actually we need some precise information about curves avoiding a divisor. The extended version is some kind of FS category, whose HH* may not have nice "closed-string" cousins. Jan 11 '20 at 16:24