Homological projective duality conjecture for higher-dimensional Pfaffian varieties

Let $W$ be a vector space of dimension $n$, and consider the Grassmannian

$$X = \operatorname{Gr}(2,W) \subset \PP(\Lambda^2W).$$

Let

$$Y = \operatorname{Pf}(2\lfloor n/2\rfloor - 2,W^*) \subset \PP(\Lambda^2W^*)$$

be the corresponding Pfaffian variety. A noncommutative resolution of singularities of $Y$ is a pair $(Y,\CR)$ consisting of $Y$ and its noncommutative resolution algebra $\CR$.

Higher-dimensional Pfaffian HPD conjecture. There exists a noncommutative resolution of singularities $(Y,\CR)$ of $Y$ which is Homologically Projectively Dual to $X$.

The conjecture extends the constructions available for $\dim W=6,7$ to the case $\dim W>7$. If true, it would yield descriptions of derived categories of Pfaffian hypersurfaces of all degrees; the paper does not establish the general construction or the conjecture.