Hybrid models for homological projective duals and noncommutative resolutions

Abstract

We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\to \mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding $f$. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space $B$ or, equivalently, as the derived category of the sheaf of $\mathcal{A}$-modules on $B$, where $\mathcal{A}$ is an $A_{\infty }$ algebra. This latter interpretation corresponds to a noncommutative resolution of $B$. We compute explicitly the algebra $\mathcal{A}$ by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an $A_{\infty }$ algebra with a cyclic group. Then we apply our results to the HPD of $f$ corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in ${\mathbb{P}}^n$.

50 pages, LaTeX; v3: references added, several improvements on the examples and the section on complete intersections. Published version

References in corpus (1)

• Matrix factorisations and open topological string theory

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