Hybrid models for homological projective duals and noncommutative resolutions

TL;DR

This paper explores hybrid models as homological projective duals of Fano manifolds, revealing their structure as noncommutative resolutions and computing associated algebras explicitly.

Contribution

It introduces a framework connecting homological projective duality with noncommutative resolutions via $A_{ abla}$-algebras and computes these algebras for specific hybrid models.

Findings

Hybrid models correspond to noncommutative resolutions of base spaces.

The algebra $\\mathcal{A}$ often forms a smash product with a cyclic group.

Explicit computations are provided for certain Veronese and Fano complete intersection embeddings.

Abstract

We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ 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 $\mathbb{P}^{n}$.