The homological projective dual of Sym^2 P(V)

TL;DR

This paper explores the homological projective duality of Sym^2 P(V), linking derived categories of certain complete intersections to Clifford algebra modules, and provides new insights into Calabi-Yau threefolds.

Contribution

It introduces a new homological projective duality framework for Sym^2 P(V) using GIT and matrix factorisations, connecting derived categories to Clifford modules.

Findings

Established a duality between Sym^2 P(V) and Clifford module categories.

Derived categories of certain Calabi-Yau threefolds are equivalent.

Provided a new proof for a known Calabi-Yau derived equivalence.

Abstract

We study the derived category of a complete intersection X of bilinear divisors in the orbifold Sym^2 P(V). Our results are in the spirit of Kuznetsov's theory of homological projective duality, and we describe a homological projective duality relation between Sym^2 P(V) and a category of modules over a sheaf of Clifford algebras on P(Sym^2 V^vee). The proof follows a recently developed strategy combining variation of GIT stability and categories of global matrix factorisations. We begin by translating D^b(X) into a derived category of factorisations on an LG model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of nonbirational Calabi-Yau 3-folds have equivalent derived categories.