2,545 research outputs found
Matrix factorizations and singularity categories for stacks
We study matrix factorizations of a section W of a line bundle on an
algebraic stack. We relate the corresponding derived category (the category of
D-branes of type B in the Landau-Ginzburg model with potential W) with the
singularity category of the zero locus of W generalizing a theorem of Orlov. We
use this result to construct push-forward functors for matrix factorizations
with relatively proper support.Comment: 29 page
Lifts of convex sets and cone factorizations
In this paper we address the basic geometric question of when a given convex
set is the image under a linear map of an affine slice of a given closed convex
cone. Such a representation or 'lift' of the convex set is especially useful if
the cone admits an efficient algorithm for linear optimization over its affine
slices. We show that the existence of a lift of a convex set to a cone is
equivalent to the existence of a factorization of an operator associated to the
set and its polar via elements in the cone and its dual. This generalizes a
theorem of Yannakakis that established a connection between polyhedral lifts of
a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts
of convex sets can also be characterized similarly. When the cones live in a
family, our results lead to the definition of the rank of a convex set with
respect to this family. We present results about this rank in the context of
cones of positive semidefinite matrices. Our methods provide new tools for
understanding cone lifts of convex sets.Comment: 20 pages, 2 figure
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