Twisted Vector Bundles Research Articles

Locally free twisted sheaves of infinite rank

We study twisted vector bundles of infinite rank on gerbes, giving a new point of view on Grothendieck’s famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme.We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of “very positive” infinite rank vector bundles on projective varieties, and show that an infinite rank vector bundle on a curve in a surface can be lifted to the surface away from finitely many points.

Supersymmetric Field Theories from Twisted Vector Bundles

We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for {1vert1}-dimensional EFTs from the data of a gerbe with connection.

Geometric models of twisted differential K-theory I

This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. By differential twists we will mean smooth U(1)-gerbes with connection, and we use twisted vector bundles with connection as cocycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. This paper confirms a long-standing hypothetical idea that twisted vector bundles with connection define twisted differential K-theory.

Twists over étale groupoids and twisted vector bundles

Inspired by recent papers on twisted K K -theory, we consider in this article the question of when a twist R \mathcal {R} over a locally compact Hausdorff groupoid G \mathcal {G} (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of G \mathcal {G} . We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of G \mathcal {G} . When G \mathcal {G} is an étale groupoid, we establish conditions (involving the classifying space B G B\mathcal {G} of G \mathcal {G} ) which imply that a torsion twist R \mathcal {R} over G \mathcal {G} admits a twisted vector bundle.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

Einstein–Hermitian connection on twisted Higgs bundles

Let X be a smooth projective variety over C . We prove that a twisted Higgs vector bundle ( E , θ ) on X admits an Einstein–Hermitian connection if and only if ( E , θ ) is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved in Wang [10]. Our approach is simpler.

Twisted vector bundles on pointed nodal curves

Motivated by the quest for a good compactification of the moduli space ofG-bundles on a nodal curve we establish a striking relationship between Abramovich’s and Vistoli’s twisted bundles and Gieseker vector bundles.

Twisted K-theory of differentiable stacks

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1 -gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K α i ⊗ K β j → K α + β i + j are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory ( KK-theory) of C ∗ -algebras.