Raymond F. Vozzo

The caloron correspondence and higher string classes for loop groups

Abstract We review the caloron correspondence between G -bundles on M × S 1 and Ω G -bundles on M , where Ω G is the space of smooth loops in the compact Lie group G . We use the caloron correspondence to define characteristic classes for Ω G -bundles, called string classes, by transgression of characteristic classes of G -bundles. These generalise the string class of Killingback to higher-dimensional cohomology.

The general caloron correspondence

Abstract We outline in detail the general caloron correspondence for the group of automorphisms of an arbitrary principal G -bundle Q over a manifold X , including the case of the gauge group of Q . These results are used to define characteristic classes of gauge group bundles. Explicit but complicated differential form representatives are computed in terms of a connection and Higgs field.

CIRCLE ACTIONS, CENTRAL EXTENSIONS AND STRING STRUCTURES

The caloron correspondence can be understood as an equivalence of categories between G-bundles over circle bundles and LG ⋊ρ S1-bundles where LG is the group of smooth loops in G. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an LG ⋊ρ S1-bundle.

A geometric model for odd differential K-theory☆

Odd K-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential K-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We characterise the odd Chern character and its transgression form in terms of a connection and Higgs field and discuss some applications. Our model can be seen as the odd counterpart to the Simons–Sullivan construction of even differential K-theory. We use this model to prove a conjecture of Tradler–Wilson–Zeinalian [16], which states that the model developed there also defines the unique differential extension of odd K-theory.

Equivariant bundle gerbes

We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on a unitary group and a string structure for a principal bundle. We show that the basic bundle gerbe is equivariant for the conjugation action and calculate its characteristic class; we show also that a string structure gives rise to a bundle gerbe which is equivariant for a natural action of the String 2-group.

The Smooth Hom-Stack of an Orbifold

For a compact manifold M and a differentiable stack Open image in new window presented by a Lie groupoid X, we show the Hom-stack Open image in new window is presented by a Frechet–Lie groupoid Map(M, X) and so is an infinite-dimensional differentiable stack. We further show that if Open image in new window is an orbifold, presented by a proper etale Lie groupoid, then Map(M, X) is proper etale and so presents an infinite-dimensional orbifold.

The Faddeev–Mickelsson–Shatashvili Anomaly and Lifting Bundle Gerbes

In gauge theory, the Faddeev–Mickelsson–Shatashvili anomaly arises as a prolongation problem for the action of the gauge group on a bundle of projective Fock spaces. In this paper, we study this anomaly from the point of view of bundle gerbes and give several equivalent descriptions of the obstruction. These include lifting bundle gerbes with non-trivial structure group bundle and bundle gerbes related to the caloron correspondence.

Loop groups, string classes and equivariant cohomology

We give a classifying theory for LG -bundles, where LG is the loop group of a compact Lie group G , and present a calculation for the string class of the universal LG -bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG -bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.