arXiv research

Study calorons and their constituent monopoles under collapse.

A new transform links rotating calorons to solutions of a differential equation.

We study $SU(2)$ calorons, also known as periodic instantons, and consider invariance under isometries of $S^1\times\mathbb{R}^3$ coupled with a non-spatial isometry called the rotation map. In particular, we investigate the fixed points under various cyclic symmetry groups. Our approach utilises a construction akin to…

We review the caloron correspondence between $G$-bundles on $M \times S^1$ and $ΩG$-bundles on $M$, where $ΩG$ is the space of smooth loops in the compact Lie group $G$. We use the caloron correspondence to define characteristic classes for $ΩG$-bundles, called string classes, by transgression of characteristic classes…

Study shows instantons and monopoles decompose into U(1) components.

The caloron correspondence is a tool that gives an equivalence between principal $G$-bundles based over the manifold $M \times S^1$ and principal $LG$-bundles on $M$, where $LG$ is the Fréchet Lie group of smooth loops in the Lie group $G$. This thesis uses the caloron correspondence to construct certain differential f…

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 re…

In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on $S^1\times R^3$) and solutions to Nahm's equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the s…

The moduli space of solutions to Nahm's equations of rank (k,k+j) on the circle, and hence, of SU(2) calorons of charge (k,j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P^1xP^1 trivialized at infinity with c_2=k and equipped with a flag of degree j along P^1x{0}. An explicit matrix descri…

We use bundle gerbes and their connections and curvings to obtain an explicit formula for a de Rham representative of the string class of a loop group bundle. This is related to earlier work on calorons.

The caloron correspondence can be understood as an equivalence of categories between $G$-bundles over circle bundles and $LG \rtimes_ρS^1$-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…

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 li…

We consider Chern-Simons theory on 3-manifold $M$ that is the total space of a circle bundle over a 2d base $Σ$. We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with stru…

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analyt…

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 chara…

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 t…

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection i…

We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles…