arXiv subjects

Vincent S. Schlegel

Publications and source records attributed to Vincent S. Schlegel.

3 recordsLinked to original sources

arXiv2017

Gluing Manifolds in the Cahiers Topos

We show that there is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos, one of the premier models for Synthetic Differential Geometry. This embedding is shown to have a number of nice properties, such as preservation of open covers and transverse fibre products. We develop a theory for gluing manifolds with corners in the Cahiers topos. In this setting, the result of gluing together manifolds with corners along a common face is shown to coincide with a pushout along an infinitesimally thickened face. Our theory is designed with a view toward future applications in Field Theory within the context of Synthetic Differential Geometry.

math.DG↗

arXiv2015

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 regarding a related differential extension of odd $K$-theory

math.KT↗

arXiv2013

The Caloron Correspondence and Odd Differential K-theory

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 forms called "string potentials" that play the same role as Chern-Simons forms for loop group bundles. Following their construction, the string potentials are used to define degree 1 differential characteristic classes for $ΩU(n)$-bundles. The notion of an "$Ω$ vector bundle" is introduced and a caloron correspondence is developed for these objects. Finally, string potentials and $Ω$ vector bundles are used to define an $Ω$ bundle version of the structured vector bundles of Simons--Sullivan. The "$Ω$ model" of odd differential $K$-theory is constructed using these objects and an elementary differential extension of odd $K$-theory due to Tradler et al.

math.DG↗