Higher Gauge Theory (HGT) studies categorical analogs of constructions in the geometry of connections on bundles. In this talk, I will describe an HGT analog of a phenomenon that occurs in gauge theory: a groupoid of such connections arises both as a category of functors and also from a group action. I will describe a 2-categorical analog of this phenomenon, showing a relation between global and local symmetry in categorical geometry.

Speaker: Luuk Verhoeven (Western)

"Immersions as unbounded Kasparov products: embedding spheres inside Euclidean space"

Time: 11:30

Room: MC 106

Abstract: We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on R^n+1 . We find that the resulting spectral triple for the algebra C(S^n) differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in KK_0(C, C) represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky's criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KK-theoretical level. The necessary KK-theory will be introduced as a black box.