TY - JOUR
AU - Kragh, Thomas
PY - 2013/09/01
Y2 - 2024/10/09
TI - Orientations on 2-vector Bundles and Determinant Gerbes
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 113
IS - 1
SE - Articles
DO - 10.7146/math.scand.a-15482
UR - https://www.mscand.dk/article/view/15482
SP - 63-82
AB - In a paper from 2009, a half magnetic monopole was discovered by Ausoni, Dundas, and Rognes. This describes an obstruction to the existence of a continuous map $K(ku) \to B(ku^*)$ with determinant like properties. This magnetic monopole is in fact an obstruction to the existence of a map from $K(ku)$ to $K(\mathsf{Z},3)$, which is a retract of the natural map $K(\mathsf{Z},3) \to K(ku)$; and any sensible definition of determinant like should produce such a retract. In this paper we describe this obstruction precisely using monoidal categories. By a result from 2011 by Baas, Dundas, Richter and Rognes $K(ku)$ classifies 2-vector bundles. We thus define the notion of oriented 2-vector bundles, which removes the obstruction by the magnetic monopole. We use this to define an oriented K-theory of 2-vector bundles with a lift of the natural map from $K(\mathsf{Z},3)$. It is then possible to define a retraction of this map and since $K(\mathsf{Z},3)$ classifies complex gerbes we call this a determinant gerbe map.
ER -