Variations of rational higher tangential structures

Abstract

The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the rational level and by considering Lie groups as a starting point for defining each of the higher structures, making close connection to p i -structures. We indicatively call these (rational) Spin-Fivebrane and Spin-Ninebrane structures. We study the space of such structures and characterize their variations, which reveal interesting effects whereby variations of higher structures are arranged to systematically involve lower ones. We also study the homotopy type of the gauge group corresponding to bundles equipped with the higher rational structures that we define.

String structures associated to indefinite Lie groups

“…4]). Rationally, there is essentially a unique multiplication on the connected covers of the orthogonal group [32,Prop. 3].…”

“…Similarly to [34] one can show that the canonical evaluation homomorphism Gauge(P ) → K(A, n) is a homotopy equivalence. In summary, we have: Note that a detailed treatment of the Whitehead tower in the rational case is given in [32], where the group structures are also identified.…”

“…The arguments work rationally for any k [9][32]. However in the integration from algebras to groups in the L∞ setting for the orthogonal case, one encounters difficulty for the Z/2 groups in the homotopy groups of O.…”