From String structures to Spin structures on loop spaces

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Let XX be a nn-dimensional smooth manifold, with n3n \geq 3. In a series of papers culminating in Spin structures on loop spaces that characterize string manifolds, arXiv:1209.1731{arXiv:1209.1731}, Konrad Waldorf recently gave the first rigorous proof that a String structure on XX induces a Spin structure on its loop space.

Here we give a completely independent proof of this result by working with smooth higher stacks. The unfamiliar reader, to begin, may take an initial look at motivation for sheaves, cohomology and higher stacks, even if part of the thesis itself presents smooth stacks and suggests how to describe some classical objects as bundle gerbes with and without connection by means of them.

Why did we use this approach in our proof? Because, rereading the question in this more general setting, the result can be obtained in a very natural and easy way. In particular, the crucial point in our proof is the existence of a natural morphism of smooth stacks

BSpinB 2(BU(1) conn) \mathbf{B}Spin \rightarrow {\mathbf{B}}^2({\mathbf{B}}U(1)_{conn})

refining the first fractional Pontryagin class. Once this morphism is exhibited, we show how Waldorf’s result follows from general constructions in the setting of smooth stacks.

category: reference

Last revised on June 26, 2016 at 05:59:10. See the history of this page for a list of all contributions to it.