Algebraic & Geometric Topology

Spin structures on loop spaces that characterize string manifolds

Konrad Waldorf

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Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well known that only the if part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey and Murray, recent results of Stolz and Teichner on loop spaces, and some of our own results about string geometry and Brylinski–McLaughlin transgression.

Article information

Algebr. Geom. Topol., Volume 16, Number 2 (2016), 675-709.

Received: 1 December 2013
Revised: 10 April 2015
Accepted: 27 July 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Secondary: 58B05: Homotopy and topological questions 53C08: Gerbes, differential characters: differential geometric aspects

string structures loop group transgression fusion product


Waldorf, Konrad. Spin structures on loop spaces that characterize string manifolds. Algebr. Geom. Topol. 16 (2016), no. 2, 675--709. doi:10.2140/agt.2016.16.675.

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