# Spin structures on loop spaces that characterize string manifolds

@article{Waldorf2012SpinSO, title={Spin structures on loop spaces that characterize string manifolds}, author={Konrad Waldorf}, journal={arXiv: Algebraic Topology}, year={2012} }

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, as desired… Expand

#### 19 Citations

The Spinor Bundle on Loop Space and its Fusion product

- Physics
- 2020

Given a manifold with a string structure, we construct a spinor bundle on its loop space. Our construction is in analogy with the usual construction of a spinor bundle on a spin manifold, but… Expand

String geometry vs. spin geometry on loop spaces

- Mathematics, Physics
- 2015

Abstract We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between… Expand

Connes fusion of spinors on loop space

- Mathematics, Physics
- 2020

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the… Expand

Equivalence of string and fusion loop-spin structures

- Mathematics, Physics
- 2013

The importance of the fusion relation of loops was recognized in the context of spin structures on the loop space by Stolz and Teichner and further developed by Waldorf. On a spin manifold M the… Expand

Supersymmetric Path Integrals II: The Fermionic Integral and Pfaffian Line Bundles

- Mathematics
- 2017

The Pfaffian line bundle of the covariant derivative and the transgression of the spin lifting gerbe are two canonically given real line bundles on the loop space of an oriented Riemannian manifold.… Expand

Fusion of implementers for spinors on the circle.

- Mathematics
- 2019

We consider the space of odd spinors on the circle, and a decomposition into spinors supported on either the top or on the bottom half of the circle. If an operator preserves this decomposition, and… Expand

Gerbes in Geometry, Field Theory, and Quantisation

- Mathematics, Physics
- 2021

Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes… Expand

Towards an M5‐Brane Model II: Metric String Structures

- Physics, Mathematics
- 2019

In this paper, we develop the mathematical formulation of metric string structures. These play a crucial role in the formulation of certain six-dimensional superconformal field theories and we… Expand

Supersymmetric Path Integrals I: Differential Forms on the Loop Space

- Mathematics, Physics
- 2017

In this paper, we construct an integral map for differential forms on the loop space of Riemannian spin manifolds. In particular, the even and odd Bismut-Chern characters are integrable by this map,… Expand

The Chern Character of {\theta}-summable Fredholm Modules over dg Algebras and Localization on Loop Space

- Mathematics
- 2019

We introduce the notion of a {\theta}-summable Fredholm module over a locally convex dg algebra {\Omega} and construct its Chern character as an entire cyclic cocyle in the entire cyclic complex of… Expand

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