# 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…

## 27 Citations

### The Spinor Bundle on Loop Space and its Fusion product

- Mathematics
- 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…

### Connes fusion of spinors on loop space

- Mathematics
- 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…

### The fermionic integral on loop space and the Pfaffian line bundle

- MathematicsJournal of Mathematical Physics
- 2022

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the “top degree component” of a differential form on it. In this paper, we show that a…

### Equivalence of string and fusion loop-spin structures

- 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…

### Smooth Fock bundles, and spinor bundles on loop space

- Mathematics
- 2020

We address the construction of smooth bundles of fermionic Fock spaces, a problem that appears frequently in fermionic gauge theories. Our main motivation is the spinor bundle on the free loop space…

### Bigerbes

- MathematicsAlgebraic & Geometric Topology
- 2021

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have…

### 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.…

### The Clifford algebra bundle on loop space

- Mathematics
- 2022

We construct a Cliﬀord algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that…

## References

SHOWING 1-10 OF 36 REFERENCES

### Twistor spaces and spinors over loop spaces

- Mathematics
- 2007

In this paper two natural twistor spaces over the loop space of a Riemannian manifold are constructed and their equivalence is shown in the Kählerian case. This relies on a detailed study of frame…

### The Spinor bundle on loop space

- Mathematics
- 2005

We give a definition of 6-connected covering groups String(n) → Spin(n) in terms of “local fermions” on the circle. These are certain very explicit von Neumann algebras, the easiest examples of…

### The Vanishing problem of the string class with degree 3

- MathematicsJournal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
- 1998

Abstract Let ξbe an SO(n)-bundle over a simple connected manifold M with a spin structure Q → M. The string class is an obstruction to h1 the structure group LSpin(n) of the loop group bundle LQ → LM…

### String connections and Chern-Simons theory

- Mathematics, Computer Science
- 2009

A finite-dimensional and smooth formulation of string structures on spin bundles that enables it to prove that every string structure admits a string connection and that the possible choices form an affine space is presented.

### A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS

- MathematicsJournal of the Australian Mathematical Society
- 2011

Abstract We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbes…

### String structures and the path fibration of a group

- Mathematics
- 1991

We use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class, that is the obstruction to a loop group bundle lifting to a…

### String theory and loop space index theorems

- Mathematics
- 1987

We study index theorems for the Dirac-Ramond operator on a compact Riemannian manifold. The existence of a group action on the loop space makes possible the definition of a character valued index…