String geometry vs. spin geometry on loop spaces

@article{Waldorf2015StringGV,
  title={String geometry vs. spin geometry on loop spaces},
  author={Konrad Waldorf},
  journal={Journal of Geometry and Physics},
  year={2015},
  volume={97},
  pages={190-226}
}
  • K. Waldorf
  • Published 2015
  • Mathematics, Physics
  • Journal of Geometry and Physics
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 loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin structures on the loop space and string structures on the manifold itself. The equivalence exists in two settings: in a purely topological one and in a geometrical one… Expand

Figures from this paper

String structures associated to indefinite Lie groups
  • H. Sati, H. Shim
  • Physics, Mathematics
  • Journal of Geometry and Physics
  • 2019
Abstract String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures,Expand
The Spinor Bundle on Loop Space and its Fusion product
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, butExpand
Connes fusion of spinors on loop space
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 theExpand
String structures, 2-group bundles, and a categorification of the Freed-Quinn line bundle
For a 2-group constructed from a finite group and 3-cocycle, we provide an explicit description of the bicategory of flat 2-group bundles on an oriented surface in terms of weak representations ofExpand
Principal ∞-Bundles and Smooth String Group Models
We provide a general, homotopy-theoretic definition of string group models within an ∞-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is aExpand
Rational Structures and Fractional Differential Refinements
In the following thesis, we explore the notion of rational Fivebrane structures. This is done through a combination of obstruction theory and rational homotopy theory. We show that these structuresExpand
String$\mathbf{^c}$ Structures and Modular Invariants
In this paper, we study some algebraic topology aspects of String$^c$ structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of $Spin^c(n)$. We alsoExpand
Current Groups and the Hamiltonian Anomaly
Gauge symmetry invariance is an indispensable aspect of the field-theoretic models in classical and quantum physics. Geometrically this symmetry is often modelled with current groups and currentExpand
Non-abelian gerbes and some applications in string theory
We review a systematic construction of the 2-stack of bundle gerbes via descent, and extend it to non-abelian gerbes. We review the role of non-abelian gerbes in orientifold sigma models, for theExpand
Current Groups and the Hamiltonian Anomaly
Gauge symmetry invariance is an indispensable aspect of the eld-theoretic models in classical and quantum physics. Geometrically this symmetry is often modelled with current groups and currentExpand
...
1
2
...

References

SHOWING 1-10 OF 44 REFERENCES
Spin structures on loop spaces that characterize string manifolds
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, theExpand
String connections and Chern-Simons theory
We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation isExpand
Loop Spaces, Characteristic Classes and Geometric Quantization
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematicalExpand
A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS
  • K. Waldorf
  • Mathematics
  • Journal 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 gerbesExpand
Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles
We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusionExpand
Equivalence of string and fusion loop-spin structures
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 theExpand
ON THE GEOMETRY OF FREE LOOP SPACES
We verify the following three basic results on the free loop space LM. (1) We show that the set of all points, where the fundamental form on LM is nondegenerate, is an open subset. (2) TheExpand
Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps
We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles.Expand
String structures on loop bundles
Differential geometry and topology of principal loop bundles (bundles of loop groups over loop spaces) are investigated. String structures, defined as bundle extensions corresponding to the centralExpand
Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection
We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped withExpand
...
1
2
3
4
5
...