Spin structures on loop spaces that characterize string manifolds

  title={Spin structures on loop spaces that characterize string manifolds},
  author={Konrad Waldorf},
  journal={arXiv: Algebraic Topology},
  • K. Waldorf
  • Published 2012
  • Mathematics, Physics
  • arXiv: Algebraic Topology
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
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
String geometry vs. spin geometry on loop spaces
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 betweenExpand
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
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
Supersymmetric Path Integrals II: The Fermionic Integral and Pfaffian Line Bundles
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.
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, andExpand
Gerbes in Geometry, Field Theory, and Quantisation
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 gerbesExpand
Towards an M5‐Brane Model II: Metric String Structures
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 weExpand
Supersymmetric Path Integrals I: Differential Forms on the Loop Space
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
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 ofExpand


Twistor spaces and spinors over loop spaces
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 frameExpand
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
The Vanishing problem of the string class with degree 3
Let be an SO .n/-bundle over a simply connected manifold M with a spin structure Q! M. The string class is an obstruction to lift the structure group LSpin.n/ of the loop group bundle LQ ! LM to theExpand
The Dirac-Ramond operator in string theory and loop space index theorems☆☆☆
Abstract The index of the Direc-Ramond operator is computed and analyzed. It is shown to be the extension of the Atiyah-Singer index theorem for loop space. It can also be seen as a generatingExpand
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
  • 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
String structures and the path fibration of a group
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 aExpand
Lifting Problems and Transgression for Non-Abelian Gerbes
We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen'sExpand
String theory and loop space index theorems
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 indexExpand
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