A Construction of String 2-Group Models using a Transgression-Regression Technique

@article{Waldorf2012ACO,
title={A Construction of String 2-Group Models using a Transgression-Regression Technique},
journal={arXiv: Differential Geometry},
year={2012}
}
• K. Waldorf
• Published 24 January 2012
• Mathematics
• arXiv: Differential Geometry
In this note we present a new construction of the string group that ends optionally in two different contexts: strict diffeological 2-groups or finite-dimensional Lie 2-groups. It is canonical in the sense that no choices are involved; all the data is written down and can be looked up (at least somewhere). The basis of our construction is the basic gerbe of Gawedzki-Reis and Meinrenken. The main new insight is that under a transgression-regression procedure, the basic gerbe picks up a…
Quasi-periodic paths and a string 2-group model from the free loop group
• Mathematics
• 2017
In this paper we address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group $LSpin$ (or more generally $LG$ for compact simple
Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Mathematics
• 2020
We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean
Explicit Non-Abelian Gerbes with Connections
• Mathematics
• 2022
We derive the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is a 2-group with adjustment datum. We depart from the common fake-flat connections and
The 2-Hilbert Space of a Prequantum Bundle Gerbe
• Mathematics
• 2016
We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of
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 a
Principal $\infty$-Bundles and Smooth String Group Models
We provide a general, homotopy-theoretic definition of string group models within an $\infty$-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is
Transgressive loop group extensions
A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group.
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, the
A representation of the string 2-group
• Mathematics
• 2022
We construct a representation of the string 2-group on a 2-vector space, namely on the hyperfinite type III1 von Neumann algebra. We prove that associating this representation to the frame bundle of

References

SHOWING 1-10 OF 40 REFERENCES
Central extensions of smooth 2–groups and a finite-dimensional string 2–group
We provide a model of the String group as a central extension of finite-dimensional 2‐groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a
Polyakov-Wiegmann formula and multiplicative gerbes
• Mathematics
• 2009
An unambiguous definition of Feynman amplitudes in the Wess-Zumino-Witten sigma model and the Chern-Simon gauge theory with a general Lie group is determined by a certain geometric structure on the
From loop groups to 2-groups
• Mathematics
• 2005
We describe an interesting relation between Lie 2-algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where
Higher gauge theory I: 2-Bundles
I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link
Kac-Moody Groups
This chapter is devoted to the construction of the “maximal” Kac—Moody group C associated to a Kac—Moody Lie algebra g = g(A), for A any£x£GCM, due to Tits. There are other versions of groups
Higher-Dimensional Algebra V: 2-Groups
• Mathematics
• 2003
A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this
A Smooth Model for the String Group
• Mathematics
• 2013
We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to
Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories
• Mathematics
• 2005
We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie