48 research outputs found
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 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. This new version consists of a classical
spin structure plus a certain fusion product related to loops of frames in the
manifold. We use the lifting gerbe theory of Carey-Murray, recent results of
Stolz-Teichner on loop spaces, and some own results about string geometry and
Brylinski-McLaughlin transgression.Comment: 30 pages. v2 comes with some minor corrections and improvement
Connections on non-abelian Gerbes and their Holonomy
We introduce an axiomatic framework for the parallel transport of connections
on gerbes. It incorporates parallel transport along curves and along surfaces,
and is formulated in terms of gluing axioms and smoothness conditions. The
smoothness conditions are imposed with respect to a strict Lie 2-group, which
plays the role of a band, or structure 2-group. Upon choosing certain examples
of Lie 2-groups, our axiomatic framework reproduces in a systematical way
several known concepts of gerbes with connection: non-abelian differential
cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These
relationships convey a well-defined notion of surface holonomy from our
axiomatic framework to each of these concrete models. Till now, holonomy was
only known for abelian gerbes; our approach reproduces that known concept and
extends it to non-abelian gerbes. Several new features of surface holonomy are
exposed under its extension to non-abelian gerbes; for example, it carries an
action of the mapping class group of the surface.Comment: 57 pages. v1 is preliminary. v2 is completely rewritten, former
Sections 1 and 2 have been moved into a separate paper (arxiv:1303.4663), and
the discussion of non-abelian surface holonomy has been improved and
extended. v3 is the final and published version with a few minor correction
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, and acts on the bottom half in the same
way as a second operator acts on the top half, then the fusion of both
operators is a third operator acting on the top half like the first, and on the
bottom half like the second. Fusion restricts to the Banach Lie group of
restricted orthogonal operators, which supports a central extension of
implementers on a Fock space. In this article, we construct a lift of fusion to
this central extension. Our construction uses Tomita-Takesaki theory for the
Clifford-von Neumann algebras of the decomposed space of spinors. Our
motivation is to obtain an operator-algebraic model for the basic central
extension of the loop group of the spin group, on which the fusion of
implementers induces a fusion product in the sense considered in the context of
transgression and string geometry. In upcoming work we will use this model to
construct a fusion product on a spinor bundle on the loop space of a string
manifold, completing a construction proposed by Stolz and Teichner.Comment: 49 page
Transgression of D-branes
Closed strings can be seen either as one-dimensional objects in a target
space or as points in the free loop space. Correspondingly, a B-field can be
seen either as a connection on a gerbe over the target space, or as a
connection on a line bundle over the loop space. Transgression establishes an
equivalence between these two perspectives. Open strings require D-branes:
submanifolds equipped with vector bundles twisted by the gerbe. In this paper
we develop a loop space perspective on D-branes. It involves bundles of simple
Frobenius algebras over the branes, together with bundles of bimodules over
spaces of paths connecting two branes. We prove that the classical and our new
perspectives on D-branes are equivalent. Further, we compare our loop space
perspective to Moore-Segal/Lauda-Pfeiffer data for open-closed 2-dimensional
topological quantum field theories, and exhibit it as a smooth family of
reflection-positive, colored knowledgable Frobenius algebras