scholarly journals A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS

2011 ◽  
Vol 90 (1) ◽  
pp. 129-144 ◽  
Author(s):  
KONRAD WALDORF
Keyword(s):  
Riemannian Manifold ◽  
Loop Space ◽  
Principal Bundle ◽  
Structure Group ◽  
Loop Spaces ◽  
Spin Structures ◽  
Space Formulation ◽  
Complex Spin ◽  
Bundle Gerbes ◽  
New Formulation

AbstractWe 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 represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.

EXPONENTIAL INTEGRABILITY FOR IMAGES OF ORNSTEIN–UHLENBECK OPERATORS ACTING ON CYLINDER FUNCTIONS ON LOOP SPACES

2002 ◽  
Vol 05 (04) ◽  
pp. 541-553
Author(s):  
FUZHOU GONG
Keyword(s):  
Riemannian Manifold ◽  
Loop Space ◽  
Loop Spaces ◽  
Exponential Integrability ◽  
Cylinder Function ◽  
Civita Connection ◽  
Levi Civita Connection

Let E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric (TSS) connection. Let L be the Ornstein–Uhlenbeck (O-U) operator on the loop space E, and f be a cylinder function on E. We first extend the expression of Lf, proved by Enchev and Stroock for the Levi–Cività connection, to a general TSS connection, and then prove that if [Formula: see text], ε |Lf|2 is exponential integrable for some constant ε := ε (f)>0.

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

scholarly journals Deformation quantization of principal bundles

2016 ◽  
Vol 13 (08) ◽  
pp. 1630010
Author(s):  
Paolo Aschieri
Keyword(s):  
Quantum Group ◽  
Deformation Quantization ◽  
Principal Bundle ◽  
Base Space ◽  
Structure Group ◽  
Galois Extensions ◽  
Principal Bundles ◽  
Group Of Automorphisms ◽  
Drinfeld Twist

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles and, more in general, to the deformation of Hopf–Galois extensions. First, we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next, we twist deform a subgroup of the group of automorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations, we obtain noncommutative principal bundles with noncommutative fiber and base space as well.

The Palais–Smale conditions for the Yang–Mills functional

1988 ◽  
Vol 108 (3-4) ◽  
pp. 189-200
Author(s):  
D. R. Wilkins
Keyword(s):  
Riemannian Manifold ◽  
Principal Bundle ◽  
Compact Riemannian Manifold ◽  
Hilbert Manifold ◽  
Yang Mills ◽  
Palais Smale Condition ◽  
Dimension 2

SynopsisWe consider the Yang–Mills functional denned on connections on a principal bundle over a compact Riemannian manifold of dimension 2 or 3. It is shown that if we consider the Yang–Mills functional as being defined on an appropriate Hilbert manifold of orbits of connections under the action of the group of principal bundle automorphisms, then the functional satisfies the Palais–Smale condition.

Decomposable Free Loop Spaces

1987 ◽  
Vol 39 (4) ◽  
pp. 938-955 ◽  
Author(s):  
J. Aguadé
Keyword(s):  
Topological Group ◽  
Loop Space ◽  
Base Point ◽  
Point Of View ◽  
Homotopy Equivalence ◽  
Compact Open Topology ◽  
Loop Spaces ◽  
Continuous Maps

In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

On the quotients of mapping class groups of surfaces by the Johnson subgroups

2019 ◽  
pp. 1-23
Author(s):  
TOMÁŠ ZEMAN
Keyword(s):  
Boundary Component ◽  
Loop Space ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

Abstract We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula

1973 ◽  
Vol 25 (4) ◽  
pp. 765-771
Author(s):  
Hansklaus Rummler
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Result ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Group ◽  
Parallel Translation ◽  
Result Theorem

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

scholarly journals M-BRANE MODELS AND LOOP SPACES

2012 ◽  
Vol 27 (20) ◽  
pp. 1230019 ◽  
Author(s):  
CHRISTIAN SÄMANN
Keyword(s):  
Loop Space ◽  
The Self ◽  
String Equation ◽  
Loop Spaces ◽  
Brane Model ◽  
Recent Developments ◽  
Brane Models ◽  
Space Approach

I review an extension of the ADHMN construction of monopoles to M-brane models. This extended construction gives a map from solutions to the Basu–Harvey equation to solutions to the self-dual string equation transgressed to loop space. Loop spaces appear in fact quite naturally in M-brane models. This is demonstrated by translating a recently proposed M5-brane model to loop space. Finally, I comment on some recent developments related to the loop space approach to M-brane models.

On the Homotopy-Commutativity of Loop-Spaces and Suspensions

1968 ◽  
Vol 20 ◽  
pp. 1531-1536 ◽  
Author(s):  
C. S. Hoo
Keyword(s):  
Loop Space ◽  
Loop Spaces

Let X be a space. We are interested in the homotopy-commutativity of the loop-space ΩX and the suspension ΣX, that is, in the question whether or not nil X ≦ 1, conil X ≦ 1, respectively. Let c: ΩX× ΩX ⟶ ΩX, c': ΣX ⟶ ΣX V ΣX be the commutator and co-commutator maps, respectively.

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