Loops| Loops |
Algebraic Notion
* Idea: A generalization of
a finite group, in which the binary operation is not associative.
* Examples: The smallest loops
that are not themselves groups are those of order five.
@ And physics: Frampton et al ht/01-in.
Topological Notion > s.a. Cuntz
Algebra; tiling.
$ Def: An equivalence class
of closed curves on a manifold, where two are equivalent if they differ by
retraced
segments.
$ Hoop: An equivalence class
of loops, where two differ if they have the same holonomy for all connections
in a given fiber bundle.
* Small loop: A loop
which is homotopic to a loop contained in an arbitrarily small neighborhood
of its base point.
@ Dynamics: Kondev PRL(97)
[field theory of fluctuating loops]; Arreaga et al
PRE(02)cm/01 [equilibrium
configurations with constraints].
@ Generalizations: Griego gq/95 [and applications to knot theory and quantum
gravity].
Loop Group
$ Def: For a given manifold,
it is the set of loops based at a point p in M,
with the natural, non-commutative composition 
:=
followed by .
* Topology: It is a topological
group, with either (i) in
Ueps()
if there exist curves a in and b in ,
with a in Ueps(b)
in the usual sense of curves; or (ii) based on holonomies [@ Barrett
IJTP(91)].
* Loop algebra: The Lie
algebra of a loop group.
$ For a given group:
The group of maps f : S1 → G from
the circle to a fixed finite-dimensional group G, with composition
law (fg)(s):= f(s) g(s).
@ General references: Adams 78; Pressley & Segal 86; Bars NPB(89);
Rasmussen & Weis
ht/94 [hoop
group topology].
@ Generalizations: Di Bartolo et al CMP(93)gq, PRD(95)gq/94 [extended
loop group]; Leal
PRD(02)ht [signed
points].
@ Related topics: Spallanzani CMP(01)
[relationship with hoops]; Mickelsson in(06)mp/04 [central
extension]; Frenkel & Zhu a0810 [double loop groups, gerbal representations].
Loop Space
@ General references: Adams 78; Bars NPB(89);
Morozov et al PLB(91)
[loop space geometry and supersymmetry]; Lempert JDG(93).
@ Calculus: Cattaneo et al CMP(99)
[connections]; Reiris & Spallanzani
CQG(99)
[loop derivative]; Pickrell mp/04 [invariant
measure]; Reyes JMP(07)ht/06 [operators
on loop functions]; > s.a. Paths.
@ Related topics: Wurzbacher JGP(95)
[symplectic geometry]; Sergeev TMP(08)
[compact Lie group, twistor quantization].
Loop-Related Physical Systems > s.a. loop
quantum gravity; QCD; quantum
field theory; string theory.
@ General references:
Rajeev ht/04-in
[Yang-Mills theory and loop space]; Troyer et al PRL(08) [quantum loop gas].
@ Loop transform: Abbati et al LMP(01)mp [abelian
group]; > used in quantum
gauge theory; canonical quantum gravity.
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nov 2009