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-fs.
Topological Notion
$ 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].
> Related topics: see Cuntz
Algebra; knot theory; Link Theory;
tiling.
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 α \(\circ\) β:=
α followed by β.
* Topology: It is a topological
group, with either (i) α in
Uε(β)
if there exist curves a in α and b in β,
with a in Uε(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];
Solomon a1303 [comment].
@ Representations:
Carey & Langmann in(02)-a1007 [survey, and quantum field theory].
@ 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];
Zeitlin JFA(12)-a1012 [loop ax+b group, unitary representations];
Carpi & Hillier RVMP(17)-a1509 [and non-commutative geometry].
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
@ General references:
Rajeev ht/04-conf [Yang-Mills theory and loop space];
Ferreira & Luchini NPB(12)-a1109 [and the generalized non-abelian Stokes theorems for p-form connections];
Belokurov & Shavgulidze a1109
[quantum field theories on loop space, local limit];
Afriat a1311 [on the reality of loops].
@ Statistical ensembles of loops: Troyer et al PRL(08) [quantum loop gas];
Nahum et al PRL(13) [in a 3D or higher-dimensional lattice, loop length distriution].
@ Gravity: Venkatesh a1212,
a1305 [space and dynamics of gravity from loop algebras];
Nelson & Picken ATMP(14)-a1309 [intersecting loops on a 2D torus];
> s.a. loop quantum gravity.
@ Loop transform: Abbati et al LMP(01)mp [abelian group];
> used in quantum gauge theory; canonical quantum gravity.
> Other applications: see gauge theories
[loop-based variables]; QCD; quantum field theory;
string theory.
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send feedback and suggestions to bombelli at olemiss.edu – modified 27 jan 2019