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
*U*_{ε}(*β*)
if there exist curves *a* in *α* and *b* in *β*,
with *a* in *U*_{ε}(*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* : S^{1} → *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 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-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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