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 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