Connections
on a Fiber Bundle |

**In General** > s.a. curvature; loops; Parallel Transport; projective structures.

* __General idea__: Given a manifold *M* as the base space for
some fiber bundle with structure group *G*, a connection is a path-dependent way to compare elements in the fibers
at two different points *p*, *q* ∈ *M*; This is done by assigning a group element to each path
\(p\mapsto q\) ("holonomy," the finite case), or a Lie algebra element to each displacement
\(p\mapsto p+{\rm d}p\)
("connection," the infinitesimal case).

@ __Intros and texts__: in Kobayashi & Nomizu 69; Horowitz ln(86); Darling
94.

@ __Covariant normal coordinate expansion__: Dilkes gq/95.

@ __Spin connections__: Hurley & Vandyck JPA(94), JPA(94), JPA(95) [covariant derivatives of spinors]; Gu gq/06 [simplified
calculation].

@ __Discrete__: Manton CMP(87)
[discrete bundle]; Novikov mp/03 [triangulated
manifolds]; Díaz-Marín & Zapata JMP(12)-a1101 [effective theory and measuring scales, based on coarse-graining maps]; Fernández & Zuccalli a1311 [geometric approach]; > s.a. graph invariants.

@ __And holonomy__: Anandan in(83); Rosenstock & Weatherall a1504 [categorical equivalence between holonomy maps and connections].

@ __Related topics__: Fischer CMP(87) [internal symmetry group];
Alekseevsky et
al JMP(03)m.DG/02 [manifold with Grassmann structure];
Cahen & Schwachhöfer LMP(04)
[special symplectic connections];
Gover et al CMP(08) [Yang-Mills detour complex];
Fatibene et al PRD(11)-a1011 [bundle reductions and Barbero-Immirzi connections];
> s.a. Wilson Loop.

> __Online resources__:
see Wikipedia page.

**On a Principal Fiber Bundle** > s.a. affine connection;
aharonov-bohm; gauge transformations;
holonomy.

$ __Def 1__: An assignment of a horizontal subspace of the tangent space
at each point in the bundle, preserved by the (right) action of the group [needs to be generalized];

$ __Def 2__: A locally defined Lie algebra-valued 1-form *ω*
on the principal fiber bundle, such that \(\langle\)*ω*, *X*\(\rangle\) =
0 for all horizontal vector fields *X*, with the forms defined on two neighborhoods agreeing on their overlap.

* __Parallel transport__: From a connection,
one gets a notion of parallel transport along a curve *γ* starting at an *x* in the base space *M*;
Given a *p* ∈ *π*^{–1}(*x*), lift
*γ* to its (unique) horizontal lift through *p*; Then

*D*_{a} = ∂_{a} –
*A*_{a}^{i}
*T*_{i}^{I }*g* (∂/∂*g*)
.

* __Connection form__: A
Lie algebra-valued form *A*_{a}^{i}
on the base space *M*, such that *ω* = *g*^{–1}*Ag*
+ *g*^{–1}d*g*;
It can be expressed as *A* = *σ***ω*,
where *σ* is a cross-section of the principal fiber bundle; Under a change in
coordinates *g* \(mapsto\) *g*':= *hg* in the fiber, *A*'
= *hAh*^{–1} + *h*d*h*^{–1}.

* __From holonomy__: A connection can be recovered, up to gauge, from the
holonomies around all closed curves.

* __Flat connection__: A connection is called flat if its curvature vanishes
(no torsion to compute here).

**Generalized Connections** > s.a. Cartan
Geometry; graphs; Gribov
Problem; holonomy; Parallel
Transport [over path spaces].

* __Idea__: Elements *A* ∈ Hom(Path(*M*),*G*),
that give an *A*(*e*) ∈ *G*
for each (piecewise smooth) edge *e* in a manifold *M*,
with consistency conditions reproducing the group structure of holonomies; Smooth connections are dense.

* __Result__: Given any finite
graph *γ*, there is a connection *A* such
that for all *e*_{i} in *γ*,
\(\bar A\)(*e*_{i})
= *h*_{i}(*A*) = P
exp (–∫_{ei}* A*).

@ __Over graphs__: Fleischhack CMP(00)mp,
CMP(00)mp [gauge
orbits], JGP(03)mp/00 [hyphs],
JGP(03)mp/02 [\(\cal A\) ⊂ \(\bar{\cal A}\)],
CMP(03)mp/00 [Gribov
problem]; Velhinho JGP(02)ht/00, ht/01-MG9
[groupoid approach], IJGMP(04)mp [rev],
MPLA(05)mp/04-proc
[functorial aspects]; Fleischhack mp/06 [mappings].

@ __Non-linear__: Dehnen & Vacaru GRG(03)gq/00,
Vacaru & Dehnen GRG(03)gq/00 [in
general relativity]; Bucataru DG&A(07)
[compatible with a metric]; Brinzei a0706 [second-order
geometry]; > s.a. dynamical systems.

@ __Generalized fluxes__: Sahlmann JMP(11)gq/02 [Hilbert spaces and electric flux operators]; Dittrich et al CQG(13)-a1205 [space of generalized fluxes as an inductive limit].

@ __Related topics__: Kunzinger et
al MPCPS(05)m.FA/04 [Colombeau];
Martínez et al JMP(05) [cellular decompositions and continuum limit];
Roberts & Ruzzi TAG-m.AT/06 [over
posets]; Velhinho IJGMP(09)-a0804 [transformations];
Sahlmann & Thiemann JGP(12)-a1004 [in
abelian Chern-Simons theory]; Vilela Mendes a1504 [projective limits on hypercubic lattices, mass gap]; > s.a. finsler spaces.

**(Moduli) Space of Connections** > s.a. gauge theory;
metric [examples]; quantum gauge theories;
symplectic geometry and structures.

* __Idea__: The space \(\cal A\)/\(\cal G\) of
connections modulo gauge transformations, or the generalized version \(\bar{\cal A}\)/\(\bar{\cal G}\)
(where the symbol ^{–} denotes the closure).

* __Structure__: See Atiyah and Bott's conjecture on Morse theory.

** G*

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**In Physics** > s.a. field theories; gauge theory;
gravitation; lagrangian systems;
loop quantum gravity; quantum field theories;
topological field theories.

@ __References__: Gaeta & Morando LMP(98)
[reducible connections]; Mangiarotti & Sardanashvily
00 [field theory and quantum field theory]; Dragomir et al DG&A(03)
[Yang-Mills and conjugate connections]; Aldrovandi & Barbosa mp/04 [field
strength and Wu-Yang ambiguity]; Wu & Yang IJMPA(06)
[as vector potential, history].

> __Related topics__: see bundle [gerbes];
gravitational instantons [self-dual connections]; representations
in quantum theory [Segal-Bargmann].

main page – abbreviations – journals – comments – other
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send feedback and suggestions to bombelli at olemiss.edu – modified
18 aug 2016