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 JMP(16)-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];
> s.a. Flux.

@ __2-connections__:
Bouzid & Tahiri a1803 [on a lattice];
Zucchini a1905 [in principal 2-bundles].

@ __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; supermanifolds.

**(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 closure).

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

* __G____ = SU(2)__: The
moduli space of flat connections on *M* can be parametrized by homomorphisms
π_{1}(*M*) → *G*.

@ __Characterization__:
Fischer GRG(86) ["grand superspace"];
Ashtekar & Lewandowski CQG(93)gq
[SL(2, \(\mathbb C\)) and SU(1,1), completeness of Wilson loops];
Thaddeus Top(00)
[SU(2) connections on ^{2}*M*];
Nelson & Picken LMP(02)mp/01 [flat SL(2, \(\mathbb R\)) connections on T^{2}];
Ballico et al T&A(12) [anti-self-dual connections].

@ __Differential calculus__: Ashtekar & Lewandowski JGP(95)ht/94;
Lewandowski in(94).

@ __Measure and integration__:
Ashtekar & Isham CQG(92);
Rendall CQG(93);
Baez in(94)ht/93,
LMP(94)ht/93;
Ashtekar & Lewandowski in(94)gq/93,
JMP(95)gq/94;
Lewandowski IJMPD(94)gq-in;
Baez & Sawin JFA(97)qa/95;
Marolf & Mourão CMP(95)ht/94;
Mourão et al JMP(99)ht/97 [properties];
Levy m.PR/01 [2D compact surfaces];
Fleischhack mp/01,
mp/01 [2D Yang-Mills];
Velhinho CMP(02)mp/01 [Fock measure];
Sahlmann JMP(11)gq/02;
Sengupta JGP(03) [flat];
Khatsymovsky MPLA(10)-a0912 [and path integrals].

@ __Hilbert space structure__: Okołów CQG(05)gq/04 [non-compact group].

@ __Flat connections on M__

@

**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].

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