Connections
on a Fiber Bundle| Connections
on a Fiber Bundle |
In General > s.a. curvature.
* 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 
q ("holonomy," the
finite case), or a Lie algebra element to each displacement p p +
dp ("connection," the infinitesimal case).
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 
on
the principal fiber bundle, such that 
, X
=
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 
starting at an x in the base space M;
Given a p 
–1(x), lift
to its
(unique) horizontal lift through p; Then
Da = 
a –
Aai
TiI g (/g)
.
* Connection form: A
Lie algebra-valued form Aai
on the base space M, such that = g–1Ag + g–1dg;
It can be expressed as A = 
*,
where is a cross-section
of the principal fiber bundle; Under a change in
coordinates g g':= hg in
the fiber, A'
= hAh–1 + hdh–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.
* 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 ,
there is a connection A such
that for all ei in , A-bar(ei)
= hi(A) = P
exp (–
e_i A).
@ Over graphs: Fleischhack CMP(00)mp,
CMP(00)mp [gauge
orbits], JGP(03)mp/00 [hyphs],
JGP(03)mp/02 [
-bar],
CMP(03)mp/00 [Gribov
problem]; Velhinho JGP(02)ht/00, ht/01-MG9
[groupoid
approach], IJGMP(04)mp [rev],
MPLA(05)mp/04-in
[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.
@ Related topics: Kunzinger et al MPCPS(05)m.FA/04 [Colombeau];
Martínez et al JMP(05)
[cellular decompositions and continuum limit]; Roberts & Ruzzi m.AT/06 [over
posets]; Velhinho a0804 [transformations]; > s.a. finsler
spaces.
(Moduli) Space of Connections > s.a. gauge
theory; metric [examples]; quantum
gauge theories; symplectic geometry and structures.
* Idea: The space /
of
connections modulo gauge transformations, or the generalized version -bar/-bar.
* 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 1(M)
→ G.
@ Characterization: Fischer GRG(86)
["grand superspace"]; Ashtekar & Lewandowski CQG(93)gq [SL(2,
C) and SU(1,1), completeness of Wilson loops]; Thaddeus Top(00)
[SU(2) connections on 2M];
Nelson & Picken
LMP(02)mp/01 [flat
SL(2, R) connections on T2].
@ 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]; Sengupta JGP(03)
[flat].
@ Hilbert space structure: Okolów CQG(05)gq/04 [non-compact
group].
@ Flat connections on M 2:
Gelca & Uribe CMP(03)
[SU(2) on T 2, Weyl and quantum group quantization];
Jeffrey & Ho CMP(05)m.SG/03 [non-orientable M 2];
Meusburger
JPA(06)ht [Poisson
structure and dual generators of 1(M 2)].
@ Related topics: Schweigert NPB(97)
[non-simply connected G]; Rudolph
et al JGP(02)mp/00 [space
of gauge orbits as poset].
General References and Related Concepts > s.a. loops; Wilson
Loop.
@ Intros and texts: in Kobayashi & Nomizu 69; Horowitz notes; 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]; > s.a. graph invariants.
@ Related topics: Anandan in(83) [and holonomy]; 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].
In Physics > s.a. field
theories; gauge
theory;
gravitation; lagrangian
systems; 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]; representations
in quantum
theory [Segal-Bargmann].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jun 2008