Connections on a Fiber Bundle| 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
Da = ∂a − Aai TiI g (∂/∂g) .
* Connection form: A Lie algebra-valued form
Aai on the base
space M, such that ω = g−1Ag
+ g−1dg; 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
+ 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; 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
ei in γ,
\(\bar A\)(ei)
= hi(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
π1(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 2M];
Nelson & Picken LMP(02)mp/01 [flat SL(2, \(\mathbb R\)) connections on T2];
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 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 π1(M 2)].
@ Related topics: Schweigert NPB(97) [non-simply connected G];
Rudolph et al JGP(02)mp/00 [space of gauge orbits as poset];
Aastrup & Grimstrup a1909 [non-commutative geometry over a space of connections];
> s.a. quantum groups.
In Physics > s.a. field theories; gauge
theory; 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].
> In gravity: see connection-based formulations;
gravitation; gravitational instantons [self-dual connections];
loop quantum gravity.
> 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 17 jan 2021