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, qM; 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 pπ–1(x), lift γ to its (unique) horizontal lift through p; Then

Da = ∂aAai 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].
@ 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 = 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]; > s.a. quantum groups.

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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