Holonomy

In General > s.a. connection.
* Idea: In a fiber bundle with a connection A, the change of an element in the fiber when transported along the lift \(\hat l\) of a closed curve l in the base space, l ⁽¹⁾ = l ⁽⁰⁾ H_A(l); It depends on the choice of lifting or gauge.
* Properties: It allows to reconstruct A up to gauge; For the product of two loops, H_A(l \(\circ\) m) = H_A(l) H_A(m).
* For a trivial bundle: We can write a connection form A_a, and, if F^k = dA^k − g C^k_ij Aⁱ ∧ A^j,

H_A(l) = P exp ∫_l A_a(x) dx^a = −(ig/\(\hbar\)c) ∫_l A_m^k T_k dx^m = 1 + (ig/2\(\hbar\)c) F^k_mn T_k dσ^mn + h.o.t.

* In general relativity: Using the generators of the Poincaré group, it is

H_e,Γ = P exp{−(i/\(\hbar\)) ∫_l (e^a_m P_a + \(1\over2\)Γ_m^ab M_ab) dx^m } = 1 + (i/2\(\hbar\)) (Q^a_mn P_a + \(1\over2\)R^ab_mn M_ab) dσ^mn ,

where Q^a = de^a + Γ^a_b ∧ e^b is the torsion, and is observable for open l too [@ Anandan in(93)].
@ And connections: Chi et al IM(96)dg/95; Díaz-Marín & Zapata JMP(10)-a1101 [holonomies and bundle structures]; Rosenstock & Weatherall a1504 [categorical equivalence between generalized holonomy maps and principal connections]; Cekić & Lefeuvre a2105 [results on holonomy inverse problem].
@ Of the Levi-Civita connection: Klitgaard et al CQG(20)-a2004 [3D and 4D relations with curvature integrals].
@ Other special cases: Alfaro et al JPA(03) [non-abelian, triangular paths]; Mendes mp/05 [U(1)].
> Online resources: see MathWorld page; Wikipedia page.

Holonomy Groups and Algebras
@ Holonomy groups: McInnes JMP(93) [classification, for Riemannian manifolds], JMP(93) [Einstein manifolds], JPA(97) [from curvature], CMP(99) [spin holonomy of Einstein manifolds]; Hall & Lonie CQG(00)gq/03 [and different T_mns]; Boya RACZ(06)mp [intro for physicists].
@ Holonomy algebras: Abbati & Manià JGP(02)mp [spectra]; Okołów & Lewandowski CQG(03)gq, CQG(05) [representations]; Aastrup & Grimstrup CMP(06)ht/05 [spectral triple from non-commutative algebra of loops]; Lewandowski et al CMP(06)gq/05 [representations]; Rios gq/05 [Jordan GNS]; Gryc JMP(08) [manifolds with boundaries]; Aastrup et al JNCG(09)-a0802, CMP(09)-a0807, CQG(09)-a0902-conf [and lqg], a0911 [emergent Dirac Hamiltonians]; Dziendzikowski & Okołów CQG(10)-a0912 [diffeomorphism-invariant states]; Aastrup & Grimstrup a1709 [representations], a1709 [quantum gravity and quantum Yang-Mills theory].
@ Lorentzian: Galaev DG&A(05)m.DG/03 [D < 12, algebras]; Hernandez et al JHEP(04)ht [and supersymmetry, various dimensions]; Atkins BAusMS(06)mp [reducibility, and existence of metrics]; Galaev IJGMP(06), JGP(10); Leistner JDG(07) [classification]; Galaev LMP(15)-a1110, RMS(15)-a1611 [holonomy algebra of an arbitrary Lorentzian manifold]; > s.a. tensor fields.

Variations, Generalizations > see Wilson Loop [on supermnaifolds].
* Generalized holonomy: A homomorphism \(\cal L\)₀ \(\mapsto\) G, where \(\cal L\)₀ is the loop group of a manifold.
@ General references: Kozameh & Newman PRD(85) [differential holonomies]; Lewandowski et al JMP(93); Tavares JGP(98) [generalized]; Mackaay & Picken AiM(02)m.DG/00 [abelian gerbes]; Gubser ht/02-ln [special holonomy and strings]; Lupercio & Uribe JGP(06) [gerbes over orbifolds].
@ Higher holonomy invariants: Zucchini IJGMP(16)-a1505, IJGMP(16)-a1505 [in higher gauge theory].

And Physics > s.a. geometric phase; Wilson Loop [a basic variable in some formulations of gauge theory].
* Quantum holonomy theory: A candidate for a fundamental theory based on gauge fields and non-commutative geometry.
@ In quantum theory: Cheon & Tanaka EPL(09)-a0807, Tanaka & Cheon AP(09)-a0902 [unified formulation].
@ Quantum holonomy theory: Aastrup & Grimstrup IJMPA(16)-a1404, FdP(16)-a1504, CQG(16)-a1602; Grimstrup CQG+(16); Aastrup & Grimstrup a1810 [fermionic sector]; > s.a. canonical approaches to quantum gravity.
@ And spacetime: Hall GRG(95); Bezerra & Letelier JMP(96) [conical singularities]; Rothman et al CQG(01)gq/00 [Schwarzschild-Droste geometry]; Carvalho & Furtado GRG(07) [FLRW metrics]; Viennot JMP(10)-a1003 [non-abelian geometric phases and gauge theory of gravity].
@ Holonomy of SU(2) spin connection: Jacobson & Romano CMP(93)gq/92 [holonomy group classification and conservation].
> Specific field theories: see gauge theory; loop gravity; quantum gravity in the connection representation [quantum holonomy theory].

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