Holonomy| 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 l^ of
a closed curve l in
the
base space, l (1) = l (0)HA(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, HA(l 
m)
= HA(l) HA(m).
* For a trivial bundle: We can
write a connection form Aa,
and, if Fk = dAk – g Ckij Ai 
Aj,
HA(l)
= P exp 
l Aa(x)
dxa = –(ig/
c)
l Amk Tk dxm =
1 + (ig/2c) Fkmn Tk
d
mn +
h.o.t.
* In general relativity: Using the generators of the Poincaré group, it is
H_{e,
}
= P exp{–(i/) 
l
(eam Pa + 
mab Mab)
dxm } = 1 + (i/2)
(Qamn Pa
+ Rabmn Mab)
dmn ,
where Qa = dea + ab eb is
the torsion, and is observable for
open l too [@ Anandan in(93)].
@ Special cases: Alfaro et al JPA(03)
[non-abelian, triangular paths]; Mendes mp/05 [U(1)].
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 Tmn's]; Boya RACZ(06)mp [intro
for physicists].
@ Holonomy algebras: Abbati & Manià JGP(02)mp [spectra];
Okolow & 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 a0802,
a0807.
@ 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);
Leistner JDG(07)
[classification]; > s.a. tensor fields.
Variations, Generalizations
* Generalized holonomy:
A homomorphism 
0 
G,
where 0 is
the loop group of a manifold.
@ 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].
And Physics > s.a. geometric phase; Wilson
Loop [a basic variable in some formulations
of gauge
theory and in loop 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)
[FRW metrics].
@ Holonomy of SU(2) spin connection: Jacobson & Romano CMP(93)gq/92 [holonomy
group
classification
and conservation].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
26 jul 2008