Diffeomorphisms| Diffeomorphisms |
In General
> s.a. Hole Argument; Homeotopy Group.
$ Def: A diffeomorphism is
a bijection f : M to N between differentiable
manifolds M and N under which the differentiable structure
is invariant.
* Effect on geometric quantities:
In general, under an infinitesimal diffeomorphism generated by a vector field
v, δT = \(\cal L\)v
T; For a metric and a connection
δgab = 2 ∇(a vb) , δΓmab = ∇a ∇b vm − Rm(ab) c vc .
* Descriptors: Given an infinitesimal
diffeomorphism of a manifold, xm
\(\mapsto\) xm
+ ξm(x),
the descriptors are the functions ξm(x),
i.e., the vector field generating the diffeomorphism [@ in Anderson 67].
@ Diffeomorphism groups: in Marsden 74;
Michor 80;
Banyaga 97;
> s.a. types of groups [mapping class].
@ Representations: Goldin et al JPA(83) [of \(\mathbb R\)3];
Maxera ht/98 [non-linear realizations];
Larsson gq/99 [of extended algebra].
@ Geometry: Khesin & Misiolek JMFM(05)m.DG [asymptotic directions];
Gordina & Lescot mp/05
[Ricci curvature of Diff(S1)/S1].
@ Related topics: Dolgopyat CMP(00) [mostly contracting];
Larsson a1205 [new extensions of diffeomorphism algebras].
Witt Algebra
> s.a. Virasoro Algebra.
$ Def: The infinite-dimensional
Lie algebra of Diff(S1).
* Generators: Given
by Ln
= −i exp{inθ} ∂/∂θ,
with commutation relations [Lm,
Ln] = (n−m)
Lm+n.
@ References: Schlichenmaier a1111 [second cohomology].
Types of Diffeomorphisms and Manifolds
* Large diffeomorphims: The ones
not in the component connected to the identity, which form the modular group.
* 3D: The Smale conjecture
states that the diffeomorphism group Diff(S\(^3\)) has the same
homotopy type as SO(4); In proving the Smale conjecture, A Hatcher
proposed the Generalized Smale conjecture that for all closed 3-manifolds
Diff(\(M^3\)) is homotopic to Isom(\(M^3\)), which has been proven for many
special cases; Don Witt has proposed a correspondence between the Generalized
Smale conjecture and gauge fixing in gauge/diffeomorphism invariant theories
as a procedure for proving it in such situations – the idea is to use
the fact that the diffeomorphism group is the gauge group by which one divides
to obtain the space of gauge-invariant configurations from a space of
gauge-dependent ones, and if both of the latter are known well enough
one can set up an exact sequence with which one calculates all homotopy
groups of the diffeomorphism group.
@ 1D: Banakh & Yagasaki Top(09)
[Diff(\(\mathbb R\)), topological structure];
Neretin JGM(17)-a1601 [Diff(S1), reproducing kernels and analogs of spherical functions];
Weiner CMP(17)-a1606 [Diff(S1), local equivalence of representations].
@ 2D: Thurston BAMS(88) [geometry and dynamics].
@ 3D: Bonatti et al Top(04) [gradient-like];
Benatti & Wilkinson Top(05) [transitive partially hyperbolic].
@ On compact Riemannian manifolds: Delanoë DG&A(04) [gradient rearrangement].
@ Volume-preserving: Sato EPJC(14)-a1404 [structure constants];
> s.a. general relativity [origin] and formulations;
modified versions of QED; quantum gauge theories;
spheres.
@ Generalized: Dimitrijević & Wess ht/04-talk [deformed bialgebra on non-commutative space];
> s.a. discrete gravity below; non-commutative geometry.
@ Related topics: Giacomini PRD(04)ht [Poisson algebra with spacetime bifurcations];
Harvey in Bullett et al 17 [modular group].
And Physical Theories
> s.a. canonical general relativity; Covariance [general
covariance]; symmetries; symmetry breaking.
* Quantum gravity:
Large diffeomorphisms give rise to theta sectors.
@ And gauge symmetries:
Kuchař & Stone CQG(87) [parametrised Maxwell field];
Pons et al PRD(00)gq/99 [in Ashtekar variables],
JMP(00)gq/99 [in Einstein-Yang-Mills];
> s.a. types of gauge theories
and quantum gauge theories.
@ And general relativity / quantum gravity:
Chamblin & Gibbons gq/95-proc [spacetime topology and time-orientation reversal];
Aldaya & Jaramillo IJMPA(03)gq/02 [representations of diffeomophism groups];
Larsson in(06)-a0709 [anomalies];
Samanta IJTP(09)-a0708 [Lagrangian gravity];
Dittrich ASL(09)-a0810;
Ashtekar GRG(09)-a0904-in [consequences of diffeomorphism invariance];
Farkas & Martinec JMP(11)-a1002 [extension of spatial diffeomorphisms];
Salisbury et al IJMPA(16)-a1508 [in canonical general relativity, Hamilton-Jacobi approach];
Patrascu JMP(16)-a1410 [extension to topology].
@ And discrete geometry / gravity:
Kunzinger & Steinbauer CQG(99) [distributional, pp-wave example];
Bahr & Thiemann CQG(09)-a0711 [distributional, and combinatorial lqg];
Gambini & Pullin CQG(09)-a0807 [emergent diffeomorphism invariance];
Bahr & Dittrich AIP(09)-a0909;
Baratin et al PRD(11)-a1101 [in group field theories];
Wetterich PRD(12)-a1110;
Dittrich & Steinhaus PRD(12)-a1110 [measure and triangulation independence];
Wetterich PRD(12) [on a lattice];
Dittrich a1201-proc;
Bahr et al Sigma(12) [and constraints];
Bonzom & Dittrich CQG(13)-a1304 [discrete hypersurface deformation algebras];
> s.a. lattice gravity; loop quantum gravity.
@ Large diffeomorphisms: Giulini BCP(97)gq/95 [and geons],
NPPS(97)gq;
Balachandran & de Queiroz JHEP(11)-a1109 [anomalies and mixed states].
@ Broken / partial diffeomorphism invariance: Carballo-Rubio PRD(15)-a1502 [Weyl transverse gravity, and the cosmological constant];
> s.a. Background.
@ Non-relativistic diffeomorphism invariance:
Andreev et al PRD(14) [examples and applications];
Banerjee et al PLB(14)-a1404,
IJMPA(17)-a1604 [symmetries].
@ Related topics:
Sorkin in(93) [and particle symmetries];
Pons CQG(03)gq [and phase space for generally covariant theories];
Dedushenko JHEP(10)-a1007 [diffeomorphism anomaly in quantum mechanics];
Kleppe & Nielsen a1412-conf [diffeomorphism invariance from Random Dynamics];
Pooley a1506
[diffeomorphism invariance vs background independence];
Kreimer & Yeats MPAG(17)-a1610 [quantum field theories, and renormalization];
Johns a1908
[different interpretations of active diffeomorphisms, and substantivalism].
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