Diffeomorphisms |

**In General** > s.a. Hole
Argument; Homeotopy Group.

* __Effect of geometric quantities__: In general,
δ*T* = \(\cal L\)_{v} *T*;
For a metric and a connection

δ*g*_{ab} =
2 ∇_{(a} *v*_{b)}
, δΓ^{m}_{ab}
= ∇_{a} ∇_{b}* v*^{m} – *R*^{m}_{(ab) c}*v*^{c}
.

* __Descriptors__: Given
an infinitesimal diffeomorphism of a manifold,
*x*^{m} \(\mapsto\) *x*^{m} + *ξ*^{m}(*x*),
the descriptors are the functions *ξ*^{m}(*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(S^{1})/S^{1}].

@ __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(S^{1}).

* __Generators__: Given by *L*_{n} =
–i exp{i*nθ*} ∂/∂*θ*, with
commutation relations [*L*_{m}, *L*_{n}]
= (*n*–*m*) *L*_{m+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 a1601 [Diff(S^{1}), reproducing kernels and analogs of spherical functions]; Weiner a1606 [Diff(S^{1}), 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.

@ __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, 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 a1610 [quantum field theories, and renormalization].

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