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 = ∇ab vmRm(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{i} ∂/∂θ, with commutation relations [Lm, Ln] = (nm) 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].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 17 nov 2019