Conformal Structures and Transformations

Conformal Structure > s.a. Compactification; geodesics; in physics; spacetime and models [axiomatic]; Weyl Manifold.
* Idea: A way of defining angles between line elements in a differentiable manifold (including the notion of orthogonality) and, in the Lorentzian case, infinitesimal light cones.
@ General references: Herranz & Santander JPA(02) [of some important spaces]; Nurowski JGP(05)m.DG/04 [and differential equations].
@ And duality operation: Dray et al JMP(89).

Conformal Transformation > s.a. analytic transformation (in 2D); conformal invariance; scalar-tensor theories [in cosmology].
* Idea: A transformation of the metric preserving the conformal structure.
$ Def: A transformation of the metric of the form g g' = ² g, for some (non-vanishing) function on M.
* Other geometrical quantities: In an n-dimensional manifold, if we define C^c_ab by '_a k_b = _a k_b – C^c_ab k_c, we find that

C^c_ab = ^–1 g^cd (g_{bd a} + g_{ad b} – g_abd) = 2 ^c_(ab) ln – g_ab g^cd _d ln ,

C'_abc^d = C_abc^d

R'_ab = R_ab – (n–2) ^–1 _ab – ^–1 g_ab² + 2 (n–2) ^–2 (_a ) (_b ) – (n–3) ^–2 g_ab g^mn (_m )(_n )

R' = ^–2 R – 2 (n–1) ^{–3 2} – (n–1) (n–4) ^–4 g^mn (_m ) (_n )
= ^–2 [R – 2 (n–1) ² ln – (n–2) (n–1) g^mn (_m ln ) (_n ln )]

G'_ab = G_ab – (n–2) ^–1 _ab + (n–2) ^–1 g_ab ²
+ 2 (n–2) ^–2 (_a ) (_b ) – (n–2)(n–5) ^–2 g_ab g^mn (_m ) (_n )

' ²' = ^{s–1 2} + (2s+n–2) ^s–1 g^mn (_m ) (_n ) + s ^s–3 (²)
+ s (s–3+n) ^s–4 g^mn (_m ) (_n ) ,   if   ' = ^s .

* Conformal weight: A.k.a. scaling dimension of a spinor or tensor field ; The number d such that ^–d when the metric g ²g for the field theory to be conformally invariant; If n is the spacetime dimension, d = (n–2)/2 for a scalar field, d = (n–1)/2 for a spinor field, and d = 0 for a vector field if n = 4.
@ References: in Wald 84, app D; Krantz AS(99)sep [conformal mappings, I]; Nikolov & Valchev mp/04-in [conformally invariant differential operators]; Carneiro et al G&C(04)gq [applications in general relativity].

Conformal Group > s.a. conformal invariance; killing fields [conformal killing spinor].
$ Def: The group of diffeomorphisms f : M → M such that f*g = g, for some = (x).
* In 2+1 Minkowski: It is isomorphic to SO(3,2), with 10 generators, the 3 translations P_a and 3 rotations J_ab of the Poincaré group + 3 special conformal transformations K_a + dilation D, with (semisimple) Lie algebra

[P_a, P_b] = [J_ab, D] = [K_a, K_b] = 0;   [P_a, D] = P_a;   [K_a,D] = –K_a ;

[P_a, J_bc] = _ac P_b – _ab P_c;   [K_a, J_bc] = _ac K_b – _ab K_c ;   [P_a, K_b] = J_ab + _ab D ;

[J_ab, J_cd] = _ac J_bd – _ad J_bc + _bd J_ac – _bc J_ad .

* In 2+1 Euclidean space: It is isomorphic to SO(1,4).
* In 3+1 Minkowski space: It is isomorphic to SU(2,2).
@ References: Fulton et al RMP(62); Defrise-Carter CMP(75) [conformally equivalent isometry groups]; Wheeler ht/00 [extended, by grading]; Dolan JMP(06)ht/05 [higher-D, character formulae].

Conformal Killing Vector / Tensor > s.a. [killing vectors]; solutions of general relativity.
$ Def: A generator of the conformal group, i.e., a vector field k such that _a k_b = g_ab – F_ab, with F_ab = F_[ab] the conformal bivector, and some nonsingular function; This is equivalent to _k g_ab = 2 g_ab.
* Examples: In Minkowski space, one conformal Killing vector field is the dilation vector field; The Edgar-Ludwig metric.
* Special conformal Killing vector field: A conformal Killing vector field with _ab = 0.
* Homothecy group, Killing vector: The case with = constant, respectively = constant.
* Killing vector field: A conformal Killing vector field with = 0.
@ Conformal Killing vectors: Hall GRG(88) [special cases]; Hall JMP(90) [fixed points of conformal vector fields in 4D Lorentzian manfolds]; Carot GRG(90) [general relativity solutions]; Hall et al CQG(97) [conformal vector fields]; Saifullah a0810-in [and classification of spacetimes].
@ Conformal Killing tensors: Barnes et al gq/02-in [Killing tensors from conformal Killing vectors]; Coll et al JMP(06)gq [spectral decomposition].
@ Homothecy transformations: Hall GRG(88) [with fixed points]; Hall & Steele GRG(90); Steele GRG(91)
@ Generalizations: García-Parrado JGP(06) [biconformal vector fields].

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