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conformal group
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A conformal transformation (conformal mapping) is a transformation of a space which preserves the angles between the curves. In other words, it preserves the angles infinitesimally. The conformal group of a space which has well defined notion of angles between the curves is the group of space automorphisms which are also conformal transformations.
Examples
Of euclidean space
In euclidean n n -space for n > 2 n\gt 2 a general conformal transformation is some composition of a translation, dilation, rotation and possibly an inversion with respect to a n − 1 n-1 -sphere.
For n = 2 n=2 , i.e. in a complex plane, this still holds for (the group of) global conformal transformations but one also has nontrivial local automorphisms. One has in fact infinite-dimensional family of local conformal transformations, which can be described by an arbitrary holomorphic or an antiholomorphic automorphism (in fact one writes z z and z ¯ \bar{z} as independent coordinates in the complexification ℂ 2 \mathbb{C}^2 and restricts to the real part ℝ 2 ≅ ℂ \mathbb{R}^2\cong \mathbb{C} ). This is important for CFT in 2d.
Of ℝ d , t \mathbb{R}^{d,t}
For d , t ∈ ℕ d,t \in \mathbb{N} write ℝ d , t \mathbb{R}^{d,t} for the pseudo-Riemannian manifold which is the Cartesian space ℝ d + t \mathbb{R}^{d+t} equipped with the constant metric of signature ( d , t ) (d,t) . I.e. for t = 0 t = 0 this is Euclidean space and for t = 1 t=1 this is Minkowski spacetime .
If d + t > 2 d+t \gt 2 then the conformal group of ℝ d , t \mathbb{R}^{d,t} is the orthogonal group P ( d + 1 , t + 1 ) / { ± 1 } P(d+1, t+1)/\{\pm 1\} . The connected component of the neutral element is the special orthogonal group SO ( d + 1 , t + 1 ) SO(d+1,t+1) . (e.g Schottenloher 08, chapter 2, theorem 2.9 ).
Notice that for t = 1 t= 1 this is also the anti de Sitter group , the isometry group of anti de Sitter spacetime of dimension d + 1 + t d+1+t . This equivalence is the basis of the AdS-CFT correspondence .
Cosets
(…) Möbius space (…)
(Arutyunov-Sokatchev 02 )
(…)
group symbol universal cover symbol higher cover symbol orthogonal group O ( n ) \mathrm{O}(n) Pin group Pin ( n ) Pin(n) Tring group Tring ( n ) Tring(n)
special orthogonal group SO ( n ) SO(n) Spin group Spin ( n ) Spin(n) String group String ( n ) String(n)
Lorentz group O ( n , 1 ) \mathrm{O}(n,1) \, Spin ( n , 1 ) Spin(n,1) \, \,
anti de Sitter group O ( n , 2 ) \mathrm{O}(n,2) \, Spin ( n , 2 ) Spin(n,2) \, \,
conformal group O ( n + 1 , t + 1 ) \mathrm{O}(n+1,t+1) \,
Narain group O ( n , n ) O(n,n)
Poincaré group ISO ( n , 1 ) ISO(n,1) Poincaré spin group ISO ^ ( n , 1 ) \widehat {ISO}(n,1) \, \,
super Poincaré group sISO ( n , 1 ) sISO(n,1) \, \, \, \,
superconformal group
geometric contextgauge group stabilizer subgroup local model space local geometry global geometry differential cohomology first order formulation of gravity differential geometry Lie group /algebraic group G G subgroup (monomorphism ) H ↪ G H \hookrightarrow G quotient (“coset space ”) G / H G/H Klein geometry Cartan geometry Cartan connection
examples Euclidean group Iso ( d ) Iso(d) rotation group O ( d ) O(d) Cartesian space ℝ d \mathbb{R}^d Euclidean geometry Riemannian geometry affine connection Euclidean gravity
Poincaré group Iso ( d − 1 , 1 ) Iso(d-1,1) Lorentz group O ( d − 1 , 1 ) O(d-1,1) Minkowski spacetime ℝ d − 1 , 1 \mathbb{R}^{d-1,1} Lorentzian geometry pseudo-Riemannian geometry spin connection Einstein gravity
anti de Sitter group O ( d − 1 , 2 ) O(d-1,2) O ( d − 1 , 1 ) O(d-1,1) anti de Sitter spacetime AdS d AdS^d AdS gravity
de Sitter group O ( d , 1 ) O(d,1) O ( d − 1 , 1 ) O(d-1,1) de Sitter spacetime dS d dS^d deSitter gravity
linear algebraic group parabolic subgroup /Borel subgroup flag variety parabolic geometry
conformal group O ( d , t + 1 ) O(d,t+1) conformal parabolic subgroup Möbius space S d , t S^{d,t} conformal geometry conformal connection conformal gravity
supergeometry super Lie group G G subgroup (monomorphism ) H ↪ G H \hookrightarrow G quotient (“coset space ”) G / H G/H super Klein geometry super Cartan geometry Cartan superconnection
examples super Poincaré group spin group super Minkowski spacetime ℝ d − 1 , 1 | N \mathbb{R}^{d-1,1\vert N} Lorentzian supergeometry supergeometry superconnection supergravity
super anti de Sitter group super anti de Sitter spacetime
higher differential geometry smooth 2-group G G 2-monomorphism H → G H \to G homotopy quotient G / / H G//H Klein 2-geometry Cartan 2-geometry
cohesive ∞-group ∞-monomorphism (i.e. any homomorphism ) H → G H \to G homotopy quotient G / / H G//H of ∞-action higher Klein geometry higher Cartan geometry higher Cartan connection
examples extended super Minkowski spacetime extended supergeometry higher supergravity : type II , heterotic , 11d
References
Textbook accounts include
Martin Schottenloher , The conformal group , chapter 2 of A mathematical introduction to conformal field theory , 2008 (pdf )
and (with an eye towards combination with spin geometry )
Pierre Anglès , Conformal Groups in Geometry and Spin Structures , Progress in Mathematical Physics 2008
Details on the conformal Lie algebra of conformal Killing vectors? acting on 3+1 dimensional Minkowski spacetime are spelled out for instance in
Discussion in conformal field theory
G. Arutyunov, E. Sokatchev, Conformal fields in the pp-wave limit , JHEP 0208 (2002) 014 (arXiv:hep-th/0205270 )
See also
Last revised on September 21, 2024 at 14:04:45.
See the history of this page for a list of all contributions to it.