Conformal geometry

The conformal group for the Minkowski quadratic form q(x, y) = 2xy in the plane is the abelian Lie group

${\displaystyle \operatorname {CSO} (1,1)=\left\{\left.{\begin{pmatrix}e^{a}&0\\0&e^{b}\end{pmatrix}}\right|a,b\in \mathbb {R} \right\},}$

with Lie algebra cso(1, 1) consisting of all real diagonal 2 × 2 matrices.

Consider now the Minkowski plane, ${\displaystyle \mathbb {R} ^{2}}$ equipped with the metric

${\displaystyle g=2\,dx\,dy~.}$

A 1-parameter group of conformal transformations gives rise to a vector field X with the property that the Lie derivative of g along X is proportional to g. Symbolically,

L_X g = λg   for some λ.

In particular, using the above description of the Lie algebra cso(1, 1), this implies that

1. L_X  dx = a(x) dx
2. L_X  dy = b(y) dy

for some real-valued functions a and b depending, respectively, on x and y.

Conversely, given any such pair of real-valued functions, there exists a vector field X satisfying 1. and 2. Hence the Lie algebra of infinitesimal symmetries of the conformal structure, the Witt algebra, is infinite-dimensional.

The conformal compactification of the Minkowski plane is a Cartesian product of two circles S¹ × S¹. On the universal cover, there is no obstruction to integrating the infinitesimal symmetries, and so the group of conformal transformations is the infinite-dimensional Lie group

${\displaystyle (\mathbb {Z} \rtimes \mathrm {Diff} (S^{1}))\times (\mathbb {Z} \rtimes \mathrm {Diff} (S^{1})),}$

where Diff(S¹) is the diffeomorphism group of the circle.[1]

The conformal group CSO(1, 1) and its Lie algebra are of current interest in two-dimensional conformal field theory.

A coordinate grid prior to a Möbius transformation

The same grid after a Möbius transformation

The group of conformal symmetries of the quadratic form

${\displaystyle q(z,{\bar {z}})=z{\bar {z}}}$

is the group GL₁(C) = C^×, the multiplicative group of the complex numbers. Its Lie algebra is gl₁(C) = C.

Consider the (Euclidean) complex plane equipped with the metric

${\displaystyle g=dz\,d{\bar {z}}.}$

The infinitesimal conformal symmetries satisfy

1. ${\displaystyle \mathbf {L} _{X}\,dz=f(z)\,dz}$
2. ${\displaystyle \mathbf {L} _{X}\,d{\bar {z}}=f({\bar {z}})\,d{\bar {z}},}$

where f satisfies the Cauchy–Riemann equation, and so is holomorphic over its domain. (See Witt algebra.)

The conformal isometries of a domain therefore consist of holomorphic self-maps. In particular, on the conformal compactification – the Riemann sphere – the conformal transformations are given by the Möbius transformations

${\displaystyle z\mapsto {\frac {az+b}{cz+d}}}$

where ad − bc is nonzero.

In two dimensions, the group of conformal automorphisms of a space can be quite large (as in the case of Lorentzian signature) or variable (as with the case of Euclidean signature). The comparative lack of rigidity of the two-dimensional case with that of higher dimensions owes to the analytical fact that the asymptotic developments of the infinitesimal automorphisms of the structure are relatively unconstrained. In Lorentzian signature, the freedom is in a pair of real valued functions. In Euclidean, the freedom is in a single holomorphic function.

In the case of higher dimensions, the asymptotic developments of infinitesimal symmetries are at most quadratic polynomials.[2] In particular, they form a finite-dimensional Lie algebra. The pointwise infinitesimal conformal symmetries of a manifold can be integrated precisely when the manifold is a certain model conformally flat space (up to taking universal covers and discrete group quotients).[3]

The general theory of conformal geometry is similar, although with some differences, in the cases of Euclidean and pseudo-Euclidean signature.[4] In either case, there are a number of ways of introducing the model space of conformally flat geometry. Unless otherwise clear from the context, this article treats the case of Euclidean conformal geometry with the understanding that it also applies, mutatis mutandis, to the pseudo-Euclidean situation.

The inversive model of conformal geometry consists of the group of local transformations on the Euclidean space Eⁿ generated by inversion in spheres. By Liouville's theorem, any angle-preserving local (conformal) transformation is of this form.[5] From this perspective, the transformation properties of flat conformal space are those of inversive geometry.

The projective model identifies the conformal sphere with a certain quadric in a projective space. Let q denote the Lorentzian quadratic form on Rⁿ⁺² defined by

${\displaystyle q(x_{0},x_{1},\ldots ,x_{n+1})=-2x_{0}x_{n+1}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}.}$

In the projective space P(Rⁿ⁺²), let S be the locus of q = 0. Then S is the projective (or Möbius) model of conformal geometry. A conformal transformation on S is a projective linear transformation of P(Rⁿ⁺²) that leaves the quadric invariant.

In a related construction, the quadric S is thought of as the celestial sphere at infinity of the null cone in the Minkowski space R^n+1,1, which is equipped with the quadratic form q as above. The null cone is defined by

${\displaystyle N=\left\{(x_{0},\ldots ,x_{n+1})\mid -2x_{0}x_{n+1}+x_{1}^{2}+\cdots +x_{n}^{2}=0\right\}.}$

This is the affine cone over the projective quadric S. Let N⁺ be the future part of the null cone (with the origin deleted). Then the tautological projection R^n+1,1 \ {0} → P(Rⁿ⁺²) restricts to a projection N⁺ → S. This gives N⁺ the structure of a line bundle over S. Conformal transformations on S are induced by the orthochronous Lorentz transformations of R^n+1,1, since these are homogeneous linear transformations preserving the future null cone.

Intuitively, the conformally flat geometry of a sphere is less rigid than the Riemannian geometry of a sphere. Conformal symmetries of a sphere are generated by the inversion in all of its hyperspheres. On the other hand, Riemannian isometries of a sphere are generated by inversions in geodesic hyperspheres (see the Cartan–Dieudonné theorem.) The Euclidean sphere can be mapped to the conformal sphere in a canonical manner, but not vice versa.

The Euclidean unit sphere is the locus in Rⁿ⁺¹

${\displaystyle z^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}$

This can be mapped to the Minkowski space R^n+1,1 by letting

${\displaystyle x_{0}={\frac {z+1}{\sqrt {2}}},\,x_{1}=x_{1},\,\ldots ,\,x_{n}=x_{n},\,x_{n+1}={\frac {z-1}{\sqrt {2}}}.}$

It is readily seen that the image of the sphere under this transformation is null in the Minkowski space, and so it lies on the cone N⁺. Consequently, it determines a cross-section of the line bundle N⁺ → S.

Nevertheless, there was an arbitrary choice. If κ(x) is any positive function of x = (z, x₀, ..., x_n), then the assignment

${\displaystyle x_{0}={\frac {z+1}{\kappa (x){\sqrt {2}}}},\,x_{1}=x_{1},\,\ldots ,\,x_{n}=x_{n},\,x_{n+1}={\frac {(z-1)\kappa (x)}{\sqrt {2}}}}$

also gives a mapping into N⁺. The function κ is an arbitrary choice of conformal scale.

A representative Riemannian metric on the sphere is a metric which is proportional to the standard sphere metric. This gives a realization of the sphere as a conformal manifold. The standard sphere metric is the restriction of the Euclidean metric on Rⁿ⁺¹

${\displaystyle g=dz^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots +dx_{n}^{2}}$

to the sphere

${\displaystyle z^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}$

A conformal representative of g is a metric of the form λ²g, where λ is a positive function on the sphere. The conformal class of g, denoted [g], is the collection of all such representatives:

${\displaystyle [g]=\left\{\lambda ^{2}g\mid \lambda >0\right\}.}$

An embedding of the Euclidean sphere into N⁺, as in the previous section, determines a conformal scale on S. Conversely, any conformal scale on S is given by such an embedding. Thus the line bundle N⁺ → S is identified with the bundle of conformal scales on S: to give a section of this bundle is tantamount to specifying a metric in the conformal class [g].

Another way to realize the representative metrics is through a special coordinate system on R^{n+1, 1}. Suppose that the Euclidean n-sphere S carries a stereographic coordinate system. This consists of the following map of Rⁿ → S ⊂ Rⁿ⁺¹:

${\displaystyle \mathbf {y} \in \mathbf {R} ^{n}\mapsto \left({\frac {2\mathbf {y} }{\left|\mathbf {y} \right|^{2}+1}},{\frac {\left|\mathbf {y} \right|^{2}-1}{\left|\mathbf {y} \right|^{2}+1}}\right)\in S\subset \mathbf {R} ^{n+1}.}$

In terms of these stereographic coordinates, it is possible to give a coordinate system on the null cone N⁺ in Minkowski space. Using the embedding given above, the representative metric section of the null cone is

${\displaystyle x_{0}={\sqrt {2}}{\frac {\left|\mathbf {y} \right|^{2}}{1+\left|\mathbf {y} \right|^{2}}},x_{i}={\frac {y_{i}}{\left|\mathbf {y} \right|^{2}+1}},x_{n+1}={\sqrt {2}}{\frac {1}{\left|\mathbf {y} \right|^{2}+1}}.}$

Introduce a new variable t corresponding to dilations up N⁺, so that the null cone is coordinatized by

${\displaystyle x_{0}=t{\sqrt {2}}{\frac {\left|\mathbf {y} \right|^{2}}{1+\left|\mathbf {y} \right|^{2}}},x_{i}=t{\frac {y_{i}}{\left|\mathbf {y} \right|^{2}+1}},x_{n+1}=t{\sqrt {2}}{\frac {1}{\left|\mathbf {y} \right|^{2}+1}}.}$

Finally, let ρ be the following defining function of N⁺:

${\displaystyle \rho ={\frac {-2x_{0}x_{n+1}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{t^{2}}}.}$

In the t, ρ, y coordinates on R^n+1,1, the Minkowski metric takes the form:

${\displaystyle t^{2}g_{ij}(y)\,dy^{i}\,dy^{j}+2\rho \,dt^{2}+2t\,dt\,d\rho ,}$

where g_ij is the metric on the sphere.

In these terms, a section of the bundle N⁺ consists of a specification of the value of the variable t = t(yⁱ) as a function of the yⁱ along the null cone ρ = 0. This yields the following representative of the conformal metric on S:

${\displaystyle t(y)^{2}g_{ij}\,dy^{i}\,dy^{j}.}$

Consider first the case of the flat conformal geometry in Euclidean signature. The n-dimensional model is the celestial sphere of the (n + 2)-dimensional Lorentzian space R^n+1,1. Here the model is a Klein geometry: a homogeneous space G/H where G = SO(n + 1, 1) acting on the (n + 2)-dimensional Lorentzian space R^n+1,1 and H is the isotropy group of a fixed null ray in the light cone. Thus the conformally flat models are the spaces of inversive geometry. For pseudo-Euclidean of metric signature (p, q), the model flat geometry is defined analogously as the homogeneous space O(p + 1, q + 1)/H, where H is again taken as the stabilizer of a null line. Note that both the Euclidean and pseudo-Euclidean model spaces are compact.

To describe the groups and algebras involved in the flat model space, fix the following form on R^p+1,q+1:

${\displaystyle Q={\begin{pmatrix}0&0&-1\\0&J&0\\-1&0&0\end{pmatrix}}}$

where J is a quadratic form of signature (p, q). Then G = O(p + 1, q + 1) consists of (n + 2) × (n + 2) matrices stabilizing Q : ^tMQM = Q. The Lie algebra admits a Cartan decomposition

${\displaystyle \mathbf {g} =\mathbf {g} _{-1}\oplus \mathbf {g} _{0}\oplus \mathbf {g} _{1}}$

where

${\displaystyle \mathbf {g} _{-1}=\left\{\left.{\begin{pmatrix}0&^{t}p&0\\0&0&J^{-1}p\\0&0&0\end{pmatrix}}\right|p\in \mathbb {R} ^{n}\right\},\quad \mathbf {g} _{-1}=\left\{\left.{\begin{pmatrix}0&0&0\\^{t}q&0&0\\0&qJ^{-1}&0\end{pmatrix}}\right|q\in (\mathbb {R} ^{n})^{*}\right\}}$

${\displaystyle \mathbf {g} _{0}=\left\{\left.{\begin{pmatrix}-a&0&0\\0&A&0\\0&0&a\end{pmatrix}}\right|A\in {\mathfrak {so}}(p,q),a\in \mathbb {R} \right\}.}$

Alternatively, this decomposition agrees with a natural Lie algebra structure defined on Rⁿ ⊕ cso(p, q) ⊕ (Rⁿ)^∗.

The stabilizer of the null ray pointing up the last coordinate vector is given by the Borel subalgebra

h = g₀ ⊕ g₁.