Conformal transformations of space-time are, by definition, those general coordinate transformations which preserve the angles between any two vectors, or what is the same, which scale the metric locally by an overall factor. In flat space-time of dimension D > 2, the infinitesimal conformal transformations generate a (D+1)(D+2)/2 dimensional Lie algebra isomorphic to that of the group SO(2,D)(*). In contrast, for D = 2 any ARBITRARY holomorphic mapping of the (compactified) complex plane IS angle-preserving. Thus, in complex coordinates z, \bar{z}, infinitesimal conformal transformations are generated by mappings which transform z;
z \rightarrow z + f(z),
by functions which do not depend on \bar{z};
f(z) = - \sum_{n \in \mathbb{Z}} \epsilon_{n}z^{n+1},
and by analogous mappings of \bar{z} with functions, \bar{f}(\bar{z}), which do not depend on z;
\bar{z}\rightarrow \bar{z} - \sum_{n \in \mathbb{Z}} \bar{\epsilon}_{n} \bar{z}^{n+1}.
It is easy to see that on functions of z,\bar{z}, these mappings are generated by differential operators;
\ell_{n}= - z^{n+1}\partial_{z} \ \mbox{and} \ \bar{\ell}_{n}= - \bar{z}^{n+1}\partial_{\bar{z}},
respectively. These operators satisfy the following algebra
[\ell_{m},\ell_{n}] = (m-n)\ell_{m+n},
[\bar{\ell}_{m},\bar{\ell}_{n}] = (m-n) \bar{\ell}_{m+n},
[\ell_{m}, \bar{\ell}_{n}] = 0.
In mathematical terms, this means that both the \ell_{n} and the \bar{\ell}_{n} span an infinite-dimensional Lie algebra known as the Witt algebra, moreover, these two algebras are combined as a direct sum. Thus the algebra of the local conformal group \mathcal{D}^{\infty}(2) is a direct sum of two commuting Witt algebras. It is easy to show that the Witt algebra \Span\{\ell_{n}\} is the same as the algebra of infinitesimal diffeomorphisms of the circle S^{1} (try to prove this “easy to show” thing!). Thus, the local conformal group in 2D space-time is (essentially)
\mathcal{D}^{\infty}(2) = \Diff (S^{1}) \times \Diff (S^{1})
The Global Conformal Group in Two Dimensions; the Complex Mobius Group:
Let me remark that the general 2D conformal transformations are neither globally well defined nor invertible, even on the Riemann sphere S^{2}= \mathbb{C}\cup \{\infty\} (think of it as the z plane plus a point at infinity). This is because the vector fields
V(z) = \sum_{n}a_{n}\ell_{n},
generating holomorphic transformations, are globally defined only if a_{n}=0 for n < -1 and n > 1. Namely, the action of (-z^{n+1}\partial_{z}) is non-singular at z = 0 only for n \geq -1, and non-singular at z = \infty (where it acts as (w^{1-n}\partial_{w}) with w = 1/ z) only for n \leq 1. Thus, in order to obtain a global group of well-defined and invertible conformal transformations on the Riemann sphere, we must demand that the infinitesimal transformation \delta z = \epsilon (z) does not have a pole at the “point at infinity”. In terms of a new coordinate system w = 1/z, the “point at infinity” is just an ordinary point, namely the origin w = 0.
Now,
\delta w = - \frac{\delta z}{z^{2}} = - \frac{\epsilon (z)}{z^{2}},
is non-singular at w = 0 if and only if \epsilon / z^{2} is finite for z \rightarrow \infty. Thus, \epsilon (z) must be quadratic polynomial in z. We therefore conclude that the global group of 2D infinitesimal conformal transformations have the form
\delta z = \alpha + \beta z + \gamma z^{2},
with \alpha , \beta , \gamma being arbitrary complex parameters. These transformations integrate to the so-called complex Mobius group (or the complex projective group);
z \rightarrow \frac{az + b}{cz + d}, \ \ \ (1)
where a,b,c,d are complex parameters and ad – bc = 1. Thus, the global conformal group in 2D is parametrized by 6 real parameters; being isomorphic to SO(1,3) \approx SL(2,\mathbb{C}) / Z_{2}. To understand why SL(2,\mathbb{C}) is the group of such transformations, note that a 2 \times 2 complex matrix can act on a 2-dimensional complex vector space;
<br />
\left( \begin{array}{c} v_{1} \\ v_{2} \\ \end{array}\right) \rightarrow \left( \begin{array}{rr} a & b \\ c & d \end{array}\right) \left( \begin{array}{c} v_{1} \\ v_{2} \\ \end{array}\right) \ \ (2)<br />
If we think of v_{1} and v_{2} as homogeneous coordinates for CP^{1}, then v_{1}/v_{2} \equiv z is the stereographic projection from CP^{1} to the z plane. From eq(2), we see that z transforms as in eq(1). Thus we learn that SL(2,\mathbb{C}) acts as the group of conformal mappings of the complex plane (including the point at infinity).
The Real Mobius Group SL(2,\mathbb{R}) \approx SO(2,1):
Instead of the Riemann sphere, which is a closed surface of genus zero, we can consider an open Riemann surface given by an upper half-plane. In this case we obtain the real projective group SL(2, \mathbb{R}) as a subgroup of the full global conformal group
SO(2,2) \approx SL(2,\mathbb{R})_{1} \times SL(2,\mathbb{R})_{2}.
This is precisely the conformal group that one would get when “naively” extrapolating from D > 2 to D = 2.
The apparent similarity between the open case and the holomorphic part of the closed case extends to a similar relation in string theory between open and closed strings; the doubling of degrees of freedom in going from open strings to closed strings; SL(2, \mathbb{C}) has twice as many parameters as SL(2,\mathbb{R}).
Each infinite-dimensional Witt algebra \Span\{\ell_{n}\} contains a finite-dimensional subalgebra, \Span \{\ell_{-1},\ell_{0},\ell_{+1}\}, isomorphic to sl(2). When the field of scalars is restricted from \mathbb{C} to \mathbb{R}, we obtain the real form sl(2,\mathbb{R)}. Exponentiating the action of \{\ell_{-1},\ell_{0},\ell_{1}\} on the real axis y, gives the finite transformations:
Translation;
T(\lambda_{-1}) = e^{\lambda_{-1}\ell_{-1}}: \ y \rightarrow y + \lambda_{-1}
Scale;
S(\lambda_{0}) = e^{\lambda_{0}\ell_{0}}: \ y \rightarrow e^{\lambda_{0}}y
Special Conformal transformations;
e^{\lambda_{1}\ell_{1}}: \ y \rightarrow \frac{y}{1 - \lambda_{1}y}
Combining these transformations leads to the general form of an SL(2, \mathbb{R}) transformation;
y \rightarrow \frac{ay + b}{cy + d}, \ \ ad – bc = 1 \ \ (3)
This means that the 2 \times 2 real matrices of the group SL(2, \mathbb{R}) have natural non-linear action on ONE real variable y! Does this surprise you? I hope not.
Since a, b, c and d are real, eq(3) clearly maps the real axis to itself. If we analytically continue to complex values of y, then SL(2, \mathbb{R}) transformations map the upper half y-plane to itself. Indeed, one can show that SL(2, \mathbb{R}) is the group of all one-to-one mappings of the upper half-plane to itself; y \rightarrow \bar{y}(y), with \bar{y} being an analytic function of y. In some region such transformation is conformal. Therefore, we may say that the mappings of the upper half-plane onto itself form a subgroup of the full conformal group, namely the real projective group SL(2, \mathbb{R}). And I should stop here.
Sam
(*) See
www.physicsforums.com/showthread.php?t=172461
