Complex Coordinates and Local Conformal Symmetry

Goal

The previous modules developed conformal field theory in general spacetime dimension $d$. In $d>2$, conformal symmetry is powerful but finite-dimensional: the conformal group is generated by translations, rotations, dilatations, and special conformal transformations, and its Lie algebra is $\mathfrak{so}(d,2)$ in Lorentzian signature or $\mathfrak{so}(d+1,1)$ in Euclidean signature.

Two dimensions are different. On a two-dimensional Euclidean surface, every holomorphic coordinate transformation is locally conformal. This promotes the finite-dimensional global conformal algebra into an infinite-dimensional local algebra. The basic two-dimensional slogan is

$$\boxed{ \text{orientation-preserving local conformal maps in } d=2 \quad\Longleftrightarrow\quad z\mapsto f(z)\text{ holomorphic.} }$$

This single fact is the engine behind Virasoro symmetry, exact minimal models, modular invariance, WZW models, worldsheet string theory, and much of AdS$_3$/CFT$_2$.

The purpose of this page is to build the complex-coordinate language carefully. The stress tensor, central charge, Virasoro algebra, and radial quantization will come in the next pages.

Complex coordinates on the Euclidean plane

Let the Euclidean plane have Cartesian coordinates $(x^1,x^2)$ and metric

$$ds^2=(dx^1)^2+(dx^2)^2.$$

Define complex coordinates

$$z=x^1+i x^2, \qquad \bar z=x^1-i x^2.$$

Equivalently,

$$x^1=\frac{z+\bar z}{2}, \qquad x^2=\frac{z-\bar z}{2i}.$$

The corresponding derivatives are the Wirtinger derivatives

$$\partial\equiv \partial_z =\frac{1}{2}\left(\partial_1-i\partial_2\right), \qquad \bar\partial\equiv \partial_{\bar z} =\frac{1}{2}\left(\partial_1+i\partial_2\right).$$

They obey

$$\partial z=1, \qquad \partial \bar z=0, \qquad \bar\partial z=0, \qquad \bar\partial \bar z=1.$$

The metric becomes

$$ds^2=dz\,d\bar z.$$

If one writes $ds^2=g_{ab}dx^a dx^b$ with coordinates $(z,\bar z)$, then

$$g_{z\bar z}=g_{\bar z z}=\frac12, \qquad g_{zz}=g_{\bar z\bar z}=0,$$

and the inverse metric satisfies

$$g^{z\bar z}=g^{\bar z z}=2, \qquad g^{zz}=g^{\bar z\bar z}=0.$$

The area form is

$$d^2x=dx^1 dx^2=\frac{i}{2}\,dz\wedge d\bar z.$$

The flat Laplacian is especially simple:

$$\partial_1^2+\partial_2^2=4\partial\bar\partial.$$

This factor of $4$ is a tiny detail that causes a surprising number of wrong factors in two-dimensional CFT calculations. Keep it close; it will keep you out of several algebraic ditches.

On the real Euclidean plane, $\bar z$ is the complex conjugate of $z$. But in many CFT calculations it is useful to complexify and temporarily treat $z$ and $\bar z$ as independent variables. The physical real surface is recovered by imposing $\bar z=z^*$ at the end.

Why holomorphic maps are conformal

A coordinate transformation is conformal if it preserves the metric up to a local Weyl factor:

$$ds^2\mapsto ds'^2=\Omega(z,\bar z)^2 ds^2.$$

Consider an orientation-preserving change of coordinates

$$z' = w(z,\bar z), \qquad \bar z'=\bar w(z,\bar z).$$

The transformed line element is

$$ds'^2=dw\,d\bar w.$$

For the transformation to be conformal, $dw\,d\bar w$ must contain only $dz\,d\bar z$ and no $(dz)^2$ or $(d\bar z)^2$ terms. This gives the Cauchy-Riemann condition

$$\bar\partial w=0, \qquad \partial \bar w=0.$$

Thus the orientation-preserving local conformal maps are

$$w=f(z), \qquad \bar w=\bar f(\bar z),$$

where $f$ is holomorphic and $\bar f$ is antiholomorphic. On the real surface, $\bar f$ is the complex conjugate of $f$.

For such a map,

$$dw=f'(z)dz, \qquad d\bar w=\bar f'(\bar z)d\bar z,$$

so

$$ds'^2=f'(z)\bar f'(\bar z)dz\,d\bar z.$$

On the real surface this is

$$ds'^2=|f'(z)|^2 ds^2.$$

The local Weyl factor is therefore

$$\Omega(z,\bar z)=|f'(z)|.$$

Geometrically, near a point $z_0$,

$$f(z_0+dz)=f(z_0)+f'(z_0)dz+O(dz^2).$$

Multiplication by the complex number $f'(z_0)$ is a scale times a rotation:

$$f'(z_0)=\rho e^{i\alpha}, \qquad \rho=|f'(z_0)|, \qquad \alpha=\operatorname{arg} f'(z_0).$$

A scale changes lengths but not angles; a rotation also preserves angles. Hence a holomorphic map with $f'(z_0)\neq0$ is locally conformal at $z_0$.

A holomorphic map $w=f(z)$ is locally $dw=f'(z_0)dz$. Since $f'(z_0)=\rho e^{i\alpha}$, it acts on tangent vectors by a scale $\rho$ and a rotation $\alpha$, preserving the angle $\theta$ between them.

This is the two-dimensional miracle. In higher dimensions, the conformal Killing equation is overdetermined and only has finitely many solutions. In two dimensions, the Cauchy-Riemann equation has infinitely many local solutions.

Local versus global conformal transformations

Not every holomorphic function is a globally well-defined one-to-one map of the Riemann sphere. This distinction matters.

A local conformal transformation is a holomorphic change of coordinate in a patch:

$$z\mapsto f(z), \qquad f'(z)\neq0$$

inside the patch. It may have poles, branch points, or fail to be invertible globally. Local transformations are what give two-dimensional CFT its infinite-dimensional symmetry algebra.

A global conformal transformation of the Riemann sphere is a holomorphic bijection from the sphere to itself. These are exactly the Möbius transformations

$$z\mapsto \frac{az+b}{cz+d}, \qquad ad-bc\neq0.$$

Because multiplying $a,b,c,d$ by a common nonzero constant gives the same map, the group is projective. It is usually written as

$$PSL(2,\mathbb C)=SL(2,\mathbb C)/\mathbb Z_2.$$

This is the same finite-dimensional global conformal group that one expects from the general $d$-dimensional analysis, since

$$PSL(2,\mathbb C)\simeq SO^+(3,1).$$

The infinite-dimensional enhancement is therefore not the group of globally defined conformal maps on the sphere. It is the algebra of locally holomorphic coordinate transformations.

That subtlety is worth saying bluntly:

$$\boxed{ \text{Global conformal symmetry is finite. Local conformal symmetry is infinite.} }$$

The power of two-dimensional CFT comes from learning how the local symmetry is represented quantum mechanically. The answer is the Virasoro algebra.

Infinitesimal local conformal transformations

An infinitesimal local conformal transformation has the form

$$z' = z+\epsilon(z), \qquad \bar z' = \bar z+\bar\epsilon(\bar z),$$

where $\epsilon(z)$ is holomorphic and $\bar\epsilon(\bar z)$ is antiholomorphic.

Expand the holomorphic vector field in Laurent modes:

$$\epsilon(z)=\sum_{n\in\mathbb Z}\epsilon_n z^{n+1}.$$

It is conventional to introduce the holomorphic vector-field generators

$$\ell_n=-z^{n+1}\partial_z, \qquad n\in\mathbb Z.$$

Similarly,

$$\bar\ell_n=-\bar z^{n+1}\partial_{\bar z}.$$

They satisfy

$$[\ell_n,\ell_m]=(n-m)\ell_{n+m},$$ $$[\bar\ell_n,\bar\ell_m]=(n-m)\bar\ell_{n+m},$$

and

$$[\ell_n,\bar\ell_m]=0.$$

This is the classical local conformal algebra, also called the Witt algebra, together with its antiholomorphic copy.

The global holomorphic subalgebra is generated by

$$\ell_{-1}=-\partial_z, \qquad \ell_0=-z\partial_z, \qquad \ell_1=-z^2\partial_z.$$

These generate, respectively, translations, dilatations/rotations, and special conformal transformations in the $z$ coordinate. The barred generators do the same for $\bar z$.

A quick dictionary is

Generator	Infinitesimal map	Meaning
$\ell_{-1}$	$z\mapsto z+a$	translation
$\ell_0$	$z\mapsto \lambda z$	scale/rotation
$\ell_1$	$z\mapsto z+bz^2$	special conformal

The algebra generated by $\ell_{-1},\ell_0,\ell_1$ is $\mathfrak{sl}(2)$:

$$[\ell_0,\ell_{-1}]=\ell_{-1}, \qquad [\ell_0,\ell_1]=-\ell_1, \qquad [\ell_1,\ell_{-1}]=2\ell_0.$$

In the quantum theory, these classical vector fields become operator generators $L_n$ and $\bar L_n$. The commutator acquires a central extension:

$$[L_n,L_m]=(n-m)L_{n+m}+\frac{c}{12}n(n^2-1)\delta_{n+m,0}.$$

That is the Virasoro algebra. We will derive its origin from the stress tensor on the next page. For now, the important point is that $c$ vanishes from the global subalgebra because

$$n(n^2-1)=0 \qquad \text{for } n=-1,0,1.$$

So global conformal symmetry survives as an honest $SL(2)$ subalgebra even when the full local algebra has a central charge.

Primary fields and conformal weights

In higher dimensions, a primary operator is labeled by its scaling dimension $\Delta$ and its spin representation under rotations. In two dimensions, rotations and dilatations combine naturally into holomorphic and antiholomorphic weights.

A local field $\phi(z,\bar z)$ is called a primary field of weights $(h,\bar h)$ if under a finite conformal map

$$z' = f(z), \qquad \bar z'=\bar f(\bar z),$$

it transforms as

$$\phi'(z',\bar z') = \left(\frac{dz'}{dz}\right)^{-h} \left(\frac{d\bar z'}{d\bar z}\right)^{-\bar h} \phi(z,\bar z).$$

The ordinary scaling dimension and spin are

$$\Delta=h+\bar h, \qquad s=h-\bar h.$$

To see this, take a real scale transformation

$$z'=\lambda z, \qquad \bar z'=\lambda \bar z.$$

Then

$$\phi'(z',\bar z')=\lambda^{-(h+\bar h)}\phi(z,\bar z) =\lambda^{-\Delta}\phi(z,\bar z).$$

For a rotation

$$z'=e^{i\theta}z, \qquad \bar z'=e^{-i\theta}\bar z,$$

one obtains

$$\phi'(z',\bar z')=e^{-i\theta(h-\bar h)}\phi(z,\bar z) =e^{-is\theta}\phi(z,\bar z).$$

Thus $h$ and $\bar h$ are more refined labels than $\Delta$ and $s$:

$$h=\frac{\Delta+s}{2}, \qquad \bar h=\frac{\Delta-s}{2}.$$

Examples:

Field	Weights	Comment
scalar primary $\phi$	$(h,h)$	spin $s=0$, dimension $\Delta=2h$
holomorphic current $J(z)$	$(1,0)$	conserved chiral current
antiholomorphic current $\bar J(\bar z)$	$(0,1)$	opposite chirality
stress tensor $T(z)$	roughly $(2,0)$	not quite primary when $c\neq0$
stress tensor $\bar T(\bar z)$	roughly $(0,2)$	antiholomorphic partner

The phrase “roughly” for $T$ is deliberate. The stress tensor behaves like a field of holomorphic weight $2$ under global conformal transformations, but under general local transformations it picks up an anomalous Schwarzian derivative when $c\neq0$. That is exactly why the central charge matters.

Infinitesimal transformation of a primary

Let

$$z'=z+\epsilon(z), \qquad \bar z'=\bar z+\bar\epsilon(\bar z).$$

Using the finite transformation law and expanding to first order gives the active variation at the same coordinate point:

$$\delta_{\epsilon,\bar\epsilon}\phi(z,\bar z) = - \left[ \epsilon(z)\partial +h\partial\epsilon(z) + \bar\epsilon(\bar z)\bar\partial +\bar h\bar\partial\bar\epsilon(\bar z) \right] \phi(z,\bar z).$$

This formula is one of the most useful formulas in the subject. It says that a primary is not just dragged along by the vector field. It also carries a local scale weight, encoded by $h\partial\epsilon+\bar h\bar\partial\bar\epsilon$.

In terms of modes, the holomorphic action on a primary is represented by the differential operator

$$\mathcal L_n^{(h)} = -z^{n+1}\partial_z-h(n+1)z^n.$$

The antiholomorphic counterpart is

$$\bar{\mathcal L}_n^{(\bar h)} = -\bar z^{n+1}\partial_{\bar z}-\bar h(n+1)\bar z^n.$$

These satisfy the same Witt algebra:

$$[\mathcal L_n^{(h)},\mathcal L_m^{(h)}] =(n-m)\mathcal L_{n+m}^{(h)}.$$

This is the differential-operator representation of local conformal transformations on primary fields. In the quantum theory, the abstract generator $L_n$ acts on operator insertions through these differential operators plus possible contributions from descendant structure.

Two- and three-point functions

Let $z_{ij}=z_i-z_j$ and $\bar z_{ij}=\bar z_i-\bar z_j$.

For primary fields, global conformal symmetry fixes the two-point function up to normalization. For diagonal fields with the same weights,

$$\langle \phi_i(z_1,\bar z_1)\phi_j(z_2,\bar z_2)\rangle = \frac{C_{ij}}{z_{12}^{2h_i}\bar z_{12}^{2\bar h_i}}.$$

The correlator vanishes unless the two fields have compatible weights. In an orthonormal basis one often writes

$$\langle \phi_i(z_1,\bar z_1)\phi_j(z_2,\bar z_2)\rangle = \frac{\delta_{ij}}{z_{12}^{2h_i}\bar z_{12}^{2\bar h_i}}.$$

The three-point function is also fixed up to one coefficient:

$$\langle \phi_1(z_1,\bar z_1) \phi_2(z_2,\bar z_2) \phi_3(z_3,\bar z_3) \rangle = \frac{C_{123}}{ z_{12}^{h_1+h_2-h_3} z_{23}^{h_2+h_3-h_1} z_{13}^{h_1+h_3-h_2}} \frac{1}{ \bar z_{12}^{\bar h_1+\bar h_2-\bar h_3} \bar z_{23}^{\bar h_2+\bar h_3-\bar h_1} \bar z_{13}^{\bar h_1+\bar h_3-\bar h_2}}.$$

This is the two-dimensional version of the higher-dimensional statement that conformal symmetry fixes scalar two- and three-point functions. The difference is that the dependence factorizes into holomorphic and antiholomorphic powers.

For four points, there is a nontrivial cross-ratio

$$\eta=\frac{z_{12}z_{34}}{z_{13}z_{24}}, \qquad \bar\eta=\frac{\bar z_{12}\bar z_{34}}{\bar z_{13}\bar z_{24}}.$$

Global conformal symmetry fixes only the kinematic prefactor. The remaining dynamical information is a function of $(\eta,\bar\eta)$. Local conformal symmetry then reorganizes this function into Virasoro conformal blocks.

Holomorphic factorization: the first glimpse

The complex-coordinate split suggests a decomposition into holomorphic and antiholomorphic sectors. In a CFT, the stress tensor will satisfy

$$T_{z\bar z}=0,$$

up to contact terms and anomalies, while conservation gives

$$\bar\partial T_{zz}=0, \qquad \partial T_{\bar z\bar z}=0.$$

One therefore introduces

$$T(z)\sim T_{zz}(z), \qquad \bar T(\bar z)\sim T_{\bar z\bar z}(\bar z),$$

with $T$ holomorphic and $\bar T$ antiholomorphic away from insertions. The precise normalization will be fixed on the next page.

This is where two-dimensional CFT becomes qualitatively different from higher-dimensional CFT. The stress tensor is not merely one conserved operator. It generates infinitely many charges:

$$L_n\sim \oint dz\, z^{n+1}T(z), \qquad \bar L_n\sim \oint d\bar z\, \bar z^{n+1}\bar T(\bar z).$$

These charges are the quantum version of the local vector fields $\ell_n$ and $\bar\ell_n$.

AdS/CFT checkpoint

For AdS/CFT, this page matters in two different ways.

First, in AdS$_3$/CFT$_2$, the boundary CFT has two-dimensional local conformal symmetry. The resulting Virasoro symmetry is not an optional refinement; it is the organizing principle of the theory. Black-hole entropy, modular invariance, heavy-light correlators, and the Brown-Henneaux central charge all use this language.

Second, in string theory, the worldsheet theory is a two-dimensional CFT. Even when the spacetime holographic CFT lives in $d=3$ or $d=4$, the string worldsheet is governed by the complex-coordinate machinery developed here. The distinction between holomorphic and antiholomorphic sectors becomes the distinction between left- and right-moving worldsheet modes.

So this module is not a detour from AdS/CFT. It is the exact-solvable 2D technology that repeatedly reappears in holography.

Common pitfalls

The first pitfall is to identify “holomorphic” with “globally valid.” Holomorphic maps are locally conformal, but only Möbius maps are globally well-defined conformal automorphisms of the Riemann sphere.

The second is to forget the difference between $z$ and $\bar z$. On the physical Euclidean plane they are complex conjugates. In complexified CFT manipulations they are often treated as independent variables.

The third is to confuse primary and descendant fields. A primary transforms covariantly under local conformal maps. Derivatives of primaries are usually descendants, not primaries.

The fourth is to assume that $T(z)$ is an ordinary primary of weight $(2,0)$. It is not, unless $c=0$. The central charge measures precisely the failure of $T$ to transform as a primary under general local conformal maps.

Summary

Two-dimensional conformal symmetry is special because the local conformal transformations are holomorphic maps:

$$z\mapsto f(z), \qquad \bar z\mapsto \bar f(\bar z).$$

Their infinitesimal generators are

$$\ell_n=-z^{n+1}\partial_z, \qquad \bar\ell_n=-\bar z^{n+1}\partial_{\bar z},$$

with

$$[\ell_n,\ell_m]=(n-m)\ell_{n+m}, \qquad [\bar\ell_n,\bar\ell_m]=(n-m)\bar\ell_{n+m}, \qquad [\ell_n,\bar\ell_m]=0.$$

Primary fields carry weights $(h,\bar h)$:

$$\Delta=h+\bar h, \qquad s=h-\bar h.$$

Their finite transformation law is

$$\phi'(z',\bar z') = \left(\frac{dz'}{dz}\right)^{-h} \left(\frac{d\bar z'}{d\bar z}\right)^{-\bar h} \phi(z,\bar z).$$

This is the input for Virasoro symmetry, the stress-tensor OPE, minimal models, modular invariance, and AdS$_3$/CFT$_2$.

Exercises

Exercise 1: Cauchy-Riemann equations from conformality

Let

$$w=u(x^1,x^2)+iv(x^1,x^2).$$

Show that the condition

$$du^2+dv^2=\Omega(x)^2\left[(dx^1)^2+(dx^2)^2\right]$$

is equivalent, for an orientation-preserving map, to the Cauchy-Riemann equations

$$\partial_1 u=\partial_2 v, \qquad \partial_2 u=-\partial_1 v.$$

Solution

Write

$$du=u_1dx^1+u_2dx^2, \qquad dv=v_1dx^1+v_2dx^2,$$

where $u_i=\partial_i u$ and $v_i=\partial_i v$. Then

$$du^2+dv^2 =(u_1^2+v_1^2)(dx^1)^2 +2(u_1u_2+v_1v_2)dx^1dx^2 +(u_2^2+v_2^2)(dx^2)^2.$$

Conformality requires the cross-term to vanish and the two diagonal coefficients to be equal:

$$u_1u_2+v_1v_2=0,$$ $$u_1^2+v_1^2=u_2^2+v_2^2.$$

These say that the two column vectors $(u_1,v_1)$ and $(u_2,v_2)$ are orthogonal and have equal length. For an orientation-preserving map, this means the second is obtained by rotating the first by $+\pi/2$:

$$(u_2,v_2)=(-v_1,u_1).$$

Therefore

$$u_1=v_2, \qquad u_2=-v_1,$$

which are the Cauchy-Riemann equations.

Exercise 2: Local scale factor of a holomorphic map

Let $w=f(z)$ with $f'(z)\neq0$. Show that

$$ds_w^2=dw\,d\bar w=|f'(z)|^2 dz\,d\bar z.$$

Then compute the Weyl factor for $f(z)=z^n$ away from $z=0$.

Solution

Since $w=f(z)$,

$$dw=f'(z)dz.$$

On the real Euclidean plane, $\bar w=\overline{f(z)}$, so

$$d\bar w=\overline{f'(z)}d\bar z.$$

Thus

$$ds_w^2=dw\,d\bar w=|f'(z)|^2 dz\,d\bar z.$$

For $f(z)=z^n$,

$$f'(z)=nz^{n-1},$$

so the Weyl factor is

$$\Omega(z,\bar z)=|f'(z)|=n|z|^{n-1}.$$

The map is locally conformal wherever $z\neq0$. At $z=0$, the derivative vanishes for $n>1$, so the map is not locally invertible there.

Exercise 3: The Witt algebra

Using

$$\ell_n=-z^{n+1}\partial_z,$$

show that

$$[\ell_n,\ell_m]=(n-m)\ell_{n+m}.$$

Solution

Act on a test function $f(z)$. First,

$$\ell_m f=-z^{m+1}\partial f.$$

Then

$$\ell_n\ell_m f =z^{n+1}\partial\left(z^{m+1}\partial f\right) =(m+1)z^{n+m+1}\partial f+z^{n+m+2}\partial^2 f.$$

Similarly,

$$\ell_m\ell_n f =(n+1)z^{n+m+1}\partial f+z^{n+m+2}\partial^2 f.$$

Subtracting,

$$[\ell_n,\ell_m]f =(m-n)z^{n+m+1}\partial f.$$

Since

$$\ell_{n+m}=-z^{n+m+1}\partial,$$

we get

$$[\ell_n,\ell_m]f=(n-m)\ell_{n+m}f.$$

Thus

$$[\ell_n,\ell_m]=(n-m)\ell_{n+m}.$$

Exercise 4: Dimension and spin from weights

A primary field has weights $(h,\bar h)$. Use the finite primary transformation law to show that under a scale transformation it has dimension

$$\Delta=h+\bar h,$$

and under a rotation it has spin

$$s=h-\bar h.$$

Solution

For a scale transformation,

$$z'=\lambda z, \qquad \bar z'=\lambda\bar z.$$

Therefore

$$\frac{dz'}{dz}=\lambda, \qquad \frac{d\bar z'}{d\bar z}=\lambda.$$

The primary transformation law gives

$$\phi'(z',\bar z')=\lambda^{-h}\lambda^{-\bar h}\phi(z,\bar z) =\lambda^{-(h+\bar h)}\phi(z,\bar z).$$

So $\Delta=h+\bar h$.

For a rotation,

$$z'=e^{i\theta}z, \qquad \bar z'=e^{-i\theta}\bar z.$$

Then

$$\phi'(z',\bar z') =e^{-ih\theta}e^{+i\bar h\theta}\phi(z,\bar z) =e^{-i(h-\bar h)\theta}\phi(z,\bar z).$$

Hence $s=h-\bar h$, with the sign convention used in this page.

Exercise 5: Two-point function of primaries

Assume translation and rotation invariance and let

$$G(z_1,\bar z_1,z_2,\bar z_2) = \langle \phi_1(z_1,\bar z_1)\phi_2(z_2,\bar z_2)\rangle.$$

Use global conformal invariance to show that a nonzero two-point function of primary fields requires compatible weights and has the form

$$G=\frac{C_{12}}{z_{12}^{2h}\bar z_{12}^{2\bar h}}.$$

Solution

Translation invariance implies that $G$ depends only on

$$z_{12}=z_1-z_2, \qquad \bar z_{12}=\bar z_1-\bar z_2.$$

Assume

$$G=C z_{12}^{-a}\bar z_{12}^{-b}.$$

Under a scale transformation $z\to\lambda z$, primary covariance gives

$$G(\lambda z_{12},\lambda \bar z_{12}) = \lambda^{-(h_1+h_2+\bar h_1+\bar h_2)}G(z_{12},\bar z_{12}).$$

So

$$a+b=h_1+h_2+\bar h_1+\bar h_2.$$

Under rotation $z\to e^{i\theta}z$, covariance gives the spin constraint

$$a-b=(h_1+h_2)-(\bar h_1+\bar h_2).$$

Thus

$$a=h_1+h_2, \qquad b=\bar h_1+\bar h_2.$$

Invariance under the special conformal generator, or equivalently under inversion $z\to1/z$, further requires

$$h_1=h_2, \qquad \bar h_1=\bar h_2.$$

Therefore a nonzero two-point function has

$$G=\frac{C_{12}}{z_{12}^{2h}\bar z_{12}^{2\bar h}}.$$

Exercise 6: Why three points can be fixed globally

Show that the Möbius transformation

$$f(z)=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$$

maps

$$z_1\mapsto0, \qquad z_2\mapsto1, \qquad z_3\mapsto\infty.$$

Why does this explain why there is no conformal cross-ratio made from only three points?

Solution

Evaluate directly:

$$f(z_1)=0$$

because the numerator contains $z-z_1$.

At $z=z_2$,

$$f(z_2)=\frac{(z_2-z_1)(z_2-z_3)}{(z_2-z_3)(z_2-z_1)}=1.$$

At $z=z_3$, the denominator vanishes, so

$$f(z_3)=\infty.$$

Thus any three distinct points on the Riemann sphere can be mapped to $0,1,\infty$ by a global conformal transformation. Since their positions can all be removed by symmetry, no invariant cross-ratio can be built from only three points. Four points are different: after three are fixed, the fourth remains as the cross-ratio.

Further reading

For the classic treatment of this material, see Di Francesco, Mathieu, and Sénéchal, Chapter 5, especially the sections on conformal mappings, primary fields, correlation functions, and Ward identities. For the modern holographic perspective, keep this page paired with later discussions of AdS$_3$/CFT$_2$, the Cardy formula, and worldsheet CFT.