Conformal field theory in two dimensions/Conformal symmetry

Conformal transformations

Definition

On a given space or spacetime $M$ with coordinates $x^{\mu }$, distances are defined using a metric $g_{\mu \nu }$. In particular, the length of a tangent vector $v^{\mu }$ is $\|v\|={\sqrt {g_{\mu \nu }v^{\mu }v^{\nu }}}$. If we know distances, we can also compute angles. The angle $\theta$ between two tangent vectors $v^{\mu },w^{\mu }$ obeys

$\cos \theta ={\frac {g_{\mu \nu }v^{\mu }w^{\nu }}{{\sqrt {g_{\mu \nu }v^{\mu }v^{\nu }}}{\sqrt {g_{\mu \nu }w^{\mu }w^{\nu }}}}}$

For $f:M\to M$ a diffeomorphism, we define the w:pullback $f^{*}g$ of the metric by

$(f^{*}g)_{\mu \nu }=g_{\rho \sigma }(f(x)){\frac {\partial f^{\rho }}{\partial x^{\mu }}}{\frac {\partial f^{\sigma }}{\partial x^{\nu }}}$

equivalently $(f^{*}g)_{\mu \nu }(x)dx^{\mu }dx^{\nu }=g_{\mu \nu }(f)df^{\mu }df^{\nu }$. A diffeomorphism $f:M\to M$ is called an isometry if it preserves distances, equivalently if

$f^{*}g=g$.

It is called a conformal transformation if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function $\Omega :M\to \mathbb {R}$, the metrics $g_{\mu \nu }$ and $\Omega ^{2}g_{\mu \nu }$ define the same angles. So $f$ is a conformal transformation if it preserve the metric up to a scalar factor,

$\exists \Omega ,\ f^{*}g=\Omega ^{2}g$

The set of conformal transformations is called the conformal group associated to $M,g$.

Let us indicate how many functions on $M$ are needed to parametrize the objects $\Omega ,f,g$, if $d=\dim(M)$:

Object	Number of functions on $M$
Function $\Omega :M\to \mathbb {R}$	$1$
Diffeomorphism $f:M\to M$	$d$
Metric $g$ on $M$	${\frac {d(d+1)}{2}}$

Therefore, given two metrics $g_{1},g_{2}$, the condition $f^{*}g_{1}=\Omega ^{2}g_{2}$ that they are conformally equivalent involves ${\frac {d(d+1)}{2}}$ equations for $d+1$ unknowns. For $d\leq 2$, there is always a solution (under reasonable assumptions): in particular any metric on $\mathbb {R} ^{2}$ is conformally flat. For $d\geq 3$, two metrics are in general not equivalent modulo rescaling.

Similarly, the size of the conformal group depends on the dimension:

• For $d=1$, any diffeomorphism is conformal.
• For $d\geq 2$, the conformal group depends on $M,g$, with flat space $\mathbb {R} ^{d}$ having the largest possible group $SO(d+1,1)$.
• For $d\geq 3$, a generic space has only one conformal transformation: the identity.

Case of flat space

Since we live in a space that is flat to a good approximation, let us study the conformal group of $\mathbb {R} ^{d}$ with the flat metric $g_{\mu \nu }=\delta _{\mu \nu }$. This group first includes isometries:

• Translations $x^{\mu }\mapsto x^{\mu }+a^{\mu },\ \forall a\in \mathbb {R} ^{d}$.
• Rotations $x^{\mu }\mapsto R_{\nu }^{\mu }x^{\nu },\ \forall R\in SO(d)$.

There are also conformal transformations that are not isometries:

• Dilations, also known as scale transformations $x^{\mu }\mapsto \lambda x^{\mu },\ \forall \lambda \in \mathbb {R}$.
• The inversion $x^{\mu }\mapsto {\frac {x^{\mu }}{\|x\|^{2}}}$.
• Special conformal transformations $x^{\mu }\mapsto {\frac {x^{\mu }-a^{\mu }\|x\|^{2}}{1-2a_{\mu }x^{\mu }+\|a\|^{2}\|x\|^{2}}},\ \forall a\in \mathbb {R} ^{d}$.

These transformations generate the conformal group $SO(d+1,1)$. Strictly speaking this is the conformal group of ${\overline {\mathbb {R} ^{d}}}=\mathbb {R} ^{d}\cup \{\infty \}$, since we allow tranformations that send finite points to infinity and vice-versa, such as the inversion. It is natural to include the point at infinity, because it can be reached by very large dilations, in the same way as 0 is reached by very small dilations.

Two-dimensional case

Let $M$ be a two-dimensional connected, oriented manifold. A metric $g$ on $M$ is called Riemannian if it is positive definite. Let ${\bar {g}}$ be the equivalence class of $g$ modulo conformal transformations, also called a conformal structure. Then $(M,{\bar {g}})$ is called a w:Riemann surface.

A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterized by its genus $h\in \mathbb {N}$, the number of holes. For a given genus, there is a finite-dimensional moduli space ${\mathcal {M}}_{h}$ of Riemann surfaces. The conformal group of a Riemann surface depends on $h$ and also on the surface:

Genus $h$	Name	$\dim _{\mathbb {C} }{\mathcal {M}}_{h}$	Conformal group
0	Riemann sphere	$0$	$SO(3,1)\simeq PSL_{2}(\mathbb {C} )$
1	Torus	1	$\left(\mathbb {C} /\mathbb {Z} ^{2}\right)\rtimes {\text{finite}}$
$\geq 2$		$3h-3$	finite

• The Riemann sphere is conveniently parametrized by a complex coordinate $z\in \mathbb {C} \cup \{\infty \}$, with the flat metric $ds^{2}=dzd{\bar {z}}$. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group $PSL_{2}(\mathbb {C} )$, which acts as

$z\mapsto {\frac {az+b}{cz+d}}\qquad {\text{for}}\qquad \left({\begin{array}{cc}a&b\\c&d\end{array}}\right)\in PSL_{2}(\mathbb {C} )$

• The torus $\mathbb {T} ={\frac {\mathbb {C} }{\mathbb {Z} +\tau \mathbb {Z} }}$ is parametrized by a modulus $\tau \in \mathbb {H} \equiv \left\{\tau \in \mathbb {C} |\Im \tau >0\right\}$. Two toruses are conformally equivalent if their moduli are related by $\tau \mapsto {\frac {a\tau +b}{c\tau +d}}$ for $\left({\begin{array}{cc}a&b\\c&d\end{array}}\right)\in PSL_{2}(\mathbb {Z} )$ so the moduli space is ${\mathcal {M}}_{1}=\mathbb {H} /PSL_{2}(\mathbb {Z} )$. The conformal group of $\mathbb {T}$ includes translations, which are parametrized by $\mathbb {T}$ itself or equivalently by $\mathbb {C} /\mathbb {Z} ^{2}$. Depending on $\tau$, there is also a finite group of rotations, which always includes the rotation by $\pi$.

Conformal symmetry and gravitation

Conformal invariance and Weyl invariance

In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism $f$ with $g\mapsto f^{*}g$, we obtain a change of coordinates, which should leave the physics invariant.

By definition, modulo a change of coordinates, a conformal transformation amounts to a Weyl transformation of the metric $g\mapsto \Omega ^{2}g$. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for $d=2$ only: in this case it is actually topological, in the sense that the w:Einstein-Hilbert action only depends on the topology of the space, and not on the metric.

Two-dimensional quantum gravity

In $d=2$, we expect quantum gravity to be topological, like general relativity. However, quantizing general relativity in the functional integral formalism does not directly give rise to a topological theory, beause of the w:Weyl anomaly: the integration measure is not invariant under Weyl transformations of the metric. The Weyl anomaly for a rescaling by a factor $\Omega$ is found to be given by the Liouville action for $\log \Omega$, i.e. the action of w:Liouville theory: a solvable CFT that we will study in this course. As a result, we can build a quantum theory of gravity called Liouville gravity based on Liouville theory. There are in fact several possible theories, depending on the matter contents, which must be chosen such that the Weyl anomaly cancels.

String theory

String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime $Z$ using a $d=2$ worldsheet $M$ and an embedding $X:M\to Z$. The metric $G$ on $Z$ induces a metric $g_{0}=X^{*}G$ on $M$, but it is convenient to allow the worldsheet metric $g$ to be an independent dynamical object, unrelated to the induced metric $g_{0}$. Nevertheless, the physics should not depend on $g$. It is possible to fix 2 of the 3 components of $g$ using a change of coordinates on $M$, and to get $g$-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses $d=2$ CFT.

String theory provides another motivation for CFT in arbitrary $d$: the AdS/CFT correspondence, a holographic relation between a string theory in a $d+1$-dimensional Anti-de Sitter space, and a CFT in $d$ dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case.

Scale invariance and conformal invariance

The relevance of conformal field theory extends well beyond gravitational physics. It is relatively easy (but not trivial) to explain why scale invariance is important in physics. Moreover, in many cases scale invariance implies conformal invariance, although the reason is not always clear.

Scale invariance in nature

Physical properties of matter vary a lot across scales:

• At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy.
• In the solar system, Newtonian gravity with relativistic corrections is an adequate description.
• From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension.
• At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons.

Scale invariance is therefore not a fundamental symmetry of nature. When there is scale invariance, it only holds approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories.

Geometrically, a scale-invariant shape is called a w:fractal. In nature, there are many objects that are fractal over some range of scales, from coastlines to plants.

Scale invariance at phase transitions

Consider the w:Ising model: a statistical model of spins on a lattice (say square or cubic), which can take 2 values $\pm 1$. Two neighbouring spin interact: the interaction energy is lower if their values are the same. The system has 2 phases:

• An ordered phase at low temperature, where the energy dominates over the entropy, and almost all spins have the same value.
• A disordered phase at high temperature, where the entropy dominates over the energy, and spins are not correlated beyond short distances.

The lattice has 2 geometrical scales: the distance $a$ between neighbouring sites, and the length $L\gg a$ of the whole lattice. From the model's dynamics, we also define the correlation length $\xi$: the distance at which spins are correlated. The correlation length is also the typical sizes of clusters: domains where all spins are aligned. The correlation length is $\xi =O(L)$ in the ordered phase, and $\xi =O(a)$ in the disordered phase. There is a phase transition between the two phases, at a critical temperature $T=T_{c}$. At the phase transition, there coexist clusters of all possible sizes $a<\ell <L$. The system is approximately scale invariant over that range of scales.

A physical manifestation of this phenomenon is the w:critical opalescence of fluids near critical gas/liquid phase transitions. Due to the existence liquid droplets of arbitrary sizes, the fluid scatters light of arbitrary wavelengths and appears cloudy, whereas the liquid and the gas would both appear transparent.

In general, scale invariance appears in second-order phase transitions, i.e. transitions where $\xi$ becomes infinite (which means $O(L)$ in a finite system). First-order transitions such as the melting of water ice have finite $\xi$ and no scale invariance.

Scale invariance of renormalization group fixed points

There is a systematic way of reaching a scale-invariant theory, starting from any physical theory. The idea is simply to rescale the theory until it becomes invariant under rescaling. The resulting scale-invariant theory is called a fixed point of the renormalization group. It may or may not describe a phase transition.

For example, to rescale the Ising model, we can use a block spin transformation: replace a block of $3^{d}$ neighbouring spins with one spin, therefore changing $a\to 3a$. This transformation changes all the model's parameters in a way that is described by the w:renormalization group. More abstractly, in a field theory, we can similarly rescale the theory such that we obtain a similar theory where however the parameters have changed.

At a fixed point, the resulting theory is usually much simpler than the original theory. This is not only because we now have the extra symmetry of scale invariance, but also because many interactions of the original theory go to zero at the fixed point. (They are called irrelevant.) This results in w:universality: the fact that many different physical systems share the same fixed point. The fixed point to which a system flows, also called its universality class, depends only on a few features such as the symmetries.

Physics at or near the fixed point can be described using a limited number of quantities, called the w:critical exponents. For example, in the limit $L\to \infty$, the correlation length diverges at the critical temperature as $\xi (T)\propto (T-T_{c})^{-\nu }$ where $\nu$ is a critical exponent.

Scale invariance allows us to predict the existence of critical exponents, and relations between different critical exponents. However, conformal invariance can allow us to actually compute critical exponents, by solving a CFT. In CFT, critical exponents appear as the conformal dimensions of fields, and describe how fields behave under conformal transformations.

Does scale invariance imply conformal invariance?

Since conformal invariance is much more demanding than scale invariance, we might expect it to be much less common. However, among the physical systems and theories that have been studied, few are scale invariant but not conformally invariant. Finding a case of scale invariance without conformal invariance is quite rare, to the extent that it deserves particular attention. Explaining why and in which cases scale invariance implies conformal invariance is a deep question, whose answer is not fully known.

There are arguments that scale invariance implies conformal invariance under certain assumptions. These arguments are more general and convincing for $d=2$. The assumptions include w:unitarity (physics), and the existence of a w:stress-energy tensor. These assumptions are necessary, and there are counter-examples when they are relaxed. Nevertheless, there are also many CFTs that violate these assumptions. For example, critical w:percolation is described by a non-unitary CFT. It may well be that for some of these CFTs, conformal invariance follows from scale invariance. But the currently known arguments do not work in these cases.

Exercises

• C1. Show that the scale factor $\Omega$ of a conformal transformation, to the power $d$, coincides with the Jacobian of that transformation.
• C2. Show that the inversion is a conformal transformation. Write a special conformal transformation in terms of a translation and inversions. Deduce that the special conformal transformation is indeed conformal.
• C3. Show that two toruses whose moduli are related by $\tau \mapsto {\frac {a\tau +b}{c\tau +d}}$ for $\left({\begin{array}{cc}a&b\\c&d\end{array}}\right)\in PSL_{2}(\mathbb {Z} )$ are conformally equivalent.
• C4. Show that the w:Polyakov action is invariant under Weyl and conformal transformations.

COGS: The conformal group of flat space

Consider the Euclidean space $\mathbb {R} ^{d}$ with the flat metric $g_{\mu \nu }=\delta _{\mu \nu }$, and the Minkowski space $\mathbb {R} ^{d+1,1}$ with coordinates $Y=\left(y^{\mu },y^{-},y^{+}\right)$ and the flat metric $\|dY\|^{2}=\sum _{\mu =1}^{d}\left(dy^{\mu }\right)^{2}-dy^{-}dy^{+}$. Consider the diffeormorphisms

$\varphi :\left\{{\begin{array}{ccl}\mathbb {R} ^{d}&\to &\mathbb {R} ^{d+1,1}\\x^{\mu }&\mapsto &\left(x^{\mu },\|x\|^{2},1\right)\end{array}}\right.\quad ,\quad \psi :\left\{{\begin{array}{ccl}\mathbb {R} ^{d+1,1}&\to &\mathbb {R} ^{d+1,1}\\Y&\mapsto &{\frac {1}{y^{+}}}Y=\left({\frac {y^{\mu }}{y^{+}}},{\frac {y^{-}}{y^{+}}},1\right)\end{array}}\right.$

1. Check that $\varphi$ is an isometry. Is $\psi$ an isometry? Is it a conformal transformation?
2. Show that the restriction of $\psi$ to the light cone ${\mathcal {L}}=\left\{Y\in \mathbb {R} ^{d+1,1}\left|\|Y\|^{2}=0\right.\right\}$ is a conformal transformation, and that $\varphi (\mathbb {R} ^{d})\subset {\mathcal {L}}$.
3. Let $G\in SO(d+1,1)$ be an isometry of $\mathbb {R} ^{d+1,1}$, in particular $G$ is linear. Show that $\varphi ^{-1}\circ \psi \circ G\circ \varphi$ is a conformal transformation of $\mathbb {R} ^{d}$. Deduce that the conformal group of $\mathbb {R} ^{d}$ includes $SO(d+1,1)$.
4. Explicitly write the action of $G\in SO(d+1,1)$ on $x^{\mu }$.
5. In the case $d=2$, find the relation between the two different descriptions of the conformal group: $SO(3,1)$ and $PSL_{2}(\mathbb {C} )$.