Conformal Field Theory

Prerequisites: Conformal Group, Canonical Quantum Field Theory. Full path (24 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Conformal field theory is either a classical field theory or quantum field theory that is invariant under conformal transformations. In such theories, there is no mass scale and hence the theory is scale invariant, which means it looks the same no matter from how far away we look at it.

Concrete

We work in Euclidean signature in $d$ dimensions. A conformal transformation is a coordinate change that preserves angles, so the metric changes only by a local scale factor, $g_{\mu\nu}(x) \to \Omega(x)^2 g_{\mu\nu}(x)$ . Infinitesimally, $x^\mu \to x^\mu + \epsilon^\mu(x)$ is conformal if

$\partial_\mu\epsilon_\nu + \partial_\nu\epsilon_\mu = \frac{2}{d}\,(\partial\cdot\epsilon)\,\delta_{\mu\nu}.$

For $d > 2$ the solutions are translations $\epsilon^\mu = a^\mu$ , rotations $\epsilon^\mu = \omega^\mu{}_\nu x^\nu$ , dilatations $\epsilon^\mu = \lambda x^\mu$ and special conformal transformations $\epsilon^\mu = 2(b\cdot x)x^\mu - b^\mu x^2$ . The finite special conformal transformation is

$x'^\mu = \frac{x^\mu - b^\mu x^2}{1 - 2\,b\cdot x + b^2x^2},$

an inversion, a translation by $-b$ and another inversion. Together these form the conformal group, of dimension $\frac12(d+1)(d+2)$ , locally isomorphic to $SO(d+1,1)$ (or $SO(d,2)$ in Lorentzian signature).

Primary operators and correlation functions. A conformal field theory is organized by local operators $\mathcal{O}$ with a scaling dimension $\Delta$ : under $x \to \lambda x$ a scalar primary transforms as $\mathcal{O}(x) \to \lambda^{\Delta}\mathcal{O}(\lambda x)$ . Conformal symmetry fixes the two- and three-point functions of scalar primaries up to constants:

$\langle\mathcal{O}_i(x)\,\mathcal{O}_j(y)\rangle = \frac{C\,\delta_{\Delta_i\Delta_j}}{|x-y|^{2\Delta_i}},$

$\langle\mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\mathcal{O}_3(x_3)\rangle = \frac{C_{123}}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}\,|x_{23}|^{\Delta_2+\Delta_3-\Delta_1}\,|x_{13}|^{\Delta_1+\Delta_3-\Delta_2}},$

with $x_{ij} = x_i - x_j$ . Scale invariance alone gives the power of $|x-y|$ in the two-point function; invariance under special conformal transformations forces the two dimensions to be equal. Four-point functions are not fixed: they depend on the two cross-ratios $u = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}$ and $v = \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}$ , which are invariant under all conformal transformations.

For example, a free massless scalar in $d$ dimensions has $\langle\phi(x)\phi(0)\rangle \propto 1/|x|^{d-2}$ , so $\Delta_\phi = (d-2)/2$ . At the critical point of the three-dimensional Ising model the spin field has $\Delta_\sigma \approx 0.518$ , and critical exponents of the phase transition are functions of such scaling dimensions.

Energy-momentum tensor. In a local theory, conformal invariance is reflected in a traceless energy-momentum tensor, $T^\mu{}_\mu = 0$ (in flat space, up to contact terms). Then the currents $j^\mu = T^{\mu\nu}\epsilon_\nu$ are conserved for every conformal Killing vector $\epsilon$ .

Two dimensions. For $d = 2$ the equation for $\epsilon$ is the Cauchy-Riemann equation, and every holomorphic map $z \to f(z)$ is conformal wherever $f'(z) \neq 0$ . Writing $T(z) = \sum_n L_n z^{-n-2}$ , the modes obey the Virasoro algebra

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

with central charge $c$ , equivalent to the operator product expansion

$T(z)T(w) \sim \frac{c/2}{(z-w)^4} + \frac{2T(w)}{(z-w)^2} + \frac{\partial T(w)}{z-w}.$

A free boson has $c = 1$ , a free Majorana fermion has $c = \frac12$ , and the critical Ising model is the $c = \frac12$ minimal model.

Abstract

Radial quantization and the state-operator map. Choose an origin and foliate $\mathbb{R}^d$ by spheres $S^{d-1}$ centered on it. Dilatations play the role of time evolution, with $\log|x|$ as time, and the dilatation operator $D$ as Hamiltonian. Equivalently, $\mathbb{R}^d\setminus\{0\}$ is conformal to the cylinder $\mathbb{R}\times S^{d-1}$ . Inserting a local operator $\mathcal{O}$ at the origin and doing the path integral inside a sphere produces a state $|\mathcal{O}\rangle$ on that sphere, and conversely every state is obtained this way. Under this correspondence the scaling dimension is the energy on the cylinder of unit radius. Primary operators correspond to states annihilated by the special conformal generators $K_\mu$ (lowest weight states), and descendants are obtained by acting with $P_\mu$ . In a unitary (reflection positive) theory, the Hilbert space decomposes into lowest-weight representations of the conformal algebra, and positivity of norms gives the unitarity bounds

$\Delta \geq \frac{d-2}{2} \ \ (\text{scalars, } \Delta = 0 \text{ only for the identity}), \qquad \Delta \geq \ell + d - 2 \ \ (\text{symmetric traceless spin } \ell \geq 1),$

saturated by free fields and by conserved currents respectively (for $d \geq 3$ ).

Operator product expansion. The product of two operators can be expanded as

$\mathcal{O}_1(x)\mathcal{O}_2(0) = \sum_k C_{12k}\, |x|^{\Delta_k - \Delta_1 - \Delta_2}\big(\mathcal{O}_k(0) + \text{descendants}\big),$

where the descendant contributions are fixed by conformal symmetry. In radial quantization this expansion converges inside correlation functions whenever the other operators are farther from the origin. The CFT data, the spectrum $\{\Delta_k, \ell_k\}$ and the OPE coefficients $C_{ijk}$ , determine all correlation functions on flat space. Four-point functions can be expanded in two OPE channels, $(12)(34)$ and $(14)(23)$ . Equating them gives the crossing equation

$\sum_k C_{12k}C_{34k}\, g_{\Delta_k,\ell_k}(u,v) = \left(\frac{u}{v}\right)^{\Delta}\sum_k C_{14k}C_{23k}\, g_{\Delta_k,\ell_k}(v,u)$

(written here for four identical scalars of dimension $\Delta$ , where all products of coefficients are squares), with conformal blocks $g_{\Delta,\ell}$ that are known functions of the cross-ratios. Combined with unitarity, crossing gives the numerical conformal bootstrap, which has produced the most precise determinations of critical exponents of the three-dimensional Ising model.

Two dimensions. In $d = 2$ the local conformal algebra is two copies of the Witt algebra, centrally extended to Virasoro $\oplus$ Virasoro. The chiral part of a CFT is a vertex operator algebra, and rational CFTs, which have finitely many irreducible representations of their chiral algebra, are classified in many cases: for $c < 1$ the unitary theories are the minimal models with $c = 1 - \frac{6}{m(m+1)}$ , $m = 3,4,\ldots$ . Consistency on higher genus surfaces adds modular invariance of the torus partition function. Segal's axioms phrase a 2D CFT as a functor, projective because of the central charge, from the category of Riemann surfaces with parametrized boundary to Hilbert spaces.

Relation to other structures. Conformal field theories are the fixed points of the renormalization group, and relevant deformations of a CFT define RG flows. In two dimensions Zamolodchikov's $c$ -theorem states that a function equal to $c$ at fixed points decreases along such flows; in four dimensions the analogous $a$ -theorem was proven by Komargodski and Schwimmer. Through the AdS/CFT correspondence, a $d$ -dimensional CFT is conjectured to be equivalent to a quantum gravity theory on $(d+1)$ -dimensional anti-de Sitter space, whose isometry group $SO(d,2)$ is the Lorentzian conformal group.

Why is it interesting?

The modern way of understanding relativistic quantum field theories (QFTs) is through renormalization group flows away from conformal field theories (CFTs). In the parameter space of all QFTs, CFTs arise as fixed points with enhanced scale and conformal symmetry. The very ambitions programme of understanding all QFTs thus is intimately related to the classification of all consistent CFTs.https://iopscience.iop.org/article/10.1088/1361-6382/aa8003/pdf

It is an old idea in particle physics that, in some sense, at sufficiently high energies the masses of the elementary particles should become unimportant. In recent years this somewhat vague hope has acquired a more definite form in the theory of scale transformations, or dilatations.

Aspects of Symmetry: Selected Erice Lectures by Sidney Coleman

In particle physics, one longstanding hope has been that at high energies, particle masses can be neglected, so that the physics would become scale invariant. It turns out that in a local field theory, it is true, more or less in general, that scale invariance typically leads to conformal invariance. (This is because the violation of scale invariance and conformal invariance are both determined by the trace $T^\mu_\mu$ of the energy momentum tensor.)

page 621 Einstein Gravity in a Nutshell - A. Zee

It seems that all QFTs can be viewed as points along an Renormalization Flow (or RG flow, this is the name we give to the zooming process) from a ‘UV’ CFT to another ‘IR’ CFT. Renormalization flows occur when we deform the UV CFT, breaking its conformal symmetry. […] Well-defined QFTs can be viewed as either CFTs or as RG flows between CFTs. We can remove the UV cutoff from a QFT (send it to infinite energy or zero length) if it can be interpreted as an RG flow from the vicinity of a CFT fixed point. So studying the space of CFTs basically amounts to studying the space of all well-defined QFTs.

Lectures on AdS/CFT from the Bottom Up by Jared Kaplan

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