More is Universal: An Introduction to Conformal Field Theory
| Presenter | You-Chiuan (Andy) Chen (Prof. Ribhu Kaul’s group, Penn State) |
| Date | August 17, 2026 · 5:00–6:00 PM |
| Location | Davey 339 |
| Topic | An introduction to conformal field theory — Lecture I |
Lecture I of a two-part introduction to conformal field theory, told from the condensed-matter side. The organising question: a critical $\phi^4$ theory is strongly interacting and we cannot solve it — so what can symmetry alone tell us? The answer runs from the emergent scale invariance at a fixed point, through the conformal group and the correlators it fixes, to the operator product expansion and the bootstrap, where crossing symmetry plus unitarity pin the 3D Ising critical exponents to six digits without ever evaluating a path integral. A preview of radial quantization closes the session.
Part 1 — Criticality and universality
The problem: an interacting theory we cannot solve
Start from the standard continuum description of a scalar order parameter,
$$ S[\phi] = \int \mathrm{d}^dx \left[ \frac{1}{2}(\partial_\mu\phi)^2 + \frac{r}{2}\phi^2
- \frac{u}{4!}\phi^4 - h\phi \right], $$
with $r$ the tuning parameter, $u$ the interaction, $h$ an external field. At $h = 0$ the action carries a $\mathbb{Z}_2$ symmetry $\phi \mapsto -\phi$.
The classical strategy is stationary action, $\delta S/\delta\phi = 0$, giving $-\partial^2\phi + r\phi + \frac{u}{3!}\phi^3 - h = 0$, which for a uniform saddle collapses to $r\phi + \frac{u}{3!}\phi^3 - h = 0$. That provides candidate equilibrium configurations and the mean-field phase structure — and misses exactly what we care about: fluctuations around the saddle, and the physics of the continuous transition as $r \to 0$.
Scale invariance is emergent, not imposed
Write the local fluctuation $\delta\phi(x) = \phi(x) - \langle\phi\rangle$ and its correlator $G(x) = \langle\delta\phi(0)\,\delta\phi(x)\rangle$. Away from criticality a finite correlation length permits exponential decay, $G(x)\sim e^{-|x|/\xi}$. At the transition $r\to 0$, however, $\xi \to \infty$, and the RG argument says scale invariance is present at the critical point. Under $\tilde{x} = \lambda x$,
$$ [\delta\tilde\phi(\tilde x)] = \lambda^{-\Delta_\phi}[\delta\phi(x)] \qquad\Longrightarrow\qquad G(\lambda x) = \lambda^{-2\Delta_\phi} G(x), $$with $\Delta_\phi$ the scaling dimension. The exponential is gone; a power law replaces it.
Wilson–Fisher, and what universality means
The RG reorganises the problem as microscopic theory → (coarse grain and rescale) → RG flow → (long distances) → fixed point. In $d = 4-\epsilon$ the dimensionless quartic coupling flows as
$$ \beta(g) = -\epsilon g + A g^2 + \mathcal{O}(g^3), \qquad A > 0, $$with a non-Gaussian fixed point at $g_* = \epsilon/A + \mathcal{O}(\epsilon^2)$. Different microscopic systems flow to the same fixed point and therefore share the same scaling dimensions and critical exponents — universality.
Symmetry already solves part of the theory
Suppose $h=0$ and the state preserves $\mathbb{Z}_2$. Then $\langle\phi\rangle = \langle-\phi\rangle = -\langle\phi\rangle \Rightarrow \langle\phi\rangle = 0$, and more generally every odd correlator vanishes, $\langle\phi(x_1)\phi(x_2)\cdots\phi(x_{2n+1})\rangle = 0$. Exact information about observables, with the partition function never evaluated. (The conclusion assumes a $\mathbb{Z}_2$-invariant state; a chosen symmetry-broken vacuum may have $\langle\phi\rangle \neq 0$.)
Stack the symmetries and the unknowns keep shrinking:
| Symmetry | What it tells us |
|---|---|
| $\mathbb{Z}_2$ | Selection rules: odd correlators vanish |
| Translations and rotations | Correlators depend only on relative geometry |
| Scale invariance | Correlators acquire power-law behaviour |
| Conformal invariance | Two- and three-point functions strongly constrained; four-point functions obey consistency conditions |
Which sets up the question the rest of the lecture answers: can an enlarged symmetry determine the critical theory without solving the underlying field theory?
Part 2 — Conformal transformations
The defining condition
A map $\mathcal{M}: x^\mu \mapsto \tilde{x}^\mu$ is conformal when
$$ \tilde{g}_{\rho\sigma}(\tilde x)\,\frac{\partial \tilde x^\rho}{\partial x^\mu} \frac{\partial \tilde x^\sigma}{\partial x^\nu} = \Omega^2(x)\, g_{\mu\nu}(x), $$i.e. $\mathrm{d}\tilde s^2 = \Omega^2(x)\,\mathrm{d}s^2$. Because every inner product picks up the same local factor, the normalised ratio defining an angle is unchanged: lengths and areas may distort, intersection angles may not. Locally a conformal map is a rotation followed by a position-dependent rescaling.
One equation for the infinitesimal maps
Take $\tilde x^\mu = x^\mu + \epsilon^\mu(x)$ with $\Omega^2 = 1 + 2\sigma(x)$. To first order the metric condition reads $\partial_\mu\epsilon_\nu + \partial_\nu\epsilon_\mu = 2\sigma\eta_{\mu\nu}$, and its trace fixes $\sigma = \frac{1}{d}\partial_\rho\epsilon^\rho$, leaving the conformal Killing equation
$$ \partial_\mu \epsilon_\nu + \partial_\nu \epsilon_\mu = \frac{2}{d}(\partial\cdot\epsilon)\,\eta_{\mu\nu}. $$For $d \geq 3$ the integrability conditions make $\sigma$ affine, so $\epsilon^\mu$ is at most quadratic:
$$ \epsilon^\mu(x) = a^\mu + \omega^\mu{}_\nu x^\nu + \lambda x^\mu + 2(b\cdot x)x^\mu - b^\mu x^2, \qquad \omega_{\mu\nu} = -\omega_{\nu\mu}, $$with $a^\mu$ a translation, $\omega_{\mu\nu}$ a rotation, $\lambda$ a dilatation, and $b^\mu$ a special conformal transformation (SCT). In $d=2$ these generate the global conformal subgroup only — local conformal transformations there form an infinite-dimensional symmetry, which is what makes two dimensions special (and is the subject of the next lecture).
Four generators, three easy exponentials, one that is not
$$ P_\mu = -\mathrm{i}\partial_\mu, \qquad L_{\mu\nu} = \mathrm{i}\left(x_\mu\partial_\nu - x_\nu\partial_\mu\right), \qquad D = -\mathrm{i}\,x^\mu\partial_\mu, \qquad K_\mu = -\mathrm{i}\left(2x_\mu x^\nu\partial_\nu - x^2\partial_\mu\right). $$Exponentiating the first three is immediate: translations $\tilde x = x + a$ and rotations $\tilde x = Rx$ have $\Omega^2 = 1$ (they are isometries), dilatations $\tilde x = \lambda x$ have $\Omega^2 = \lambda^2$ (one constant rescaling everywhere). The SCT is the one with a position-dependent scale factor, and it is built from inversion $I : x^\mu \mapsto x^\mu/x^2$ as the composition $I \circ T_{-b} \circ I$:
$$ \tilde x^\mu = \frac{x^\mu - b^\mu x^2}{1 - 2b\cdot x + b^2x^2}, \qquad \Omega(x) = \frac{1}{1 - 2b\cdot x + b^2 x^2}. $$It is convenient to name the denominator $\sigma_b(x) = 1 - 2b\cdot x + b^2x^2$, so $\Omega = \sigma_b^{-1}$.
Part 3 — Defining a conformal field theory
Primaries transform homogeneously
A primary operator $\mathcal{O}_a(x)$ of scaling dimension $\Delta$ transforms with the local scale factor and nothing else:
$$ \tilde{\mathcal{O}}_a(\tilde x) = \Omega(x)^{-\Delta} D[R(x)]_a{}^b\,\mathcal{O}_b(x), $$the matrix $D[R(x)]$ accounting for spin. For a scalar primary this is simply $\tilde\phi(\tilde x) = \Omega(x)^{-\Delta}\phi(x)$: $\Delta$ measures how the local operator responds to a local change of length scale.
| Transformation | Coordinates | Scalar primary |
|---|---|---|
| Translation | $\tilde x^\mu = x^\mu + a^\mu$ | $\tilde\phi(\tilde x) = \phi(x)$ |
| Rotation | $\tilde x^\mu = R^\mu{}_\nu x^\nu$ | $\tilde\phi(\tilde x) = \phi(x)$ |
| Dilatation | $\tilde x^\mu = \lambda x^\mu$ | $\tilde\phi(\tilde x) = \lambda^{-\Delta}\phi(x)$ |
| Special conformal | $\tilde x^\mu = (x^\mu - b^\mu x^2)/\sigma_b(x)$ | $\tilde\phi(\tilde x) = \sigma_b(x)^{\Delta}\phi(x)$ |
Each operator insertion in an $n$-point function $G_n = \langle\phi_1(x_1)\cdots\phi_n(x_n)\rangle$ contributes one such factor,
$$ \tilde G_n(\tilde x_1,\dots,\tilde x_n) = \prod_{i=1}^{n}\Omega(x_i)^{-\Delta_i}\,G_n(x_1,\dots,x_n), $$and this becomes a constraint because the transformed and original correlators describe the same physical function.
Two points: three symmetries almost do it, the SCT finishes
For two scalar primaries with dimensions $\Delta_1,\Delta_2$ and $x_{12} = x_1-x_2$: translation invariance gives $G_{12} = F(x_{12})$, rotation invariance $F = f(|x_{12}|)$, and dilatation covariance $f(\lambda|x_{12}|) = \lambda^{-(\Delta_1+\Delta_2)}f(|x_{12}|)$ — so the only possible answer is a power law $C_{12}/|x_{12}|^{\Delta_1+\Delta_2}$.
The SCT then decides which power laws survive. Pairwise distances transform as $|\tilde x_{12}|^2 = |x_{12}|^2/[\sigma_b(x_1)\sigma_b(x_2)]$, so the candidate power law produces equal powers of $\sigma_b(x_1)$ and $\sigma_b(x_2)$, while covariance demands $\Delta_1$ and $\Delta_2$ separately. Hence
$$ \langle \phi_i(x_1)\phi_j(x_2)\rangle = \begin{cases} \dfrac{C_{ij}}{|x_{12}|^{2\Delta}}, & \Delta_i = \Delta_j = \Delta,\\[2ex] 0, & \Delta_i \neq \Delta_j, \end{cases} $$and within operators of equal dimension the two-point matrix can be diagonalised and normalised to $C_{ij} = \delta_{ij}$.
Three points: scale invariance is not enough, conformal invariance is
With three distances, the power-law ansatz $G_{123} = C_{123}/(|x_{12}|^\alpha|x_{23}|^\beta|x_{31}|^\gamma)$ gets only one equation from scale covariance, $\alpha+\beta+\gamma = \Delta_1+\Delta_2+\Delta_3$. Matching the local scale factor at each insertion under an SCT gives three: $\alpha+\gamma = 2\Delta_1$, $\alpha+\beta = 2\Delta_2$, $\beta+\gamma = 2\Delta_3$. Therefore
$$ G_{123} = \frac{C_{123}} {|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}\,|x_{23}|^{\Delta_2+\Delta_3-\Delta_1}\,|x_{31}|^{\Delta_3+\Delta_1-\Delta_2}}. $$Conformal symmetry has fixed all the position dependence of scalar two- and three-point functions. What it leaves behind are numbers: the dimensions $\Delta_i$ and the three-point (OPE) coefficients $C_{ijk}$. Together $\{\Delta_i, C_{ijk}\}$ are the CFT data — the local dynamical content of the theory.
Part 4 — OPE and the conformal bootstrap
Fusion
When two local operators approach one another, their product can be written as a sum of local operators at a single point,
$$ \mathcal{O}_i(x)\,\mathcal{O}_j(0) \sim \sum_k f_{ij}{}^k(x)\,\mathcal{O}_k(0). $$From far away two nearby insertions cannot be resolved separately, and their combined effect is reproduced by every local operator the symmetries allow. Scaling fixes the distance dependence, $f_{ij}{}^k(\lambda x) = \lambda^{\Delta_k-\Delta_i-\Delta_j}f_{ij}{}^k(x)$, so for scalars
$$ \mathcal{O}_i(x)\mathcal{O}_j(0) = \sum_k C_{ijk}\,|x|^{\Delta_k-\Delta_i-\Delta_j} \left[\mathcal{O}_k(0) + \text{descendants}\right], $$with conformal symmetry fixing the relative descendant contributions — the undetermined information is $\Delta_k$, the spin of $\mathcal{O}_k$, and $C_{ijk}$.
Symmetry restricts the channels before any calculation. In the Ising universality class, $\sigma \mapsto -\sigma$ and $\epsilon \mapsto \epsilon$, so $\sigma\times\sigma = \mathbf{1} + \sum\mathcal{O}^+$, $\sigma\times\mathcal{O}^+ = \sum\mathcal{O}^-$, $\mathcal{O}^+\times\mathcal{O}^+ = \mathbf{1} + \sum\mathcal{O}^+$; in particular $\sigma\times\sigma = \mathbf{1} + \epsilon + \epsilon' + T_{\mu\nu} + \cdots$. Note that a fusion rule specifies which symmetry sectors may appear — the OPE need not contain only finitely many operators.
Four points: the first function symmetry does not fix
Four points admit two conformal cross-ratios, and the four-point function is only determined up to a function of them:
$$ \langle\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle = \frac{\mathcal{G}(u,v)}{|x_{12}|^{2\Delta_\phi}|x_{34}|^{2\Delta_\phi}}, \qquad u = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2},\quad v = \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}. $$$\mathcal{G}(u,v)$ carries genuine dynamical information — but the OPE lets us compute it in more than one way. Fusing the pairs $(12)(34)$ (the $s$-channel) and $(14)(23)$ (the $t$-channel) are two expansions of the same object, so they must agree.
In the $(12)(34)$ channel,
$$ \mathcal{G}(u,v) = \sum_{\mathcal{O}\in\phi\times\phi} C_{\phi\phi\mathcal{O}}^2\, g_{\Delta_\mathcal{O},\ell_\mathcal{O}}(u,v), $$where the conformal block $g_{\Delta,\ell}$ packages a primary together with every descendant in its multiplet, and is fixed by conformal symmetry once $\Delta$ and $\ell$ are given. The block is universal kinematics; the spectrum and the $C_{\phi\phi\mathcal{O}}$ are the theory.
Crossing symmetry is OPE associativity
Equating the two channels gives the crossing equation
$$ 0 = \sum_{\mathcal{O}} C_{\phi\phi\mathcal{O}}^2 \left[ v^{\Delta_\phi} g_{\Delta_\mathcal{O},\ell_\mathcal{O}}(u,v)
- u^{\Delta_\phi} g_{\Delta_\mathcal{O},\ell_\mathcal{O}}(v,u) \right], $$
which is nothing but $(\mathcal{O}_1\mathcal{O}_2)\mathcal{O}_3 = \mathcal{O}_1(\mathcal{O}_2\mathcal{O}_3)$: different orders of local fusion must reconstruct exactly the same correlation functions.
The unknowns $\{\Delta_\mathcal{O}, \ell_\mathcal{O}, C_{ijk}\}$ must then satisfy several constraints simultaneously — crossing symmetry, unitarity ($C^2_{\phi\phi\mathcal{O}} \geq 0$), unitarity bounds (lower bounds on $\Delta$ at fixed spin), the internal symmetry (only allowed representations appear), and local CFT structure (the identity and the stress tensor are present). The numerical bootstrap asks: can a proposed spectrum satisfy all of them at once?
The 3D Ising island
For the critical Ising CFT, with a lowest $\mathbb{Z}_2$-odd scalar $\sigma$ and lowest $\mathbb{Z}_2$-even scalar $\epsilon$ in $\sigma\times\sigma = \mathbf{1} + \epsilon + \cdots$, crossing and unitarity applied to $\langle\sigma\sigma\sigma\sigma\rangle$ carve a sharp corner in the allowed region right at the Ising theory. Combining $\sigma$ and $\epsilon$ correlators with mild spectral-gap assumptions isolates a small allowed island:
$$ \Delta_\sigma = 0.5181489(10), \qquad \Delta_\epsilon = 1.412625(10), \qquad C_{\sigma\sigma\epsilon} = 1.0518537(41). $$Six-digit critical exponents for a strongly interacting theory, from consistency conditions alone.
Answer to the opening question
The lecture began with an interacting critical theory, $\phi^4 \to$ Wilson–Fisher fixed point. Conformal symmetry reorganises it: equations of motion $\to$ CFT data $\{\Delta_i, C_{ijk}\}$. The bootstrap then constrains that data:
$$ \text{conformal symmetry} + \text{OPE consistency} + \text{unitarity} \implies \text{universal critical data}. $$More symmetry does not merely simplify the theory — combined with consistency, it can determine sharply constrained properties of a strongly interacting critical point.
Part 5 — Radial quantization (preview of Lecture II)
The punctured plane is a cylinder
In polar coordinates with $\rho = \ln r$ and $\theta \sim \theta + 2\pi$, the punctured plane is an infinite cylinder, $\mathbb{R}^2\setminus\{0\} \cong \mathbb{R}_\rho\times S^1$, with circles of constant $r$ becoming constant-$\rho$ slices and $r\to 0 \Longleftrightarrow \rho\to-\infty$. Each constant-$\rho$ slice is a spatial circle carrying a state $|\Psi(\rho)\rangle \in \mathcal{H}_{S^1}$: a path integral over the disk inside the circle prepares it, and a path integral over an annulus evolves it outward. Taking $\rho$ as Euclidean time, outward radial evolution means increasing $\rho$.
The Hamiltonian is the dilatation operator
Translation in cylinder time is a scale transformation on the plane, $\rho \mapsto \rho + \alpha \Longleftrightarrow r \mapsto e^\alpha r$, so radial evolution is generated by $D$:
$$ |\Psi(\rho_2)\rangle = e^{-(\rho_2-\rho_1)D}|\Psi(\rho_1)\rangle, \qquad H_{\text{cyl}} = D, \qquad E_\mathcal{O} - E_0 = \Delta_\mathcal{O}. $$Scaling dimensions become measurable energy spacings. And since the origin is the infinite past of the cylinder, inserting a local operator inside the disk prepares a state,
$$ |\mathcal{O}\rangle = \lim_{r\to 0}\mathcal{O}(r\hat n)|0\rangle = \mathcal{O}(0)|0\rangle, $$the state–operator correspondence. A primary satisfies $D|\mathcal{O}\rangle = \Delta|\mathcal{O}\rangle$ and $K_\mu|\mathcal{O}\rangle = 0$; acting with $P_\mu$ builds descendants at $\Delta + n$ for level $n$. A conformal multiplet is an energy tower above a primary state.
The critical Ising chain realizes the cylinder
Take the periodic critical transverse-field Ising chain $H = -\sum_{j=1}^{N}\left(X_jX_{j+1} + Z_j\right)$, $X_{N+1} = X_1$. The ring of $N$ spins is a lattice approximation to $S^1$, so for circumference $L$,
$$ E_\alpha(L) - E_0(L) = \frac{2\pi v}{L}\Delta_\alpha + \text{finite-size corrections} \qquad\Longrightarrow\qquad \Delta_\alpha \simeq \frac{L}{2\pi v}\left(E_\alpha - E_0\right). $$The Ising CFT has three primaries — $\mathbf{1}$ ($\mathbb{Z}_2$-even, $\Delta = 0$), $\sigma$ (odd, $\tfrac{1}{8}$), $\epsilon$ (even, $1$) — and diagonalising the chain returns $\Delta_\sigma = 0.1249995$, $\Delta_\epsilon = 0.9999994$, along with the identification of the leading continuum fields of simple lattice operators, $X_j \leftrightarrow \sigma$ and $X_jX_{j+1} - Z_j \leftrightarrow \epsilon$ (up to normalisation and higher-dimension corrections).
In any dimension
Writing $x^\mu = r n^\mu$ with $n \in S^{d-1}$ gives $\mathbb{R}^d\setminus\{0\} \cong \mathbb{R}^+\times S^{d-1}$, and $\rho = \ln r$ again turns a dilatation into a translation. So: a local operator in $\mathbb{R}^d$ ↔ a state in $\mathcal{H}_{S^{d-1}}$; its scaling dimension ↔ cylinder energy above the vacuum; its rotation representation ↔ angular momentum on $S^{d-1}$. The spectrum of local CFT operators is the energy spectrum of a quantum theory living on a sphere.
Next lecture
Radial quantization in full, and two-dimensional CFT — where the local conformal algebra becomes infinite-dimensional.
Discussion points
- The whole construction rests on scale invariance being emergent at the fixed point, and on the (highly non-trivial, and not proven in general) upgrade from scale to conformal invariance.
- Conformal symmetry is kinematics: it fixes where the $x$-dependence goes, and hands back the dynamical question in the compressed form $\{\Delta_i, C_{ijk}\}$. The bootstrap is what turns consistency into numbers.
- The 3D Ising island is the cleanest advertisement for the method — but note what went into it: mixed correlators plus mild gap assumptions, not crossing symmetry alone.
- The state–operator correspondence is the bridge back to condensed matter: it makes scaling dimensions something you can extract from exact diagonalisation of a finite critical chain.
References
- D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982 (2016)
- S. Rychkov, EPFL Lectures on Conformal Field Theory in $D \geq 3$ Dimensions, arXiv:1601.05000 (2016)
- R. Blumenhagen and E. Plauschinn, Introduction to Conformal Field Theory: With Applications to String Theory, Lecture Notes in Physics 779, Springer (2009)
- J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics 5, Cambridge University Press (1996)
- F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, JHEP 08 (2016) 036, arXiv:1603.04436 — the 3D Ising island