Where to Start Learning the Conformal Bootstrap

The conformal bootstrap is a way to constrain or determine a conformal field theory using consistency conditions, rather than starting from a particular Lagrangian. Its basic ingredients are conformal symmetry, the operator product expansion, and crossing symmetry. This introduction explains how those pieces fit together and suggests a sensible path into the subject.

The most accessible starting problem is usually the four-point function of an identical scalar operator in a unitary conformal field theory in dimension three or higher. The same broad ideas apply in other dimensions, though two-dimensional conformal field theory has additional structure and its own useful approaches.

What the bootstrap uses

A conformal field theory has local operators with definite scaling dimensions and spins. The operator product expansion (OPE) says that when two operators approach each other, their product can be expressed as a sum over other local operators. Schematically,

$$\phi(x_1)\phi(x_2) = \sum_{\mathcal O} \lambda_{\phi\phi\mathcal O}\, (\text{position-dependent terms})\,\mathcal O(x_2).$$

Conformal symmetry fixes much of the position dependence. The data that remain include operator dimensions, spins, and OPE coefficients. The bootstrap asks which sets of data are compatible with all the required consistency conditions.

For a four-point function, you can apply the OPE to different pairs of operators. These are different ways to expand the same correlator, so they must agree. That requirement is crossing symmetry. In a unitary theory, the OPE coefficients for identical real scalars enter as squares, which are nonnegative. Crossing plus positivity can rule out proposed spectra or place bounds on the dimensions and coefficients of operators.

The scalar four-point crossing equation

For identical real scalar primary operators $\phi$ of scaling dimension $\Delta_\phi$, normalize the two-point function to $\langle\phi(x)\phi(0)\rangle=1/(x^2)^{\Delta_\phi}$. Conformal symmetry lets you write the four-point function as

$$\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle = \frac{G(u,v)}{(x_{12}^2 x_{34}^2)^{\Delta_\phi}},$$

where $x_{ij}^2=|x_i-x_j|^2$ and

$$u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}, \qquad v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}.$$

Here the first cross-ratio is conventionally written $u$ (not the Greek letter $\nu$). Exchanging points 1 and 3 gives the crossing equation

$$v^{\Delta_\phi}G(u,v) = u^{\Delta_\phi}G(v,u).$$

The OPE decomposes $G$ into conformal blocks, which package the contribution of a primary operator and all its conformal descendants:

$$G(u,v) = 1 + \sum_{\substack{\mathcal O\ne\mathbf{1}\\ \ell\,\mathrm{even}}} \lambda_{\phi\phi\mathcal O}^{\,2}\,g_{\Delta,\ell}(u,v).$$

The 1 is the identity operator’s contribution. For identical real scalars, only even-spin primaries appear in this sum. Each block depends on the exchanged operator’s scaling dimension $\Delta$, spin $\ell$, and the spacetime dimension. Substituting this expansion into crossing turns the problem into a constraint on the allowed operators and their OPE coefficients.

Worked example: a crossing-symmetric correlator

Consider the Wick-contracted four-point function of a generalized free scalar. Its reduced correlator is

$$G(u,v) = 1 + u^{\Delta_\phi} + \left(\frac{u}{v}\right)^{\Delta_\phi}.$$

Check crossing by exchanging $u$ and $v$ and multiplying by the required prefactor:

$$\left(\frac{u}{v}\right)^{\Delta_\phi}G(v,u) = \left(\frac{u}{v}\right)^{\Delta_\phi} \left[1+v^{\Delta_\phi}+\left(\frac{v}{u}\right)^{\Delta_\phi}\right] = \left(\frac{u}{v}\right)^{\Delta_\phi}+u^{\Delta_\phi}+1 =G(u,v).$$

So this correlator satisfies the crossing equation. This example shows the basic consistency check, but it is not the whole bootstrap calculation: to extract useful constraints, you also express the correlator in conformal blocks and impose unitarity and other conditions.

A practical route into the subject

1. Review conformal field theory basics. Learn primary operators, correlation functions, conformal blocks, and the OPE. Rychkov’s lecture notes are a structured introduction to CFT in dimensions three and higher.
2. Study the crossing equation. Simmons-Duffin’s TASI lectures develop the scalar four-point bootstrap and explain how crossing becomes a numerical optimization problem.
3. Learn what positivity adds. In the identical-scalar setup, squared OPE coefficients are nonnegative. Numerical methods use this fact to bound possible operator dimensions and test candidate theories.
4. Use a review for breadth. The review by Poland, Rychkov, and Vichi surveys the theory, numerical techniques, and applications once you know the basic setup.

You will get more from these readings if you are comfortable with linear algebra, group representations at an introductory level, and the basics of quantum field theory. You do not need to master every prerequisite before reading the first sections; the essential ideas can be learned alongside the bootstrap equations.

Common mistakes

• Treating crossing alone as the full method. Crossing is central, but useful bounds generally require unitarity, symmetry assumptions, and a way to analyze the conformal-block expansion.
• Confusing operator dimensions with OPE coefficients. Dimensions and spins describe which operators can appear; OPE coefficients describe how strongly they couple to the chosen operators.
• Assuming one equation determines a unique theory. A single four-point crossing equation often allows many solutions. Additional correlators, symmetries, and physical assumptions can sharpen the constraints.
• Expecting the name to mean one specific algorithm. “Conformal bootstrap” covers a family of analytic and numerical methods built around consistency conditions. The scalar four-point numerical bootstrap is a common entry point, not the entire subject.