One four-point function expanded two ways gives the crossing equation

One number left over: the cross-ratio

Put four identical operators of dimension $\Delta_{\mathcal O}$ at points $x_1<x_2<x_3<x_4$ on a line. Conformal symmetry can move three of them to $0$ , $1$ and infinity. The fourth then sits at the cross-ratio $z=x_{12}x_{34}/(x_{13}x_{24})$ , a number between $0$ and $1$ .

After a prefactor fixed by symmetry, the whole correlator is one function $A(z)$ . Unlike two- and three-point functions, it is not fixed. It carries information about every operator the theory contains.

Source: D. Mazáč · D. Simmons-Duffin

Two ways to expand the same function

Fusing operators 1 and 2 turns $A(z)$ into the identity plus a sum over exchanged primaries. Each term is a squared OPE coefficient times a conformal block, the contribution of one primary and all its descendants. Near $z=0$ a block starts as $z^\Delta$ , so the lightest operators dominate.

Fusing operators 2 and 3 instead gives the same kind of sum in the variable $1-z$ . It converges fastest near $z=1$ . Both expansions describe one function, so where both converge they must agree. That is the crossing equation.

k_\Delta(z)=z^\Delta\,{}_2F_1(\Delta,\Delta;2\Delta;z)=z^\Delta\Bigl(1+\tfrac{\Delta}{2}z+\dots\Bigr)

On a line the block is a hypergeometric function. Every term in its series is positive for

0<z<1

(1-z)^{2\Delta_{\mathcal O}}A(z)=z^{2\Delta_{\mathcal O}}A(1-z),\qquad A(z)=1+\sum_{\Delta}c_\Delta^2\,k_\Delta(z)

The unknowns are the dimensions

\Delta

and coefficients

c_\Delta^2

. In a unitary theory every

c_\Delta^2

is non-negative.

Four identical operators on a line. Fusing the pairs

(12)(34)

(cyan) gives the first expansion, which converges fast near

z=0

. Fusing

(23)(14)

(amber) gives the crossed one, fast near

z=1

. Crossing says both describe the same function. Schematic.

Source: D. Mazáč · D. Simmons-Duffin

Why this is hard to satisfy

Each side is an infinite sum, and the same list of dimensions and coefficients must work on both. Move one dimension a little and the left side changes in one way near $z=0$ while the right side changes in a different way. Other operators would have to shift to compensate, and positivity of $c_\Delta^2$ limits how they can.

This rigidity is the whole idea of the bootstrap. Instead of computing a theory from a Lagrangian, one asks which lists of operators can satisfy crossing at all.

Source: D. Simmons-Duffin · S. Rychkov

Worked example: checking the free fermion at one point

A one-dimensional free fermion of dimension $\Delta_\psi=1/2$ has the closed form below. At $z=0.3$ it gives $A=1.128571$ . The exchanged operators have $\Delta=2,4,6,\dots$ with $c^2=1$ , $1/10$ , $1/126$ and so on.

Sum the first channel: $k_2(0.3)=0.126948$ , so identity plus one operator gives $1.126948$ . Adding $k_4(0.3)/10$ gives $1.128555$ , and the third operator brings it to $1.128571$ . Three terms reach six digits.

The crossed channel needs $A(0.7)$ . The closed form gives $2.633333$ , but the series converges slowly there: one operator gives $2.4157$ , five give $2.63331$ , and ten are needed for six digits. Multiplying by $0.3/0.7$ returns $1.128571$ . The two expansions agree.

Now move the first operator from $2$ to $2.35$ and keep everything else. The first channel at $z=0.3$ falls to $1.0900$ and the crossed one rises to $1.1633$ . One misplaced operator breaks crossing by about 7 percent.

A_\psi(z)=1+\frac{z}{1-z}-z=1+k_2(z)+\tfrac{1}{10}\,k_4(z)+\tfrac{1}{126}\,k_6(z)+\dots

Both forms are exact. The second is the first-channel expansion that the worked example sums term by term.

Source: D. Gaiotto · D. Mazáč

Try it in the Lab

Make the two sums agree, then break them

Open the free-fermion preset and drag the second point along the line. Check that both channel readouts stay equal.
Predict what the shifted-operator preset will do to the mismatch curve, then select it.
Pick the preset with too few operators. Note where on the line the mismatch grows, and explain why from the series.

Exercises with explained answers

Using $k_\Delta(z)\approx z^\Delta(1+\Delta z/2)$ , estimate $k_2(0.3)$ . How close is it to $0.126948$ ?
Show the answer
$0.09\times1.3=0.117$ . The next term in the series, $\tfrac{9}{10}z^2$ times $z^2$ , adds $0.0073$ and brings it to $0.1243$ ; the rest makes up the remaining two percent.
Check the free-fermion crossing equation at $z=0.5$ without any sums.
Show the answer
At $z=0.5$ both sides are $0.5\,A(0.5)$ , so crossing holds automatically. This is why $z=0.5$ is the natural point for derivatives of the crossing equation.
Why does keeping only two exchanged operators work well near $z=0$ but fail near $z=1$ ?
Show the answer
Each block grows as $z^\Delta$ , so near $z=0$ heavy operators are suppressed. Near $z=1$ the blocks grow large and the series converges slowly, so the missing terms matter in the crossed channel.

Sources and what we recomputed

D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982v1 (2016)
Version 1. Eqs. (157)-(158): the four-point function as a sum over conformal multiplets, one conformal block each. Eqs. (164)-(168): the Casimir equation for blocks and their closed form in two dimensions, built from the one-variable function k. Eqs. (194)-(199), Sections 10.5-10.7: the crossing equation for identical scalars as a sum of vectors, the derivative functionals and the assumptions behind the scalar-gap bound. Fig. 26: 2D crossing vectors. Fig. 27 and Table 1: 2D bounds for derivative cutoffs 6 to 28 with the kink at the Ising point. Fig. 28: the 3D bound with its kink.
D. Mazáč, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, arXiv:1611.10060v1; JHEP 04 (2017) 146
Version 1. Eqs. (2.12)-(2.17): the four-point function of identical operators on a line, the cross-ratio, the one-variable block and the crossing equation. Eq. (2.18) and Fig. 2: the numerical gap bound, which the text says converges to twice the external dimension plus one as derivatives are added. Eqs. (2.19)-(2.23): the generalized free fermion that saturates it and the zeros of the extremal functional. Fig. 1: the action of a typical extremal functional.
D. Gaiotto, D. Mazáč and M. F. Paulos, Bootstrapping the 3d Ising twist defect, arXiv:1310.5078v2; JHEP 03 (2014) 100
Version 2. Section 4.1: one-dimensional generalized free fields. Eqs. (4.17) and (4.20) give the exchanged dimensions and OPE coefficients of the fermionic and bosonic solutions; the course resums them against the closed-form correlators.
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, arXiv:1601.05000v2 (2016)
Version 2. Section 1.1: fixed points and scale invariance at a critical point. Section 1.3.4: how primary operators transform. Section 2.2: two- and three-point functions of primary scalars. Sections 3.1.3-3.1.4: the state-operator correspondence and the cylinder picture. Section 3.3: the operator product expansion. Section 4.3.1, Eqs. (4.14)-(4.16): the minimal models and the central charge and spin and energy dimensions of the 2D Ising model.