FR37 · Conformal bootstrap

Crossing Symmetry: One Four-Point Function, Two Expansions

Does the same operator data give the same correlator when you expand around either pair of points? Yes, when the operator data are consistent. A four-point function can be built by fusing either pair of operators first, and the two sums must match at every point. The exact data of a free fermion or free boson pass this test everywhere; moving one operator, changing one coefficient or keeping too few terms makes the two sums disagree.

Edit the operators of a 1D free field and compare the s- and t-channel sums of its four-point function. Consistent data agree at every cross-ratio.

Expanded around the first pair $A_s(0.30)=1.13$

Expanded around the other pair $A_t(0.30)=1.13$

Largest crossing mismatch $\max|R|<10^{-12}$

First exchanged operator $\Delta_1=2,\;c_1^2=1$

Break crossing on purpose

External dimension

0.500

\Delta_{\mathcal O}

Shift of the first operator

0

\delta\Delta_1

Scale of its coefficient

1

c_1^2/c_{1,\mathrm{free}}^2

Operators kept in each sum

40

N_{\mathrm{op}}

Drag the second point along the line, or drag the flashing operator on the dimension axis.

Model, method and sources

$R(z)=(1-z)^{2\Delta_{\mathcal O}}A_s(z)-z^{2\Delta_{\mathcal O}}A_s(1-z)$

Blocks are summed from the hypergeometric series. The sum keeps the chosen number of exchanged operators; the free-field closed form is the reference. The residual is sampled on 161 cross-ratios.

Gaiotto, Mazáč & Paulos (2013), Eqs. 4.17, 4.20 · Mazáč (2016), Eqs. 2.15–2.21

Derive the two expansions

Model assumptions

Four identical operators sit on a line: the first at the origin, the second at a movable point, the third at one and the fourth at infinity. The s-channel sums conformal blocks for the operators exchanged between the first pair; the t-channel sums the same data for the other pairing. The default data are the exact operator dimensions and OPE coefficients of a generalised free fermion or boson (Gaiotto, Mazáč and Paulos 2013, Eqs. 4.17 and 4.20); the dashed curve is the closed-form correlator. Editing the leading operator or cutting the sum short gives data that no longer solve crossing. Everything is computed; no measured data are shown.

Two expansions of one correlator

Crossing residual, in powers of ten

More numbers and how they are computed

Free-field closed form at the same point: $0$
Where the mismatch is largest: $0$
Second exchanged operator: $0$
Operators kept in each sum: $0$
External operator dimension: $0$

The exported file lists every operator with its edited and free-field dimension and coefficient, plus the two sums, the closed form and the residual on 161 cross-ratios.

Start with the essentials

Focus question: Does the same operator data give the same correlator when you expand around either pair of points?
One-sentence intuition: The free fermion and free boson data make the two sums agree at every cross-ratio. Moving one operator, changing one coefficient or keeping too few terms breaks the agreement, and the mismatch is largest where one expansion converges slowly.

Core mathematical model

Conformal block on a line

k_\Delta(z)=z^{\Delta}\,{}_2F_1(\Delta,\Delta;2\Delta;z)

Each exchanged operator of dimension $\Delta$ contributes its block, weighted by the squared OPE coefficient $c^2$ .

Crossing equation

(1-z)^{2\Delta_{\mathcal O}}\Bigl[1+\sum_{\Delta}c_\Delta^2\,k_\Delta(z)\Bigr]=z^{2\Delta_{\mathcal O}}\Bigl[1+\sum_{\Delta}c_\Delta^2\,k_\Delta(1-z)\Bigr]

The left side expands around $z=0$ , the right side around $z=1$ . The Lab plots their difference as the crossing residual.

Free fermion data

\Delta_j=2\Delta_{\mathcal O}+2j+1,\qquad c_j^2=\frac{2\,(2\Delta_{\mathcal O})_{2j+1}^2}{(2j+1)!\,(4\Delta_{\mathcal O}+2j)_{2j+1}}

These closed forms resum to $A(z)=1+\bigl(\tfrac{z}{1-z}\bigr)^{2\Delta_{\mathcal O}}-z^{2\Delta_{\mathcal O}}$ .

Common difficulties

Agreement at one point is not crossing

Typical misconceptionIf the two channels agree at the symmetric point, the data are consistent.

Better mental modelAt $z=\tfrac12$ the residual vanishes for any data by symmetry. Crossing is a statement about every $z$ ; its odd derivatives at the symmetric point carry the content.

Run the experiment

01
Watch both channels
Let the second point sweep along the line.
What to observe: The cyan bars shrink to a few operators near the first point, the amber bars near the third. Both totals stay equal.
02
Move one operator
Drag the flashing operator sideways, or choose Move one operator.
What to observe: The residual grows away from the symmetric point and the badge reports a mismatch.
03
Cut the sum short
Keep only two operators.
What to observe: The s-channel still works near the first point, but fails as the second point approaches the third.

Crossing Symmetry: One Four-Point Function, Two Expansions

One correlator, two expansions

Core mathematical model

Conformal block on a line

Crossing equation

Free fermion data

Common difficulties

Agreement at one point is not crossing

Run the experiment

Watch both channels

Move one operator

Cut the sum short

Related experiments and articles

From a critical chain to the kink in the 2D bound