Conformal Cross-Ratios and Four-Point Functions

From Three to Four Points: The Emergence of Invariants

Why is the four-point function qualitatively different from the three-point function? The reason lies in a simple counting argument. The conformal group $SO(d+1,1)$ is large enough to map any three generic points in ${\mathbb{R}}^d$ to three chosen locations (for example, $0$, $1$, and $\infty$). This is why a three-point function, after accounting for the appropriate scaling behavior, evaluates to a single number, $C_{123}$.

However, with four points, we run out of symmetry. We can use the conformal group to fix the positions of $x_1$, $x_2$, and $x_4$, but the position of the fourth point, $x_3$, remains a variable. Its remaining coordinates, relative to the fixed points, cannot be changed by any conformal transformation. These remaining degrees of freedom must be expressible as combinations of the four points that are, by themselves, invariant under the entire conformal group.

Constructing Conformal Invariants: Cross-Ratios

To build these invariants, let's recall how squared distances transform under a general conformal transformation $x\to x^{\prime }$:

The conformal factor $\Omega (x)$ prevents a simple distance from being invariant. However, if we take a ratio of products of four such distances, the $\Omega$ factors can cancel out. This leads to the definition of conformal cross-ratios. For four points $x_1,x_2,x_3,x_4$, we can form two independent, conformally invariant cross-ratios:

Any other cross-ratio you might construct (e.g., by permuting the points) can be expressed in terms of $u$ and $v$. These two variables capture all the conformally invariant information contained in the geometry of four points.

The Structure of the Four-Point Function

Since the cross-ratios $u$ and $v$ are the only conformally invariant quantities we can build from four points, the four-point function of scalar primary operators must be expressible in terms of them. The general structure is a "kinematic" prefactor, which ensures the correct transformation properties under scaling and other conformal transformations, multiplied by an arbitrary function of the cross-ratios.

The standard form is:

where the exponents are determined by the scaling dimensions of the operators:

• $a=({\Delta }_1-{\Delta }_2)/2$
• $b=({\Delta }_3-{\Delta }_4)/2$
• $c=({\Delta }_1+{\Delta }_2)/2$
• $d=({\Delta }_3+{\Delta }_4)/2$

The function $\mathcal{G}(u,v)$ is where the real physics lies. It is not determined by conformal symmetry alone and contains the detailed dynamical information of the specific CFT under study. It characterizes the interactions between the operators. Finding $\mathcal{G}(u,v)$ is a central goal in solving a CFT.

The Case of 2D CFT

The structure is particularly elegant in two dimensions. Your familiarity with complex analysis will be useful here. In 2D, we use complex coordinates $z=x^1+i{x}^2$ and $\bar{z}=x^1-i{x}^2$. The conformal group factorizes into holomorphic and anti-holomorphic transformations. The four-point function then depends on a single complex cross-ratio and its conjugate:

(Note: $z$ here denotes the cross-ratio, not a generic coordinate).

The four-point function of primary fields $V_{{\Delta }_i}(z_i,{\bar{z}}_i)$ (where ${\Delta }_i$ now represents the total scaling dimension) can be written as an arbitrary function of $z$ and $\bar{z}$, times a kinematic factor fixed by the global Ward identities.

The resource presents the result as:

where the prefactor depends on the choice of which points are used to construct it, but the essential point remains: the correlator is reduced to an unknown function of a single complex variable $z$. The function $G(z)$ in the notes is actually a function $G(z,\bar{z})$ in a general 2D CFT.

This function $G(z,\bar{z})$ is the key that unlocks the dynamics of the theory. The powerful techniques of the conformal bootstrap, which we will explore in detail later, are based on demanding that this function has consistent properties under different ways of calculating it.

Conclusion

Today we've bridged the gap between the fully-determined low-point functions and the dynamically rich higher-point functions. The four-point function is the first place where a theory's specific properties, beyond its operator spectrum, come into play.

Key Takeaways:

• Unlike 2- and 3-point functions, the 4-point function is not completely fixed by conformal symmetry.
• The freedom in the 4-point function is captured by a function $\mathcal{G}$ that depends on conformally invariant cross-ratios.
• For four points in any dimension $d$, there are two independent cross-ratios, $u$ and $v$. In 2D, these combine into a single complex cross-ratio $z$.
• The full four-point function is a product of a universal kinematic prefactor (determined by the scaling dimensions) and the theory-dependent function $\mathcal{G}(u,v)$.
• This function $\mathcal{G}(u,v)$ contains the essential dynamical data of the CFT and is the primary target of modern bootstrap methods.

We have now established the kinematic arenas for 2-, 3-, and 4-point functions. But how do we relate them? How can we use our knowledge of the $C_{ijk}$ coefficients to learn something about the function $\mathcal{G}(u,v)$? The answer lies in one of the most fundamental concepts in QFT: the Operator Product Expansion (OPE). In our next lesson, we will introduce the OPE and see how it provides a way to systematically build higher-point functions from lower-point ones.