Friday 4 September 2020

Does this covariant function belong to some 2d CFT?

In conformal field theory, correlation functions of primary fields are covariant functions of the fields’ positions. For example, in two dimensions, a correlation function of N diagonal primary fields must be such that
$$\begin{aligned} F(z_1,z_2,\cdots , z_N) = \prod_{j=1}^N {|cz_j+d|^{-4\Delta_j}}  F\left(\tfrac{az_1+b}{cz_1+d},\tfrac{az_2+b}{cz_2+d},\cdots , \tfrac{az_N+b}{cz_N+d}\right) \ , \end{aligned}$$
where zj ∈ ℂ are the fields’ positions, Δj ∈ ℂ their conformal dimensions, and $\left(\begin{smallmatrix} a& b \\ c& d \end{smallmatrix}\right)\in SL_2(\mathbb{C})$ is a global conformal transformation. In addition, there are nontrivial relations between different correlation functions, such as crossing symmetry. But given just one covariant function, do we know whether it belongs to a CFT, and what can we say about that CFT?

In particular, in two dimensions, do we know whether the putative CFT has local conformal symmetry, and if so what is the Virasoro algebra’s central charge?

Since covariance completely fixes three-point functions up to an overall constant, we will focus on four-point functions i.e. N = 4. The stimulus for addressing these questions came from the correlation functions in the Brownian loop soup, recently computed by Camia, Foit, Gandolfi and Kleban. (Let me thank the authors for interesting correspondence, and Raoul Santachiara for bringing their article to my attention.)

Doesn’t any covariant function belong to multiple 2d CFTs?

In conformal field theory, any correlation function can be written as a linear combination of s-channel conformal blocks. These conformal blocks are a particular basis of smooth covariant functions, labelled by a conformal dimension and a conformal spin. (I will not try to say preciely what smooth means.) In two dimensions, we actually have a family of bases, parametrized by the central charge c, with the limit c = ∞ corresponding to global conformal symmetry rather than local conformal symmetry.

Therefore, any smooth covariant function can be written as a linear combination of blocks, for any value of the central charge c. In this sense, we can probably find some 2d CFT where our function can be interpreted as a correlation function, since we have complete freedom to cook up the CFT’s other correlation functions in order to satisfy crossing symmetry.

This is a priori not terribly interesting: in the absence of more constraints, the space of consistent CFTs is vast, and presumably includes an unchartered jungle of atrociously complicated creatures. To obtain meaningful results, we have to look for decompositions into conformal blocks that obey more constraints.

The importance of being simple: the case of minimal models

Let us consider a four-point function in a minimal model of 2d CFT. This can be decomposed into finitely many conformal blocks with the true central charge c of the model, schematically
$$\begin{aligned} F(z_j) = \sum_{i=1}^n D_i \mathcal{F}^{c}_{\Delta_i}(z_j)\mathcal F^c_{\bar \Delta_i}(\bar z_j) \ , \end{aligned}$$
where Di are position-independent structure constants, Δc(zj) is a Virasoro conformal block, and $\Delta_i,\bar \Delta_i$ are left- and right-moving conformal dimensions, with $\Delta_i=\bar \Delta_i$ if the model is diagonal.

What happens if we insist on rewriting the same function in terms of conformal blocks with another central charge c′≠c? Up to simple prefactors, the Virasoro conformal block is a function of the cross-ratio $z=\frac{z_{12}z_{34}}{z_{13}z_{24}}$ such that
$$\begin{aligned} \mathcal{F}^c_\Delta(z) = z^{\Delta - \Delta_1-\Delta_2}\sum_{k=0}^\infty \alpha_k z^k \ , \end{aligned}$$
where α0 = 1, and the coefficients α1, α2, ⋯ are functions of c, Δ and Δ1, Δ2, Δ3, Δ4. Therefore, we have relations of the type
$$\begin{aligned} \mathcal{F}^c_\Delta(z) = \sum_{m=0}^\infty f_m \mathcal{F}^{c'}_{\Delta+m}(z)\ ,\end{aligned}$$
where the coefficients fm depend on c, c′,Δ and Δ1, Δ2, Δ3, Δ4. Therefore, the decomposition of our correlation function into conformal blocks becomes an infinite sum. The correct central charge c is singled out as the only value that leads to a finite decomposition.

In the case of minimal models, we actually do not need to go that far for finding the correct central charge. The allowed conformal dimensions Δ1, Δ2, Δ3, Δ4 indeed belong to a finite set that depends on c, called the Kac table. Moreover, correlation function obey Belavin-Polyakov-Zamolodchikov partial differential equations, which also betray the value of c. But the simplicity of the decomposition into conformal blocks is a criterion that also works in more general cases.

Simple but not minimal: the case of Liouville theory

Liouville theory is a solved 2d CFT with a continuous spectrum, so there is no Kac table for quickly finding the central charge. Four-point functions are of the type
$$\begin{aligned} F(z_j) = \int_{\frac{c-1}{24}}^\infty d\Delta\ D_\Delta \mathcal{F}^c_{\Delta}(z_j)\mathcal{F}^c_{\Delta}(\bar z_j)\ .\end{aligned}$$
We may be tempted to deduce the value of c from the lower bound of integration of Δ. In terms of the cross-ratio z, we have
$$\begin{aligned} \frac{c-1}{24} = \Delta_1+\Delta_2 +\lim_{z\to 0}\frac{\log F(z)}{2\log |z|}\ .\end{aligned}$$
However, this requires that we a priori know the spectrum of Liouville theory, with its lower bound on Δ. Moreover, in case we are determining c numerically, the convergence is slow, and the factor $\frac{1}{24}$ further degrades the precision.

In this example too, let us see what happens if we rewrite the four-point function in terms of blocks for a wrong central charge c:
$$\begin{aligned} F(z_j) = \int_{\frac{c-1}{24}}^\infty d\Delta\ D_\Delta \sum_{m,\bar m=0}^\infty f_m f_{\bar m} \mathcal{F}^{c'}_{\Delta+m}(z_j) \mathcal{F}^{c'}_{\Delta+\bar m}(\bar z_j)\ .\end{aligned}$$
Rewriting this as a sum over the spin $s=m-\bar m$ and $\Delta'=\Delta+m+\bar m$, we obtain a decomposition of the type
$$\begin{aligned} F(z_j) = \int_{\frac{c-1}{24}}^\infty d\Delta' \sum_{s\in\mathbb{Z}} D'_{\Delta', s} \mathcal{F}^{c'}_{\Delta'+\frac{s}{2}}(z_j) \mathcal{F}^{c'}_{\Delta'-\frac{s}{2}}(\bar z_j)\ ,\end{aligned}$$
for some structure constants DΔ′,s which are combinations of DΔ and fm. With the true central charge c we had a diagonal decomposition, with the same dimension Δ for the left- and right-moving conformal blocks. With a wrong central charge c we now have arbitrary integer spins s ∈ ℤ, and therefore a more complicated decomposition. Just like in the case of minimal models, the true central charge is singled out by the simplicity of the decomposition.

Families of covariant functions

Let us now consider a family of covariant functions, with dimensions Δ1, Δ2, Δ3, Δ4 that are arbitrary complex numbers instead of having specific values. (This is almost the case in the article by Camia et al, whose four dimensions are subject to only one constraint.) We can now investigate whether four-point structure constants factorize. In the example of Liouville theory, this factorization reads
$$\begin{aligned} D_\Delta(\Delta_1,\Delta_2,\Delta_3,\Delta_4) = C_\Delta(\Delta_1,\Delta_2) C_\Delta(\Delta_3,\Delta_4)\ ,\end{aligned}$$
where CΔ(Δ1, Δ2) is a three-point structure constant. This factorization has no reason to hold in the case of arbitrary covariant functions, so does it allow us to distinguish CFT fron non-CFT?

Not really: factorization holds only provided the CFT’s spectrum contains only one copy of each representation of the Virasoro algebra. In more complicated situations, the states are characterized not only by the conformal dimension Δ, but also by an additional multiplicity parameter μ, and the decomposition of four-point structure constants into three-point structure constants reads
$$\begin{aligned} D_\Delta(\Delta_1,\Delta_2,\Delta_3,\Delta_4) = \sum_\mu C_\Delta^\mu(\Delta_1,\Delta_2) C_\Delta^\mu(\Delta_3,\Delta_4)\ .\end{aligned}$$
If we do not impose restrictions on the values of μ, four-point structure constants can always be decomposed in this way, and any family of covariant functions belongs to some 2d CFT, for any value of the central charge. However, when varying the central charge, we may find a value such that the four-point structure constants factorize, or decompose into particularly small numbers of terms.

Higher symmetry algebras

If no value of the central charge leads to a simple decomposition of our covariant function, it is not yet time to despair: we may just be considering the wrong symmetry algebra. We have been decomposing covariant functions into Virasoro conformal blocks, but this has no reason to be meaningful if the true symmetry algebra is larger than the Virasoro algebra. Larger algebras that are often encountered include affine Lie algebras, W-algebras, or several copies of the Virasoro algebra.

In practice, using the wrong symmetry algebra is similar to using the wrong central charge, and leads to overly complicated decompositions. Moreover, an irreducible representation of a larger symmetry algebra typically contains many representations of the Virasoro algebra, whose conformal dimensions differ by integers. This leads to nontrivial multiplicity parameters μ.


Given a covariant function, the question of whether it belongs to some CFT is probably meaningless. The interesting problem is rather to look for a central charge and symmetry algebra such that the decomposition into conformal blocks is simple. Depending on the case, a spectrum may be considered simple if it is finite, or diagonal, or in any case smaller than what we would get for generic central charges. If this succeeds, there is a chance that our covariant function belongs to a CFT that is interesting and meaningful.

Coming back to the article by Camia et al, the true central charge is a priori known, but the decomposition into Virasoro conformal blocks for this central charge does not look particularly simple. The four-point structure constants do not factorize unless we introduce nontrivial multiplicities, and the spectrum contains many conformal dimensions that differ by integers. So the four-point functions can probably not be understood in terms of the Virasoro algebra. Maybe one should look for a larger symmetry algebra.

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