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.