Superintegrability ofd-Dimensional Conformal Blocks

Abstract

We observe that conformal blocks of scalar four-point functions in a d-dimensional conformal field theory can be mapped to eigenfunctions of a two-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two coupled Pöschl-Teller particles. Their interaction, whose strength depends smoothly on the dimension d, is known to be superintegrable. Our observation enables us to exploit the rich mathematical literature on Calogero-Sutherland models in deriving various results for conformal field theory. These include an explicit construction of conformal blocks in terms of Heckman-Opdam hypergeometric functions. We conclude with a short outlook, in particular, on the consequences of integrability for the theory of conformal blocks.

Long multiplet bootstrap

“…Alternatively, it would be of interest to extend the approach proposed in [103] to the case of superconformal groups. Quite generally, it leads to a reformulation of conformal Casimir equations as eigenvalue equations for certain Calogero-Sutherland Hamiltonians, in agreement with [104]. As was shown at the example of three-dimensional fermionic seed blocks in [103], the reformulation in terms of Calogero-Sutherland models is very universal and in particular works for spinning blocks as well as for scalars.…”

“…Hence, one would expect that a universal set of Casimir equations for long multiplets of superconformal groups can be derived in any dimension. Moreover, by exploiting the integrability of Calogero-Sutherland Hamiltonians it should be possible to develop a systematic solution theory [104,105], without the need for an Ansatz that decomposes superblocks in terms of bosonic ones.…”

Anatomy of geodesic Witten diagrams

“…The closed form expressions of G O ∆,J (u, v) for even d-dimensions have been solved explicitly in terms of hypergeometric functions using quadratic Casimir operators [8,20]; more recently the precise connections of G O ∆,J (u, v) with the eigenfunctions of quantum integrable systems have also been established for arbitrary d-dimensions in [21,22]. Now imagine these external scalar primary operators {O ∆ i } are inserted at the boundary of AdS d+1 at points {P i }, and use γ 12 and γ 34 to denote the geodesics connecting the points P 1,2 and P 3,4 respectively, the four point scalar geodesic Witten diagram is defined through the double integral (see figure 1): Here −∞ < λ, λ < +∞ are the line parameters of γ 12 and γ 34 which we integrate along with, in terms of bulk AdS d+1 coordinates X A (λ) andX A (λ ), the two geodesics are given by following curves:…”

The gravity dual of Lorentzian OPE blocks

“…More recently, Lorentzian CFT's have attracted renewed interest in light of the novel development in analytic bootstrap in lightcone [20][21][22], the Regge limits of correlation functions [23][24][25][26][27], and the remarkable applications in deriving averaged null energy condition in flat space [28,29] and the Lorentzian inversion formula [30,31]. Conformal blocks play a vital role also there as a probe of causal relationships between operators in Minkowski space, whose structures have been deeply tied to a class of quantum integrable systems in the past few years [32][33][34].…”