Calogero-Sutherland Model and Bulk-Boundary Correlations in Conformal Field Theory

We show that, in any conformal field theory, the weights of all bulk primary fields that couple to N φ2,1 fields on the boundary are given by the spectrum of an N -particle Calogero-Sutherland model. The corresponding correlation function is simply related to the N -particle wave function. Applications are discussed to the minimal models and the non-unitary O(n) model. The quantum Calogero-Sutherland (C-S) model has proved to be ubiquitous in theoretical physics. It has arisen in various ways in conformal field theory (CFT) in the past [1]. In this note, we point out a very direct connection. This was originally discovered [2] in the course of developing a multiparticle generalisation of Schramm-Loewner Evolution (SLE) [3], which, largely through the work of Lawler, Schramm and Werner (LSW) [4], has recently enlarged our perspective on conformally invariant random processes. However, the connection to the C-S model may be derived independently from SLE, using the basic principles of CFT, and it will now be presented in this way. The set-up is as follows: suppose we have a CFT in the interior of the unit disc |z| < 1, with a conformal boundary condition, and consider in particular the correlation function 〈φ(e1) . . . φ(eN ) Φ(0)〉 = 〈θ1, . . . , θN |Φ〉 (1) Address for correspondence 1 of N boundary fields φ with a single primary bulk field Φ at the origin. (Of course, any correlation function in a simply connected region with N boundary fields and a single bulk field at an interior point may be related to this by a conformal mapping.) In the second expression we have written this correlation function in the operator formulation of CFT, using radial quantisation: here |Φ〉 is a highest weight state of the holomorphic and antiholomorphic Virasoro algebras, and |θ1, . . . , θN〉 is a boundary state, lying in an N -dimensional subspace of the full Hilbert space of the CFT. Let us now suppose that φ is a primary field which is degenerate at level 2: it is a φ2,1 (or φ1,2) field in the Kac classification. As shown many years ago by Belavin, Polyakov and Zamolodchikov [5], this implies that correlation functions such as (1) satisfy second-order differential equations. In this case we shall show that these imply the C-S equation. First fix some notation: parametrise the central charge by c = 1− 6(4− κ)/4κ, so that the boundary scaling dimension of φ is h2,1 = (6 − κ)/2κ, and the null vector condition is (L−2 − (κ/4)L 2 −1)|φ2,1〉 = 0. Define the N -particle C-S hamiltonian with parameter β by HN(β) ≡ − 1 2 N