Abstract
In this paper we develop further the relation between conformal four-point blocks involving external spinning fields and Calogero-Sutherland quantum mechanics with matrix-valued potentials. To this end, the analysis of [1] is extended to arbitrary dimensions and to the case of boundary two-point functions. In particular, we construct the potential for any set of external tensor fields. Some of the resulting Schrödinger equations are mapped explicitly to the known Casimir equations for 4-dimensional seed conformal blocks. Our approach furnishes solutions of Casimir equations for external fields of arbitrary spin and dimension in terms of functions on the conformal group. This allows us to reinterpret standard operations on conformal blocks in terms of group-theoretic objects. In particular, we shall discuss the relation between the construction of spinning blocks in any dimension through differential operators acting on seed blocks and the action of left/right invariant vector fields on the conformal group.
Highlights
There is another area of theoretical physics that was born roughly around the same time as the conformal bootstrap, namely the study of Calogero-Sutherland Hamiltonians
In this paper we develop further the relation between conformal four-point blocks involving external spinning fields and Calogero-Sutherland quantum mechanics with matrix-valued potentials
We shall discuss the relation between the construction of spinning blocks in any dimension through differential operators acting on seed blocks and the action of left/right invariant vector fields on the conformal group
Summary
We will review the construction of the space of conformal four-point blocks as a space of equivariant functions on the group. Given these data we can induce an irreducible representation π∆,μ of the conformal group G It can be realised on the following space of equivariant functions. Using the equivariance law we can reconstruct any function f ∈ ΓG(∆/,Nμ)DR on the conformal group G from the values it assumes on representatives of the N DR-orbits. In this sense, we can think of the space (2.2) as a space of Vμ-valued functions on the d-dimensional quotient G/N DR. In order to construct the space of such invariants we employ theorem 9.4 of [47] It states that the tensor product of two principal series representations (2.2) can be realised on the following space of equivariant functions πi ⊗ πj ∼= Γ(Gπ/iK,πj) with (2.5).
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