From spinning conformal blocks to matrix Calogero-Sutherland models

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.

Superconformal blocks: general theory

“…the vector spaces V i are 1-dimensional, the number M − 1 counts the number of so-called nilpotent conformal invariants. Our derivation of the formula (1.1) in section 2 requires to extend a theorem for the tensor product of principal series representations of the conformal algebra from [45] to superconformal algebra g. The analysis in section 2 is similar to the discussion in [19,20] except that we will adopt an algebraic approach and characterize functions on the group through their infinite sets of Taylor coefficients at the group unit. That has the advantage that we can treat bosonic and fermionic variables on the same footing.…”

“…The index G/P on Θ may seem a bit unnatural since the right hand side only involves g or its universal enveloping algebra. Nevertheless we will continue to label the space Θ by groups and cosets thereof to keep notations as close as possible to those used in [19,20], except that now we use the spaces Θ of Taylor expansions rather than spaces Γ of functions. For later use we also note that our definition (2.8) implies that any element ϕ of Θ satisfies…”

“…In order to determine the space Θ of these invariants one starts by computing the tensor product of two principal series representations [19,20]. In the case of bosonic conformal groups, the relevant mathematical theorem was found in [45], see theorem 9.2.…”

“…which is a specialization of the correlators (4.13) and (4.17) at the same time. Hence the relevant block that satisfies 20) in addition to the constraints (4.18), so it must be G (2) . The other special case addresses correlators of the form ϕφϕφ , (4.21)…”

Reduction of a bi-Hamiltonian hierarchy on $$T^*\mathrm{U}(n)$$ to spin Ruijsenaars–Sutherland models

“…See, for example, the reviews [29,36,46] and references therein. The spin extensions of these models [19,47,24] are also important, and are currently subject to intense studies [2,8,13,14,21,33,34,38].…”