8037 1711.02022 10.1007/JHEP04(2018)052 52 52 From spinning conformal blocks to matrix Calogero-Sutherland models 01 Mathematical Sciences 0101 Pure Mathematics Regular Article - Theoretical Physics 1 29 2018 4 11 2018 1 30 2018 3 26 2018 4 11 The Author(s) 2018 Volker Schomerus volker.schomerus@desy.de Evgeny Sobko evgenysobko@gmail.com 0000 0004 0492 0453 grid.7683.a DESY Hamburg, Theory Group Notkestraße 85 22607 Hamburg Germany 0000 0004 1936 9377 grid.10548.38 Nordita, Stockholm University and KTH Royal Institute of Technology Roslagstullsbacken 23 SE-106 91 Stockholm Sweden 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. Keywords Conformal and W Symmetry Conformal Field Theory Differential and Algebraic Geometry Field Theories in Higher Dimensions ArXiv ePrint: 1711.02022