Superconformal blocks in diverse dimensions and $BC$ symmetric functions

Abstract

We uncover a precise relation between superblocks for correlators of superconformal field theories (SCFTs) in various dimensions and symmetric functions related to the BC root system. The theories we consider are defined by two integers (m, n) together with a parameter θ and they include correlators of all half-BPS correlators in 4d theories with N = 2n supersymmetry, 6d theories with (n, 0) supersymmetry and 3d theories with N = 4n supersymmetry, as well as all scalar correlators in any non SUSY theory in any dimension, and conjecturally various 5d, 2d and 1d superconformal theories. The superblocks are eigenfunctions of the super Casimir of the superconformal group whose action we find to be precisely that of the BC m|n Calogero-Moser-Sutherland (CMS) Hamiltonian. When m = 0 the blocks are polynomials, and we show how these relate to BC n Jacobi polynomials. However, differently from BC n Jacobi polynomials, the m = 0 blocks possess a crucial stability property that has not been emphasised previously in the literature. This property allows for a novel supersymmetric uplift of the BC n Jacobi polynomials, which in turn produces the (m, n; θ) superblocks. Superblocks defined in this way are related to Heckman-Opdam hypergeometrics and are non polynomial functions. A fruitful interaction between the mathematics of symmetric functions and SCFT follows, and we give a number of new results on both sides. One such example is a new Cauchy identity which naturally pairs our superconformal blocks with Sergeev-Veselov super Jacobi polynomials and yields the CPW decomposition of any free theory diagram in any dimension.

Higher-dimensional symmetry of AdS2×S2 correlators

“…The above formulae in fact generalise very nicely in two ways (see [48] for more details). All higher order Casimirs for 4-point functions in an SU(m, m|2n) CFT for arbitrary m, n (where we are interested in m = n = 1 for the 1d CFT case and m = n = 2 for the N = 4 SYM case) are given in terms of supersymmetric functions of the differential operators D ± z defined in (3.10).…”

“…The bosonic case (n = 0) of this result follows from the results of Shimeno [50] for BC m hypergeometric functions (see theorem 2.1) following the identification of the higher order Casimirs for blocks with the defining system of higher order differential equations for these hypergeometric functions in [48]. The supersymmetric generalisation is then naturally given by the above expression.…”

“…We then evaluated the action of these operators on the log u part of the 10d function D4411 expanded into 4d superblocks. In fact, since the 4d blocks are eigenfunctions of the Casimirs and the eigenvalues are explicitly known [48] this becomes simple algebra. The resulting anomalous dimensions were then compared with the results for these anomalous dimensions found in [12].…”

“…The supersymmetric generalisation is then naturally given by the above expression. The corresponding eigenvalues for superblocks of this Casimir are also given in [48].…”

BCJ relations in ${AdS}_5 \times S^3$ and the double-trace spectrum of super gluons

“…The existence of such formulas for disconnected graphs interpolating between different theories turns out to be a particular case of a more general formula for all freetheory diagrams which can be proved through a Cauchy identity [41].…”