Superconformal blocks: general theory

Abstract

In this work we launch a systematic theory of superconformal blocks for four-point functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number N of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solutions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact. We illustrate the general theory at the example of d = 1 dimensional theories with N = 2 supersymmetry for which we recover known superblocks. The paper concludes with an outlook to 4-dimensional blocks with N = 1 supersymmetry.

Superconformal surfaces in four dimensions

“…Defect operators can be organized in representations of this preserved subalgebra. Representations of su(1, 1|1), and a convenient superspace formalism, have been known for a long time [58][59][60] (see also [61][62][63] for the computation of the superblocks). However, here we are interested in the coupling between bulk and defect degrees of freedom and, in order to fully exploit the symmetries of the problem, we find it more convenient to work in components.…”

A multipoint conformal block chain in d dimensions

“…For instance, explicit expressions are known for four-point scalar conformal blocks in general spacetime dimensions [8][9][10][11][12][13]. A variety of techniques have been developed for computing four-point conformal blocks involving external and internal exchanged operators in arbitrary representations of the Lorentz group in closed-form, integral or efficient series expansions; a partial list includes various recursive methods [11,[13][14][15][16][17][18][19][20][21], shadow formalism [22], use of differential operators [23][24][25][26][27][28][29][30][31][32][33][34][35], Wilson line constructions [36][37][38], integrability methods [39][40][41][42] and holographic geodesic diagram techniques [43][44][45][46][47][48][49][50][51]…”

The superconformal equation

“…The goal of our work is to extend all this to the case of superconformal symmetry. In [52] we have constructed the Casimir equations for superconformal symmetries of type I. The form of these equations allows us to compute superblocks systematically as finite sums of spinning bosonic blocks.…”