The superconformal equation

Abstract

Crossing symmetry provides a powerful tool to access the non-perturbative dynamics of conformal and superconformal field theories. Here we develop the mathematical formalism that allows to construct the crossing equations for arbitrary four-point functions in theories with superconformal symmetry of type I, including all superconformal field the- ories in d = 4 dimensions. Our advance relies on a supergroup theoretic construction of tensor structures that generalizes an approach which was put forward in [1] for bosonic theories. When combined with our recent construction of the relevant superblocks, we are able to derive the crossing symmetry constraint in particular for four-point functions of arbitrary long multiplets in all 4-dimensional superconformal field theories.

Crossing symmetry for long multiplets in 4D $$ \mathcal{N} $$ = 1 SCFTs

“…Let us now briefly describe the outline of this paper. In section 2 we adapt the formalism developed in [33] to the computation of the crossing factor M st (α i ) that appears in eq. (1.4).…”

“…In [32,33] we launched a programme to resolve this issue of superconformal partial wave decompositions for long operators, at least for superconformal algebras of type I, for which the (internal) R-symmetry group U contains an abelian factor U(1). In particular, all superconformal algebras in d = 4 are of type I.…”

“…Once these tensor structures are found, one derives and solves Casimir differential equations for a basis of superconformal blocks. We addressed the latter step in [33] after we had dealt with the former in [32]. Both parts of the construction are heavily based on tools from group theory and harmonic analysis, following an earlier group theoretic approach to spinning conformal blocks that was developed in [34][35][36].…”

“…In these equations, the relevant tensor structures only enter through the quotient of the sand t-channel quantities. This quotient, which we dubbed the crossing factor in [33], is a matrix of conformal invariants, i.e. it only depends on cross ratios.…”

“…The example we will discuss here is relevant for the study of 4-dimensional N = 1 superconformal field theories and concerns correlations functions G 4 (x i ) = R(x 1 )φ 2 (x 2 )R(x 3 )ϕ 4 (x 4 ) , (1.1) of a chiral field ϕ 4 , an anti-chiralφ 2 and two identical long multiplets R with real scalar superprimary component R. For this example we shall derive the full set of long multiplet crossing equations. Once all the algebraic dust of [32] and [33] has settled, the equations take a fully explicit form that is ready-to-use.…”