Conformal group theory of tensor structures

Abstract

The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are functions of cross ratios only, and the correlation functions that depend on insertion points in the d-dimensional Euclidean space. Here we develop an entirely group theoretic approach to tensor structures, based on the Cartan decomposition of the conformal group. It provides us with a new universal formula for tensor structures and thereby a systematic derivation of crossing equations. Our approach applies to a ‘gauge’ in which the conformal blocks are wave functions of Calogero-Sutherland models rather than solutions of the more standard Casimir equations. Through this ab initio construction of tensor structures we complete the Calogero-Sutherland approach to conformal correlators, at least for four-point functions of local operators in non-supersymmetric models. An extension to defects and superconformal symmetry is possible.

Six-point conformal blocks in the snowflake channel

“…Several techniques have been developed over the years for the computation of four-point conformal blocks, which are the simplest blocks. For example, various methods use Casimir equations [1][2][3], the shadow formalism [4][5][6], the weight-shifting formalism [7,8], integrability [9][10][11][12][13], AdS/CFT [14][15][16][17][18][19], and the OPE [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Another important reason why four-point conformal blocks have been studied extensively is the conformal bootstrap [36,37], a way of constraining the CFT data solely from consistency of correlation functions under associativity.…”

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

“…Finding explicit formulas for the various factors requires some work even for bosonic theories [36]. But depending on the precise setup, some shortcuts are possible, see below.…”

“…Here A, B, C, D are m × m blocks of g b and how to extract Cartan coordinates from them was explained in [36]. The elements that we want to decompose are…”

“…The middle factor g (0) is an element of the underlying Lie group G (0) = G bos × U . We use Cartan coordinates for G bos as in [36] and some arbitrary coordinates for the internal symmetry group U . Now a short computation of right-invariant vector fields leads to the following expression for the Laplacian, see e.g.…”

“…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]. With the choice of tensor structures and coordinates this approach JHEP04(2021)130 provides, the Casimir differential equations for the blocks assume a universal form of a matrix Schrödinger problem of Calogero-Sutherland type.…”