Gaudin models and multipoint conformal blocks III: comb channel coordinates and OPE factorisation

Abstract

We continue the exploration of multipoint scalar comb channel blocks for conformal field theories in 3D and 4D. The central goal here is to construct novel comb channel cross ratios that are well adapted to perform projections onto all intermediate primary fields. More concretely, our new set of cross ratios includes three for each intermediate mixed symmetry tensor exchange. These variables are designed such that the associated power series expansion coincides with the sum over descendants. The leading term of this expansion is argued to factorise into a product of lower point blocks. We establish this remarkable factorisation property by studying the limiting behaviour of the Gaudin Hamiltonians that are used to characterise multipoint conformal blocks. For six points we can map the eigenvalue equations for the limiting Gaudin differential operators to Casimir equations of spinning four-point blocks.

Universal spinning Casimir equations and their solutions

“…With the appropriate factor Ξ, the wave functions Ψ eigenfunctions of the universal Casimir operators that were derived in section 2. We shall also review briefly how these universal Casimir operators for spinning four-point functions were used recently in [32] to study the OPE limit of multipoint functions with more that N = 4 scalar field insertions. This application illustrates nicely the power of universality.…”

“…In the next section, we will construct weight-shifting operators for spinning Calogero-Sutherland Hamiltonians, which provide a recursive construction of the blocks, very much in the spirit of [3,6]. For the remainder of this section, we will focus on an application in the context of multipoint blocks, [32], which illustrates the usefulness of universality.…”

“…Six point function with external scalars in the comb channel. The z i , zi , w i and Υ type of cross ratios are naturally associated with one particular internal leg or vertex of the OPE diagram, see [32] for details.…”

“…Conformal block expansions of multipoint correlators are only useful if one has sufficient control over the OPE limit of the individual blocks so that one can distinguish the dynamical OPE coefficients from the values the blocks take in these limits. Within the integrability based approach to multipoint blocks we have just sketched, the issue of OPE limits was addressed in [32]. In this work, a new set of multipoint cross ratios was introduced that extends the familiar variables z, z Dolan and Osborn introduced for N = 4 point correlators.…”

Multipoint correlators on the supersymmetric Wilson line defect CFT

“…In this work we compute multipoint correlators using standard perturbative techniques. One of our motivations is the multipoint conformal bootstrap in general, and in particular, recent developments in the study of multipoint correlators in CFT [27][28][29][30][31][32][33][34]. Although multipoint correlators are often technically challenging, 1d CFTs offer the advantage that the number of cross-ratios is reduced and the kinematics simplifies.…”