Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

Abstract

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation

“…On the other hand, there are a few cases for which one obtains well-known differential operators. For M = 3 with two spinning fields φ 1 , φ 3 , the unique vertex differential operator was shown in [34] to coincide with the lemniscatic elliptic Calogero-Moser-Sutherland Hamiltonian discovered by Etingof, Felder, Ma and Veselov in [35]. The most wellknown system appears for M = 4 when all the fields φ i are scalar.…”

“…The derivation follows the same steps we outlined in the discussion of four-point blocks in the first subsection. But in contrast to the case of N = 4, the leading term γ of the power series expansion in z r , zr is no longer constant but rather an eigenfunction of a single variable vertex differential operator for an STT-STT-scalar three-point function which we constructed and analysed in [34]. The latter was shown to arise in the OPE limit of the five-point vertex operator which acts on all the five cross ratios before taking the limit.…”

“…The polynomial cross ratios are then introduced in the second subsection. Next, in the third subsection, we discuss the OPE coordinates for N = 5 where there is a single qualitatively new invariant that was introduced in [34] already. The fourth subsection contains the construction of a new invariant that is attached to the central link of the six-point comb channel diagram.…”

“…when the four points x 1 x 2 x 3 x 4 or x 2 x 3 x 3 x 5 lie on a line. In section 4.2 of [34] we also used these coordinates to analyse the OPE limit of our Gaudin differential operators. This analysis showed clearly how the variable w 1 is naturally associated with the degree of freedom that described the choice of tensor structures at the internal vertex of a five-point OPE diagram.…”

Selected Topics in Analytic Conformal Bootstrap: A Guided Journey

“…This has been shown to be valid in any number of spacetime dimensions. This program has been further carried out for spinning external operators [351][352][353], for blocks in defect CFTs [63,354], for superconformal blocks [355,356] and for conformal blocks of multi-point correlators [357][358][359][360].…”