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.
Highlights
The N -point function of a fundamental field in a conformal field theory contains a tremendous amount of dynamical information about the theory, more than any finite set of 4-point functions could ever capture
It is worth to pause our analysis of the single variable vertex operators for a moment and to explain how this differential operator is related to the vertex operator for a 5-point function in d ≥ 3 that we worked out in [30]
We have constructed the fourth order differential equation that characterizes all vertices of the three types listed in eq (2.6), each of which appear in OPE diagrams of scalar N -point functions in d-dimensional conformal field theory
Summary
The N -point function of a fundamental field in a conformal field theory contains a tremendous amount of dynamical information about the theory, more than any finite set of 4-point functions could ever capture. In that work it was pointed out that the Casimir operators which Dolan and Osborn used to characterize and construct scalar 4-point blocks could be reinterpreted as eigenvalue equations for the Hamiltonians of an integrable 2-particle hyperbolic Calogero-MoserSutherland (CMS) model associated with the root system BC2. The reduction from both OPE limits and shadow integrals are explained in more detail in subsection 4.2 They highlight the key role that is played by the lemniscatic CMS model in developing a theory of multi-point blocks, at least in the comb channel of 3- and 4-dimensional conformal field theories. For the example of the STT-STT-scalar vertex, we explain the precise relation between the (reduced) single variable vertex operators and the vertex differential operators in multi-point functions constructed in [30], both through shadow integrals and OPE limits. The paper concludes with a discussion and overview of subsequent steps, along with a list of open problems
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