Abstract
The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d.
Highlights
Higher correlations with more than N = 3 field insertions possess an intricate dependence on the insertion points of the local fields and their simplicity only becomes visible after expanding correlators in a basis of conformal blocks
In this paper we have initiated the construction of multipoint conformal blocks for correlation functions of N scalar fields
We constructed a number of independent commuting differential operators on N -point functions that matches the number of conformally invariant cross ratios
Summary
In the first subsection we shall show that the construction of commuting differential operators for scalar N -point blocks can be reduced by rather elementary arguments to the construction of commuting differential operators for 3-point functions of spinning fields. Constructing sufficiently many commuting vertex differential operators that act on such cross ratios of the 3-point function requires more powerful technology from integrability which we shall turn to in the second subsection. The basic construction provides nv(d) of such commuting operators and sufficiently many even for the most generic vertices For special vertices, such as those appearing in the comb channel with external scalars, there exist linear relations between these operators. These are the subject of the third subsection
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