Light-Cone Bootstrap of Higher Point Functions and Wilson Loop Duality

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.

Section: Discussionmentioning

confidence: 80%

Section: Jhep10(2021)139mentioning

confidence: 99%

See 1 more Smart Citation

Gaudin models and multipoint conformal blocks: general theory

Burić¹,

Lacroix

Mann³

et al. 2021

“…Holographic representations of higher-point conformal blocks were constructed in the works [26][27][28], and dimensional reduction formulae for higher-point scalar exchange blocks were recently derived in [29]. In addition, the null polygon limit of multi-point correlators was later explored in [30]. Further, a connection to Lauricella systems was made in [31], while a connection to Gaudin integrable models was proposed in [32].…”

Section: Jhep10(2021)160mentioning

confidence: 99%

Recursion relations for 5-point conformal blocks

Poland

Prilepina

2021

We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation value of the energy operator in bilocal states created by two scalars.

“…In fact, it seems likely that multi-point blocks in the light-cone limit can be accessed with little additional effort. Such blocks have featured recently in [62]. Outside the light-cone limit, the powerful tools of integrable systems, and in particular the method of separation of variables [63,64], may become relevant.…”

Section: Jhep11(2021)182mentioning

confidence: 99%

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

Burić¹,

Lacroix

Mann³

et al. 2021

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.