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arXiv:2009.11882 · Physical Review Letters

From Gaudin Integrable Models to d -Dimensional Multipoint Conformal Blocks

Ilija Burić, Sylvain Lacroix, Jeremy A. Mann, Lorenzo Quintavalle, Volker Schomerus

Published Jan 12, 2021• 44 citations
arXivHTMLPDF

Abstract

In this work, we initiate an integrability-based approach to multipoint conformal blocks for higher-dimensional conformal field theories. Our main observation is that conformal blocks for N-point functions may be considered as eigenfunctions of integrable Gaudin Hamiltonians. This provides us with a complete set of differential equations that can be used to evaluate multipoint blocks.

DOI
10.1103/physrevlett.126.021602

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In this work, we initiate an integrability-based approach to multipoint conformal blocks for higher-dimensional conformal field theories. Our main observation is that conformal blocks for N-point functions may be considered as eigenfunctions of integrable Gaudin Hamiltonians. This provides us with a complete set of differential equations that can be used to evaluate multipoint blocks.
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.
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.
Conformal blocks are the central ingredient of bootstrap programme to constructing d-dimensional conformal field theories. I will show that such blocks can be mapped to eigenfunctions of a 2-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two coupled P\”oschl-Teller particles. Their interaction, whose strength depends smoothly on the dimension ddd, is known to be integrable. Several consequences of this observation will be sketched before I conclude with an outlook on new perspectives for the bootstrap programme.
Abstract In this paper, we recall Richardson’s solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.
In this paper, we recall Richardson's solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.