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arXiv:1806.09703 · Journal of High Energy Physics

Calogero-Sutherland approach to defect blocks

Mikhail Isachenkov, Pedro Liendo, Yannick Linke, Volker Schomerus

Published Oct 1, 2018• 41 citations
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Abstract

Abstract Extended objects such as line or surface operators, interfaces or boundaries play an important role in conformal field theory. Here we propose a systematic approach to the relevant conformal blocks which are argued to coincide with the wave functions of an integrable multi-particle Calogero-Sutherland problem. This generalizes a recent observation in [1] and makes extensive mathematical results from the modern theory of multi-variable hypergeometric functions available for studies of conformal defects. Applications range from several new relations with scalar four-point blocks to a Euclidean inversion formula for defect correlators.

DOI
10.1007/jhep10(2018)204

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Abstract Extended objects such as line or surface operators, interfaces or boundaries play an important role in conformal field theory. Here we propose a systematic approach to the relevant conformal blocks which are argued to coincide with the wave functions of an integrable multi-particle Calogero-Sutherland problem. This generalizes a recent observation in [1] and makes extensive mathematical results from the modern theory of multi-variable hypergeometric functions available for studies of conformal defects. Applications range from several new relations with scalar four-point blocks to a Euclidean inversion formula for defect correlators.
Extended objects such as line or surface operators, interfaces or boundaries play an important role in conformal field theory. Here we propose a systematic approach tothe relevant conformal blocks which are argued to coincide with the wave functions of an integrable multi-particle Calogero-Sutherland problem. This generalizes a recentobservation for four-point blocks and makes extensive mathematical results from the modern theory of multi-variable hypergeometric functions available for studies ofconformal defects. Applications range from several new relations with scalar four-point blocks to a Lorentzian inversion formula for defect correlators.
Abstract In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a. defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects from the knowledge of the defect expansion. The defect correlator admits a number of conformal cross-ratios depending on their dimensionality. We find the differential equation obeyed by the conformal block and solve them in certain special cases.
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 We present a Lorentzian inversion formula valid for any defect CFT that extracts the bulk channel CFT data as an analytic function of the spin variable. This result complements the already obtained inversion formula for the corresponding defect channel, and makes it now possible to implement the analytic bootstrap program for defect CFT, by going back and forth between bulk and defect expansions. A crucial role in our derivation is played by the Calogero-Sutherland description of defect blocks which we review. As first applications we obtain the large-spin limit of bulk CFT data necessary to reproduce the defect identity, and also calculate one-point functions of the twist defect of the 3d Ising model to first order in the ϵ-expansion.
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.
We study various questions related to operators with spin in quantum conformal field theory in dimensions higher than two. In particular, we classify conformally-invariant tensor structures which appear in correlation functions of local operators and develop tools for computation of conformal blocks which contribute to these functions. We study the crossing equations for four-point functions using numerical and analytical techniques. Finally, we explore the question of analytic continuation of local operators in spin, which leads us to a simple proof of a generalized Lorentzian inversion formula.