Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2009.07674

All Global One- and Two-Dimensional Higher-Point Conformal Blocks

Jean-François Fortin, Wen-Jie Ma, Witold Skiba

Published Sep 16, 2020• 3 citations
arXivHTMLPDF

Abstract

We introduce a full set of rules to directly express all $M$-point conformal blocks in one- and two-dimensional conformal field theories, irrespective of the topology. The $M$-point conformal blocks are power series expansion in some carefully-chosen conformal cross-ratios. We then prove the rules for any topology constructively with the help of the known position space operator product expansion. To this end, we first compute the action of the position space operator product expansion on the most general function of position space coordinates relevant to conformal field theory. These results provide the complete knowledge of all $M$-point conformal blocks with arbitrary external and internal quasi-primary operators (including arbitrary spins in two dimensions) in any topology.

DOI
10.48550/arxiv.2009.07674

Related papers

Higher-point conformal blocks in the comb channel

Jean-François Fortin, Wen-Jie Ma, Witold Skiba

Abstract We compute M -point conformal blocks with scalar external and exchange operators in the so-called comb configuration for any M in any dimension d. Our computation involves repeated use of the operator product expansion to increase the number of external fields. We check our results in several limits and compare with the expressions available in the literature when M = 5 for any d, and also when M is arbitrary while d = 1.

arXiv:1911.11046•Journal of High Energy Physics•Jul 1, 2020•54 citations

Virasoro conformal blocks in closed form

Eric Perlmutter

Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.

•May 27, 2022•65 citations

Gaudin models and multipoint conformal blocks: general theory

Ilija Burić, Sylvain Lacroix, Jeremy A. Mann +2 more

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.

arXiv:2105.00021•Journal of High Energy Physics•Oct 18, 2021•46 citations

Virasoro conformal blocks in closed form

Eric Perlmutter

Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide two new closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. The other is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.

arXiv:1502.07742•Journal of High Energy Physics•Aug 1, 2015•104 citations

Gaudin Models and Multipoint Conformal Blocks: General Theory

Ilija Burić, Sylvain Lacroix, Jeremy A. Mann +2 more

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.

•Jan 1, 2021•6 citations

Polyakov blocks for the 1D conformal field theory mixed-correlator bootstrap

Kausik Ghosh, Apratim Kaviraj, Miguel F. Paulos

We introduce manifestly crossing-symmetric expansions for arbitrary systems of 1D CFT correlators. These expansions are given in terms of certain Polyakov blocks which we define and show how to compute efficiently. Equality of operator product expansion and Polyakov block expansions leads to sets of sum rules that any mixed correlator system must satisfy. The sum rules are diagonalized by correlators in tensor product theories of generalized free fields. We show that it is possible to do a change of a basis that diagonalizes instead mixed correlator systems involving elementary and composite operators in a single field theory. As an application, we find the first nontrivial examples of optimal bounds, saturated by the mixed correlator system ϕ,ϕ2 in the theory of a single generalized free field. Published by the American Physical Society 2024

arXiv:2307.01257•Physical review. D/Physical review. D.•Mar 29, 2024•10 citations
We introduce a full set of rules to directly express all MMM-point conformal blocks in one- and two-dimensional conformal field theories, irrespective of the topology. The MMM-point conformal blocks are power series expansion in some carefully-chosen conformal cross-ratios. We then prove the rules for any topology constructively with the help of the known position space operator product expansion. To this end, we first compute the action of the position space operator product expansion on the most general function of position space coordinates relevant to conformal field theory. These results provide the complete knowledge of all MMM-point conformal blocks with arbitrary external and internal quasi-primary operators (including arbitrary spins in two dimensions) in any topology.
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.
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.
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide two new closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. The other is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.
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 introduce manifestly crossing-symmetric expansions for arbitrary systems of 1D CFT correlators. These expansions are given in terms of certain Polyakov blocks which we define and show how to compute efficiently. Equality of operator product expansion and Polyakov block expansions leads to sets of sum rules that any mixed correlator system must satisfy. The sum rules are diagonalized by correlators in tensor product theories of generalized free fields. We show that it is possible to do a change of a basis that diagonalizes instead mixed correlator systems involving elementary and composite operators in a single field theory. As an application, we find the first nontrivial examples of optimal bounds, saturated by the mixed correlator system ϕ,ϕ2 in the theory of a single generalized free field. Published by the American Physical Society 2024