# Conformal bootstrap

The conformal bootstrap is a non-perturbative mathematical method to constrain and solve conformal field theories, i.e. models of particle physics that exhibit similar properties at different levels of resolution.[1]

In physics, the conformal bootstrap is used to study the space of all possible quantum field theories.

## Overview

Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an algebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and conformal blocks.

The main ideas of the conformal bootstrap were formulated in the 1970s by the Soviet physicist Alexander Polyakov[2] and the Italian physicists Sergio Ferrara, Raoul Gatto and Aurelio Grillo.[3] Other early pioneers of this idea were Gerhard Mack and Ivan Todorov.

In two dimensions, the conformal bootstrap was demonstrated to work in 1983 by Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov.[4] Many two-dimensional conformal field theories were solved using this method, notably the minimal models and the Liouville field theory.

In higher dimensions, the conformal bootstrap started to develop following the 2008 paper by Riccardo Rattazzi, Slava Rychkov, Erik Tonni and Alessandro Vichi.[5] The method was since used to obtain many general results about conformal and superconformal field theories in three, four, five and six dimensions. Applied to the conformal field theory describing the critical point of the three-dimensional Ising model, it produced the world's most precise predictions for its critical exponents.[6][7][8]

## Current research

The international Simons Collaboration on the Nonperturbative Bootstrap unites researchers devoted to developing and applying the conformal bootstrap and other related techniques in quantum field theory.[9]

## History of the name

The modern usage of the term "conformal bootstrap" was introduced in 1984 by Belavin et al.[4] In the earlier literature, the name was sometimes used to denote a different approach to conformal field theories, nowadays referred to as the skeleton expansion or the "old bootstrap". This older method is perturbative in nature,[10][11] and is not directly related to the conformal bootstrap in the modern sense of the term.

## References

1. ^ "Using the 'Bootstrap,' Physicists Uncover Geometry of Theory Space | Quanta Magazine". Quanta Magazine. Retrieved 2018-01-03.
2. ^ Polyakov, A. M. (1974). "Nonhamiltonian approach to conformal quantum field theory". Zh. Eksp. Teor. Fiz. 66: 23–42.
3. ^ Ferrara, S.; Grillo, A. F.; Gatto, R. (1973). "Tensor representations of conformal algebra and conformally covariant operator product expansion". Annals of Physics. 76: 161–188. Bibcode:1973AnPhy..76..161F. doi:10.1016/0003-4916(73)90446-6.
4. ^ a b Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B. (1984). "Infinite conformal symmetry in two-dimensional quantum field theory". Nuclear Physics B. 241 (2): 333–380. Bibcode:1984NuPhB.241..333B. doi:10.1016/0550-3213(84)90052-X. ISSN 0550-3213.
5. ^ Rattazzi, Riccardo; Rychkov, Vyacheslav S.; Tonni, Erik; Vichi, Alessandro (2008). "Bounding scalar operator dimensions in 4D CFT". JHEP. 12: 031. arXiv:0807.0004. Bibcode:2008JHEP...12..031R. doi:10.1088/1126-6708/2008/12/031.
6. ^ El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents". Journal of Statistical Physics. 157 (4–5): 869–914. arXiv:1403.4545. Bibcode:2014JSP...tmp..139E. doi:10.1007/s10955-014-1042-7.
7. ^ Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap". Journal of High Energy Physics. 2015 (6): 1–31. arXiv:1502.02033. Bibcode:2015JHEP...06..174S. doi:10.1007/JHEP06(2015)174. ISSN 1029-8479.
8. ^ Kadanoff, Leo P. (April 30, 2014). "Deep Understanding Achieved on the 3d Ising Model". Journal Club for Condensed Matter Physics.
9. ^ https://www.simonsfoundation.org/mathematics-and-physical-science/news-announcements/foundation-announces-simons-collaboration-on-the-non-perturbative-bootstrap/
10. ^ Migdal, Alexander A. (1971). "Conformal invariance and bootstrap". Phys. Lett. B37: 386–388. Bibcode:1971PhLB...37..386M. doi:10.1016/0370-2693(71)90211-5.
11. ^ Parisi, G. (1972). "On self-consistency conditions in conformal covariant field theory". Lettere al Nuovo Cimento. 4S2: 777–780. doi:10.1007/BF02757039.