Theconformal bootstrap is anon-perturbative mathematical method to constrain and solveconformal field theories, i.e. models ofparticle physics orstatistical physics that exhibit similar properties at different levels of resolution.[1]
Unlike more traditional techniques ofquantum field theory, conformal bootstrap does not use theLagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as thescaling dimensions of the local operators and theiroperator 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 analgebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants andconformal blocks.
The main ideas of the conformal bootstrap were formulated in the 1970s by the Soviet physicistsAlexander Polyakov[2][3] and Alexander Migdal[4][5] and the Italian physicistsSergio Ferrara,Raoul Gatto [it] andAurelio Grillo.[6] Other early pioneers of this idea wereGerhard Mack [de] andIvan Todorov [bg].
In two dimensions, the conformal bootstrap was demonstrated to work in 1983 byAlexander Belavin,Alexander Polyakov andAlexander Zamolodchikov.[7] Manytwo-dimensional conformal field theories were solved using this method, notably theminimal models and theLiouville field theory.
In higher dimensions, the conformal bootstrap started to develop following the 2008 paper byRiccardo Rattazzi,Slava Rychkov,Erik Tonni andAlessandro Vichi.[8] The method was since used to obtain many general results about conformal andsuperconformal field theories in three, four, five and six dimensions. Applied to the conformal field theory describing thecritical point of the three-dimensionalIsing model, it produced the most precise predictions for itscritical exponents.[9][10][11]
The internationalSimons Collaboration on the Nonperturbative Bootstrap unites researchers devoted to developing and applying the conformal bootstrap and other related techniques in quantum field theory.[12]
The modern usage of the term "conformal bootstrap" was introduced in 1984 by Belavin et al.[7] In the earlier literature, the name was sometimes used to denote a different approach to conformal field theories, nowadays referred to as theskeleton expansion or the "old bootstrap". This older method is perturbative in nature,[13][14] and is not directly related to the conformal bootstrap in the modern sense of the term.
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