nLab conformal bootstrap

Idea

The conformal bootstrap program (Belavin-Polyakov-Zamolofchikov 84) is an attempt to construct and classify conformal field theories non-perturbatively by axiomatizing the properties of their operator product expansion/correlation functions.

The conformal bootstrap was proposed in the 1970s as a strategy for calculating the properties of second-order phase transitions. After spectacular success elucidating two-dimensional systems, little progress was made on systems in higher dimensions until a recent renaissance beginning in 2008 (Poland-Simmons-Duffin 16).

The generalization of the conformal bootstrap to superconformal field theories has the potential to provide, via AdS/CFT, a precise and detailed construction of large-N and asymptotically AdS string/M-theory.

Related concepts

• Wightman axioms, conformal net

• S-matrix

References

General

• Alexander Belavin, Alexander Polyakov, Alexander Zamolodchikov, (1984). Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Physics B. 241 (2): 333–380 (1984)

• David Poland, David Simmons-Duffin, The conformal bootstrap, Nature Physics 535–539 (2016) doi:10.1038/nphys3761

• David Simmons-Duffin, TASI Lectures on the Conformal Bootstrap (arXiv:1602.07982)

• David Poland, Slava Rychkov, Alessandro Vichi, The Conformal Bootstrap: Numerical Techniques and Applications, Rev. Mod. Phys. 91, 15002 (2019) (arXiv:1805.04405)

• Shai Chester, Weizmann Lectures on the Numerical Conformal Bootstrap (arXiv:1907.05147)

• Johan Henriksson, Analytic Bootstrap for Perturbative Conformal Field Theories (arXiv:2008.12600)

• Pedro Liendo, Zhengwen Liu, Junchen Rong, The old conformal bootstrap revisited (arXiv:2108.07295)

See also:

• Wikipedia, Conformal bootstrap

• Zechuan Zheng, Bootstrap Method in Theoretical Physics [arXiv:2401.00350]

For gauge theories:

• Zhijin Li, David Poland, Searching for gauge theories with the conformal bootstrap (arXiv:2005.01721)

For Liouville theory:

• Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Conformal bootstrap in Liouville Theory (arxiv:2005.11530)

On manifolds with corners:

• António Antunes, Conformal Bootstrap near the edge (arXiv:2103.03132)

Comparison to experiment – superfluid-transition:

• Shai Chester, Walter Landry, Junyu Liu, David Poland, David Simmons-Duffin, Ning Su, Alessandro Vichi, Carving out OPE space and precise $O(2)$ model critical exponents, JHEP 06 (2020) 142 (arXiv:1912.03324)

  exposition in:

  Slava Rychkov, Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly, Journal Club for Condensed Matter Physics recommendation 2020 (pdf, doi:10.36471/JCCM_January_2020_02)

Solving the bootstrap constraints via machine learning:

• Gergely Kántor, Vasilis Niarchos, Constantinos Papageorgakis, Solving Conformal Field Theories with Artificial Intelligence (arXiv:2108.08859)

Relation to reflection positivity:

• Leandro Lanosa, Mauricio Leston, Mario Passaglia, Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT (arXiv:2112.00232)

Superconformal bootstrap

For superconformal field theory, such as D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM, D=6 N=(1,0) SCFT, D=6 N=(2,0) SCFT:

• Christopher Beem, Madalena Lemos, Pedro Liendo, Leonardo Rastelli, Balt C. van Rees, The $N=2$ superconformal bootstrap (arXiv:1412.7541)

• Christopher Beem, Madalena Lemos, Leonardo Rastelli, Balt C. van Rees, The $(2,0)$ superconformal bootstrap (arXiv:1507.05637)

• Christopher Beem, Leonardo Rastelli, Balt C. van Rees, More $N=4$ superconformal bootstrap (arXiv:1612.02363)

• Luis F. Alday, Shai M. Chester, Pure anti-de Sitter supergravity and the conformal bootstrap [arXiv:2207.05085]

In AdS/CFT

Discussion of superconformal bootstrap in view of AdS/CFT, hence as a precise and detailed construction of large-N but also small N and asymptotically AdS string theory/M-theory:

extra dimensions:

• Luis Alday, Eric Perlmutter, Growing Extra Dimensions in AdS/CFT (arXiv1906.01477)

string scattering amplitudes :

• Luis Alday, Agnese Bissi, Eric Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP06(2019) 010 (arXiv:1809.10670)

Discussion of small N effects in M-theory using the conformal bootstrap:

• Nathan B. Agmon, Shai Chester, Silviu S. Pufu: Solving M-theory with the Conformal Bootstrap, J. High Energ. Phys. 2018 159 (2018). [arXiv:1711.07343, doi:10.1007/JHEP06(2018)159]

• Shai Chester, Bootstrapping M-theory, PhD thesis, Princeton (2018) [pdf, ark:88435/dsp01kw52jb833]

Specifically for the D=6 N=(2,0) SCFT on the M5-brane via AdS7/CFT6:

• Shai Chester, Eric Perlmutter, M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture, J. High Energ. Phys. (2018) 2018 116 (2018) [arXiv:1805.00892, doi:10.1007/JHEP08(2018)116]

  [p 2:] On the other hand, given our utter lack of a complete description of M-theory, the bulk is not terribly useful for determining finite aspects of the dual CFT. However, we can turn this problem around using the modern perspective of the conformal bootstrap, which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work.

• Luis Alday, Shai Chester, Himanshu Raj, 6d $(2,0)$ and M-theory at 1-loop, Journal of High Energy Physics 2021 133 (2021) [arXiv:2005.07175]

Specifically for the D=3 SCFT (BLG-model, ABJM model) on the M2-brane via AdS4/CFT3

• Nathan B. Agmon, Shai Chester, Silviu S. Pufu: The M-theory Archipelago, J. High Energ. Phys. 2020 10 (2020) [doi:10.1007/JHEP02(2020)010, arXiv:1907.13222]

• Damon J. Binder, Shai Chester, Max Jerdee, Silviu S. Pufu: The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes, J. High Energ. Phys. 2021 83 (2021) [arXiv:2011.05728, doi:10.1007/JHEP05(2021)083]

• Shai M. Chester, Ross Dempsey, Silviu S. Pufu: Higher-derivative corrections in M-theory from precision numerical bootstrap [arXiv:2412.14094]