nLab conformal bootstrap



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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.



See also:

For gauge theories:

For Liouville theory:

On manifolds with corners:

Comparison to experimentsuperfluid-transition:

Solving the bootstrap constraints via machine learning:

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:


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:

string scattering amplitudes :

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

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)(2,0) CFT and the Chiral Algebra Conjecture, J. High Energ. Phys. (2018) 2018: 116 (arXiv:1805.00892)

    from 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)(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

Last revised on January 2, 2024 at 19:27:18. See the history of this page for a list of all contributions to it.