The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

Abstract

Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and O(N ) models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.

Crossing antisymmetric Polyakov blocks + Dispersion relation

“…4. Fermionic/Multiple correlator bootstrap: Crossing antisymmetry also appears in interesting problems like bootstrapping correlators of fermions in higher d or a set of correlators with unequal scalars [36,37,38]. Our results are hence applicable to these cases (see e.g.…”

Machine Learning Symmetry

“…In particular, one starts with a few minimal assumptions about a putative CFT, such as the spectrum of light operators. Then crossing symmetry of four-point functions, along with unitarity, strongly constrain the possible dynamics of such theories, see [37] for a review. These developments have opened up the possibility of exploring the landscape of these theories by robust numerical methods.…”

Mixed Correlator Dispersive CFT Sum Rules

“…Moreover, the presence of the double discontinuity suppresses doubletwist operators thereby allowing us to probe non-perturbative features of CFTs. Finally, they enjoy positivity properties due to the double-zeros that make them desirable for the numerical 2 Our convention differs from that of [29,30]: for equal operators, the u-channel identity is always present. Therefore, they define their sum rules to be normalized as follows:…”

“…The success of the modern conformal bootstrap stands upon the pillars of high precision numerical methods [1,2], and deeper understanding of the analytic structure of the bootstrap equations (see [3,4] and references therein). Numerical methods have led to high precision measurements of critical exponents of conformal field theories (CFTs) such as the 3D Ising [5][6][7][8] and the O(N ) models [9][10][11], while analytical methods have proven essential to the study of quantum gravity in Anti-de Sitter (AdS) spacetime [12][13][14][15][16][17] and perturbative CFTs [18,19].…”