Solving Conformal Field Theories with the Functional Bootstrap

Project Information

FUNBOOTS

Grant agreement ID: 101043588

DOI

10.3030/101043588

EC signature date 10 May 2022

Start date 1 October 2022

End date 30 September 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 950 625,00

EU contribution

€ 1 950 625,00

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 1 - FUNBOOTS (Solving Conformal Field Theories with the Functional Bootstrap)

Reporting period: 2022-10-01 to 2025-03-31

Summary of the context and overall objectives of the project

Conformal Field Theories (CFTs) have a wide range of experimental and theoretical applications: describing classical and
quantum critical phenomena, where they determine critical exponents; as low (or high) energy limits of ordinary quantum
field theories; and as theories of quantum gravity in disguise via the AdS/CFT correspondence.
Unfortunately, most interesting CFTs are strongly interacting and difficult to analyse. On the one hand, perturbative and
renormalization group methods usually involve approximations that are hard to control and which require difficult
resummations. On the other hand, numerical simulations of the underlying systems are difficult near the critical point and can
access only a limited set of observables.
The conformal bootstrap program is a new approach. It exploits basic consistency conditions which are encoded into a
formidable set of bootstrap equations, to map out and determine the space of CFTs. A longstanding conjecture states that
these equations actually provide a fully non-perturbative definition of CFTs. In this project we will develop a groundbreaking
set of tools -- analytic extremal functionals -- to extract information from the bootstrap equations. This Functional Bootstrap
has the potential to greatly deepen our understanding of CFTs as well as to determine incredibly precise bounds on the
space of theories. Our main goals are

A) to fully develop the functional bootstrap for the simpler and mostly unexplored one-
dimensional setting, relevant for critical systems such as spin models with long-range interactions and line defects in
conformal gauge theories, leading to analytic insights and effective numerical solutions of these systems;

B) to establish functionals as the default technique for higher dimensional applications by developing the formalism, obtaining general analytic bounds and integrating into existing numerical workflows to obtain highly accurate determinations of critical
exponents

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The FunBootS team has worked on a variety of developments and applications of analytic functionals, which are tools to deepen our understanding of Conformal Field Theories via crossing equations. The main studies and results include:
1. The construction of a complete set of sum rules relevant for bootstrapping arbitrary systems of correlation functions in 1D CFTs. This means that the analytic functional bootstrap can now be performed in full generality for a variety of systems, such as long range models, line defects, and QFTs in AdS2 space.
2. The construction of the first complete set of sum rules for reconstructing local bulk operators in AdS2 in terms of the boundary CFT data. These sum rules were constructed by demanding that the operators satisfy causality, i.e. they are behave as local objects.
3. The mathematical characterization and construction of vast families of sum rules for the locality problem of the previous point. This illuminates the meaning of the analytic functionals and suggests further avenues of progress for the crossing equation case. It allows also for the automatic reconstruction of local operators dual to interacting CFT spectra.
4. A systematic study of tensor product analytic functional bases in higher dimensions. This work shows that such bases vastly outperform traditional bootstrap methods in most cases, and sets the stage for widespread application.
5. Traditional bootstrap study of the long range Ising model in various spacetime dimensions, by imposing new constraints owing to the nature of the free dual scalar field in AdS space.
6. Analytic bootstrap and new supersymmetry constraints on line defects in N=4 SYM and ABJM models.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The most significant publications so far are:
1. “Polyakov blocks for the 1D CFT mixed correlator bootstrap”, K. Ghosh, A. Kaviraj, M. F. Paulos
2. “Bootstrapping bulk locality. Part I: Sum rules for AdS form factors”, N. Levine, M. F. Paulos
3. “Numerical conformal bootstrap with analytic functionals and outer approximation”, K. Ghosh, Z. Zheng.

The above publications significantly advance the state of the art. In the 1) the first complete set of sum rules with analytic functionals was proposed for 1D CFTs, and shown to lead to new rigorous bounds saturated by known theories. In 2) the first set of complete sum rules for locality of bulk AdS operators was constructed. This allows to constrain CFTs by demanding that they have (or not) dual local AdS descriptions, and reconstruct local AdS observables from the boundary in a non-perturbative and rigorous fashion. Finally in 3), a complete, positive, set of analytic functionals was used to derive bounds in CFTs in general spacetime dimension, showing that traditional approaches are outperformed.