The Analytic Conformal Bootstrap

Projektinformationen

ACB

ID Finanzhilfevereinbarung: 787185

DOI

10.3030/787185

Projekt abgeschlossen

EK-Unterschriftsdatum 16 April 2018

Startdatum 1 Dezember 2018

Enddatum 31 Mai 2024

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 2 171 483,00

EU-Beitrag

€ 2 171 483,00

Koordiniert durch

THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
United Kingdom

Periodic Reporting for period 4 - ACB (The Analytic Conformal Bootstrap)

Berichtszeitraum: 2023-06-01 bis 2024-05-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

The aim of the present proposal is to establish a research team developing and exploiting innovative techniques to study conformal field theories (CFT) analytically. These techniques are motivated by an analytic version of the conformal bootstrap program: compute observables by analysing the full implications of symmetries and consistency conditions. Our approach does not rely on a Lagrangian description but on very basic properties. As such it applies to any CFT, offering a unified framework to study generic CFTs analytically. The initial implementation of this program has already led to striking new results and insights for both, Lagrangian and non-Lagrangian CFTs.

The overarching aims of my team will be: To develop an analytic bootstrap program for CFTs in general dimensions; To complement these techniques with more traditional methods and develop a systematic machinery to obtain analytic results for generic CFTs; and to use these results to gain new insights into the mathematical structure of the space of quantum field theories.
The proposal will bring together researchers from different areas. The objectives in brief are:

1) Develop an alternative to Feynman diagram computations for Lagrangian CFTs.
2) Develop a machinery to compute loops for QFT on AdS, with and without gravity.
3) Develop an analytic approach to non-perturbative N=4 SYM and other CFTs.
4) Determine the space of all CFTs.
5) Gain new insights into the mathematical structure of the space of quantum field theories.

Success in this research will lead to a completely new unified way to view and solve CFTs, with a huge impact in several branches of physics and mathematics.

The proposal lies in the interface of several branches of theoretical physics and mathematics.It is important for society in the sense that it advances our understanding of some of the most fundamental questions of theoretical physics.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

We have performed research along the proposed directions, making substantial progress in all of them. The project has produced 65 publications. The team was composed by myself and six PDRAs ( Bae, Connor, Koloğlu, Hansen, Silva and Bhardwaj). In addition, four Ph.D. students (van Loon, Henriksson, Ferrero, Jorge Diaz and Virally) funded mostly by the department took part in the tasks of the project. Furthermore, we have also collaborated with external researchers. The main results can be summarised as follows.

R1. We developed an alternative to Feynman diagrams for the critical O(N) model, and extended these methods to the one-dimensional case, which describes specific observables (such as Wilson lines with operator insertions) in higher dimensional conformal field theories.

R2. We developed very powerful techniques for the study and computation of holographic correlators. These culminated with the computation of all tree-level four point correlators in all maximally super-symmetric conformal field theories, in three, four and six dimensions.

R3. The techniques of R2 were extended to compute higher point correlators, in particuar five and six, in full fledge theories.

R4. The results obtained in R2 served as the basis for the computation of loops for QFT on AdS, with and without gravity, which was carried out for several theories, in several computations.

R5. Developed a machinery to compute the AdS Virasoro-Shapiro amplitude, as a curvature expansion around flat space. More precisely, the four-point graviton scattering for string theory on AdS_5 x S^5. This machinery combines tools from the conformal bootstrap, number theory and integrability, with intuition from world-sheet string theory.

R6. Made substantial progress in the understanding of generalised symmetries (and in particular non-invertible symmetries) and their role in quantum field theory.

These results were very well received by the community, and were disseminated in all major conferences in the field, including bootstrap, amplitudes, integrability and strings conferences.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

All the results mentioned above went well beyond the state of the art. For instance

.- In our work with Henriksson and van Loon in R1 we have computed, for the first time ever, several observables to higher order in 1/N for the critical O(N) models. Already to order 1/N^2 there are observables that are interesting in connection with several conjecture in the literature. R1 also includes one-dimensional results by Ferrero and collaborators, which offer analytic solutions to numerical results by integrability methods. This opened the way to combining these two powerful techniques, allowing to compute certain observables to "all-orders". This gives another example of exactly computable observables in the context of AdS/CFT.

.- Our work with Zhou in R2 solved a two decades old problem, and set the stage for much further progress in many directions. For instance, it allowed loop computations in several theories, mentioned in R4. In particular our work with Chester and Raj includes quantum loop results for full fledge non-lagrangian theories, such as the 6d (2,0) theory. The first result of its kind. Furthermore, R2 also includes an extension to less super-symmetric theories, setting up the framework to study gluon scattering amplitudes on AdS spaces holographically. This was immediately followed by a large volume of work.

.- The results R3 are among the first holographic computations of higher point functions, equivalent to scattering amplitudes on AdS. Scattering amplitudes in flat space display beautiful features, and R3 allows an exploration in the case of curved backgrounds.

.- R5 provides an algorithm for the computation of the four-point graviton amplitude for type IIB string theory in AdS5 × S5, as a curvature expansion around flat space. At the moment, there is no other technique to perform this computation. This development is paving the way for a new understanding of string theory on curved backgrounds, and attempts are being made at reproducing this result by string field theory techniques.

.- The classification by Bhardwaj for 5d CFTs, in R6, solves a long standing problem for a particular class of theories. More generally, progress in R6 has changed our view on symmetries in quantum field theory. This is by now a very active field, and some of the publications in R6 are standard references on the subject.

Exchange diagram in AdS