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The Analytic Conformal Bootstrap

Periodic Reporting for period 2 - ACB (The Analytic Conformal Bootstrap)

Reporting period: 2020-06-01 to 2021-11-30

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.
We have performed research along the proposed directions, making substantial progress in all of them. The project has so far produced 20 publications (plus some in the arXiv, to be submitted) with much more work in progress.

The team is composed by myself and five PDRAs ( Bae, Connor, Koloğlu, Hansen and Bhardwaj). In addition, four Ph.D. students (van Loon, Henriksson, Ferrero and Jorge Diaz) funded by the department took part in the tasks of the project. Furthermore, we have also collaborated with external researchers. Some of the main results achieved so far are the following:

.- WIth Henriksson and van Loon we have developed an alternative to Feynman diagrams for the critical O(N) model [1], making progress regarding objective 1.

.- These techniques were extended to the one-dimensional case by Ferrero and collaborators in [9,17]. The one-dimensional case is interesting because it describes specific observables (such as Wilson lines with operator insertions) in higher dimensional conformal field theories.

.- With Zhou we have developed a constructive method to compute all tree-level four point correlators in all maximally super-symmetric conformal field theories, in three, four and six dimensions [7,8]. These correlators were furthermore studied by Zhou, Connor and Ferrero [19]. This was an open problem for over two decades, and sets the stage for objective 2 for a variety of theories. These correlators have the interpretation of graviton scattering in AdS. Furthermore, in collaboration with Zhou, Connor and Ferrero we extended these techniques to non-maximally supersymmetric CFTs [5]. These correlators have the interpretation of gluon scattering in AdS.

.- With Bae, and Bae and Jorge Diaz we made progress in the study of 2d CFTs, and putative duals of pure gravity on three dimensional anti-de-Sitter space time [6,12]. This is a very interesting example, where one can study the question of whether pure gravity can be a consistent quantum theory in three dimensions. Our approach is purely analytic, along the lines of objective 3, although the CFT is 'lower-dimensional'.

.- Bhardwaj has been working in the classification of CFTs, objective 4, and in particular the rich structure the flavour symmetry of 5d CFTs [13,14], and the presence of generalised symmetries for 4d N=2 super conformal field theories [15].

.- With Kologlu we started the analytic study of correlators at finite temperature, by bootstrap techniques [10]. This and most of the other developments are small steps towards objective 5.
.- In our work with Henriksson and van Loon 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. We expect connections and results along these lines.

.- The one-dimensional results by Ferrero and collaborators offered analytic solutions to numerical results by integrability methods. In the future I expect results combining these two powerful techniques. Ideally, one would be able to compute these observables to "all-orders". This would give another example of exactly computable observables in the context of AdS/CFT.

.- Our work with Zhou regarding correlators in maximally supersymmetric CFTs solves a two decades old problem, and sets the stage for much further progress in those directions. Immediate expected results are one-loop results for loops in AdS, specially in 3d and 6d. Some preliminary results, for a very specific correlation, have already been obtained by myself in collaboration with Chester and Raj, see [18]. This extend loop results for the 4d case in [4,11]. Results in 3d were also made public at the time of this report, but the publication just appeared in the arXiv. Furthermore, our work with Zhou, Connor and Ferrero, pioneers the study of gluon amplitudes in AdS. The study/computation of loops in this context is one of the expected results. Furthermore, we expect to elucidate hidden structures by studying such correlators/amplitudes in AdS (both in the case of maximally and non-maximally super-symmetric theories). The structures we are searching for would mimic much of the beautiful structure for amplitude in flat space.

.- In our work with Bae and Jorge-Diaz we studied the torus partition function of the relevant conformal field theories. This goes beyond the state of the art, in the sense that the CFTs we studied are non-rational, and we developed new methods to study them. We are at the moment studying more general observables, such as correlators, and higher genus partition functions, and expect results along these lines.

.- The classification by Bhardwaj for 5d CFTs solves a long standing problem for a particular class of theories. In the future we expect to apply conformal bootstrap techniques to these theories, compute observables in them and understand better their mathematical structure.

.- The finite temperature correlators constructed with Kologlu are the first non-trivial analytic example of such correlators. They are solutions in the so called AdS phase. In the future we expect results in the more interesting black-hole phase.
Exchange diagram in AdS