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Conformal Bootstrap Methods and their applications

Final Report Summary - BOOTSTRAP (Conformal Bootstrap Methods and their applications)

This project is concerned with the conformal bootstrap, a set of methods, mostly numerical, that are used to study conformal field theories. Such theories describe critical phenomena. The original plan for the scientific work is split into two parts.

In the first part, “Applications and Theoretical development of the conformal bootstrap techniques”, we proposed to apply the methods of the conformal bootstrap to several cases of interest. One application was the study of supersymmetric conformal field theories. These are interesting both for theoretical as well as practical reasons. Theoretically supersymmetry gives us greater control over the theory which then allows for a precise comparison with numerical results. Practically, it is known that such theories have experimental applications as they can describe certain edge modes on the surface of topological insulators (Grover, Sheng, Vishawanath, Science 344), which makes them especially appealing to study. This work resulted in two publications in international peer-reviewed journals, described in section 3 of this note. In another application of the bootstrap methodology, the researcher studied the fractal Ising model. This is the critical theory underlying the Ising model on a fractal lattice. He has shown that if such a theory exists at all, then it must almost certainly be non-unitary. This surprising result was obtained by deriving bounds on conformal dimensions of operators for CFTs in several dimensions between one and two. This study resulted in a publication on the Journal of High Energy Physics.

In other work, the author has developed the theory of logarithmic conformal field theories. Although such models had been studied to a large extent in two dimensions, their systematic description in higher dimensions is new. This serves as important preliminary work for future numerical bootstrap applications. Similar remarks apply to work done studying the long range Ising model, a model with a conformally invariant fixed point which should also be able to be tackled in future numerical studies.

A surprising new application was to the study of the S-matrix. In what is perhaps one of the most exciting outcome of this project, the author has shown in collaboration with others that it is possible to obtain constraints on massive quantum field theories using methodology inspired by that of the conformal field theories. This resulted in two publications and more are forthcoming.

The second part of this project was concerned with the development and publication of the numerical methods of the conformal bootstrap. This was achieved with the publication of the JuliBootS package ( the first publicly available package for bootstrap computations. The package was developed in the Julia programming language which allows for great flexibility combined with the speed and efficiency of Python and C. The package was accompanied by a bootstrap review which appeared in ([arXiv:1412.4127]). In what concerns the development of the numerical methods, the author has introduced the technique of extremal flows, which can improve numerical efficiency by several orders of magnitude. This technique introduces the notion of extremality of a spectrum to continuously deform approximate, numerical solutions to crossing symmetry. In many cases this is a vast improvement in current techniques.