Periodic Reporting for period 2 - CFT-MAP (Charting the space of Conformal Field Theories: a combined nuMerical and Analytical aPproach) Reporting period: 2019-11-01 to 2021-04-30 Summary of the context and overall objectives of the project 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. A prototypical example is boiling water at the critical pressure and temperature. In addition, conformal field theories sit at the centre of our modern understanding of quantum field theory, describing the asymptotic behaviour of quantum systems at infinitesimally small or large distances.The description of this theories in a quantitative way is often very difficult: due to the absence of typical scales, all the degrees of freedom of the theory interact together, giving rise to a strong dynamics which eludes any description as a perturbation of simpler and solvable system.For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using only symmetries and other consistency conditions. This approach, which takes the name of Conformal Bootstrap, was extremely successful in two dimensions and has led to exact solutions, as in the case of the planar Ising model. On the other hand, only recently we understood how to formulate the conformal bootstrap in higher dimensions and concretely apply it to extract quantitative results. This project explores new directions in this field and develops more efficient numerical techniques and complementary analytical tools. By using these techniques CFT-MAP is able to scan the space of possible conformal field theories and identify where and how they can exist. CFT- MAP will connect high energy theory and the study of phase transitions. Besides the innovative methodologies, a fundamental outcome of CFT-MAP will be a word record determination of critical exponents in second phase transition, together with additional information that allows an approximate reconstruction of a quantum field theory in the proximity of a conformal field theory. 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 first part of the project focused on studying correlation functions of operators present in any quantum field theory, such as global symmetry conserved currents and the energy momentum tensor and combining these techniques with existing ones. The main objective of this study was to assess the feasibility of the method and compare the new approach to existing ones, both in terms of precision of the results obtained and in terms of the computing resources needed to perform the study. The outcome is encouraging and suggests to push this approach further. Along the way, the team developed a various algorithm to efficiently deal with large scale bootstrap problems and used it to settle a longstanding tension between experiment results and simulation for a particular observable describing the transition from fluid to superfluid in Helium. Similarly, these techniques allowed to rigorously establish the instability of isotropic magnates under cubic perturbations, another long-lasting problem.CFT-MAP also produced the most precise determination to date for many observables, such as critical exponents, for many theory. This includes supersymmetric Ising model, the O(2) model and O(3) model.The computation of some of these quantities is extremely hard, if not unfeasible, with other methods ad disposal. CFT-MAP also started the exploration of more exotic phase transitions, characterised various symmetry breaking patterns.Finally, CFT-MAP introduced a new analytical tool, a dispersion relation, that will be combined with the numerical approach in the near feature. 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) In the remaining part of the project, CFT-MAP will build on previous results and consider more constraining setups. The inclusion of correlation function of the stress tensor will be completed and its impact assessed. The techniques developed in the first part will allow to realize a detailed chart of the space of conformal invariant systems and extract very precise determinations of many observables. Alternative ways to impose consistency condition on conformal field theories are under investigation. In particular, the combination of the dispersion relation, obtained during the first period, with the numerical techniques is expected to give rise to a new powerful methodology.