A conceptual understanding of strongly correlated electronic materials is the main challenge of modern Condensed Matter theory. To address the phase diagrams of high-Tc superconducting cuprates or strongly frustrated quantum magnets or the physics of (Abelian or non-Abelian) fractional quantum Hall states one needs a reliable treatment of simple paradigmatic microscopic models such as extended Hubbard or frustrated Heisenberg models. The time is ripe to expect rapid progress in the field. On one hand, new promising methods based on quantum entanglement concepts - so-called tensor network (TN) methods - have just started to be applied to Condensed Matter problems with unprecedented success. Such methods also provide for the first time the correct framework for understanding topological order (which goes beyond Landau paradigm) present in very important classes of materials such as quantum spin liquids and fractional quantum Hall states. On the other hand, developments in Atomic Physics of cold atom "quantum simulators" will provide direct confrontations to the theoretical approaches and complement Solid State Physics experiments. Our ambitious project aims to develop innovating cutting-edge TN methods on two parallel complementary fronts: first, by designing simple paradigmatic wave functions to provide a deep understanding of the major phases of correlated systems (Spin liquids, fractional quantum Hall states and topological insulators, etc...) and, secondly, by attacking directly the key low-dimensional microscopic models using novel efficient TN optimization tools. The word-class expertise in state-of-the-art numerical and analytical techniques of the two PI’s Didier Poilblanc (DP) and Nicolas Regnault (NR) together with (i) potentially strong interactions with some of the members of DP’s Strongly Correlated Electron group (LPT-Toulouse) like Matthieu Mambrini or NR’s close collaborators like Benoit Estienne at LPTHE (Paris), and (ii) thanks to existing collaborations with the most renowned experts in the field of TN will be the keys to carry on successfully this ambitious project and solve the paradigmatic microscopic models essential to the scientific community working on correlated materials.

An outstanding challenge of modern physics is to unify General Relativity and Quantum Mechanics into a consistent framework. String theory is an important step towards this goal. It offers a coherent framework that overcomes the perturbative divergences that plagued previous attempts to quantize gravity and which naturally incorporates quantum gauge theories that can describe all other non-gravitational forces. Moreover, the study of black holes, dualities, and holography has led to substantial evidence about the nonperturbative consistency of the theory. However, there are two major hurdles in making further progress. 1) Even though we have some idea about the dynamics of string theory, we do not know which phase (compactification) of the theory may correspond to the real world. 2) We do not possess a powerful enough microscope, such as a super-LHC, that can directly access the short distance UV structure of the theory. To make further progress we need new ideas. A productive research strategy is to focus on universal features that must hold in all phases of the theory and to use statistical reasoning to learn indirectly about the UV properties from the IR properties. One such requirement is that thermodynamic IR properties of a black hole must have statistical interpretation in terms of an ensemble of quantum states. It is a universal and highly stringent requirement since it must hold in all phases and for all black holes. Much of earlier work on the subject has been in the limit when the size of the black hole (or the area of the event horizon) is large which corresponds to classical gravity. The subleading finite size quantum corrections to the entropy are extremely interesting precisely as an IR window into the UV physics. Our goal will be to study these finite size quantum corrections to probe the short distance structure of the theory. There is a natural extension of these questions to black branes (an extended membrane generalization of black holes) and the associated AdS/CFT holography which gives a broader context to frame this enquiry. With this physical guiding principle, our two main organizing principles will be `Quantum Holography’ and `New Symmetries’ as we explain below. I) Quantum Holography: Holography is one of the most important physical insights to emerge from the study of black holes which implies that the number of degrees of freedom in a quantum theory of gravity scales with area and not with volume as one might naively expect. A concrete realization of holography within string theory has led to the remarkable quantum equivalence between a theory with gravity described by strings moving in Anti de Sitter (AdS) space and a theory without gravity described by conformal quantum field theory (CFT) in one less dimension. While there is substantial nontrivial evidence for this `AdS/CFT Duality’ over the past decades, much of it is at the level of semi-classical gravity. Our goal will be to develop methods to study AdS/CFT duality at the quantum level. II) New Symmetries: Recent studies have revealed unexpected new symmetries of string theory. In particular, the nonperturbative spectrum of microstates corresponding to stable black holes in string theory has revealed surprising hints of a symmetry based on Borcherds algebras--a vast generalization of Lie algebras that underlie known gauge symmetries of particle physics. For a large class of black holes, this spectrum of microstates exhibits a `hidden’ modular symmetry and a deep connection with the fascinating mathematics of mock modular forms. Our goal will be to understand the physical origin and implications of these new symmetries. We will achieve these goals by creatively combining new techniques of localization, topological string theory, worldsheet methods and new mathematics such as mock modular forms. Recent work of the three PIs has already played a leading role in developing these techniques.

Recently, conformal bootstrap techniques have made great strides in tackling strongly-coupled problems in theoretical physics. Using this approach we have already produced world-leading predictions for three-dimensional Ising critical exponents and several other experimentally as well as theoretically important results. In this proposal we aim to develop several novel bootstrap methods that will greatly enhance the predictive power of this approach. These new methods have the potential to revolutionize the field by reducing the bootstrap from a very difficult numerical problem to one of simple linear algebra. This would render the field much more accessible and allow for much faster exploration and development of new ideas. These new methods may also help shed light on tantalizing new features of the three-dimensional Ising model unearthed by existing bootstrap studies. We will also explore several interesting classes of theories that have not yet been studied using the bootstrap such as two-dimensional theories with Virasoro symmetry and long-range theories in various dimensions. Finally we will apply the ideas of the bootstrap to holographic conformal field theories (in particular at finite temperature) in order to understand what underlies holographic dualities.