## Final Report Summary - GAUGE/GRAVITY (The Gauge/Gravity Duality and Geometry in String Theory)

The most important conceptual breakthrough that emerged from string theory is Maldacena's conjectured duality between quantum field theory and gravity, now known as gauge/gravity duality. This implies that quantum theory and gravity, instead of being conflicting, are in fact equivalent descriptions of the same Physics. The broad aim of this project was to extend the gauge/gravity duality beyond the current state of the art. More specifically, the objectives outlined comprised the following. The systematic study of supersymmetric backgrounds of string theory comprising an Anti de Sitter (AdS) space-time has been pursued as a method for exploring supersymmetric conformal field theories (SCFTs) in various dimensions. This includes the description of the geometry characterising the most general M-theory backgrounds, holographically dual to three dimensional SCFTs. Several novel deformations of AdS geometries have also been obtained, along with their holographic interpretation in terms of deformations of SCFTs. Moreover, a novel approach was introduced, that allows one to combine F-theory (which is a mathematical framework useful for making contact with particle physics) with holography for a class of three-dimensional AdS geometries. Another theme of the project was that of studying supersymmetric gauge theories on curved manifolds employing techniques of localization. Novel exact results in supersymmetric field theories were obtained employing the localization method, including an original extension of this method to the context of non-compact manifolds. The parallel development of exact field theory results and of the dual gravitational solutions of increasing sophistication was one of the distinctive aspects of the project. Pursuing this idea, a number of precision tests of the gauge/gravity duality has been obtained. One of the surprising outcomes of the study of gauge theories on curved manifold has been the introduction of a novel observable, dubbed supersymmetric Casimir energy, that is a special type of "vacuum energy" characterising universally supersymmetric field theories in four dimensions. This quantity has turned out to be relevant for a variety of problems. For example, it prompted a critical reexamination of the technique of holographic renormalization in the presence of supersymmetry, which subsequently led to the discovery of a supersymmetry anomaly. After benefitting from a one-year extension, the project culminated with two remarkable discoveries. One publication presented an extremal problem that has been proposed as the geometric dual of an extremization principle governing two-dimensional SCFTs (c-extremization); this result is a substantial extension of the influential work on Sasakian geometry carried out by the PI in collaboration with J. Sparks and S.T.-Yau more than a decade ago. This is a new formidable tool for studying a wide class of gauge/gravity duals, as it allows to determine exact properties of gravitational solutions bypassing the daunting task of solving the complicated generalised Einstein's equations. Moreover, it has an independent mathematical interest, as it opens up the path to investigations of a novel class of manifolds, much richer than the Sasakian class. Another article addressed and solved a long-standing open problem in the gauge/gravity duality, namely the explanation of the Bekenstein-Hawking entropy of certain supersymmetric black holes, in terms of a dual field theory computation. The relevant computation was successfully identified, building on the conspicuous list of examples of holographic dualities in which one has analytic control on the field theory side, thanks to the localization method. In particular, it turned out that the supersymmetric Casimir energy, introduced in previous work, plays a key role in providing a microscopic description of the the black hole physics. While these two pioneering works are the climax of the ERC project, they also constitute the seed for exciting future research. Indeed, the PI is vigorously pursuing these research directions, and he is starting to understand how these findings are likely to be instances of a broad set extremization principles governing SCFTs, supersymmetric black holes, as well as a class of problems in Riemannian geometry.