Final Report Summary - GAUGEGRAVSYM (Extended Symmetries in Gauge and Gravity Theories) N=4 supersymmetric Yang-Mills (N=4 SYM) theory is a four-dimensional quantum field theory with a long list of remarkable properties. A very important one is that its classical conformal symmetry is exact even at the quantum level. Moreover, it is apparently dual to a string theory on the AdS₅×S⁵ background geometry by means of the celebrated AdS/CFT correspondence. Much of the recent progress involving this and similar models is related to exploiting its conjectured exact integrability in the planar limit. Integrability is a hidden feature of some physics models which enhances their symmetries substantially and makes them particularly tractable. For integrable systems there exist a host of methods to conveniently compute many relevant observables.The aim of the ERC project was to put the recent rapid progress in integrability and scattering amplitudes within the AdS/CFT correspondence on a solid foundation, and to construct new tools to get access to the observables of these models. Another goal was to study the applicability of methods of integrability to more realistic models as well as hidden symmetries to alternative models of gravity.We established a novel notion of integrability for planar superconformal gauge theories as invariance of the action under an infinite-dimensional symmetry algebra known as a Yangian quantum algebra. We proved that classical planar N=4 SYM indeed possesses this Yangian symmetry and showed that it applies to several key observables such as Wilson loops and correlation functions of the fields. It remains to generalise the results to other relevant observables and to understand potential quantum anomalies. A central object in quantum algebras related to integrable models is the universal R-matrix. For the extended psl(2|2) superalgebra, which is closely related to the worldsheet scattering matrix, we constructed the R-matrix and understood some of its unusual features. It remains to lift these structures to a quantum affine algebra of psl(2|2) to make direct contact with the scattering matrix.We explored and constructed integrable deformations of AdS/CFT superstring worldsheet theories, in particular q-deformations, Yang-Baxter deformations and twists. We studied under what conditions the deformations describe supergravity backgrounds and therefore can serve as proper superstring theories. We also showed how the various deformations are related by Poisson-Lie duality.We enriched the ongoing study of scattering amplitudes in gauge theories by mathematical technique like multi-loop integration-by-parts identities, syzygies and the global duality theorem in algebraic geometry. As applications, we found an algebraic way to extract amplitudes from the scattering equations. We also developed a systematic procedure for computing maximal unitarity cuts of multiloop Feynman integrals based on the Baikov representation. Our work on gravitational theories featured several studies of asymptotic symmetries and elaborated on the puzzling notion of energy and related conserved charges. In particular we related BMS super-translations, Newman-Penrose charges, Aretakis charges, Penrose quasi-local energy in certain classes of spacetimes such as Kerr-Schild metrics. Towards understanding aspects of modified gravity theories, we studied bimetric theory in detail. We made progress in establishing a hidden symmetry related a partially massless mode in a particular version of this model. This model resembles conformal gravity, albeit with no unphysical ghost modes, and the additional symmetry may potentially improve the behaviour at the quantum level. We also demonstrated that the model can realistically be applied to cosmology, in particular its additional spin-2 mode could well serve as a dark matter particle.
