## Periodic Report Summary 2 - 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 is 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 is to study the applicability of methods of integrability to more realistic models as well as hidden symmetries to alternative models of gravity.

So far, we have 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. We also showed that the same Yangian symmetry applies to several key observables such as Wilson loops and correlation functions of the fields. We are working to generalise the results to other relevant observables and to loop level.

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 have constructed the R-matrix and understood some of its unusual features. We are now trying to lift these structures to a quantum affine algebra of psl(2|2) in order to directly derive the worldsheet scattering matrix.

We extend the partial results from integrability with non-integrable terms via the study of scattering amplitudes and mathematical methods like multi-loop integration-by-parts (IBP) identities, syzygies and the global duality theorem in algebraic geometry. As an application, we found an algebraic way to extract amplitudes from the so-called CHY scattering equations.

Towards understanding aspects of gravity, we have studied bimetric theory in detail. We have 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 have 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.

The aim of the ERC project is 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 is to study the applicability of methods of integrability to more realistic models as well as hidden symmetries to alternative models of gravity.

So far, we have 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. We also showed that the same Yangian symmetry applies to several key observables such as Wilson loops and correlation functions of the fields. We are working to generalise the results to other relevant observables and to loop level.

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 have constructed the R-matrix and understood some of its unusual features. We are now trying to lift these structures to a quantum affine algebra of psl(2|2) in order to directly derive the worldsheet scattering matrix.

We extend the partial results from integrability with non-integrable terms via the study of scattering amplitudes and mathematical methods like multi-loop integration-by-parts (IBP) identities, syzygies and the global duality theorem in algebraic geometry. As an application, we found an algebraic way to extract amplitudes from the so-called CHY scattering equations.

Towards understanding aspects of gravity, we have studied bimetric theory in detail. We have 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 have 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.