## Final Report Summary - RICCIFLATHOLOGRAPHY (Holography for Ricci-flat and asymptotically flat spacetimes)

General arguments indicate that theories describing the gravitational force can be related to theories without gravity in one dimension less. These arguments rest on an old, insightful observation by Bekenstein: the maximum entropy of a region of space does not scale with the volume of the region, but only with the area of its boundary, the upper bound being reached when the system collapses to a black hole. After this, 't Hooft and Susskind argued that it must be possible to describe all phenomena taking place within that region by non-gravitating degrees of freedom living on some bounding surface, carrying naturally the correct number of degrees of freedom.

This fascinating idea, the Holographic Principle, gives us a new, deep insight into the nature of gravitational forces, and is believed to be key in the quest for a quantum theory of gravity, with the potential to change the paradigm for physical reality.

The first concrete, and to date best understood, embodiment of the Holographic Principle is the AdS/CFT correspondence, that links asymptotically anti-de Sitter (AdS) spacetimes (i.e. with a negative cosmological constant) to some non-gravitating field theory enjoying conformal invariance - a conformal field theory, or CFT - and living on the conformal boundary, that has one dimension less, of the spacetime.

The peculiar structure of these spacetimes grants better control over the mathematics description, and allows to develop very effective tools: this brought a good understanding of AdS black holes, a precise fluid/gravity correspondence, and we now have a full, non-pertubative formulation of quantum gravity for AdS_5 in the form of a four dimensional conformal super-Yang-Mills theory. Gravitational techniques are now commonly applied to tackle field theoretic questions, such as the quark-gluon plasma, strongly interacting condensed matter systems, and quantum information problems. Following the inverse path, the dual SYM is currently our best window to investigate fundamental questions such as the emergence of spacetime, causality, and shed some light on the role of singularities.

However, the general arguments for holography do not depend on the spacetime structure at infinity, and there should be a holographic description of asymptotically flat spacetimes, which are commonly used to describe particle physics phenomena and astrophysical objects. With this project, we made a small step towards such a holographic formulation of gravity for asymptotically flat spacetimes.

The starting point is the AdS/Ricci-flat correspondence: a mathematical map between a class of asymptotically AdS spacetimes and a class of Ricci-flat (RF) spacetimes. The latter are solutions to the Einstein field equations in vacuum (with a vanishing cosmological constant), such as Minkowski spacetime, in which the ordinary quantum field theories are defined, and Schwarzschild and Kerr metrics, that describe space-time outside stars and black holes. The idea is to use this map to bridge the formulation of holography in AdS to Minkowski spacetime.

An obstruction to a direct application of this idea comes from the detailed workings of the map. It requires that the AdS spacetime contains a torus of arbitrary dimension, and similarly that the RF spacetime contains a sphere of arbitrary dimension. These internal manifolds are swapped by the map, and must be kept completely frozen.

The researcher has studied general linear perturbations around the simplest incarnation of the AdS/RF correspondence, that links AdS on a torus to Minkowski spacetime, to understand how to bypass these limitations. While perturbations that preserve the symmetries of the torus in AdS, and of the related sphere in Minkowski, are covered by the AdS/RF correspondence, the ones that deform these compact manifolds go beyond its original scope and are vital for many applications.

They have been analyzed by means of a full Kaluza-Klein reduction of AdS on a torus and of Minkowski on a sphere, obtaining thus their spectrum. As expected, only the zero modes are be mapped one to one between the two sides of the correspondence. The modes obtained by unfreezing the torus and the sphere do not match, despite their similarities.

To overcome this obstacle, both compactification manifolds have been split in two parts. One part, of fixed dimension, is allowed to fluctuate, while the other part, of arbitrary dimension, is kept rigid. The AdS/RF map can be performed over the rigid parts of the compactifications manifolds, leading to a partial unfreezing of the torus and the sphere. This generalizes the AdS/RF map to such modes. In particular, it allows to describe any axisymmetric perturbation of Minkowski spacetime, and relate them to AdS perturbations.

To extend it further, the researcher additionally took a limit for which the dimension of the rigid part of the compactification manifold tends to zero. This allows to completely unfreeze all modes and clarifies an important aspect of the correspondence: holography on asymptotically flat spacetimes appears to be defined on half-spaces, the boundary of which correspond to a brane housing the dual theory.

As a byproduct of this analysis, we also obtained a class of theories with matter on which the AdS/RF map applies.

The researcher has also developed holographic techniques to tackle planar AdS black holes with a dilaton and axion fields that break translational invariance in the boundary directions. These black holes can undergo a phase transition by spontaneously growing scalar hair and are potentially interesting for condensed matter applications. One of such black hole families under investigation, carrying an extra magnetic charge, was discovered by the researcher and his collaborators. He has shown that there are conserved charges associated to the axionic fields and that they enter the first law of thermodynamics, and studied the phases of these black holes. Currently, he is extending this study to Robinson-Trautman extensions of these solutions.

This project consists in purely theoretical work and is thus not expected to have any socio-economical impact and wider societal implications. The researcher has nevertheless contributed in the training of a PhD student by co-supervising her.

The results of this research marked a first step towards the formulation of holography for asymptotically flat spacetimes, and will lead to a better understanding of gravitational forces and quantum field theory in general. As such, the work is expected to prepare the ground for further developments in this area by the scientific community.

This fascinating idea, the Holographic Principle, gives us a new, deep insight into the nature of gravitational forces, and is believed to be key in the quest for a quantum theory of gravity, with the potential to change the paradigm for physical reality.

The first concrete, and to date best understood, embodiment of the Holographic Principle is the AdS/CFT correspondence, that links asymptotically anti-de Sitter (AdS) spacetimes (i.e. with a negative cosmological constant) to some non-gravitating field theory enjoying conformal invariance - a conformal field theory, or CFT - and living on the conformal boundary, that has one dimension less, of the spacetime.

The peculiar structure of these spacetimes grants better control over the mathematics description, and allows to develop very effective tools: this brought a good understanding of AdS black holes, a precise fluid/gravity correspondence, and we now have a full, non-pertubative formulation of quantum gravity for AdS_5 in the form of a four dimensional conformal super-Yang-Mills theory. Gravitational techniques are now commonly applied to tackle field theoretic questions, such as the quark-gluon plasma, strongly interacting condensed matter systems, and quantum information problems. Following the inverse path, the dual SYM is currently our best window to investigate fundamental questions such as the emergence of spacetime, causality, and shed some light on the role of singularities.

However, the general arguments for holography do not depend on the spacetime structure at infinity, and there should be a holographic description of asymptotically flat spacetimes, which are commonly used to describe particle physics phenomena and astrophysical objects. With this project, we made a small step towards such a holographic formulation of gravity for asymptotically flat spacetimes.

The starting point is the AdS/Ricci-flat correspondence: a mathematical map between a class of asymptotically AdS spacetimes and a class of Ricci-flat (RF) spacetimes. The latter are solutions to the Einstein field equations in vacuum (with a vanishing cosmological constant), such as Minkowski spacetime, in which the ordinary quantum field theories are defined, and Schwarzschild and Kerr metrics, that describe space-time outside stars and black holes. The idea is to use this map to bridge the formulation of holography in AdS to Minkowski spacetime.

An obstruction to a direct application of this idea comes from the detailed workings of the map. It requires that the AdS spacetime contains a torus of arbitrary dimension, and similarly that the RF spacetime contains a sphere of arbitrary dimension. These internal manifolds are swapped by the map, and must be kept completely frozen.

The researcher has studied general linear perturbations around the simplest incarnation of the AdS/RF correspondence, that links AdS on a torus to Minkowski spacetime, to understand how to bypass these limitations. While perturbations that preserve the symmetries of the torus in AdS, and of the related sphere in Minkowski, are covered by the AdS/RF correspondence, the ones that deform these compact manifolds go beyond its original scope and are vital for many applications.

They have been analyzed by means of a full Kaluza-Klein reduction of AdS on a torus and of Minkowski on a sphere, obtaining thus their spectrum. As expected, only the zero modes are be mapped one to one between the two sides of the correspondence. The modes obtained by unfreezing the torus and the sphere do not match, despite their similarities.

To overcome this obstacle, both compactification manifolds have been split in two parts. One part, of fixed dimension, is allowed to fluctuate, while the other part, of arbitrary dimension, is kept rigid. The AdS/RF map can be performed over the rigid parts of the compactifications manifolds, leading to a partial unfreezing of the torus and the sphere. This generalizes the AdS/RF map to such modes. In particular, it allows to describe any axisymmetric perturbation of Minkowski spacetime, and relate them to AdS perturbations.

To extend it further, the researcher additionally took a limit for which the dimension of the rigid part of the compactification manifold tends to zero. This allows to completely unfreeze all modes and clarifies an important aspect of the correspondence: holography on asymptotically flat spacetimes appears to be defined on half-spaces, the boundary of which correspond to a brane housing the dual theory.

As a byproduct of this analysis, we also obtained a class of theories with matter on which the AdS/RF map applies.

The researcher has also developed holographic techniques to tackle planar AdS black holes with a dilaton and axion fields that break translational invariance in the boundary directions. These black holes can undergo a phase transition by spontaneously growing scalar hair and are potentially interesting for condensed matter applications. One of such black hole families under investigation, carrying an extra magnetic charge, was discovered by the researcher and his collaborators. He has shown that there are conserved charges associated to the axionic fields and that they enter the first law of thermodynamics, and studied the phases of these black holes. Currently, he is extending this study to Robinson-Trautman extensions of these solutions.

This project consists in purely theoretical work and is thus not expected to have any socio-economical impact and wider societal implications. The researcher has nevertheless contributed in the training of a PhD student by co-supervising her.

The results of this research marked a first step towards the formulation of holography for asymptotically flat spacetimes, and will lead to a better understanding of gravitational forces and quantum field theory in general. As such, the work is expected to prepare the ground for further developments in this area by the scientific community.