## Periodic Reporting for period 2 - HoloHair (Information Encoding in Quantum Gravity and the Black Hole Information Paradox)

Okres sprawozdawczy: 2022-04-01 do 2023-09-30

Holography is the idea that a quantum theory of gravity in some spacetime has an equivalent, or “dual”, description in terms of a lower-dimensional (non-gravitational) quantum theory defined at the boundary of the spacetime. In this sense information about bulk physical processes is encoded “holographically” in the boundary theory. The holographic principle has been spectacularly successful in spacetimes with a negative cosmological constant. Anti-de Sitter spacetime (AdS) is a space of negative curvature effectively describing systems in isolation. It has a time-like boundary with one less spatial dimension where standard rules of quantum field theory apply but with a larger symmetry group than the usual Poincaré group: besides translations and Lorentz transformations there are scale transformations and “angle-preserving” special conformal transformations. The boundary turns out to be described by a conformal field theory (CFT) whose symmetries match precisely the isometries of AdS! This AdS/CFT correspondence has revolutionized string theory and high energy quantum field theory over the last 25 years.

How generally does the idea of holography apply? What about spacetimes with positive or vanishing cosmological constant such as de Sitter and Minkowski spacetimes? A significant amount of my recent research, and some of my future plans, are devoted to developing a holographic principle for asymptotically flat (Minkowski) spacetimes which are a good approximation for many processes in our universe. While a daunting task, known laws of (low-energy) physic offer us some clues of how and where to begin.

The S-matrix, describing the transition probability between an initial and a final state in a scattering process, is the basic observable in quantum gravity in asymptotically flat spacetimes. Usually we compute it in basis of asymptotic energy eigenstates described by a plane wave basis which makes translation symmetry manifest. What does the S-matrix look like when instead making Lorentz symmetry (rotations and boosts) manifest? Recasting the S-matrix in a basis of boost eigenstates reveals that it shares similarities with correlation functions in a conformal field theory (CFT). In particular, universal features that arise in energetically soft and collinear limits of the scattering particles take a natural form in CFT language, and spacetime (asymptotic) symmetries can be understood as generated by special CFT operators. This together with novel insights into the infrared structure of gravity and gauge theories from soft theorems, asymptotic symmetries and memory effects have culminated in a promising proposal for a holographic principle in asymptotically flat spacetimes: Celestial holography conjectures that quantum gravity in spacetimes with flat asymptotics is dual to a conformal field theory on the celestial sphere at null infinity. This proposal differs significantly from our most studied example of holography relating string theory on a d+1 dimensional AdS space to a CFT on its d dimensional time-like boundary. In asymptotically flat spacetimes, the boundary is light-like, or null, and the proposed celestial CFT is co-dimension two. Moreover, while AdS/CFT naturally arises from a top-down construction in string theory the approach towards establishing celestial holography is currently bottom-up (though it would be extremely interesting to find a string theory embedding).

The first key step in this endeavour is the identification of the symmetries that govern both sides of the proposed celestial holographic duality. Universal properties of scattering amplitudes at low energies turn out to have a symmetry origin: Weinberg’s soft graviton theorem is related to BMS supertranslation symmetries which are infinitely enhanced translations in space and time that act at the null boundary of asymptotically flat spacetimes. When recast in a basis of boost (rather than energy) eigenstates soft theorems take the form of correlation function of conserved operators on the co-dimension two celestial sphere. The classification of all such conserved operators as well as the construction of their associated conserved charges has been achieved by my ERC group using the powerful toolkit of CFT. Building on these results we can ask: what are (the implications of) all the symmetries and how do they constrain bulk scattering, what axioms or principles should we impose on the dual celestial CFT and can we bootstrap it, how in the dual theory do we describe non-trivial bulk geometries such as black holes and their dynamics...? Insight into such questions could lead to a non-perturbative formulation of quantum gravity in asymptotically flat spacetimes.

Indeed, a litmus test of any proposed holographic duality is to account for non-perturbative processes such as the formation and evaporation of black holes and achieving a statistical interpretation of the Bekenstein-Hawking entropy of black holes - an incredibly challenging task. To resolve the microphysics we also need a better understanding of the information encoding, storage and flow between the horizon and the far asymptotic region where Hawking radiation escapes to. This includes determining the role and implications of “soft hair” at the horizon of black holes which is a consequence of conservation laws associated to BMS symmetries and a potential relation to fuzzballs which have been proposed as the fundamental microscopic description of black holes. Key insights into the statistical nature of black hole entropy are obtained via the correspondence principle for black holes and strings. As one adiabatically decreases the string coupling, a black hole makes a transition to a state of weakly coupled strings with the same mass. At the correspondence point the Bekenstein-Hawking entropy of the black hole is comparable to the string entropy: this gives a statistical interpretation of black hole entropy in terms of the degeneracy of string states. This correspondence has been developed in the 1990s for static black holes and was recently refined. An imminent goal of my ERC group is to establish this correspondence for general rotating black holes.

The study of information encoding in quantum gravity and in black holes is at the heart of this ERC project, carried out by the research group of Dr. Andrea Puhm at the Center for Theoretical Physics of Ecole Polytechnique, Palaiseau, France. Our focus is on developing a holographic principle for asymptotically flat spacetimes, examining the symmetries and memory effects in celestial CFTs and their constraints on bulk physics, describing bulk geometries holographically, determining the implications of soft hair on black hole horizons, and formulating a microscopic description of black holes via string theory.

Soft theorems of quantum field theory reveal universal properties of gauge theories and gravity in the infrared. Universal phenomena often have a symmetry origin. We set out to systematically determine the symmetries corresponding to the low-energy physics of gauge theory and gravity using the framework of “celestial holography” which suggests a holographic duality between quantum gravity in asymptotically flat spacetimes and a non-gravitational theory defined on the celestial sphere at its boundary. The universality from soft theorems can then be understood from the symmetries of a “celestial conformal field theory” (CCFT) on the codimension-two sphere at the null boundary of the spacetime. Our key result is the classification of all CCFT operators associated to soft theorems and the identification of the symmetries they generate. We extended this to supersymmetric theories and to general spacetime dimensions. Our work reveals a tension between the infinite enhancement in two-dimensional CCFTs related to BMS symmetries, named after Bondi, van der Burg, Metzner and Sachs, and the finite symmetry group in higher-dimensional CCFTs.

* Celestial amplitude relations

Scattering amplitudes in gauge theory and gravity obey hidden relations known as double copy. We showed that celestial amplitudes obey “celestial double copy” relations that generalize the famous BCJ relations, named after Bern, Carrasco and Johansson, of momentum-space amplitudes to operator-valued statements. This gives further support to the expectation that these hidden relations are fundamental properties of scattering amplitudes and paves the way towards a general curved space double copy. The structure of celestial amplitudes beyond tree level is largely unknown. We remedied this for the case of N=4 super Yang-Mills theory where we computed the all-loop celestial four gluon amplitude. We, moreover, found that the famous momentum-space factorization into the tree-level amplitude times an infrared factor persists also for celestial amplitudes albeit in the form of an exponential differential operator acting on the tree-level amplitude.

* Celestial holography and black holes

One of the key ingredients of the AdS/CFT correspondence is the relation between the gravitational partition function in the bulk and the generating functional of correlation functions for the theory on the boundary. Motivated to look for the same principle for the S-matrix in asymptotically flat spacetimes we showed that the boundary on-shell action for general backgrounds becomes the generating functional for tree-level correlation functions in celestial CFT. An important goal in celestial holography which we have initiated is to describe non-trivial asymptotically flat backgrounds in CCFT and their effects on celestial scattering amplitudes. We proposed a generalization of conformal primaries and showed that this includes a large class of physically interesting metrics such as ultra-boosted black holes and shockwaves. These exact conformal primary solutions lend themselves to suitable backgrounds for celestial amplitudes. Celestial two-point correlators on backgrounds are already non-trivial (they encode information about the background and couplings) and have the desirable feature that the they exist at generic operator positions (unlike their flat space counterparts). We then investigated celestial wave scattering on a variety of backgrounds in gauge theory and gravity including Coulomb backgrounds, Schwarzschild black holes, their ultraboosted limits given by electromagnetic and gravitational shockwaves, as well as their spinning counterparts. We also derived (conformal) Faddeev-Kulish dressings for particle-like backgrounds which remove all infrared divergent terms in the two-point functions to all orders in perturbation theory.

* Black holes microphysics and soft hair

Acting with asymptotic symmetry transformation on the horizon of black holes implants soft hair/memory. Besides the BMS supertranslations we investigated the role of a new set of “dual” supertranslations, and its effect on the symmetry algebra on black hole horizons. We have furthermore investigated effects that generate or pinch off horizons : the quantum backreaction of scalar fields in three-dimensional de Sitter space, where classical black holes are known not to exist, generates a black hole horizon, while the Gregory-Laflamme instability of black strings, which causes the horizon to pinch off in finite time and form naked singularities, is shown to persist in Anti-de Sitter space where "black tsunamis” arise and the formation of the bulk naked singularity can be studied in the boundary CFT. The framework of AdS/CFT braneworld holography gives a new perspective on dynamical evaporation of a black hole as the classical evolution in time of a black hole in an Anti-de Sitter braneworld where the evolution of the Page curve of the radiation shows entanglement islands appearing and then shrinking thus supporting a unitarity evolution.

Beyond the classification of soft operators in celestial CFTs in general dimensions achieved by my ERC group we plan to address a variety of questions to establish the properties of celestial CFTs and test the proposed holographic duality with quantum gravity in asymptotically flat spacetimes. Currently, we lack a good understanding of the implications of all Goldstone and memory operators in the infinite tower of conserved soft operators that we identified, as well as the corrections of the soft symmetry algebra from quantum loops. Moreover, we are in urgent need of a list of axioms or principles that we should impose on celestial CFTs. This would allow us to apply the “bootstrap” philosophy that the observables of a CFT can be fixed by requiring some general axioms. An important goal for any holographic proposal is to establish a correspondence between bulk geometries and boundary states and their dynamics. We already initiated such a program by identifying the boundary generators for special bulk geometries and examining celestial correlation functions on a variety of gauge and gravity backgrounds. We plan to continue our investigations of celestial holography on backgrounds including black holes.

* Correspondence principle between black holes and strings

Recent works on the correspondence principle between static black holes and fundamental strings revealed new insights into the microscopic nature of black holes. Motivated by these results we set out to extend the correspondence principle to rotating black holes in general spacetime dimensions. Surprisingly, this has not been achieved so far even though the correspondence principle for static black holes including charges had been put forward in the 1990s! The correspondence states that as one adiabatically decreases the string or gravitational coupling (Newton’s constant), a black hole makes a transition to a state of weakly coupled strings with the same mass, and offers a statistical interpretation of black hole entropy in terms of the degeneracy of string states. Rotating black holes in higher dimensions have a much more complicated structure than their non-rotating counterparts - they can have different shapes, distinct horizon topologies, and there are dynamical instabilities - the correspondence principle for rotating black holes will thus be much richer.