Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - Emergent-BH (Emergent spacetime and maximally spinning black holes)

Teaser

\"Black holes are fascinating objects. The extreme violence of a star’s collapse, the strength of the tidal forces, the formation of a surface - the horizon - of no return, are certainly powerful images. But even from a theoretical point of view, black holes are very...

Summary

\"Black holes are fascinating objects. The extreme violence of a star’s collapse, the strength of the tidal forces, the formation of a surface - the horizon - of no return, are certainly powerful images. But even from a theoretical point of view, black holes are very striking. To make thermodynamics compatible with the fact that matter can fall into a black hole, the latter needs to be assigned an immense entropy, proportional to the area of its horizon. Moreover, Stephen Hawking showed that, quantum-
mechanically, black holes evaporate, emitting thermal radiation. This appears to lead to violations of quantum mechanics, the main pillar of our understanding of microscopic physics. Thus, black holes reveal a powerful tension between general relativity and quantum mechanics, known as the black hole information paradox.

The fact that the black hole entropy scales as an area, rather than a volume has lead t’Hooft to propose that quantum gravity is “holographic”, i.e. gravitational physics in a given region of space-time can be entirely encoded in a non-gravitational theory - a \"\"field theory\"\" - residing at the boundary of that region- just like in a hologram, where a three-dimensional image can be fully reconstructed from data on a two-dimensional surface. As in the hologram, the dictionary between gravity and the boundary field theory is very complicated, and one can think of gravity and the extra dimension it occupies as “emergent”. Thus, in holography, space is not a fundamental concept. This also has profound implications for our own universe, where it is expected that the time direction itself is emergent. Holography can resolve the tension between gravity and quantum mechanics through subtle non-local effects.

There has been significant progrees in understanding the emergence mechanism in certain special contexts such as gravity in hyperbolic spacetimes, which occur frequently in the context of string theory. However, very little is known about it for the
spacetime backgrounds relevant to the real world, due mainly to our lack of knowledge of the underlying field theories, for which a completely new framework needs to be developed.

The goal of this project is to uncover the fundamental nature of spacetime and gravity in our universe by: i) formulating and working out the properties of the relevant lower-dimensional field theories and ii) studying the mechanism by which spacetime and gravity emerge from them. The main idea is to address these problems not in the most general cosmological setting, which is hard, but by concentrating on the near-horizon regions of maximally spinning black holes, for which the dual field theories greatly simplify and can be studied using a combination of conformal field theory and string theory methods. This program has seen significant progress in the past year and a half with the discovery of certain solvable toy models that exhibit exactly the needed features. To study the emergence mechanism, I plan to adapt the tools that were successfully used to understand emergent gravity in hyperbolic spacetimes - such as holographic quantum entanglement and conformal bootstrap - to this new framework.\"

Work performed

The work on this project has extended in several directions.

i) The emergence of spacetime is best understood in the context of the AdS/CFT correspondence, where local operators in the conformal field theory (CFT) are promoted to propagating fields in a one higher-dimensional spacetime known as anti de-Sitter
(AdS). This map is well-understood around the vacuum, but there is no first-principles derivation of the map for non-trivial states in the CFT (such as thermal states, which correspond to black holes in AdS), in the sense that one needs information from the bulk to write down a CFT operator that corresponds to a propagating bulk field. In [2*], I tried to establish a more direct relation between the bulk field and the boundary operators, by showing that the geodesic integral of an appropriately
defined bulk field operator, which includes all its gravitational dressing, equals the Virasoro block contribution of its dual operator to the boundary OPE.

ii) The main goal of this ERC project is to develop the holographic correspondence for spacetimes other than AdS. This is a difficult question because in most cases it is not known what kind of field theory is dual to gravity in such spacetimes. A possibly tractable problem is to understand holography for the kind of spacetimes appearing in the near-horizon region of maximally spinning black holes. In this case, the boundary QFT is a non-local deformation of a CFT, termed “dipole CFT”. Before 2017, the only known concrete example of a dipole CFT was a particular null non-local deformation of N=4 super YangMills theory. This theory is integrable at plan. ar level (has an infinite number of conserved quantities), and one can use the powerful tools of integrability to understand the holographic correspondence. In [1], we mapped the computation of anomalous dimensions in the SL(2) sector of dipole-deformed N=4 SYM to a spin chain with twisted boundary conditions, and showed that for long operators, the field theory and supergravity results match at one loop. This setup opens the possibility to compute many other interesting quantities in this theory.

iii) While null dipole-deformed N=4 SYM is an extremely precious concrete example to study, its relevance to understanding maximally spinning black holes is somewhat lessened by the fact that it is a four-dimensional model, whereas black holes are described by field theories in two dimensions. In [3], I constructed what could be the simplest concrete example of a two-dimensional dipole CFT. In a similar spirit to the more widely discussed TTÌ„bar deformation, it is defined by deforming a usual two-dimensional CFT by an composite irrelevant operator JTÌ„bar constructed from the CFT stress tensor and a U (1) current. Even though the deformation is irrelevant, the spectrum and thermodynamics can be worked out exactly for finite deformation parameter.

The next step was to derive [2] the holographic interpretation of a generic JTbar -deformed CFT with large central charge. We showed that the holographic boundary data are determined by applying a canonical transformation to the holographic data
(source and expectation value of the stress tensor and of the U (1) current) of the original CFT. We also showed that the energy spectrum of black holes computed with this dictionary precisely matches the non-trivial energy spectrum in the field-
theory, computed with very different methods. This is a first proof of principle that the type of canonical transformations used to deal with multitrace deformations in AdS/CFT can be used to do precision holography, even when the operators involved are irrelevant. This work also clarifies the meaning of works by other groups on the holographic significance of certain alternative boundary conditions for AdS3. We also found some very interesting infinite-dimensional enhancement of the symmetries of this model, which are reminiscent of theinfinite-dimensional symmetries found in extremal black ho

Final results

I think the most exciting progress has been to find a concrete toy model (JTbar deformed CFTs) of a two-dimensional dipole CFT, where many quantities can be computed exactly and expectations about the behaviour of the theory can be tested. By the end of this project, I expect to have completely understood all aspects of JTbar deformed CFTs, both from the point of view of field theory and from that of the holographic duality. An important point is that since we have full control over both sides of the duality, we can check in detail what happens to the holographic dictionary. This may be a first instance of precision holography involving a non-asymptotically AdS spacetime, and I expect it to teach us many lessons about how our intuition should be modified in such cases.

Another line of research I expect to make concrete progress on is that of generalization of JTbar deformed CFTs to more complicated theories. Yet another direction that I am starting to explore now with my group is that of the study of emergence in this model, mostly using the conformal bootstrap in an adapted basis. I am also planning to pursue the study of null dipole-deformed N=4 SYM, where a complementary perspective on the behaviour of dipole theories can be obtained.