Periodic Reporting for period 5 - QBH (Quantum Black Holes: A macroscopic window into the microstructure of gravity)
Período documentado: 2022-09-01 hasta 2023-02-28
Black holes are astrophysical objects that are formed by the collapse of very massive stars. They create extremely strong gravitational fields and their complete description requires the theory of general relativity as well as the principles of quantum mechanics. These two fields, each well-established by experiments, form the basis of modern physics. However, combining them into one consistent theory of quantum gravity has proved to be very difficult and has remained an outstanding challenge for the last fifty years. The project aims to use black holes as a theoretical laboratory to address the following important questions:
(a) How does quantum gravity quantitatively differ from classical general relativity?
(b) How do we construct analytically calculable models of microscopic quantum gravity?
The reason black holes can give us insights into quantum gravity stems from the fact that they have thermodynamic entropy, as was shown by Bekenstein and Hawking. This suggests that black holes are made up of many microscopic states, just like an ordinary gas, and understanding the properties of these states would teach us about the microscopic theory of quantum gravity that governs the behavior of these states. One of the objectives of the project is to extract detailed information about the deviations from classical general relativity in the full quantum theory of gravity. In particular, the project focusses on supersymmetric black holes wherein the objective is to calculate an all-order formula to sum up all the quantum corrections to black hole entropy for a large class of black holes.
In order to test such a formula, the framework of string theory is used where one can, in principle, independently count the number of microscopic states in the Hilbert space of the black hole. In practice this turns out to be a very subtle problem because of the so-called wall-crossing phenomenon. The project aims to establishes the framework of mock modular forms to overcome this problem. Mock modular forms are functions that were discovered by S. Ramanujan about a hundred years ago in the completely different context of number theory. A second main objective of the project is to explore the consequences of mock modularity on the microscopic theory of gravity.
At its conclusion, the project has shown how the discrete nature of quantum black holes emerges from the collective interactions of the underlying quantum-statistical system. In doing so, it has established the first prototype of a quantum black hole in which we obtain the integer dimension of the underlying Hilbert space through the gravitational path integral. In addition, the project has also found explicit models of quantum gravity wherein one can explain the emergence of collective phases and their transitions.
(a) How does quantum gravity quantitatively differ from classical general relativity?
(b) How do we construct analytically calculable models of microscopic quantum gravity?
The reason black holes can give us insights into quantum gravity stems from the fact that they have thermodynamic entropy, as was shown by Bekenstein and Hawking. This suggests that black holes are made up of many microscopic states, just like an ordinary gas, and understanding the properties of these states would teach us about the microscopic theory of quantum gravity that governs the behavior of these states. One of the objectives of the project is to extract detailed information about the deviations from classical general relativity in the full quantum theory of gravity. In particular, the project focusses on supersymmetric black holes wherein the objective is to calculate an all-order formula to sum up all the quantum corrections to black hole entropy for a large class of black holes.
In order to test such a formula, the framework of string theory is used where one can, in principle, independently count the number of microscopic states in the Hilbert space of the black hole. In practice this turns out to be a very subtle problem because of the so-called wall-crossing phenomenon. The project aims to establishes the framework of mock modular forms to overcome this problem. Mock modular forms are functions that were discovered by S. Ramanujan about a hundred years ago in the completely different context of number theory. A second main objective of the project is to explore the consequences of mock modularity on the microscopic theory of gravity.
At its conclusion, the project has shown how the discrete nature of quantum black holes emerges from the collective interactions of the underlying quantum-statistical system. In doing so, it has established the first prototype of a quantum black hole in which we obtain the integer dimension of the underlying Hilbert space through the gravitational path integral. In addition, the project has also found explicit models of quantum gravity wherein one can explain the emergence of collective phases and their transitions.
The main results achieved so far are the following:
(1) Explanation of the discrete nature of a quantum black hole.
The discrete nature of any quantum-statistical system is encoded in its Hilbert space. A foundational open question is to explain the emergence of this Hilbert space starting from the gravitational field variables. The results of the project explain how to calculate these effects in the simplest signature of discreteness of a BH – the dimension of its Hilbert space. This should clearly be a positive integer, but it has been a long-standing problem as to how such an integer arises from continuum gravity. The results of the project show, for the first time, how to calculate this integer by summing an infinite series of quantum-mechanical effects in the black hole.
(2) Development of new techniques to calculate functional integrals in gravity.
The functional integral is a fundamental object in the description of quantum theories. Calculating this integral for quantum gravity has been very difficult so far. The results of the project rigorously establish a new technique, called supersymmetric localization, to calculate the functional integral for supersymmetric observables in a class of gravitational theories. This technique has proved to be a powerful method to calculate functional integrals in supersymmetric quantum field theories, but so far it was not known how to apply this to gravitational theories where the metric also fluctuates. A new formalism to do this has been developed in the project which uses a mix of two classical methods in field theory (the background field method and the BRST formalism). This formalism was then used to calculate the quantum entropy of a variety of black holes in four and five dimensions.
(3) Exploring microscopic symmetry of black holes in string theory.
There is a very powerful symmetry underlying black holes in string theory called modular symmetry, and more subtle variations of it called mock modular symmetry. These are symmetries that first appeared in the context of number theory. The work done so far in the project has clarified the consequences of modularity and mock modularity in quantum field theory and gravity. In addition, the consequence of mock modularity on the scattering of black holes has been clarified. New structures related to quantum modularity have been discovered.
In addition to these main results, the project work has obtained results in broader aspects of holography and quantum gravity. There have also been cross-disciplinary discoveries with impact on pure mathematics, including: new relations between entanglement entropy of certain two-dimensional quantum systems with finite size and at finite temperature and classical higher-genus theta functions; the discovery of an infinite set of new relations between theta functions of higher genus and ordinary Jacobi theta functions; the unexpected relations of L-functions in number theory and black holes in AdS5 string theory; super Yang-Mills theory and the infinite-wedge representation.
These results have been published in 32 peer-reviewed journal articles and 4 preprints which are available on the physics arxiv. They have been disseminated to the wider community in more than 50 talks given by the team members at international conferences and workshops.
(1) Explanation of the discrete nature of a quantum black hole.
The discrete nature of any quantum-statistical system is encoded in its Hilbert space. A foundational open question is to explain the emergence of this Hilbert space starting from the gravitational field variables. The results of the project explain how to calculate these effects in the simplest signature of discreteness of a BH – the dimension of its Hilbert space. This should clearly be a positive integer, but it has been a long-standing problem as to how such an integer arises from continuum gravity. The results of the project show, for the first time, how to calculate this integer by summing an infinite series of quantum-mechanical effects in the black hole.
(2) Development of new techniques to calculate functional integrals in gravity.
The functional integral is a fundamental object in the description of quantum theories. Calculating this integral for quantum gravity has been very difficult so far. The results of the project rigorously establish a new technique, called supersymmetric localization, to calculate the functional integral for supersymmetric observables in a class of gravitational theories. This technique has proved to be a powerful method to calculate functional integrals in supersymmetric quantum field theories, but so far it was not known how to apply this to gravitational theories where the metric also fluctuates. A new formalism to do this has been developed in the project which uses a mix of two classical methods in field theory (the background field method and the BRST formalism). This formalism was then used to calculate the quantum entropy of a variety of black holes in four and five dimensions.
(3) Exploring microscopic symmetry of black holes in string theory.
There is a very powerful symmetry underlying black holes in string theory called modular symmetry, and more subtle variations of it called mock modular symmetry. These are symmetries that first appeared in the context of number theory. The work done so far in the project has clarified the consequences of modularity and mock modularity in quantum field theory and gravity. In addition, the consequence of mock modularity on the scattering of black holes has been clarified. New structures related to quantum modularity have been discovered.
In addition to these main results, the project work has obtained results in broader aspects of holography and quantum gravity. There have also been cross-disciplinary discoveries with impact on pure mathematics, including: new relations between entanglement entropy of certain two-dimensional quantum systems with finite size and at finite temperature and classical higher-genus theta functions; the discovery of an infinite set of new relations between theta functions of higher genus and ordinary Jacobi theta functions; the unexpected relations of L-functions in number theory and black holes in AdS5 string theory; super Yang-Mills theory and the infinite-wedge representation.
These results have been published in 32 peer-reviewed journal articles and 4 preprints which are available on the physics arxiv. They have been disseminated to the wider community in more than 50 talks given by the team members at international conferences and workshops.
One unexpected development that has come out of the project, which has received a lot of attention, is the calculation of the microscopic entropy of supersymmetric black holes in AdS5. This has been an unsolved problem for the last 15 years or so, and our breakthrough results have been published in three articles, followed by six other articles expanding various aspects. As part of these developments, we found new asymptotic formulas for the growth of microscopic states in 4-dimensional super Yang-Mills theory, similar to the influential Cardy-formula in 2-dimensional conformal field theories.
These results have led to invitations to speak at the annual Strings conference, and to give lectures at specialized schools for PhD students and early-career researchers.
These results have led to invitations to speak at the annual Strings conference, and to give lectures at specialized schools for PhD students and early-career researchers.