Report
Teaser, summary, work performed and final results
Periodic Reporting for period 2 - QBH (Quantum Black Holes: A macroscopic window into the microstructure of gravity)
Teaser
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...
Summary
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 uses a discovery of the PI that 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.
Work performed
The main results achieved so far are two-fold:
(1) Macroscopic quantum entropy of black holes. The quantum corrections to the entropy of a supersymmetric black hole can be calculated using functional integrals in the near-horizon geometry of the black hole. The primary achievement of the project so far has been to lay down the foundations to calculate such functional integrals exactly, using the technique of supersymmetric localization applied to supersymmetric theories of gravity (supergravity). 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 applied to calculate the quantum entropy of a variety of black holes in four dimensions. Investigations have now begun to understand a corresponding family of five-dimensional black holes.
(2) 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. One achievement has been the discovery of a new three-way relation between physics, geometry, and number theory. This discovery relates two-dimensional quantum field theories on non-compact spaces that are relevant in the analysis of black holes, a class of manifolds called squashed toric manifolds, and newly-defined objects in number theory called higher-depth mock modular forms. In addition, the consequence of mock modularity on the scattering of black holes has been clarified.
In addition to these main results, the project work has also led to new relations between entanglement entropy of certain two-dimensional quantum systems with finite size and at finite temperature and classical higher-genus theta functions. In particular, this has led to the discovery of an infinite set of new relations between theta functions of higher genus and ordinary Jacobi theta functions.
Final results
The new formalism that has been developed in the project to calculate exact functional integrals is used, as explained in the summary above, to calculate the quantum entropy of supersymmetric black holes, whose near-horizon geometry has a universal factor which is Anti de Sitter (AdS) space in two dimensions. It turns out that the formula actually has a much broader scope and can be applied to calculate supergravity functional integrals in spaces with asymptotically Anti de Sitter boundary conditions in higher dimensions as well. This means that the formalism can lead to an exact version of the AdS/CFT correspondence. This correspondence is a remarkable proposal which relates gauge theories and quantum gravity, and has potential applications to a variety of physical situations including the quark-gluon plasma, superconductivity, fluid dynamics, and quantum phase transitions. However, little is known about how the correspondence works beyond the semiclassical limit, and the work of the proposal allows to extend this to the quantum regime.
Progress along the two main lines of research of the project proposal is expected to continue. Along the first line of research, the quantum entropy of five-dimensional black holes should be calculated, which will lead to an understanding of the degrees of freedom in M-theory. Along the second line, the relation between mock modular forms and two- and four-dimensional quantum field theories is expected to be further clarified.
One surprising and unexpected development that has come out of the project, which has received a lot of recent 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 the first results have been published as two recent preprints. This direction is expected to lead to many interesting formulas in the next few years, and we expect that we will have a dramatically new understanding of the microscopic properties of supersymmetric AdS5 black holes by the end of the project. In addition we expect that the phase structure of gravitational configurations in asymptotically AdS5 theories of gravity will be calculated. A successful completion of these ideas is expected to open up a lot of new research in this direction, the details of which are difficult to predict at this stage.