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.
