Quantum Black Holes: a macroscopic window into the microstructure of gravity

Black holes are astrophysical objects that are formed by the collapse of very massive stars. They are surrounded by a one-way surface, called the event horizon, from inside which nothing - not even light - can escape (thus giving them their name). In the 1980s, J. Bekenstein, S. Hawking, and other theoretical physicists, while investigating the properties of black holes using the theory of general relativity, discovered that black holes have thermodynamic properties like temperature and entropy. In particular, they could associate a precise entropy intrinsically to black holes, thus suggesting that a black hole is made up of many microscopic states, just like the gas in a room.

This thermodynamic behaviour of black holes is a precious clue in unravelling the microscopic structure of quantum gravity, an important unsolved problem in theoretical physics. In the late 19th and early 20th century, a precise understanding of the thermodynamics of gases had led physicists towards basic principles of quantum mechanics. With that analogy in mind, today we have that high precision computations of quantum black hole entropy provide a new window into the fundamental microscopic theory of gravity and its deviations from classical general relativity - this is the main setting for this research proposal. Traditional methods of quantum field theory have proved to be not well-suited to perform these computations. Two recent breakthroughs in the recent work of the PI and his collaborators establish new ground for progress.

On one front, a new method to sum up all perturbative quantum contributions to the entropy of a large class of black holes has been developed. This gives rise to the first exactly solvable model of a quantum black hole. On a second front, a long-standing theoretical obstacle called the wall-crossing problem has been cleared on the microscopic description of black holes in string theory. The newly-developed field of mock modular forms is shown to be the correct framework to address questions of exact black hole entropy. This makes a large class of microscopic models amenable to analytic control, many of which were previously beyond reach.

These developments open up a new line of research that will be pursued along two intersecting avenues. A first aim is to extend the computations of exact quantum black hole entropy towards models of realistic black holes. A second aim is to investigate the deeper origins of mock modular symmetry, so as to advance the theoretical understanding of quantum black holes. As a concrete application, a third aim is to establish that newfound group-theoretical structures called "moonshine" symmetries are physically realised in quantum black holes, thus opening up connections between two exciting fields of research previously thought to be distinct. Together, the broad goal is to explain black hole microstructure through systematic computations of exact quantum entropy, and to investigate its consequences on the fundamental microscopic theory of gravity.

Apart from the direct academic impact mentioned above, I envision this proposal to have an economical and societal impact in two directions:
A) in enhancing the research capacity and knowledge skills of organisations, and
B) in raising the public awareness of science.

A) Enhancing research capacity of organisations.
There are two concrete ideas that stem from novel applications of the methodology and techniques that I use. The benefit groups include curiosity-driven researchers as well as applied/industrial researchers.

1. Algorithms for discrete mathematics: The tool of modular forms used in this proposal is also useful in other problems arising in diverse fields like physics, geometry, arithmetic, and algebra, as well as in practical applications like crystallography (study of lattices in solid state) and cryptography (on which internet security is based). An important practical starting point for applications is having an explicit list of functions with modular properties of various kinds.
Mock modular forms are a new type of functions of this kind. The first examples of mock modular forms were given by Ramanujan about 90 years ago, but it was only in 2002 that S. Zwegers in his PhD thesis at Utrecht University gave them a proper definition. Other examples have appeared since then, scattered in the literature of diverse fields (percolation, combinatorics). However, there has been no systematic classification of these examples so far and no efficient algorithmic method to produce new examples. My proposal is expected to lead to such an algorithm, which will be useful to these researchers.

2. Computer algebra packages: My proposed research involves intensive calculations on special types of mathematical functions, which are performed on the computer using symbolic manipulation packages PARI/GP (modular forms computations) and Mathematica (symbolic manipulation of continuous functions). The modern efficient method used in such situations is programming in modules, i.e. small software packages each designed to solve a generic algebraic problem, which are then tied together by users depending on their specific needs. My research methodology involves such programming and I plan to disseminate the resulting packages. This will benefit the community of scientists and engineers who use heavy computer calculations for their research.

B) Raising public awareness of science
The topic of black holes has managed to capture the public imagination very effectively, and it is a very good subject in which to disseminate the ideas that drive cutting-edge research. It is also an excellent topic to explain to the public at large about the concept of fundamental curiosity-driven research, to demystify the pathways of research so as to bring it closer to the experience of problem solving in everyday life. I intend to use my expertise in this area to advantage for effective outreach to the public.