QBH

A new window into quantum gravity Three of the four fundamental forces are described within the standard model of particle physics, yet the challenge of describing gravity within the framework of quantum mechanics remains unresolved. Studies of the thermodynamic behaviour of black holes can reveal important insights into the microstructure of gravity, as Dr Sameer Murthy of the QBH project explains The theoretical basis of the standard model of particle physics is quantum mechanics, a framework which describes how elementary particles behave at a fundamental level. Three of the four fundamental forces which govern the interactions of particles are described within the standard model – the strong, electromagnetic and weak forces – yet the problem of writing down the quantum theory of gravity remains unsolved. “Bringing together gravity and quantum mechanics is a major challenge in physics. So far there has been no experiment in which researchers have been able to probe the quantum properties of gravity,” says Dr Sameer Murthy, the Principal Investigator of the QBH project. With no direct experimental guide, Dr Murthy is using a novel approach in the QBH project. “I want to use black holes and their thermodynamic properties as a guide to understand the quantum properties of gravity,” he outlines.

A historical analogy can be drawn here with the work of physicists in the nineteenth century. At the time, researchers were studying the thermodynamics of gases; while they could make gross measurements on a container filled with gas, scientists didn’t understand the microscopic properties.

“They could measure things like heat transfer, pressure, temperature and entropy, the macroscopic, large-scale variables. But what you want to understand is the microscopic properties – what is the nature of the constituents?

How do they interact? What are their properties?” explains Dr Murthy. From measurements of entropy – a measure of disorder of a system, or the number of ways in which a system could exist –

to gain new insights into the structure of gravity. “By thinking carefully about the macroscopic phenomena, we can deduce non-trivial aspects of the microscopic physics,” he outlines.

We are now in a situation similar to that faced by nineteenth century physicists – we know that a black hole is made up of something. We don’t know what it is, but we can compute various macroscopic quantities scientists were able to learn about the microscopic properties of gases, indeed they were able to deduce fundamental quantum physics concepts, like the indistinguishability of elementary particles and the fundamental cutoff on energy excitations. Now Dr Murthy aims

Black holes Black holes are regions of space-time that are surrounded by one-way surfaces called event-horizons. According to classical general relativity nothing can come out from behind this horizon, not even light, yet the findings of Jacob Bekenstein and

Mock theta functions, discovered by the brilliant Indian mathematician S. Ramanujan in 1920, seem to make an unexpected but important appearance in describing aspects of the physics of black holes and quantum gravity in string theory (see Page 3).

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Illustration: CXC/M. Weiss.

Stephen Hawking seem to run contrary.

“They found that a black hole behaves like a thermodynamic object when one takes quantum mechanics into account. It has temperature and entropy, and emits thermal radiation – in a sense it behaves like a cylinder of gas. This was a remarkable finding,” says Dr Murthy. This suggests that a black hole is actually made up of many microscopic states, so they could be used to probe quantum gravity. “We are now in a situation similar to that faced by nineteenth century physicists – we know a black hole is made up of something. We don’t know its microscopic constituents, but we can compute various macroscopic quantities,” continues Dr Murthy.

Bekenstein and Hawking developed a formula to calculate the thermodynamic entropy of a black hole, effectively the number of ways in which it can exist, now Dr Murthy aims to use this to probe their microscopic structure. This research again builds on historical foundations. “As Boltzmann taught us, entropy is really a measure of the number of microscopic ways in which a macroscopic state of a system can exist. If we know the entropy exactly, we know the most basic fact of the quantum behaviour of a system, the dimension of the possible space of quantum states of the system,” he says.

These ideas of Boltzmann underlie all of physics and are key to understanding the relationship between the micro and the

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macro worlds. He gave the precise mathematical formulation of how the various constituents of the microscopic world come together to form what looks like a macroscopic object, when viewed from afar without a sharp lens. “If we can move away from the macroscopic approximations of Boltzmann, we begin to get back details about the microscopic world,” says Dr. Murthy.

Together with his collaborators Prof. A. Dabholkar and Dr. J. Gomes, Dr. Murthy applied this idea to a specific type of black hole to extract clues about its microscopic structure. “We considered a certain type of black hole, and, based on a beautiful idea of A. Sen, we developed new methods to compute the quantum corrections to the thermodynamic entropy of black holes,” he explains. This gave researchers a basis to understand quantum effects in black holes at an unprecedented level of accuracy; Dr Murthy is now looking to verify this approach. “We have developed this new technique to compute quantum corrections to black hole entropy, but what are we learning about the black hole? How can we check that our results are correct?

As this is a new technique, it’s important to check it,” he stresses.

String theory This is where a theoretical framework called string theory comes in. In string theory one can describe the microscopic constituents

of a certain class of black holes. “If you understand what the black hole molecules are, then you can count how many states they can be in, and check if the Boltzmann equation also applies to quantum gravity,” says Dr Murthy. Researchers A. Strominger and C. Vafa, and A. Sen were able to show in the 90’s that the Bekenstein-Hawking entropy formula could be explained via the Boltzmann equation, as the logarithm of the number of microscopic states of the system – in the same way that entropy is explained in the rest of physics. Dr Murthy and his colleagues wanted to investigate this further. “We said; ‘ok, let’s test this beyond the approximation that the system size is very large’. We now know how to extend the Bekenstein-Hawking formula and sum up all the corrections to it, so let’s try to see if, in these examples, we can get not just an approximate agreement, but an exact agreement,” he says.

The number of configurations of the molecules in a black hole is an important quantity to understand in quantum gravity, and it is these numbers that Dr Murthy and his colleagues have been working to compute. “We have managed to compute the number of microscopic states of a black hole, starting from the semi-classical approximation of Bekenstein and Hawking, making it better and better, until it’s exactly correct,” explains Dr Murthy. Researchers can then look to gain deeper insights into the microscopic structure of quantum gravity.

EU Research

At a glance Full Project Title Quantum Black Holes: A macroscopic window into the microstructure of gravity (QBH)

String theory gives us two pictures of black holes - the traditional macroscopic picture, and a “dual” microscopic picture in terms of the fundamental objects of the theory called strings and branes. The number of microscopic constituents turn out to arrange themselves in extremely interesting patterns governed by deep underlying symmetry structures. Studying them systematically gives us clues about quantum effects in gravity.

“The fact that we can compute these numbers means we now have some information about how those quanta are reached, about how the quanta are arranged in a black hole, and about how forces interact to make these quanta. This is very exciting,” continues Dr Murthy. “Now that we have derived these numbers, we want to look for patterns.

Pursuing these ideas in quantum gravity led us to some intriguing relations with an unexpected field, the theory of numbers, which deals with the properties and relationships of positive integers.”

The research of earlier number theorists and mathematicians holds importance here, in particular that of G.H Hardy and Srinivasa Ramanujan, a renowned Indian mathematician who left a vast number of results following his death. “Hardy and Ramanujan were trying to answer questions in combinatorics like; ‘if you take number N, how many ways can it be divided out into smaller numbers?’” says Dr Murthy.

This eventually resulted in the development of an analytic approximation to solve these types of problems, called the Hardy-Ramanujan-Rademacher method, an important tool in analytic number theory. “What was very curious was that the mathematical process by which classical gravity approximates the quantum degeneracy of states of a black hole is exactly the same as the process that Hardy and Ramanujan found,” explains Dr Murthy. “We think there is a deep connection between black holes, string theory, and number theory.”

This connection was realized in a remarkable development which related black holes and another Ramanujan discovery called mock theta functions, a type of function he investigated late in his

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life and wrote about in his last letter to Hardy. While Ramanujan gave several examples of these mock theta functions in this last letter, he didn’t provide a precise definition, leaving later generations of mathematicians many puzzles to ponder.

“Mathematicians have worked very hard to try and discern what he could have meant,” says Dr Murthy. The problem was cracked in 2002 by Dutch PhD student Sanders Zwegers, who discovered the key idea for a theory of mock-theta functions that included all the examples of Ramanujan, which led to many new developments in number theory. A four-year collaboration between Dr Murthy, Prof. Atish Dabholkar, and Prof. Don Zagier then revealed a completely unexpected connection between black holes in string theory and mock theta functions.

The power of the theory of theta functions comes from an underlying symmetry called modular symmetry. A mock-theta function doesn’t quite have the same symmetry as an ordinary theta function, but it still inherits that symmetry in a very subtle way. It turned out that a large class of black holes in string theory also display exactly the same symmetry structure as a mock theta function, thus opening up new ways of thinking about black holes as well as mock theta functions. One intriguing possibility, that both physicists and mathematicians are excited about, is a link to unexpected discrete group-theoretical structures called ‘moonshine symmetries’. “We could now enlarge the story of exact black hole entropy from just one example to a large class of black holes in string theory,” says Dr Murthy.

“Mock theta functions seem to be the correct mathematical basis that encodes the set of quantum states of these black holes.”

Project Objectives The first major aim of the project is to construct a systematic treatment of quantum effects in black hole entropy – which tells us, quantitatively, what fundamental quantum gravity could be. A second major aim is to advance the theoretical understanding of quantum black holes by identifying the microscopic symmetry (and symmetry breaking) principles of quantum gravity, and in particular, investigating the deeper origins of mock modular symmetry.

Project Funding Funded by the European Commission, ERC consolidator grant. Total Budget: EUR 1,759,064.00.

Contact Details Dr Sameer Murthy Department of Mathematics, King’s College London The Strand, London WC2R2LS T: +44 20 7848 2219 E: sameer.murthy@kcl.ac.uk W: https://nms.kcl.ac.uk/sameer.murthy “Quantum black holes, wall crossing, and mock modular forms”, A. Dabholkar, S. Murthy, D. Zagier, (appendix by M. Cheng). arXiv:1208.4074 [hep-th]. To be published in Cambridge monographs in mathematical physics, (Cambridge University Press).

“Localization & Exact Holography” Atish Dabholkar, Joao Gomes, Sameer Murthy. arXiv:1111.1161 [hep-th] 10.1007/JHEP04(2013)062 JHEP 1304 (2013) 062.

“Quantum black holes, localization and the topological string” Atish Dabholkar, Joao Gomes, Sameer Murthy. arXiv:1012.0265 [hep-th] 10.1007/JHEP06(2011)019 JHEP 1106 (2011) 019.

Dr Sameer Murthy

Sameer graduated from the Indian Institute of Technology Bombay, and got his PhD at Princeton University under the supervision of Nathan Seiberg. He subsequently held a research position at the Abdus Salam ICTP Trieste, a Marie Curie fellowship at the University of Paris, and a senior post-doctoral research position at Nikhef Amsterdam where he was awarded the NWO VIDI research grant by the Dutch organization for scientific research. In September 2013 he moved to King’s college where he is currently a Reader in Theoretical Physics. In 2016 he was awarded the ERC consolidator grant for a project on Quantum Black holes.

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