Gravity and Quantum Field Theory
We focus on the mathematical structures that underpin fundamental physical theories including quantum field theories such as Chern—Simons and Yang—Mills theories as well as gravitational theories such as supergravity and string theory.
Research leads
Dr Jan Gutowski
Associate Professor
Dr Andrea Prinsloo
Lecturer Teaching Track
Professor Alessandro Torrielli
Professor
Dr Martin Wolf
Associate Professor
Overview
Quantum field theory is a framework that combines the laws of quantum mechanics and special relativity in a self-consistent manner and underpins most of theoretical physics today.
General relativity is a highly successful theory describing gravity at large scales such as the solar system. So far, however, it has remained incompatible with the laws of quantum mechanics.
String theory has emerged as the leading candidate for a theory describing all the fundamental forces of nature, including gravity, in a single unified self-consistent quantum mechanical framework. It assumes that all particles are different harmonics of small vibrating strings, much in the way the different harmonics of a guitar string correspond to different musical notes. Surprisingly, even though it started as a theory of just strings, it also accommodates consistently in a non-perturbative manner higher-dimensional extended objects called branes.
There is a rich and fruitful interplay between quantum field theory and string theory, and mathematics. New and sophisticated mathematical techniques have rendered many problems in string theory and quantum field theory tractable and, in certain cases, even exactly soluble.
Conversely, string theory has also led to striking advances and conjectures in mathematics. It is this synthesis between theoretical physics and mathematics that forms the foundation of our research.
Research projects
Selected PhD theses
- V Gautam, Partial confinement, matrix degrees of freedom, and holography, supervisors: M Hanada and A Torrielli
- R D'Onofrio, Singularities in fluid dynamics: from the classical approach to Monge-Ampere geometry, supervisors: G Ortenzi, I Roulstone, and T Bridges
- R Delmasso, Killing spinors, extensions and metric diagonalization in pseudo-Riemannian geometry, supervisors: D Conti and J Grant
- D Farotti, Supergravity, supersymmetry, and black holes, supervisor: J Gutowski
- D Bielli, Non-Abelian T-duality in superspace, supervisors: S Penati and M Wolf
- L Wyss, Strings and spins in deformed AdS/CFT, supervisor: A Torrielli
- L Raspolini, Higher gauge theory, BV formalism, and self-dual theories from twistor space, supervisor: M Wolf
- T Macrelli, Homotopy algebras, gauge theory, and gravity, supervisor M Wolf
- R Sisca, Heterotic vacua and their universal geometry, supervisor: J McOrist
- A Fontanella, Black horizons and integrability in string theory, supervisor: J Gutowski
- A Pittelli, Dualities and integrability in low dimensional AdS/CFT, supervisors: A Torrielli and M Wolf
- F Nieri, Integrable structures in supersymmetric gauge theories, supervisor: S Pasquetti
Upcoming seminars
Our group holds regular meetings covering a wide range of topics in modern quantum sciences. This includes both fundamental, mathematical, and practical aspects.
Black holes and supergravity
Supergravity is the low-energy limit of string theory, which is the most promising candidate theory capable of unifying quantum theory with general relativity. Supergravity also has solutions which describe supersymmetric black holes. Supersymmetry is a mathematical extension of the standard symmetries, such as rotations, and is a key property of string theory.
Hawking discovered that holes satisfy thermodynamic laws analogous to the laws satisfied by gases. Thermodynamic entropy is derived by examining statistical averages of microstates associated with gas molecules. For black holes in supergravity, the microstates come from higher-dimensional branes. Superposing many branes in well-defined configurations produces black holes so counting the branes gives the black hole entropy.
Systematic classifications of supersymmetric solutions have been constructed, which produced novel examples of higher dimensional black holes. In four dimensions there are particularly strong uniqueness theorems, which constrain possible black holes. However, these uniqueness theorems do not apply in higher dimensions. For example, in five dimensions, there are black rings, and even black Saturn solutions, and more complicated black holes with non-trivial topology outside the event horizon.
New spinorial geometry techniques have been used to construct classifications of solutions with more than the minimal amount of possible supersymmetry, which happens near to the event horizon of a supersymmetric black hole. This will provide useful new tools for investigating the geometry and physics of higher dimensional black holes.
Geometry and black hole physics
It is to be expected that more exotic black objects exist in higher dimensions, which is particularly relevant in the context of higher-dimensional supergravity theories associated with string theory. Notably, interesting geometric structures exist near to the event horizons of such black holes. The topology of these solutions plays a key role in the analysis.
For event horizons which are smooth, compact (i.e. have a finite area) and have no boundary, it can be shown that such solutions have certain symmetries.
Amongst other things, we analyse these types of solutions to gain new insights into the relationship between geometry and black hole physics, and there may also be interesting applications to condensed matter systems using conjectured gauge/string dualities.
Classifying anti-de Sitter supergravity solutions
We are interested in the classifications of anti-de Sitter supergravity solutions, which are particularly important in the context of the AdS/CFT duality.
In the case of two-dimensional anti-de Sitter solutions, the topology of the internal space plays a key role in determining the amount of supersymmetry preserved by these solutions.
Furthermore, combining this analysis with the homogeneity theorem for highly supersymmetric supergravity solutions, and the constraints obtained from the super-Jacobi identities in the associated superalgebras, produces strict conditions on the types of possible highly supersymmetric anti-de Sitter geometries.
There are many remaining open questions regarding the general classification of supergravity solutions, which are also of particular relevance in the context of black holes.
Classical and quantum integrability
Models may be integrable classically, or quantum mechanically, and various techniques exist that allow one to express the solution to these models in a remarkably compact form. These range from the Bethe ansatz, devised by Bethe in 1931 in order to solve the Heisenberg spin chain, to integral formulae and algebraic formulations, which can encompass infinite and finite-size problems.
A significant development in quantum integrability was the development of quantum inverse scattering methods. Often the hidden symmetries behind the exact solvability of these models organise themselves in infinite-dimensional quantum groups. Remarkable applications of quantum groups and integrable systems to the AdS/CFT duality were discovered by Minahan and Zarembo in 2002, who observed the emergence of integrable structures in the spectral problem of the maximally supersymmetric Yang—Mills theory in four dimensions.
Understanding quantum integrable systems
Quantum groups, Hopf algebras, and their representation theory provide an ideal framework for dealing with quantum integrable systems. The procedure of the so-called algebraic Bethe ansatz, which originates from the simultaneous diagonalisation of the quantum Hamiltonian and the higher commuting charges by means of creation and annihilation operators, can be reformulated in terms of the representation theory of a particular Hopf algebra based on the algebra known as RTT.
This in turn, in the case of rational spin chains, leads directly to the theory of Yangians. The mathematics of Yangians and their deformations has had a vast impact in supersymmetric Yang—Mills theory, both in the study of the spectral problem and of amplitudes and correlation functions.
We develop and analyse all these structures, and there are now many exciting new directions for integrability and quantum group theory with exciting cross-disciplinary aspects, e.g. in twistor geometry.
Using twistor geometry to study integrable systems
Since their discovery by Penrose, twistors have provided deep insights into gauge and gravity theories, and especially into integrable theories. The key idea is to replace space-time as a background for physical processes by an auxiliary space, called twistor space. Differentially constrained data on space-time, such as solutions to field equations, are then encoded in terms of holomorphic data on twistor space, such as cohomology groups. This allows one to classify solutions to certain problems. Two key examples are instantons in Yang—Mills and gravity theories in four dimensions. From a mathematical point of view, differential geometry and algebraic geometry lie at the heart of twistor geometry.
In view of applying twistors in quantum physics, a major breakthrough was made when, in late 2003, Witten proposed twistor strings as a dual formulation of perturbative maximally supersymmetric Yang—Mills theory in four dimensions. This is another instance of a gauge/string duality. Since then it has become clear that twistors have a dramatic impact on perturbative gauge and gravity theories, for instance, by providing an explanation of the observed simplicity of particle scattering amplitudes. This approach has led to many fruitful developments in perturbative quantum field theory.
We use the ideas of twistor geometry to study integrable system in the context of gauge and gravity theories as well as in fluid dynamics. We also use twistorial methods in the context of higher structures to shed light on the properties of scattering amplitudes and to analyse higher-dimensional superconformal field theories.
Gauge/string dualities
A powerful tool in the study of string theory is known as string theory/gauge theory dualities. These dualities can address certain questions in gauge theory that are out of reach of perturbative field theory. The basic idea is that string theory in d+1 space-time dimensions arises holographically from gauge theory in d dimensions, just like a two-dimensional hologram can appear to have a third-dimension when correctly illuminated.
The original example of such a duality, first proposed by Maldacena in 1997, is known as anti-de Sitter space/conformal field theory duality (AdS/CFT duality). In its original formulation this duality states the full (quantum) equivalence of type IIB superstring theory on a five-dimensional anti-de Sitter space, a space of constant negative curvature, and maximally supersymmetric Yang—Mills theory on its four-dimensional conformal boundary. As such, it relates a theory with gravity (the string theory) to a theory without gravity (the Yang—Mills theory).
The AdS/CFT duality has been generalised to various other settings and has found many applications.
String theory/gauge theory dualities
Recent important developments that use these ideas include the construction of gauge theory scattering amplitudes and attempts of providing microscopic descriptions of phenomena in condensed matter physics and fluid dynamics (gravity/fluid correspondence).
Still we do not know yet why and how this duality can work. For instance, we do not know how the geometry and the local excitations on the gravity side are encoded in the dual non-gravitational theory.
Part of our research involves trying to understand these issues.
Higher structures
Categorified or higher mathematical structures appear very naturally within string theory and M-theory. For instance, the Kalb-Ramond two-form B-field is part of the connection on what is called a higher principal bundle, which is the categorification of the notion of a connection on a circle bundle. Moreover, string field theory, the second quantisation of string theory, is fundamentally based on homotopy Maurer—Cartan theory, the generalisation of Chern—Simons theory to L∞-algebras which are ∞-categorifications of the notion of a Lie algebra.
The standard approach to quantising a theory makes use of either the path integral approach or the Becchi—Rouet—Stora—Tyutin formalism. While in normal circumstances this is sufficient, when gauge symmetries only close on-shell, one has to use a more general quantisation procedure known as the Batalin—Vilkovisky quantisation. Importantly, the Batalin—Vilkovisky formalism is extremely useful in analysing mathematical structures of quantum systems, even when standard quantisation methods are applicable.
The homotopy algebra perspective
The Batalin—Vilkovisky formalism associates to a field theory an L∞-algebra. This, in turn, implies that the field theory can be understood as a higher homotopy Maurer—Cartan theory.
Recently, it has been realised that the L∞-algebra encodes not only classical physics such the field content, the equations of motion, and all the symmetries but also everything about perturbative quantum physics via homological perturbation theory. We have demonstrated how powerful recursive methods of constructing scattering amplitudes can be understood in terms of homotopy algebras, and in fact, exist for any theory. Ultimately, this leads to the homotopy algebraic perspective on perturbative quantum field theory, which implies a dictionary between physical concepts and algorithms and mathematical notions and constructions.
Generally, our research focuses on understanding all aspects of perturbative quantum field theory in terms of homotopy algebras.
Higher gauge theory
Principal bundles with connections are mathematical gadgets that are central to formulating gauge theories. The prime example in this context is Yang—Mills theory which, in turn, is the fundamental building block of the Standard Model of Particle Physics.
Higher principal bundles are certain generalisations, also known as categorifications, of principal bundles. In particular, the connections on such bundles are described by higher differential forms, contrary to principal bundles where they are described by differential one-forms. Higher principal bundles with connections are central to formulating of what is known as higher gauge theory.
We develop and analyse higher structures, such as higher differential geometry and homotopy algebras, with an aim of gaining a better understanding of higher gauge theory. We also make use of twistor geometry in our approach to higher gauge theory.