Introduction and Background
Over the years, string theory has emerged as the most promising candidate theory that could yield a unified description of all the fundamental forces in nature, including gravity, in a single quantum mechanical framework. It assumes that all particles we observe arise as excitations of tiny strings, similar to the strings of a violin whose excitations produce all the different sounds. The theory brings together many elegant mathematical ideas from a variety of fields.
(i) String theory/gauge theory 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 prime example of such a duality, proposed by Juan Maldacena in 1997, is known as Anti-de Sitter space/Conformal Field Theory duality (AdS/CFT duality). In its original formulation, it states the full (quantum) equivalence of type IIB superstring theory on a five-dimensional AdS space and maximally supersymmetric Yang—Mills theory on the four-dimensional conformal boundary of this AdS space. 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. 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).
For a brief introduction to this subject, please see Jan de Boer's lecture notes entitled Introduction to the AdS/CFT correspondence (PDF).
(ii) Twistor geometry. Since their discovery by Sir Roger Penrose in 1967, twistors have provided deep insights into gauge and gravity theories as they yield geometric transformations, such as the Penrose transform. 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. Key examples are instantons (finite-action solutions) 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, Edward Witten proposed twistor-strings as a dual formulation of perturbative maximally supersymmetric Yang—Mills theory in four dimensions, and, as such, it is another instance of a string theory/gauge theory 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 and it will be exciting to see what the future holds.
For an introduction to twistor geometry and its applications to gauge theory scattering amplitudes, you are invited to read my review article entitled A first course on twistors, integrability and gluon scattering amplitudes (PDF).
NB: The Penrose transform is one of the far reaching generalisations of the so-called Radon transform, such as Cormack's generalisation that led to his Nobel Prize in Medicine for developing the theoretical underpinnings of computer assisted tomography.
My current work focuses on string and gauge theories in the context of string theory/gauge theory dualities with an emphasis on so-called integrable theories, and on the fruitful interplay of mathematics and physics in these theories. Integrable theories are a special type of theories that admit a sufficient number of conserved quantities for their complete solution to be found exactly (at least in principle). Despite being integrable, such theories are still very complicated and to give explicit solutions for quantities of interest is hard. One of the approaches to studying such theories is twistor geometry. I also work on various topics related to higher mathematical structures emerging in string and M-theory such as higher gauge theory. The latter is a generalisation of ordinary gauge theory that makes use of higher category theory and which allows to incorporate higher degree gauge potentials in a systematic and geometric fashion. Apart from an appealing mathematical subject, higher gauge theory might provide solutions to important problems in M-theory. Concretely, my current interests include:
- Twistor Geometry and Applications to Differential Geometry
- Higher Gauge Theory and Category Theory, and Membranes in String and M-Theory
- Integrability and Hidden Symmetries in String and Gauge Theory
- String Theory/Gauge Theory Dualities
- Instantons and Solitons
- Supergravity Theories
- Geometry, Monge—Ampere Structures, and Fluid Dynamics
A detailed list of publications can be found on my publications page.
If you are a student and interested in becoming a PhD student under my supervision, please drop me an e-mail.