# Mathematical and theoretical physics

## Classical and quantum integrable systems

**Staff: **Dr Livia Ferro, Dr Tomasz Lukowski, Dr Vidas Regelskis, Dr Charles Young

The concept of a classical integrable model with finitely many degrees of freedom dates back to Liouville. More recently, the notion of integrability has been extended to quantum-mechanical systems (examples include Gaudin models and Toda chains) and to field theories with infinitely many degrees of freedom. The prototypical example of an integrable field theory is the KdV equation, which models non-linear waves in shallow water. Other important examples include Toda field theories and non-linear sigma models.

An interesting question is how to quantize a given classical integrable field theory. In recent years many interesting dualities between integrable models have been discovered. Both the ODE/IM and AdS/CFT correspondences can be seen as dualities between classical and quantum integrable models.

We work on topics including:

- Integrable Field Theories
- Ordinary Differential Equation (ODE)/Integrable Model (IM) correspondence
- Integrability in Gauge Theory and String Theory; AdS/CFT correspondence
- Gaudin Models and the Bethe Ansatz

## Strings, supergravity and geometry

**Staff: **Dr Livia Ferro, Dr Tomasz Lukowski, Dr Charles Strickland-Constable

String theory is the most developed theory of quantum gravity, as well as providing deep insights into non-perturbative aspects of quantum field theory via the AdS/CFT correspondence. It also has deep connections with many interesting areas of pure mathematics, such as differential and algebraic geometry, topology, algebra and even number theory, and research in string theory has gone hand-in-hand with many exciting developments in those areas.

Strings 'see' the spacetime geometry very differently to point particles in more conventional theories, which give rise to exotic features such as string dualities - equivalences between seemingly very different string theory scenarios. Indeed, a full formulation of the theory is currently beyond our reach, and most researchers believe that we will have to develop many new mathematical concepts to provide one, including a much better understanding of this 'stringy geometry'.

One strand of our research is focused on the development of a new notion of geometry, known as generalised geometry, which gives an elegant geometrical reformulation of the low energy field theory limit of string theory, and includes some of the features of string dualities, hinting that it may be the first step towards a better understanding of the geometry of the full theory. The mathematical technology is also very useful for studying supersymmetric solutions and consistent truncations of supergravity, both of which are also important for the AdS/CFT correspondence.

## Scattering amplitudes in gauge theories

**Staff:** Dr Livia Ferro, Dr Tomasz Lukowski

Scattering amplitudes are prominent physical quantities which describe the interactions between particles. A traditional method to compute them perturbatively are Feynman rules derived from the Lagrangian. This standard approach has made possible huge progress in giving theoretical predictions relevant for experiments at high-energy colliders, such as the Large Hadron Collider at CERN. However, there is strong evidence that more fundamental, underlying principles have yet to be discovered. This is very apparent for integrable Quantum Field Theories, such as maximally supersymmetric Yang-Mills theory (MSYM), where new insights are provided by, for instance, infinite dimensional symmetries and novel geometric formulations for amplitudes.

In particular, our research focuses on:

- Amplituhedron in planar MSYM
- positive geometries
- Yangian symmetry and integrability
- extensions of these methods to supergravity amplitudes