## Periodic Reporting for period 4 - IQFT (Integrable Structures in Quantum Field Theory)

Reporting period: 2020-04-01 to 2021-09-30

The IQFT project aims to study quantum field theory. Quantum field theory forms the foundation of our understanding of elementary particle physics and provides the theoretical framework for the interpretation of data from collider experiments.

While quantum field theory is an old subject, over the last decade new features have begun to emerge which hint that there are new ways to understand it. Astonishingly simple formulas have been found to describe the scattering of particles but these formulas are often the result of very long calculations when approached with textbook methods. This suggests that there are new ways to formulate the problem which look radically different from the standard textbook discussions of quantum field theory. Exploring such alternative formulations is a principle aim of this project.

Enormous amounts of effort go into performing the calculations of scattering amplitudes needed to make precise predictions for collider experiments. New techniques to handle such calculations are therefore much needed. This research will allow us to greatly improve on existing efforts to make predictions for scattering processes and therefore maximise the theoretical benefit from the huge experimental effort going on at colliders such as the LHC at CERN.

In addition this research makes connections to areas of mathematics and therefore helps to bring together different academic communities working in fundamental science.

While quantum field theory is an old subject, over the last decade new features have begun to emerge which hint that there are new ways to understand it. Astonishingly simple formulas have been found to describe the scattering of particles but these formulas are often the result of very long calculations when approached with textbook methods. This suggests that there are new ways to formulate the problem which look radically different from the standard textbook discussions of quantum field theory. Exploring such alternative formulations is a principle aim of this project.

Enormous amounts of effort go into performing the calculations of scattering amplitudes needed to make precise predictions for collider experiments. New techniques to handle such calculations are therefore much needed. This research will allow us to greatly improve on existing efforts to make predictions for scattering processes and therefore maximise the theoretical benefit from the huge experimental effort going on at colliders such as the LHC at CERN.

In addition this research makes connections to areas of mathematics and therefore helps to bring together different academic communities working in fundamental science.

The IQFT research team have been working on many ideas in quantum field theory. Particular emphasis has been placed on studying aspects of the maximally supersymmetric gauge theory in four dimensions: N=4 super Yang-Mills theory. This theory is a close cousin to quantum chromodynamics (QCD) which is the theory describing the strong nuclear interactions and an important ingredient in understanding proton-proton collisions at the LHC. By studying simpler models such as N=4 super Yang-Mills theory we gain greater theoretical control while retaining central features that we wish to study - namely that the results of different calculations are significantly simpler that the textbook approaches would suggest. Understanding and exploiting these features is a key line of investigation.

In the area of scattering amplitudes, new results have been obtained by extending and improving the analytic bootstrap programme. This is a framework for calculating scattering amplitudes based on understanding their general analytic structure. At least in the case of the simpler model of N=4 SYM these methods are vastly more powerful than textbook Feynman diagram calculations. Even more importantly they have lead us to discover new mathematical structures which control the analytic behaviour in suprprising ways. For example, the set of possible singularities of the amplitudes seems to be controlled in certain instances by a mathematical structure known as a cluster algebra. My team has explored and greatly extended the relationship between cluster algebras and scattering amplitudes in N=4 SYM, an insight we hope will lead to general lessons for quantum field theory.

We have also been studying a particular kind of high energy limit known as multi-Regge kinematics. In this regime we have made enormous progress in understanding the mathematical structure of scattering processes - leading us to a very general but compact formula for the scattering of any number of particles of any type at any interaction strength in N=4 super Yang-Mills theory. Such a powerful formula will, we hope, lead us to expressions in such high energy regimes for other models and perhaps even make contact directly with QCD.

Another theme is the study of correlation functions in conformal field theory. A surprising discovery in this area is that our bootstrap methods are also applicable to the study of quantum gravity within the framework of the AdS/CFT correspondence. This correspondence states that quantum field thories without gravity are equivlent to theories which do include gravitational interactions in a type of (negatively) curved space known as anti-de-Sitter space. In the case of N=4 super Yang-Mills theory the gravitational theory is actually a string theory. By using general mathematical properties of N=4 super Yang-Mills theory we have found that we are able to study the regime of strong coupling with a large number of colours (or particle species), instead of the three species which describe QCD in our world. The analytic bootstrap ideas are surprisingly powerful in this regime and have led us to explicit expressions to the quantum corrections to gravitational scatting in anti-de-Sitter space. Such expressions are not currently tractable by more direct textbook methods. By further exploring this line of research we may be able to obtain greater theoretical control over aspects of quantum gravity and string theory than currently available.

In the area of scattering amplitudes, new results have been obtained by extending and improving the analytic bootstrap programme. This is a framework for calculating scattering amplitudes based on understanding their general analytic structure. At least in the case of the simpler model of N=4 SYM these methods are vastly more powerful than textbook Feynman diagram calculations. Even more importantly they have lead us to discover new mathematical structures which control the analytic behaviour in suprprising ways. For example, the set of possible singularities of the amplitudes seems to be controlled in certain instances by a mathematical structure known as a cluster algebra. My team has explored and greatly extended the relationship between cluster algebras and scattering amplitudes in N=4 SYM, an insight we hope will lead to general lessons for quantum field theory.

We have also been studying a particular kind of high energy limit known as multi-Regge kinematics. In this regime we have made enormous progress in understanding the mathematical structure of scattering processes - leading us to a very general but compact formula for the scattering of any number of particles of any type at any interaction strength in N=4 super Yang-Mills theory. Such a powerful formula will, we hope, lead us to expressions in such high energy regimes for other models and perhaps even make contact directly with QCD.

Another theme is the study of correlation functions in conformal field theory. A surprising discovery in this area is that our bootstrap methods are also applicable to the study of quantum gravity within the framework of the AdS/CFT correspondence. This correspondence states that quantum field thories without gravity are equivlent to theories which do include gravitational interactions in a type of (negatively) curved space known as anti-de-Sitter space. In the case of N=4 super Yang-Mills theory the gravitational theory is actually a string theory. By using general mathematical properties of N=4 super Yang-Mills theory we have found that we are able to study the regime of strong coupling with a large number of colours (or particle species), instead of the three species which describe QCD in our world. The analytic bootstrap ideas are surprisingly powerful in this regime and have led us to explicit expressions to the quantum corrections to gravitational scatting in anti-de-Sitter space. Such expressions are not currently tractable by more direct textbook methods. By further exploring this line of research we may be able to obtain greater theoretical control over aspects of quantum gravity and string theory than currently available.

The results described briefly above are very much beyond the state of the art in term if what is currently understood about interacting four-dimensional quantum field theories. The challenge for the future will be to see if even greater theoretical control can be gained, firstly in the simpler model of N=4 super Yang-Mills theory and then in more general quantum field theories with the aim of helping to descibe better QCD processes themselves.

We are actively investigating the consequences of the relationship to cluster algebras to the anaytic structure of scattering amplitudes. We hope to exploit this structure to obtain further explciit results and to learn general lessons about singularities in quantum field theory amplitudes.

Work on understanding quantum gravity corrections is also ongoing - we aim to improve our understanding of such quantities and learn general lessons about conformal field theories and string theories.

We are actively investigating the consequences of the relationship to cluster algebras to the anaytic structure of scattering amplitudes. We hope to exploit this structure to obtain further explciit results and to learn general lessons about singularities in quantum field theory amplitudes.

Work on understanding quantum gravity corrections is also ongoing - we aim to improve our understanding of such quantities and learn general lessons about conformal field theories and string theories.