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Details of Grant 

EPSRC Reference: EP/V053787/1
Title: Integrable models and deformations of vertex algebras via symmetric functions
Principal Investigator: Wood, Dr SJ
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Sch of Mathematics
Organisation: Cardiff University
Scheme: Standard Research
Starts: 01 November 2021 Ends: 31 October 2025 Value (£): 316,853
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
EP/V053728/1
Panel History:
Panel DatePanel NameOutcome
17 May 2021 EPSRC Mathematical Sciences Prioritisation Panel May 2021 Announced
Summary on Grant Application Form
Mathematical structures or physical laws are called scale invariant, if they do not depend on length scales, that is they are left invariant by rescaling parameters. This phenomenon is observed in various mathematical and physical settings. In mathematics, fractals form a prime example - regardless of the magnification of a section of a fractal curve, one always finds a self-similar structure. In physics, the phenomenon occurs in statistical mechanics at so-called phase critical points. An example is water at its critical point (at 374 C and 218 times standard atmospheric pressure). It is at this critical point where there ceases to be any distinction between the gaseous and liquid states of water. In quantum field theory, for example the Standard Model of Particle Physics, one also encounters scale invariance, when one restricts one's attention to massless particles, such as photons (the quanta of light). In most cases scale invariance is part of a larger symmetry known as conformal invariance - invariance of the mathematical equations describing a physical system with respect to transformations which preserve angles yet need not preserve lengths.

The mathematical models used to describe such systems are called conformal field theories. They are of great interest to both mathematics and physics due to their remarkable amount of symmetry, which often elevates them to membership of the exceedingly small set of exactly solvable models, thereby enabling deeper insights into physical phenomena. One is also interested in understanding what happens when a length scale is suddenly introduced, for example, by a particle acquiring nonzero rest mass - an event which must have occurred at some point in our universe after the big bang. Some physical models retain large amounts of symmetry, despite conformal invariance itself being lost, and can thus still be solved exactly. Such models are called integrable.

These integrable models and conformal field theories offer highly non-trivial idealisations of more complicated models of the world. Thus their study can teach us much about the fundamental properties of nature. Advancing the understanding of such theories is thus not just an interesting mathematical problem in its own right, it is also an opportunity to build further bridges between theoretical physics and cutting-edge mathematical research. In the long run, such advances will provide a key step towards a complete understanding of universality classes in condensed matter physics, and dualities in quantum field theory and superstring theory.

This is an intradisciplinary project bridging mathematical physics and pure mathematics. The main objects of study will be the conformal field theories and integrable models constructable from a famous algebra called the Heisenberg algebra or free boson algebra. Though conformal field theories and integrable models can be very different, the presence of the Heisenberg algebra leads to them sharing a number of mathematical features. The main aims of this project are to bridge these two types of theories (so that insights from one side can be used to learn as much as possible from the other side), to give a uniform construction of all such theories, and to elucidate their deeper structures.
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Organisation Website: http://www.cf.ac.uk