Higher gauge theory and higher-dimensional integrability

Integrable models in mathematical physics can be characterised by the property that the equations governing their dynamics are exactly solvable, without having to resort to approximations. This makes integrability a very desirable property from a mathematical point of view. At the same time, it turns out that integrability is also a very rare property. It is therefore quite remarkable that, despite being few and far between, integrable models have always been found to play a ubiquitous role in fundamental physics, from gravitational orbits and the hydrogen atom to the modern theories of quantum fields, critical phenomena and strings. Among the already scarce number of mathematical models which are integrable, an overwhelming majority describe physical systems either in 1 dimension (with only one time dimension and no continuous spatial dimensions) or in 2 dimensions (with one time dimension and one space dimension). For this reason, it is widely believed that integrability is a phenomenon of low-dimensional systems. Indeed, although a handful of famous examples of integrable models in 3 and even 4 dimensions exist, there is currently no available universal mathematical framework for studying integrable systems in 3 dimensions and above. Developing a better understanding of the property of integrability in higher dimensions is therefore highly desirable and the key to making transformative progress in this field, especially given that most realistic physical systems are 4-dimensional, with one time dimension and three spatial dimensions. Over the past five years, the development of two radically new frameworks has triggered tremendous progress in our conceptual understanding of 2-dimensional integrable models, which as a consequence has led to an ever-growing list of such models. The first is of a more geometric nature and is based on a certain 4-dimensional gauge field theory, known as `4-dimensional Chern-Simons theory', in which the two additional dimensions of spacetime are used to encode the integrable structure of the 2-dimensional integrable model. The second is more algebraic in nature and encodes the integrable structure of the 2-dimensional integrable model in terms of a so-called `affine Gaudin model'. The project team has made significant contributions to both of these frameworks. The central aim of this project is to develop a novel and comprehensive theory of higher-dimensional integrable models using techniques from higher gauge theory. Our approach to constructing and analysing integrable models in D+1 dimensions (with one time dimension and arbitrary D spatial dimensions) will be based on the study of higher gauge-theoretic versions of 4-dimensional Chern-Simons theory living in D+3 dimensions, and their connections to D-dimensional generalisations of affine Gaudin models. This work, which lies at the intersection between the well-established field of integrability and the modern field of higher-categorical structures, will pave the way for a systematic study of integrable models in higher dimensions by providing the necessary long-sought-after framework rooted in higher gauge theory. At the same time, the rich structures exhibited by higher-dimensional integrable models will provide valuable inspiration and guidance for the further development of new and existing higher-categorical concepts, especially in the context surrounding factorisation algebras.