QM Research Seminar: Homological Algebra and Integrability

Speaker: Ryan Cullinan (​University of York).

Abstract: 
Integrable systems provide a pragmatic point in the landscape of physical theories. Such systems are characterised by their solvability, which is usually underpinned by the possession of an infinite-dimensional symmetry group, the existence of solitonic solutions, or admittance of an algebro-geometric description of the system.

Given the abundance of models deemed ‘integrable’ in one of the aforementioned senses, it becomes essential to ask what universal characteristics these models share that enable their integrability. A systematic understanding of Lax integrability for integrable field theories in two dimensions was provided by the introduction of a topological holomorphic theory in four dimensions called four-dimensional Chern-Simons theory, which illuminates many of the superficially mysterious aspects of a system's integrability by describing the data of a two-dimensional integrable field theory in the language of four-dimensional gauge theory.

​​​​​​​In this talk, we will describe the refinements required to build on this paradigm by studying a topological-holomorphic theory in five dimensions called five-dimensional 2-Chern-Simons. The aforementioned theory is a 2-categorical generalisation of its four-dimensional counterpart, describing Lax integrable field theories in three dimensions via the medium of higher categorical gauge theory. Bringing together results from homotopical and homological algebra, Hodge theory, and the BV formalism, we will outline the subtleties in defining the theory and in attempting to chart this new landscape of 'higher Lax integrable' theories.