To link integrable models and vertex operators algebras in CFT via symmetric functions

This is a collaborative project between Dr Simon Wood (Cardiff University) and Prof Christian Korff (University of Glasgow) and and their international network of collaborators. 

There are many interesting examples of vertex operator algebras (and their associated conformal field theories) and quantum integrable models that can be constructed from the Heisenberg algebra. In the conformal field theory literature these theories are usually referred to as the free boson (with various qualifiers). The Heisenberg algebra has a natural realisation in terms of the ring of symmetric functions (in terms of multiplication and differential operators). A consequence of this is that, for each example of a vertex operator algebra or quantum integrable model constructed from the Heisenberg algebra, there is a distinguished basis of the ring of symmetric functions which encodes many of the properties of the vertex operator algebra or integrable model. Therefore, symmetric functions, with their rich combinatorics, provide a powerful computational tool for analysing and understanding vertex operator algebras and integrable models. They also provide a crucial guiding principle for going beyond vertex operator algebras to so-called deformed vertex operator algebras.

Objective 1: Endow the deformed Virasoro algebra with the structure of a deformed vertex operator algebra in full generality by studying the role that the basis of Macdonald functions (and some of its special limits) plays in this context.

Objective 2: Construct Macdonald functions via quantum integrable statistical lattice models and use their underlying algebraic structures (Yang-Baxter algebras) to study the deformed Virasoro and deformed vertex operator algebras.

Objective 3: Combine the results of Objectives 1&2 to study open questions within the representation theory of deformed vertex operator algebras and quantum integrable systems.

In addition to the primary research objectives of the proposal we will also train early career researchers (employed through the grant) in highly intradisciplinary and active subjects: conformal field theory, integrable models, representation theory, symmetric functions, vertex operator algebra, etc. We will support them by teaching them the concepts from distinct mathematical communities and help them to build networks within and across these communities. The organisation of a workshop and summer school will further support the development of early-career researchers and will foster collaborationsand cross-fertilisation between different mathematical disciplines.