Vertex Algebras Associated with 2- and 4-dimensional Conformal Field Theories

• Applicant: Dr. Sven Möller
• Subject Area: Mathematics ; Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 460925688

Quantum field theories describe the physics of subatomic particles. Conformal field theories (CFTs) are particularly symmetric field theories that admit a mathematically rigorous axiomatisation in two dimensions by means of vertex operator algebras (VOAs). VOAs are also the algebraic structures at the centre of remarkable connections (dubbed "moonshine") between group theory, the mathematical description of symmetries, and modular forms, certain complex functions that arise in number theory. Recently, it was discovered that certain supersymmetric four-dimensional CFTs relevant in string theory correspond to VOAs as well (4d/2d-correspondence).While there are well-studied families of examples, often motivated from physics, and despite some prominent applications in mathematics, the understanding of many essential aspects of VOAs is still quite limited. The proposed research plan aims to study the structure and representation theory of VOAs with the goal of charting the landscape of two- and four-dimensional CFTs.One focus will be on orbifold problems. Given a VOA the aim is to describe the substructure of this VOA that is invariant under a group of symmetries of this VOA, and which may then be extended again to obtain a new VOA. Such constructions are extremely important as they provide non-classical examples of VOAs (such as the famous moonshine module). To understand orbifold theory, it is necessary to study representations of VOAs, which are objects similar to but more general than VOAs and on which VOAs act. The representation theory of a VOA and in particular how the representations "fuse" together to form new representations can be efficiently encoded in (modular) tensor categories, which will play an important role in this project.The second focus of this research proposal is on VOAs that arise in the 4d/2d-correspondence from certain supersymmetric four-dimensional CFTs, namely those with an "enhanced" supersymmetry. The main goal is to prove a recent conjecture by the physicists Bonetti, Meneghelli and Rastelli that these theories correspond to VOAs associated with certain reflection groups. As a first step, these VOAs have to be rigorously constructed, which can probably achieved by generalising a realisation due to Adamović. A proof of this conjecture is important as it demonstrates that VOAs may serve as an organising principle for a rich subclass of four-dimensional theories. The VOAs arising in this context are also worth investigating from a mathematical point of view and possess many interesting (geometric) features like their associated varieties due to Arakawa.

• DFG Programme: Independent Junior Research Groups