# Algebraic Properties of Superconformal Field Theories

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

This project falls within the EPSRC Mathematical Physics Research Area

Conformal Field Theories (CFTs) are an important class of Quantum Field Theories (QFTs) that enjoy a large enhancement of their spacetime symmetries. Conformal field theories which preserve supersymmetry enjoy an even larger group of symmetries and are known as Superconformal Theories (SCFTs). While QFTs have seen great success in a range of applications, their analysis is often limited by a reliance on perturbative methods, which fail in the regime of strong coupling. The algebraic properties of CFTs and SCFTs, however, allow for the use of powerful non-perturbative methods.

Specifically, conformal field theories are equipped with an Operator Product Expansion (OPE), which is an algebraic operation in the space of local operators. The OPE allows a product of local operators to be expanded in a power series of primary operators and their descendants. To ensure well-definedness of correlation functions, the OPE algebra must be associative. Enshrining this requirement of associativity as a key feature of a CFT has led to the idea of the Conformal Bootstrap, which seeks to constrain the data that defines a valid CFT.

In two-dimensional CFTs, the OPE algebra is highly constrained by the presence of infinite chiral subalgebras in terms of which the rest of the OPE algebra is organized. Here one finds rich algebraic structures such as Virasoro and Kac-Moody vertex operator algebras. In [1], it was shown that such chiral algebras also arise cohomologically in four-dimensional N=2 superconformal field theories, which has led to the discovery of a bevy of important and surprising consequences for the four-dimensional physics such as novel central charge bounds and connections to modular forms.

The power of the OPE algebra also allows one to analyse theories which may not have Lagrangian descriptions. For instance, there are the theories of Class S, which are the low-energy limit of a six dimensional superconformal theory compactified on a Riemann Surface. With a few exceptions, these theories are not described by a Lagrangian field theory. In [2], the chiral algebra correspondence of [1] was used to investigate the structure of the space of Class S theories. By associating the chiral algebra of a Class S theory with its UV curve (a two-dimensional manifold), it is possible to use the methods of generalized topological quantum field theory to capture some of the structure of Class S theories. This approach was subsequently used to great effect to define rigorously the chiral algebras of Class S theories in [3].

The aim of this project will be to exploit the novel mathematical structures present in chiral algebras of Class S to elucidate their structure and extend the analysis of these models that has been performed to date. Some concrete goals include a clarification of the physical interpretation of the results of Arakawa for non-simply laced Lie algebras, a generalization of those results to the case of class S theories involving general twisted punctures. These targets are at the cutting edge of the emerging interface between superconformal field theory and vertex operator algebras.

Conformal Field Theories (CFTs) are an important class of Quantum Field Theories (QFTs) that enjoy a large enhancement of their spacetime symmetries. Conformal field theories which preserve supersymmetry enjoy an even larger group of symmetries and are known as Superconformal Theories (SCFTs). While QFTs have seen great success in a range of applications, their analysis is often limited by a reliance on perturbative methods, which fail in the regime of strong coupling. The algebraic properties of CFTs and SCFTs, however, allow for the use of powerful non-perturbative methods.

Specifically, conformal field theories are equipped with an Operator Product Expansion (OPE), which is an algebraic operation in the space of local operators. The OPE allows a product of local operators to be expanded in a power series of primary operators and their descendants. To ensure well-definedness of correlation functions, the OPE algebra must be associative. Enshrining this requirement of associativity as a key feature of a CFT has led to the idea of the Conformal Bootstrap, which seeks to constrain the data that defines a valid CFT.

In two-dimensional CFTs, the OPE algebra is highly constrained by the presence of infinite chiral subalgebras in terms of which the rest of the OPE algebra is organized. Here one finds rich algebraic structures such as Virasoro and Kac-Moody vertex operator algebras. In [1], it was shown that such chiral algebras also arise cohomologically in four-dimensional N=2 superconformal field theories, which has led to the discovery of a bevy of important and surprising consequences for the four-dimensional physics such as novel central charge bounds and connections to modular forms.

The power of the OPE algebra also allows one to analyse theories which may not have Lagrangian descriptions. For instance, there are the theories of Class S, which are the low-energy limit of a six dimensional superconformal theory compactified on a Riemann Surface. With a few exceptions, these theories are not described by a Lagrangian field theory. In [2], the chiral algebra correspondence of [1] was used to investigate the structure of the space of Class S theories. By associating the chiral algebra of a Class S theory with its UV curve (a two-dimensional manifold), it is possible to use the methods of generalized topological quantum field theory to capture some of the structure of Class S theories. This approach was subsequently used to great effect to define rigorously the chiral algebras of Class S theories in [3].

The aim of this project will be to exploit the novel mathematical structures present in chiral algebras of Class S to elucidate their structure and extend the analysis of these models that has been performed to date. Some concrete goals include a clarification of the physical interpretation of the results of Arakawa for non-simply laced Lie algebras, a generalization of those results to the case of class S theories involving general twisted punctures. These targets are at the cutting edge of the emerging interface between superconformal field theory and vertex operator algebras.

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513295/1 | 01/10/2018 | 30/09/2023 | |||

2272671 | Studentship | EP/R513295/1 | 01/10/2019 | 31/03/2023 | Sujay Nair |