Algebraic Foundations of Supersymmetric Quantum Field Theory

Project Information

SCFTAlg

Grant agreement ID: 864828

DOI

10.3030/864828

EC signature date 5 February 2020

Start date 1 September 2020

End date 31 August 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 963 921,00

EU contribution

€ 1 963 921,00

Coordinated by

THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
United Kingdom

Periodic Reporting for period 3 - SCFTAlg (Algebraic Foundations of Supersymmetric Quantum Field Theory)

Reporting period: 2023-09-01 to 2025-02-28

Summary of the context and overall objectives of the project

Quantum field theory (QFT) is a ubiquitous framework for describing quantum systems involving large numbers of interacting degrees of freedom. It is the lingua franca of particle physics, condensed matter physics, early universe cosmology, quantum gravity, and more, and increasingly, it is also a source of deep and surprising mathematical ideas in areas ranging from analytic number theory to enumerative geometry to low-dimensional topology and further.

The best developed and most successful tools for studying quantum field theories leverage perturbative methods and apply when the systems in question describe some (quasi-)particle excitations subject to local, weak interactions. The greatest challenges of the moment in the formal study of QFT revolve around the issue of strong coupling and non-perturbative quantum dynamics. Indeed, many of the most puzzling and important physical phenomena, from the strong nuclear force to high temperature superconductivity to quantum gravity and black hole physics (via the holographic principle), call for an improved treatment of strongly coupled QFT.

It is important to identify and pursue simplifying assumptions other than weak coupling that can be made about a QFT to facilitate progress. Two such assumptions are conformal invariance and supersymmetry. Indeed, superconformal symmetry turns out to be an incredibly potent combination: conformal invariance provides an underlying algebraic foundation in the form of the operator product expansion, and supersymmetry endows this algebraic structure with a number of remarkable properties that not only improve its tractability but provide connections to many areas of modern mathematics. Furthermore, superconformal field theories (SCFTs) are deeply intertwined with contemporary work in string theory and M-theory, granting them significant importance in their own right.

The overarching aim of this ERC project is to develop intrinsically non-perturbative techniques for analysing SCFTs on the basis of their distinctive algebraic structure, and to leverage these techniques in the pursuit of various high-level goals in the study of SCFTs such as:
• Constraining the landscape of putative SCFTs.
• Classifying SCFTs for spacetime dimension greater than three.
• Organising the space of SCFTs and their observables according to non-perturbative algebraic principles.
• Solving specific SCFTs at the level of their supersymmetric operator algebras.
• Predicting new mathematical structures using quantum field theoretic techniques.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

In the first period of this project, the team has made progress on a variety of problems connected to strongly coupled supersymmetric quantum field theories, and especially their algebraic structure.

Perhaps the area where that has seen the most progress is in connection to the "free field realisation" of certain supersymmetric algebras of observables (taking the form of vertex operator algebras). The free field realisations in question can be understood as encoding the structure of observables in a strongly-coupled SCFT in terms of the low-energy (weakly coupled) degrees of freedom of the theory in a different "Higgsed" phase where conformal symmetry is spontaneously broken. This perspective has proven incredibly fruitful, and we have been able to develop a much more systematic understanding of the mathematics underpinning these free field realisations in various settings and in doing so, extending them to a larger range of systems than we previously anticipated. In this area, the main results in the first period have been:
1. The free field realisation of the "chiral universal centraliser" that describes the class S theory associated to an unpunctured sphere.
2. The rigorous analysis of boundary vertex algebras for A-twisted three-dimensional N=4 Abelian gauge theories leading to a derivation of free field realisations that are covariant with respect to outer automorphism symmetries encoding the symmetry of the Coulomb branch of those same theories.
3. The development of a new and systematic approach to the free field realisation of affine Kac-Moody vertex algebras and their associated W-algebras in type A that trivialise the operation of quantised Drinfel'd-Sokolov reduction.

Another area of significant progress has been the identification and study of new algebraic structures arising in twisted supersymmetric quantum field theory and string theory setups. Two significant accomplishments in this direction have been:
1. The proposal of a novel algebraic structure controlling the instanton partition functions of G2 manifolds and the development of concrete computational tools to allow for the computation of those partition functions.
2. The rigorous construction of vertex algebras out of BPS algebras of a large class of toric Calabi-Yau three-fold geometries.

Other work done in the first period includes:
- Elucidating the quasi-modular structure of superconformal indices of class S superconformal field theories.
- Uncovering new and subtle algebraic-geometric features of classical Higgs branch chiral rings of "bad" three-dimensional gauge theories.
- Identifying large classes of novel isomorphisms between irregular Hitchin systems of type A, with the consequence that the set of "Argyres-Douglas" type vertex algebras is under much better control than previously.
- Explaining the interplay between duality of three-dimensional SCFTs and generalised/higher form symmetries.
- Developing the systematic understanding of three- and four-dimensional quantum systems that arise by compactification from two dimensions higher.
- New applications of the nascent "twisted holography" framework to understand the algebra of operators in certain four-dimensional N=1 gauge theories.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Essentially all results mentioned above go beyond the state of the art, with the work on free field realisations and on new algebraic structures being of particular note.

In the second period of the project, the team will make a concerted effort to formalise the free-field technology for studying vertex algebras as well as considering the finite analogue in the context of quantised Higgs and Coulomb branches for three-dimensional N=4 SCFTs. The work on BPS algebras of Calabi-Yau threefolds will be exploited to attack the still-mysterious connection between Coulomb branch BPS particles and the associated vertex algebra of four-dimensional N=2 SCFTs. We also plan to return to the problem of characterising the constraints of unitarity of three- and four-dimensional SCFTs on their associated cohomological operator algebras.