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.
