Quantum theory and the continuum: The case of maximal supersymmetry.

Research Project, 2011 – 2013

Finding a rigorous definition of a consistent four-dimensional quantum field theory remains an outstanding problem. Empirical as well theoretical evidence indicates that this would have to be a non-abelian Yang-Mills theory, and the maximally supersymmetric (N=4) version presently appears to offer the best hope for progress. In contrast to most models with less or no supersymmetry, this theory is conformally invariant and characterized by a dimensionless coupling parameter even at the quantum level. Furthermore, there is by now substantial evidence for an equivalence known as S-duality, which relates theories with different values of this parameter. But this property, which should play an important role in a rigorous definition of the theory, is not at all manifest in the usual formulation. A promising idea, which makes S-duality manifest, is to consider this theory as the low-energy limit of a larger theory, known as six-dimensional (2,0)-theory, compactified on a small two-torus. The latter theory is a new and rather abstract kind of quantum theory; in particular it does not admit a Lagrangian formulation. This project, which is an extension of a previous grant, aims at continuing the systematic study of these maximally supersymmetric quantum theories, including an intermediate formulation in terms of five-dimensional Yang-Mills theory, with the ultimate goal of arriving at a proper definition of them.

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