Partial differential equations and quantum fields on supermanifolds

• Applicant: Professor Dr. Alexander Schenkel
• Subject Area: Mathematics
• Term: from 2014 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 252118965

One of the most indispensable tools in modern theoretical physics is quantum field theory (QFT). Due to its immense physical success and its rich mathematical structure, QFT has grown over the past decades into an established field of mathematics and mathematical physics. One of the leading candidates for a mathematical formulation of QFT is algebraic QFT. On the one hand, it provides a conceptual foundation by emphasizing the relevant structure of QFTs -- the algebras of observables localized in regions of spacetime. On the other hand, it provides a set of axioms which any reasonable model should fulfill -- the Haag-Kastler axioms. In recent years the field of algebraic QFT has made tremendous developments, most notably the extension of the Haag-Kastler framework from Minkowski spacetime to generic globally hyperbolic Lorentzian manifolds and, connected to this, the rigorous perturbative methods for QFTs on such manifolds.The goal of my project is to develop the important extension of algebraic QFT to the realm of supergeometry, as well as to construct explicit models of perturbatively interacting QFTs on supermanifolds. Due to the tight interplay between geometry, analysis and algebraic QFT, this research goal will lead additionally to novel and interesting problems in the analysis of partial differential equations on supermanifolds and in the analysis of singularities of superdistributions, which we shall study in this project as well. The motivation for my research is twofold: Firstly, supersymmetry and supergeometry have become major guiding principles in many modern approaches to fundamental physics. The rigorous techniques for QFTs on supermanifolds to be developed in my research will find an immediate application to such fields. Secondly, earlier investigations have shown that perturbatively interacting supersymmetric QFTs enjoy remarkable renormalization properties. It is of major importance to understand the reason for such features in a conceptually clean framework, such as perturbative algebraic QFT. In the long run, this may eventually pave the way for constructing such models in full mathematical rigor.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Host: Professor Dr. Richard J. Szabo