Partial differential equations and quantum fields on supermanifolds
Final Report Abstract
Quantum field theory (QFT) is among the most indispensable tools of modern theoretical physics. It has a wide range of applications reaching from elementary particle physics over cosmology to solid state physics. An important class of QFTs are those that are defined on Lorentzian manifolds, which are models for spacetime in Einstein’s theory of general relativity. Such models may be captured within the relatively modern axiomatic approach called locally covariant QFT, which has been proposed by Brunetti, Fredenhagen and Verch in 2001 and since then actively developed further. The basic idea of locally covariant QFT is to formalize a QFT in terms of a functor which coherently assigns observable algebras (i.e. measurable quantities of a QFT) to Lorentzian spacetimes. The main goal of my research fellowship was to extend the framework of locally covariant QFT to the context of supergeometry. The focus thus was on QFTs which are defined on suitable categories of superspacetimes, which I will call in the following super-QFTs. Super-QFTs play an important role in modern theoretical and mathematical physics, where they are used to develop extensions of the standard model of particle physics and to study low-energy properties of string theory. An axiomatic approach to super-QFTs, e.g. obtained by generalizing locally covariant QFT to supergeometry, would provide the conceptual and mathematical foundations for such studies of super-QFTs and thus is a well-motivated development. In a joint work with T.-P. Hack and F. Hanisch, we performed a thorough study of noninteracting models of super-QFTs that are defined on super-Cartan supermanifolds, which are models for superspacetime in supergravity. Our results revealed the necessity of techniques from enriched category theory in order to capture the important concept of supersymmetry transformations in a functorial formulation of super-QFTs. This motivated us to propose and develop an axiomatization of super-QFTs in terms of enriched functors from a suitably enriched category of superspacetimes to an enriched category of superalgebras, for which we constructed explicit examples. The perturbative construction of an interacting super-QFT requires a deep analysis and understanding of the singularities of distributions arising in the underlying free theory. To efficiently capture and analyze the singularities of distributions on supermanifolds one requires a supergeometric generalization of microlocal analysis. In a joint work with C. Dappiaggi, H. Gimperlein and S. Murro, we developed the foundations for microlocal analysis on supermanifolds. In particular, we defined and studied a supergeometric generalization of the wavefront set of a distribution, which enabled us to detect polarization information of superdistributions. This has led to a refined pullback theorem for superdistributions, which in particular establishes criteria when two superdistributions may be multiplied, and to a refined propagation of singularities theorem for solutions to supergeometric field equations. Together with our results on non-interacting super- QFTs, this establishes the foundations for future developments on perturbatively interacting super-QFTs. Besides these developments on super-QFTs, we made major progress in understanding the mathematical structure of quantum gauge theories. In a joint work with Becker, Benini and Szabo, we developed a quantization of differential cohomology theories that is manifestly covariant under Abelian duality, leading in particular to a rigorous proof of existence of such dualities. In a joint work with Benini and Szabo, we initiated the development of a homotopy theoretic generalization of locally covariant QFT. Our results showed that this approach has a great potential to solve the recently discovered inconsistencies between locally covariant QFT and gauge theory.
Publications
- “Abelian duality on globally hyperbolic spacetimes,” Commun. Math. Phys.
C. Becker, M. Benini, A. Schenkel and R. J. Szabo
(See online at https://doi.org/10.1007/s00220-016-2669-9) - “Wavefront sets and polarizations on supermanifolds”
C. Dappiaggi, H. Gimperlein, S. Murro and A. Schenkel
- “Homotopy colimits and global observables in Abelian gauge theory,” Lett. Math. Phys. 105, no. 9, 1193 (2015)
M. Benini, A. Schenkel and R. J. Szabo
- “Supergeometry in locally covariant quantum field theory,” Commun. Math. Phys. 342, no. 2, 615 (2016)
T. P. Hack, F. Hanisch and A. Schenkel
