Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 036, 29 pages      arXiv:2212.02956      https://doi.org/10.3842/SIGMA.2024.036
Contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär for his 60th birthday

Categories of Lagrangian Correspondences in Super Hilbert Spaces and Fermionic Functorial Field Theory

Matthias Ludewig
Fakultät für Mathematik, Universität Regensburg, Germany

Received May 19, 2023, in final form April 16, 2024; Published online April 24, 2024

Abstract
In this paper, we study Lagrangian correspondences between Hilbert spaces. A main focus is the question when the composition of two Lagrangian correspondences is again Lagrangian. Our answer leads in particular to a well-defined composition law in a category of Lagrangian correspondences respecting given polarizations of the Hilbert spaces involved. As an application, we construct a functorial field theory on geometric spin manifolds with values in this category of Lagrangian correspondences, which can be viewed as a formal Wick rotation of the theory associated to a free fermionic particle in a curved spacetime.

Key words: Lagrangians; correspondences; functorial field theory; Clifford algebras.

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