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.

pdf (515 kb)   tex (37 kb)  

References

1. Ammann B., Dahl M., Humbert E., Harmonic spinors and local deformations of the metric, Math. Res. Lett. 18 (2011), 927-936, arXiv:0903.4544.
2. Atiyah M.F., Bott R., Shapiro A., Clifford modules, Topology 3 (1964), 3-38.
3. Bär C., Ballmann W., Guide to elliptic boundary value problems for Dirac-type operators, in Arbeitstagung Bonn 2013, Progr. Math., Vol. 319, Birkhäuser, Cham, 2016, 43-80, arXiv:1307.3021.
4. Bär C., Ginoux N., Pfäffle F., Wave equations on Lorentzian manifolds and quantization, ESI Lect. Math. Phys., European Mathematical Society (EMS), Zürich, 2007, arXiv:0806.1036.
5. Bär C., Strohmaier A., An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary, Amer. J. Math. 141 (2019), 1421-1455, arXiv:1506.00959.
6. Bernal A.N., Sánchez M., On smooth Cauchy hypersurfaces and Geroch's splitting theorem, Comm. Math. Phys. 243 (2003), 461-470, arXiv:gr-qc/0306108.
7. Booß-Bavnbek B., Wojciechowski K.P., Elliptic boundary problems for Dirac operators, Math. Theory Appl., Birkhäuser, Boston, MA, 1993.
8. Dai X., Freed D.S., $\eta$-invariants and determinant lines, J. Math. Phys. 35 (1994), 5155-5194.
9. Freed D.S., Lectures on field theory and topology, CBMS Reg. Conf. Ser. Math., Vol. 133, American Mathematical Society, Providence, RI, 2019.
10. Halmos P.R., Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381-389.
11. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
12. Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
13. Ludewig M., Roos S., The chiral anomaly of the free fermion in functorial field theory, Ann. Henri Poincaré 21 (2020), 1191-1233, arXiv:1909.04212.
14. Palais R.S., Seminar on the Atiyah-Singer index theorem, Ann. of Math. Stud., Vol. 57, Princeton University Press, Princeton, NJ, 1965.
15. Schommer-Pries C.J., The classification of two-dimensional extended topological field theories, Ph.D. Thesis, University of California, Berkeley, 2009, arXiv:1112.1000.
16. Stolz S., Teichner P., What is an elliptic object?, in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., Vol. 308, Cambridge University Press, Cambridge, 2004, 247-343.
17. Stolz S., Teichner P., Supersymmetric field theories and generalized cohomology, in Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proc. Sympos. Pure Math., Vol. 83, American Mathematical Society, Providence, RI, 2011, 279-340, arXiv:1108.0189.