fermion in nLab

Supergeometry of fermion fields

Discussion of the classical mechanics of the spinning particle or of classical field theory with fermion fields (possibly but not necessarily super-symmetric) as taking place in supergeometry:

via (possibly infinite-dimensional) supermanifolds:

• Felix A. Berezin, M. S. Marinov: Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics, Annals of Physics 104 2 (1977) 336-362 [doi:10.1016/0003-4916(77)90335-9, pdf, pdf]

  reprinted in Appendix I of: Alexandre A. Kirillov (ed.): Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6]

• Thomas Schmitt: The Cauchy Problem for Classical Field Equations with Ghost and Fermion Fields [arXiv:hep-th/9607133]

• Thomas Schmitt: Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?, Rev. Math. Phys. 9 (1997) 993-1052 [doi:10.1142/S0129055X97000348, arXiv:hep-th/9607132].

• Thomas Schmitt: Supermanifolds of classical solutions for Lagrangian field models with ghost and fermion fields, Sfb 288 Preprint No. 270 [hep-th/9707104, inspire:445574]

• Daniel Freed, What are fermions?, Lecture 1 in: Five lectures on supersymmetry, AMS (1999) [ISBN:978-0-8218-1953-1, spire:517862]

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, chapter 3 of: Advanced classical field theory, World Scientific (2009) [doi:10.1142/7189]

• Gennadi Sardanashvily, Grassmann-graded Lagrangian theory of even and odd variables, [arXiv:1206.2508]

• Gennadi Sardanashvily W. Wachowski: SUSY gauge theory on graded manifolds [arXiv:1406.6318, spire:1302860]

• Viola Gattus, Apostolos Pilaftsis, Supergeometric Approach to Quantum Field Theory, CORFU2023, PoS 463 (2024) 156 [doi:10.22323/1.463.0156, arXiv:2404.13107]

• Viola Gattus, Apostolos Pilaftsis: Supergeometric Quantum Effective Action [arXiv:2406.13594]

and more generally via smooth super sets:

• Urs Schreiber: §3.1 in: Higher Prequantum Geometry, chapter in: New Spaces for Mathematics and Physics, Cambridge University Press (2021) [doi:10.1017/9781108854399.008, arXiv:1601.05956]

Discussion with focus on supersymmetry:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2.4 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]

• Pierre Deligne, Daniel Freed: Supersolutions, in: Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence (1999) 357-366 [arXiv:hep-th/9901094, ISBN:978-0-8218-2014-8, web version]

• Daniel Freed, Classical field theory and Supersymmetry, IAS/Park City Mathematics Series 11 (2001) [pdf, pdf]

and specifically in the context of super- string theory (regarding worldsheets as super Riemann surfaces):

• Eric D'Hoker, Duong Phong, Complex geometry and supergeometry, Current Developments in Mathematics 2005 (2007) 1-40 [arXiv:hep-th/0512197, euclid.cdm/1223654523]