superalgebra and (synthetic ) supergeometry
Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object in there.
As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.
As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superschemes.
From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.
For more see at geometry of physics – supergeometry.
Supergeometry may defined over an arbitrary commutative ring (Schwarz and Shapiro 06).
duality between algebra and geometry
in physics:
Historically influential general considerations:
Yuri Manin, New Dimensions in Geometry, in: Arbeitstagung Bonn 1984, Lecture Notes in Mathematics 1111, Springer (1985) 59–101 [doi:10.1007/BFb0084585, pdf]
Felix A. Berezin (edited by Alexandre A. Kirillov): Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6]
Survey:
Yuri Manin, Chapter 4 in: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]
John Lott, Torsion constraints in supergeometry, Comm. Math. Phys. 133 (1990) 563-615 [doi:10.1007/BF02097010]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]
Pierre Deligne, John Morgan, Notes on Supersymmetry (following Joseph Bernstein), in: Quantum Fields and Strings, A course for mathematicians, 1, Amer. Math. Soc. Providence (1999) 41-97 [ISBN:978-0-8218-2014-8, web version, pdf]
Ivan Mirković, Notes on Super Math, in Quantum Field Theory Seminar, lecture notes (2004) [pdf, pdf]
Introductory lecture notes:
Gennadi Sardanashvily, Lectures on supergeometry [arXiv:0910.0092, inspire:832789]
Discussion via functorial geometry:
Claudio Carmeli, Lauren Caston, Rita Fioresi: Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, EMS (2011) [ems:elm/97, doi:10.4171/097, epdf, short draft: arXiv:0710.5742]
Lectures on supergeometry highlighting methods from algebraic geometry:
Discussion of traditional algebraic geometry for super-schemes:
The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in
Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)
Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48
and in
A summary/review is in the appendix of
Anatoly Konechny, Albert Schwarz,
On -dimensional supermanifolds, in Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (Dmitry Volkov memorial volume), Lecture Notes in Physics 509, Springer (1998) [arXiv:hep-th/9706003, doi:10.1007/BFb0105247]
Theory of -dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486
Generalization to supergeometry over an arbitrary commutative ring, in particular -adic rings, is given in
A review of all this as geometry in the topos over the category of superpoints is in
Formulation in terms of synthetic differential supergeometry is in
For many more references see at supermanifold.
Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in
especially in the contribution
The appendix there
means to sort out various sign conventions of relevance.
Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in
For more on this see at superalgebra.
Discussion related to G-structure and Killing spinors includes
Generalization of Artin's representability theorem to supergeometry:
For AQFT-like discussion of supersymmetric field theory:
See also:
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:
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):
Last revised on November 12, 2024 at 10:45:50. See the history of this page for a list of all contributions to it.