superalgebra and (synthetic ) supergeometry
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A supermanifold is a space locally modeled on Cartesian spaces and superpoints.
There are different approaches to the definition and theory of supermanifolds in the literature. The definition
is popular. The definition
has been argued to have advantages, see also the references at super ∞-groupoid.
See at geometry of physics – supergeometry the section Supermanifolds.
We discuss a description of supermanifolds that goes back to (Berezin & Leites 1975).
A supermanifold $X$ of dimension $p|q$ is a ringed space $(|X|, O_X)$ where
the topological space $|X|$ is second countable space, Hausdorff space,
$O_X$ is a sheaf of commutative super algebras that is locally on small enough open subsets $U \subset |X|$ isomorphic to one of the form $C^\infty(\mathbb{R}^p) \otimes \wedge^\bullet \mathbb{R}^q$.
A morphism of supermanifolds is a homomorphism of ringed spaces (…).
Forgetting the graded part by projecting out the nilpotent ideal in $O_X$ (i.e. applying the bosonic modality) yields the underlying ordinary smooth manifold $X_{red}$.
One just writes $C^\infty(X)$ for the super algebra $O_X(X)$ of global sections.
With the obvious morphisms of ringed space this forms the category SDiff of supermanifolds.
For $E \to X$ a smooth finite-rank vector bundle the manifold $X$ equipped with the Grassmann algebra over $C^\infty(X)$ of the sections of the dual bundle
is a supermanifold. This is usually denoted by $\Pi E$.
In particular, let $\mathbb{R}^{p+q} \to \mathbb{R}^p$ be the trivial rank $q$ vector bundle on $\mathbb{R}^p$ then one writes
for the corresponding supermanifold.
(Batchelor’s theorem)
Every supermanifold is isomorphic to one of the form $\Pi E$ where $E$ is an ordinary smooth vector bundle.
(Batchelor 1979 reviewed in Batchelor 1984, §1.13; Rogers 2007, §8.2)
Nevertherless, the category of supermanifolds is far from being equivalent to that of vector bundles: a morphism of vector bundles translates to a morphism of supermanifolds that is strictly homogeneous in degrees, while a general morphism of supermanifolds need not be of this form.
But we have the following useful characterization of morphisms of supermanifolds:
There is a natural bijection
so the contravariant embedding of supermanifolds into superalgebra is a full and faithful functor.
Composition with the standard coordinate functions on $\mathbb{R}^{p|q}$ yields an isomorphism
The first statement is a direct extension of the classical fact that smooth manifolds embed into formal duals of R-algebras.
We discuss a description of supermanifolds that goes back to (DeWitt 92) and (Rogers 2007).
(…)
Let $SuperPoint$ be the category of superpoints. And $Sh(SuperPoint) = PSh(SuperPoint)$ its presheaf topos.
We discuss a definition of supermanifolds that characterizes them, roughly, as manifolds over this base topos. See (Sachse) and the references at super ∞-groupoid.
See also this post at Theoretical Atlas.
Let
be the sheaf topos over superpoints. Let
be the canonical continuum real line under the restricted Yoneda embedding of supermanifolds and equipped with its canonical internal algebra structure, hence by prop. the presheaf of algebras which sends a Grassmann algebra to its even subalgebra, as discussed at superalgebra.
A superdomain is an open subfunctor (…) of a locally convex $\mathbb{K}$-module.
This appears as (Sachse, def. 4.6).
We now want to describe supermanifolds as manifolds in $SuperSet$ modeled on superdomains.
Write SmoothMfd for the category of ordinary smooth manifolds.
A supermanifold is a functor $X : SuperPoint^{op} \to SmoothMfd$ equipped with an equivalence class of supersmooth atlases.
A morphism of supermanifolds is a natural transformation $f : X \to X'$, such that for each pair of charts $u : U \to X$ and $u' : U' \to X'$ the pullback
can be equipped with the structture of a Banach superdomain such that $p_1$ and $p_2$ are supersmooth (…)
This appears as (Sachse, def. 4.13, 4.14).
The categories of supermanifolds defined as locally ringed spaces, def. and as manifolds over superpoints, def. are equivalent.
This appears as (Sachse, theorem 5.1). See section 5.2 there for a discussion of the relation to the DeWitt-definition.
Via the formally dual superalgebra of the super-function algebras on supermanifolds
Discussion from the point of view of functorial geometry:
Claudio Carmeli, Lauren Caston, Rita Fioresi, Mathematical Foundations of Supersymmetry, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( ISBN:978-3-03719-097-5, arXiv:0710.5742)
Henning Hohnhold, Stephan Stolz, Peter Teichner: Super manifolds: an incomplete survey, Bulletin of the Manifold Atlas (2011) 1-6 [webpage, pdf, pdf]
Henning Hohnhold, Matthias Kreck, Stephan Stolz, Peter Teichner, Sections 2-3 of: Differential forms and 0-dimensional supersymmetric field theories, Quantum Topology 2 1 (2011) 1–41 [doi:10.4171/QT/12, pdf]
Felix A. Berezin, Dimitry A. Leites: Supermanifolds, (Russian) Dokl. Akad. Nauk SSSR 224 3 (1975) 505-508; English transl.: Soviet Math. Dokl. 16 5 (1975) 1218-1222 [mathnet:dan39282]
Marjorie Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979) 329-338 [doi:10.1090/S0002-9947-1979-0536951-0]
Dimitry A. Leites, Introduction to the Theory of Supermanifolds, Russ. Math. Surv. 35 1 (1980) [doi:10.1070/RM1980v035n01ABEH001545, MathNet, iop, pdf]
Marjorie Batchelor, Graded Manifolds and Supermanifolds in: Mathematical Aspects of Superspace, NATO ASI Series 132, Springer (1984) 91-134 [doi:10.1007/978-94-009-6446-4_4]
Yuri Manin, §4.1 in: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]
I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace CRC Press (1998) [ISBN:10.1201/9780367802530]
Pierre Deligne, John Morgan, Ch 2 in: 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, Sec 2 in: Notes on Super Math, in Quantum Field Theory Seminar, lecture notes (2004) [pdf, pdf]
A more general variant of this in the spirit of locally algebra-ed toposes:
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 $(k \oplus l|q)$-dimensional supermanifolds, in: Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Lecture Notes in Physics, 509, Springer 1998 (arXiv:hep-th/9706003)
Anatoly Konechny, Albert Schwarz, Theory of $(k \oplus l|q)$-dimensional supermanifolds, Sel. math., New ser. 6 (2000) 471 - 486 (doi:10.1007/PL00001396)
Albert Schwarz, I. Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)
A review with more emphasis on the relevant category theory/topos theory is in
The site of formal duals not just to Grassmann algebras but to all super-infinitesimally thickened points is discussed in (Konechny-Schwarz) above and also in
Katsumi Yagi: Super manifolds, Osaka J. Math. 25 4 (1988) 909-932 [euclid:ojm/1200781174]
Bryce DeWitt, Supermanifolds, Monographs on Mathematical Physics, Cambridge University Press (1984, 1992) [doi:10.1017/CBO9780511564000]
Alice Rogers, Supermanifolds: Theory and Applications, World Scientific (2007) [doi:10.1142/1878]
(Rogers 2007, Ch. 1 that the smooth-manifold-of-(infinite-dimensional)-Grassmann-algebras approach (the “concrete approach”) is identical to the sheaf-of-ringed-spaces approach (the “algebro-geometric” approach) and that this equivalence is shown in Chapter 8. DeWitt seems unsure of this, but is writing more than 20 years earlier, before the ringed-space approach has been fully developed.)
Alice Rogers, Aspects of the Geometrical Approach to Supermanifolds, in: Mathematical Aspects of Superspace, NATO ASI Series 132, Springer (1984) 135-148 [doi:10.1007/978-94-009-6446-4_5]
Discussion with an eye towards integration over supermanifolds:
Discussion of global properties:
See also:
Yuri Manin, Topics in noncommutative geometry, Princeton Univ. Press 1991.
Pierre Deligne, P. Etingof, Daniel Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and Edward Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Veeravalli Varadarajan, The Concept of a Supermanifold and Elementary Theory of Supermanifolds, Chapters 2 and 4 in: Supersymmetry for mathematicians: An introduction, Courant Lecture Notes in Mathematics 11, American Mathematical Society (2004) [doi:10.1090/cln/011]
Alberto S. Cattaneo, Florian Schaetz, Introduction to supergeometry [arxiv/1011.3401]
There are many books in physics on supersymmetry (most well known is by Wess and Barger from 1992), but they emphasise more on the supersymmetries rather than on (the superspace and) supermanifolds. They should therefore rather be listed under supersymmetry.
See also pdf
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 9, 2024 at 12:28:36. See the history of this page for a list of all contributions to it.