# Supermanifolds — elementary introduction?

I am looking for an elementary but mathematically precise introductory text on supermanifolds in a modern differential geometric setting.

Elementary in the sense that there is plenty of motivation for the concepts and methods, and that these are explained in some detail with simple examples featuring only few bosonic and one fermionic coordinates, for example $$R^{1|1}$$ or $$\operatorname{OSp}(1|1)$$.

• There is an old review by Leites but I don't know if it has been superseded by a better or more recent reference. You can also look at the two IAS volumes on QFT edited by Deligne et al. Apr 19, 2017 at 13:53
• @AbdelmalekAbdesselam: Did you mean the following paper? math.northwestern.edu/~spoho/pdf/Leites.pdf Apr 19, 2017 at 14:05
• yes that's the one Apr 19, 2017 at 18:12

There is a short elementary survey by Hohnhold, Stolz, and Teichner: Super manifolds: an incomplete survey.

• Thanks. Reference 3 there is the source recommended above by Abdelmalek Abdesselam. Apr 21, 2017 at 20:47

Some further references, that might be of interest for your purposes:

• You can see at this article and the book Supermanifolds Theory and Applications by Alice Rogers. The article discusses -among others- the relation between the De Witt approach to supermanifolds and the approaches and definitions of Kostant and Leites (see also lower in this list). In the book, several topics of interest in physics, such as for example super Lie groups, the super Poincare group, Grassman algebras, $N=1$ supersymmetry, supergravity, topics and applications from string theory etc are studied from a modern algebraic-geometric point of view.
• The book Supermanifolds by Bryce DeWitt.
• The article Introduction to the theory of supermanifolds, D A Leites 1980 Russ. Math. Surv. 35, 1. (this has also been mentioned in the comments aboven and is cited here for the sake of completeness).
• The conference paper Graded Manifolds, Graded Lie Theory, and Prequantization, B. Kostant, Lect.Notes Math. 570 (1977) 177-306, in "Bonn 1975, Proceedings, Differential Geometrical Methods In Mathematical Physics", Berlin 1977. This paper states the definitions and basic notions in a more general setting (see also the work of Leites mentioned above) -at least in my understanding- but contains an extreme wealth of information and lots of detailed proofs which are not easy to be found in other sources.
• The article The structure of supermanifolds, Marjorie Batchelor, Trans. Amer. Math. Soc. 253 (1979), 329-338 (also cited as ref [1] in the reference provided in the answer by user Dmitri Pavlov).

Finally, it is interesting to mention the Wikipedia pages on Supermanifolds and Graded manifolds which attempt to discuss the relations between the various definitions met in the literature.

Some further (further) references:

(Don't be deceived by the title of the last reference: the first four chapters are about superalgebra and supergeometry, not about SUSY.)

• Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional is expected to be published on May 30, and will (according to the book description) include an introduction to supergeometry.
– Théo
Feb 26, 2019 at 15:30

Supergeometry in mathematics and physics by Kapranov (arXiv, 34 pages, submitted on 22 Dec 2015, last revised 2 Apr 2018). Abstract:

This is a chapter for a planned collective volume entitled "New spaces in mathematics and physics" (M. Anel, G. Catren Eds.). The first part contains a short formal exposition of supergeometry as it is understood by mathematicians. The second part discusses aspects of supergeometry that are used by physicists in relation to supersymmetry. Finally, the third part is an attempt to uncover homotopy-theoretic roots of the super formalism.

On the afterthought, I decided to add here earlier work by Manin - since Kapranov acknowledges his guidance, but also since it is still quite informative (I think), and has not been mentioned here so far.

Chapters 3 - 5 of his "Gauge Field Theory and Complex Geometry", as well as Chapter B of the appendix to the second (1997) edition by Merkulov provide a self-contained exposition of superalgebra and supergeometry, with a description of physical applications.

"Topics in Noncommutative Geometry", especially chapters 2 and 3, provide, I believe, sort of a continuation of the above.

I would like to add Riemannian supergeometry by Oliver Goertsches. Abstract:

Motivated by Zirnbauer in J Math Phys 37(10):4986–5018 (1996), we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g., geodesics, isometry groups and invariant metrics on Lie supergroups and homogeneous superspaces.

There is also:

Bartocci, Bruzzo, Hernández-Ruipérez, The geometry of supermanifolds (1991)

I don't know if it's "elementary" in your sense.

Alice Rogers Supermanifolds is a rigorous introduction to supermanifolds in the geometric and algebraic approaches with the emphasis on the geometric. She also discusses applications to physics such as the Wess-Zumino model, the basic example of a 4d N=1 supersymmetric theory as well as BV quantisation of gauge theories which is naturally formulated using supermanifolds. She also discusses applications to maths, for example super Lie groups & algebras. Obviously these are important to physics too, for example, the Wess-Zumino model is constructed using the super-Poincare group.

Gijs Tuynmams, Supermanifolds and Supergroups is more of a reference. He rigorously develops manifold theory over a graded commutative ring which specialises to the real, complex and super manifolds. There's much less motivation and applications here than in Roger's text. There are various definitions of super manifolds, not all equivalent, and he focuses on the de Witt's $$H^\infty$$ supermanifolds on the geometric approach and the Kostant-Leites sheaf-theretic on the algebraic approach. Roger's also develops the former, but focuses on a broader class she calls de Witt $$G^\infty$$ supersmooth functions.

I also found Cattaneo & Schatz's Introduction to Supergeometry useful. They not only discuss supermanifolds but also graded manifolds, an even analogue of supermanifolds of which examples are Poisson manifolds and Courant algebroids.

I'd say of these three, Roger's is the most elementary in the sense you are asking for. Certainly I found the text approachable even when I knew next to nothing about rigorous differential geometry.