nLab
supergeometry

Contents

Idea

Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object \mathbb{R} 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 superscheme?s.

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.

References

Some historically influential general considerations are in

Introductory lecture notes include

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

and in

  • V. Molotkov., Infinite-dimensional 2 k\mathbb{Z}_2^k-supermanifolds , ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

  • Anatoly Konechny and Albert Schwarz,

    On (kl|q)(k \oplus l|q)-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)

    Theory of (kl|q)(k \oplus l|q)-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486 A summary/review is in the appendix of

  • Albert Schwarz, I- Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

A review of all this as geometry in the topos over the category of superpoints 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

  • Sign manifesto (pdf)

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.

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Revised on October 9, 2013 08:59:07 by Urs Schreiber (89.204.139.247)