# nLab geometry of physics -- supergeometry and superphysics

Contents

this entry is one chapter of “geometry of physics

previous chapter: manifolds and orbifolds

next chapter: BPS charges

Presently this entry is under construction. It is being incrementally expanded as this lecture series progresses: From the Superpoint to T-Folds.

# Contents

In Klein geometry and Cartan geometry the fundamental geometric concept is the symmetry group $G$ of the local model space, which is then recovered as some coset space $G/H$. These symmetry groups $G$ are reflected in their categories of representations $Rep(G)$, which are certain nice tensor categories. In terms of physics via Wigner classification, the irreducible objects in $Rep(G)$ label the possible fundamental particle species on the spacetime $G/H$. Hence if we regard the tensor category $Rep(G)$ as the actual fundamental concept, then the natural question is that of Tannaka reconstruction: Given any nice tensor category, is it equivalent to $Rep(G)$ for some symmetry group $G$? For rigid tensor categories in characteristic zero subject only to a mild size constraint this is answered by Deligne's theorem on tensor categories: all of them are, but only if we allow $G$ to be a “supergroup”. This we discuss in the first section below.

## Superalgebra

this section is at geometry of physics – superalgebra

## Supergeometry

this section is at geometry of physics – supergeometry

## Spacetime supersymmetry

this section is at geometry of physics – supersymmetry

## Fundamental super $p$-branes

this section is at geometry of physics – fundamental super p-branes

## References

The following first mentions

that might usefully be held on to during the seminar. Then I list

with refernces to original results and to reviews of these. Then I list pointers to my own work with collaborators on

### General accounts

An excellent general textbook for our purposes is

This is written by physicists in physics style, but the development is careful and thorough, and the “geometric perspective” in the title is nothing but the perspective of higher super Cartan geometry in slight disguise. See also at D'Auria-Fré formulation of supergravity.

Lecture notes closely related to the seminar are in

### More specialized literature

• Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

building on his general work on Tannakian categories

A brief survey is in

and a more comprehensive texbook account is in chapter 9.11 of

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf

))

The observation that supergeometry is naturally regarded as ordinary geometry inside the sheaf topos over superpoints is due to

• Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)

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

Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

A nice account is in

Useful discussion of Majorana spinors and the induced supersymmetry algebras includes

The close relation between supersymmetry and division algebras was first observed in

A clean survey is in

• John Baez, John Huerta, Division algebras and supersymmetry I, in R. Doran, G. Friedman, Jonathan Rosenberg(eds.) Superstrings, Geometry, Topology, and $C^\ast$-algebras, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (arXiv:0909.0551)

and the discussion of the spinor bilinear pairings from this perspective is in

The seminal analysis of torsion of G-structures is due to

• Victor Guillemin, The integrability problem for $G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544–560. (JSTOR)

Discussion of torsion of G-structures in the context of supergeometry (supertorsion) is in

• John Lott, The Geometry of Supergravity Torsion Constraints Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

An elegant construction of 11-dimensional supergravity, right in the spirit of super Cartan geometry, is due to

This is the main original result on which the D'Auria-Fré formulation of supergravity is based, as laid out in CDF.

The observation that the equations of motion of bosonic solutions of 11-dimensional supergravity are equivalent simply to vanishing of the supertorsion is due to

Discussion of Fierz identities includes

The classification of the invariant super Lie algebra cocycles on super-Minkowski spacetime, hence that of super p-branes without gauge fields on their worldvolume, is due to

The extension of this classification to D-branes and to the M5-brane using extended super Minkowski spacetime is due to

• C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137)

• Makoto Sakaguchi, IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)

The M5-brane cocycle on the “M2-brane extended super-Minkowski spacetime” that appears here has in fact been observed, as a cocycle, all the way back in D’Auria-Fré 82. But there it was seen just as a means for constructing 11-dimensional supergravity. That it indeed gives the higher WZW term in the Green-Schwarz type action functional that defines the fundamental M5-brane has been argued in

The observation that super p-branes on curved super spacetimes require definite globalization of super Lie algebra cocycles from Minkowski spacetime over the supermanifold is due to

The formulation of topological T-duality is due to

and in an alternative form due to

The suggestion that there ought to be “T-folds” or “doubled geometry” is due to

The mathematical formalization of this idea in terms of principal 2-bundles for the T-duality 2-group was claimed in

• Thomas Nikolaus, T-Duality in K-theory and elliptic cohomology, talk at String Geometry Network Meeting, Feb 2014, ESI Vienna (website)

### Higher super Cartan geometry

The following articles develop the higher super Cartan geometry that we give an exposition of in the second part of the seminar.

The mathematical foundation of higher supergeometry:

The general idea of The brane bouquet and the general construction of higher WZW terms from higher $L_\infty$-cocycles:

The homotopy-descent of the M5-brane cocycle and of the type IIA D-brane cocycles:

The derivation of supersymmetric topological T-duality, rationally, and of the higher super Cartan geometry for super T-folds:

The derivation of the process of higher invariant extensions that leads from the superpoint to 11-dimensional supergravity:

Last revised on July 27, 2019 at 10:23:43. See the history of this page for a list of all contributions to it.