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.
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.
this section is at geometry of physics – superalgebra
this section is at geometry of physics – supergeometry
this section is at geometry of physics – supersymmetry
this section is at geometry of physics – fundamental super p-branes
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
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
Deligne's theorem on tensor categories is due to
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
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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
José Figueroa-O'Farrill, Majorana spinors (pdf)
The close relation between supersymmetry and division algebras was first observed in
A clean survey is in
and the discussion of the spinor bilinear pairings from this perspective is in
The seminal analysis of torsion of G-structures is due to
Discussion of torsion of G-structures in the context of supergeometry (supertorsion) is in
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
A. Candiello, K. Lechner, Duality in Supergravity Theories, Nucl.Phys. B412 (1994) 479-501 (arXiv:hep-th/9309143)
Paul Howe, Weyl Superspace, Physics Letters B
Volume 415, Issue 2, 11 December 1997, Pages 149–155 (arXiv:hep-th/9707184)
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, José M. Izquierdo, C. Pérez Bueno, The geometry of branes and extended superspaces, Nucl. Phys. B 567 (2000) 293-330 [arXiv:hep-th/9904137, doi:10.1016/S0550-3213(99)00512-X]
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
Eric Bergshoeff, Ergin Sezgin, Paul Townsend, Superstring actions in $D = 3, 4, 6, 10$ curved superspace, Phys.Lett., B169, 191, (1986) (spire)
Eric Bergshoeff, Ergin Sezgin, Paul Townsend, Supermembranes and eleven dimensional supergravity, Phys.Lett. B189 (1987) 75-78, In Mike Duff, (ed.), The World in Eleven Dimensions 69-72 (pdf, spire)
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
Chris Hull, A Geometry for Non-Geometric String Backgrounds, JHEP0510:065,2005 (arXiv:hep-th/0406102)
Chris Hull, Doubled geometry and T-folds JHEP0707:080,2007 (arXiv:hep-th/0605149)
The mathematical formalization of this idea in terms of principal 2-bundles for the T-duality 2-group was claimed in
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 June 19, 2023 at 14:43:26. See the history of this page for a list of all contributions to it.