nLab geometry of physics -- supergeometry and superphysics

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.

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

• General accounts

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

• Special literature

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

• Higher super Cartan geometry

General accounts

An excellent general textbook for our purposes is

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

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

• Urs Schreiber, Structure Theory for Higher WZW Terms, lectures at Higher Structures in String Theory and Quantum Field Theory, ESI Vienna, November 30-December 4, 2015

More specialized literature

Deligne's theorem on tensor categories is due to

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

building on his general work on Tannakian categories

• Pierre Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195. (pdf)

A brief survey is in

• Victor Ostrik, Tensor categories (after P. Deligne) (arXiv:math/0401347)

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

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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

• Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)

Useful discussion of Majorana spinors and the induced supersymmetry algebras includes

• CDF, II.7.3

• José Figueroa-O'Farrill, Majorana spinors (pdf)

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

• Taichiro Kugo, Paul Townsend, Supersymmetry and the division algebras, Nuclear Physics B, Volume 221, Issue 2 (1983) p. 357-380. (spires, pdf)

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

• John Baez, John Huerta, Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (arXiv:1003.34360)

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

• Riccardo D'Auria, Pietro Fré; Geometric Superravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 (pdf)

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

• CDF, II.8

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

• Anna Achúcarro, Jonathan Evans, Paul Townsend, David Wiltshire, Super $p$-Branes, Phys. Lett. B 198 (1987) 441 (spire)

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

• Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys.Rev.Lett. 78 (1997) 4332-4334 (arXiv:hep-th/9701149)

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

• Peter Bouwknegt, Jarah Evslin, Varghese Mathai, T-Duality: Topology Change from H-flux, Commun.Math.Phys.249:383-415,2004 (hep-th/0306062)

and in an alternative form due to

• Ulrich Bunke, P. Rumpf, Thomas Schick, The topology of $T$-duality for $T^n$-bundles, Rev. Math. Phys. 18, 1103 (2006). (arXiv:math.GT/0501487)

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

• 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:

• Urs Schreiber, differential cohomology in a cohesive topos, Thesis, (v1 arXiv:1310.7930, v2)

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018, (arXiv:1308.5264)

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory, to appear in Journal of Geometry and Physics (arXiv:1606.03206)

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, T-Duality from super Lie n-algebra cocycles for super p-branes (arXiv:1611.06536)

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

• John Huerta, Urs Schreiber, M-Theory from the Superpoint