A locally supersymmetric SO(10,2)SO(10,2) invariant action for D=12D=12 supergravity

We present an action for $N=1$ supergravity in $10+2$ dimensions, containing the gauge fields of the $OSp(1|64)$ superalgebra, i.e. one-forms $B^{(n)}$ with $n$=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino $\psi$. The vielbein and spin connection correspond to $B^{(1)}$ and $B^{(2)}$ respectively. The action is not gauge invariant under the full $OSp(1|64)$ superalgebra, but only under a subalgebra ${\tilde F}$ (containing the $F$ algebra $OSp(1|32)$), whose gauge fields are $B^{(2)}$, $B^{(6)}$, $B^{(10)}$ and the Weyl projected Majorana gravitino ${1 \over 2} (1+\Gamma_{13}) \psi$. Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply $\int STr ({\bf R}^6 {\bf \Gamma})$ where ${\bf R}$ is the $OSp(1|64)$ curvature supermatrix two-form, and ${\bf \Gamma}$ is a constant supermatrix involving $\Gamma_{13}$ and breaking $OSp(1|64)$ to its ${\tilde F}$ subalgebra. The action includes the usual Einstein-Hilbert term.Comment: LaTeX, 13 pages. Added a reference, a Table in Appendix A for the gamma commutations in d=12, and corrected eq. (4.14) for the Einstein-Hilbert term; v4: corrected formulas (A.3), (A.4) and (A.10), modified last paragraph of Section 5, added acknowledgement

Differential calculi on finite groups

A brief review of bicovariant differential calculi on finite groups is given, with some new developments on diffeomorphisms and integration. We illustrate the general theory with the example of the nonabelian finite group S_3.Comment: LaTeX, 16 pages, 1 figur

U_q(N) Gauge Theories

Improving on an earlier proposal, we construct the gauge theories of the quantum groups $U_q(N)$. We find that these theories are consistent also with an ordinary (commuting) spacetime. The bicovariance conditions of the quantum differential calculus are essential in our construction. The gauge potentials and the field strengths are $q$-commuting ``fields", and satisfy $q$-commutation relations with the gauge parameters. The transformation rules of the potentials are given explicitly, and generalize the ordinary infinitesimal gauge variations. The $q$-lagrangian invariant under the $U_q(N)$ variations has the Yang-Mills form \Fmn^i \Fmn^j g_{ij}, the ``quantum metric'' $g_{ij}$ being a generalization of the Killing metric.Comment: 7pp., plain TeX, DFTT-74/9

Gravity on Finite Groups

Gravity theories are constructed on finite groups G. A self-consistent review of the differential calculi on finite G is given, with some new developments. The example of a bicovariant differential calculus on the nonabelian finite group S_3 is treated in detail, and used to build a gravity-like field theory on S_3.Comment: LaTeX, 26 pages, 1 figure. Corrected misprints and formula giving exterior product of n 1-forms. Added note on topological actio

Higher form gauge fields and their nonassociative symmetry algebras

We show that geometric theories with $p$-form gauge fields have a nonassociative symmetry structure, extending an underlying Lie algebra. This nonassociativity is controlled by the same Chevalley-Eilenberg cohomology that classifies free differential algebras, $p$-form generalizations of Cartan-Maurer equations. A possible relation with flux backgrounds of closed string theory is pointed out.Comment: 8 pages, LaTeX. Shortened review part on extended Lie derivatives and free differential algebras, added computation of the Jacobiator for D=11 supergravity, added references. Matches published version on JHE

OSp(1|4) supergravity and its noncommutative extension

We review the OSp(1|4)-invariant formulation of N=1, D=4 supergravity and present its noncommutative extension, based on a star-product originating from an abelian twist with deformation parameter \theta. After use of a geometric generalization of the Seiberg-Witten map, we obtain an extended (higher derivative) supergravity theory, invariant under usual OSp(1|4) gauge transformations. Gauge fixing breaks the OSp(1|4) symmetry to its Lorentz subgroup, and yields a Lorentz invariant extended theory whose classical limit \theta --> 0 is the usual N=1, D=4 AdS supergravity.Comment: 20 pages, LaTeX. Added fields and curvatures at first order in \theta. Matches published version in Phys. Rev.

Chern-Simons supergravities, with a twist

We discuss noncommutative extensions of Chern-Simons (CS) supergravities in odd dimensions. The example of D=5 CS supergravity, invariant under the gauge supergroup SU(2,2|N), is worked out in detail. Its noncommutative version is found to exist only for N=4.Comment: LaTeX, 12 pages. Matches published version on JHE