On G/H geometry and its use in M-theory compactifications

The Riemannian geometry of coset spaces is reviewed, with emphasis on its applications to supergravity and M-theory compactifications. Formulae for the connection and curvature of rescaled coset manifolds are generalized to the case of nondiagonal Killing metrics. The example of the N^{010} spaces is discussed in detail. These are a subclass of the coset manifolds N^{pqr}=G/H = SU(3) x U(1)/U(1) x U(1), the integers p,q,r characterizing the embedding of H in G. We study the realization of N^{010} as G/H=SU(3) x SU(2)/U(1) x SU(2) (with diagonal embedding of the SU(2) \in H into G). For a particular G-symmetric rescaling there exist three Killing spinors, implying N=3 supersymmetry in the AdS_4 \times N^{010} compactification of D=11 supergravity. This rescaled N^{010} space is of particular interest for the AdS_4/CFT_3 correspondence, and its SU(3) x SU(2) isometric realization is essential for the OSp(4|3) classification of the Kaluza-Klein modes.Comment: 12 page

Supergravity in the group-geometric framework: a primer

We review the group-geometric approach to supergravity theories, in the perspective of recent developments and applications. Usual diffeomorphisms, gauge symmetries and supersymmetries are unified as superdiffeomorphisms in a supergroup manifold. Integration on supermanifolds is briefly revisited, and used as a tool to provide a bridge between component and superspace actions. As an illustration of the constructive techniques, the cases of $d=3,4$ off-shell supergravities and $d=5$ Chern-Simons supergravity are discussed in detail. A cursory account of $d=10+2$ supergravity is also included. We recall a covariant canonical formalism, well adapted to theories described by Lagrangians $d$-forms, that allows to define a form hamiltonian and to recast constrained hamiltonian systems in a covariant form language. Finally, group geometry and properties of spinors and gamma matrices in $d=s+t$ dimensions are summarized in Appendices.Comment: LaTeX, 65 pages, 2 Tables, 1 figure. v2: included Figure missing in v1, ref.s added. v3: added missing term in eq. (9.3). v4: eq. (9.41) corrected. Matches published version. v5: added missing terms in eq.s (7.21), (7.24), (7.27), (9.38), added a paragraph in Sect. 11, added re

The complete N=3 Kaluza Klein spectrum of 11D supergravity on AdS_4 x N^{010}

We derive the invariant operators of the zero-form, the one-form, the two-form and the spinor from which the mass spectrum of Kaluza Klein of eleven-dimensional supergravity on AdS_4 x N^{010} can be derived by means of harmonic analysis. We calculate their eigenvalues for all representations of SU(3)xSO(3). We show that the information contained in these operators is sufficient to reconstruct the complete N=3 supersymmetry content of the compactified theory. We find the N=3 massless graviton multiplet, the Betti multiplet and the SU(3) Killing vector multiplet.Comment: 1+50 pages, LaTe

The structure of N=3 multiplets in AdS_4 and the complete Osp(3|4) X SU(3) spectrum of M-theory on AdS_4 X N^{010}

In this paper, relying on previous results of one of us on harmonic analysis, we derive the complete spectrum of Osp(3|4) X SU(3) multiplets that one obtains compactifying D=11 supergravity on the unique homogeneous space N^{0,1,0} that has a tri-sasakian structure, namely leads to N=3 supersymmetry both in the four-dimensional bulk and on the three-dimensional boundary. As in previously analyzed cases the knowledge of the Kaluza Klein spectrum, together with general information on the geometric structure of the compact manifold is an essential ingredient to guess and construct the corresponding superconformal field theory. This is work in progress. As a bonus of our analysis we derive and present the explicit structure of all unitary irreducible representations of the superalgebra Osp(3|4) with maximal spin content s_{max}>=2.Comment: Latex2e, 13+1 page

The Lagrangian of q-Poincare' Gravity

The gauging of the q-Poincar\'e algebra of ref. hep-th 9312179 yields a non-commutative generalization of the Einstein-Cartan lagrangian. We prove its invariance under local q-Lorentz rotations and, up to a total derivative, under q-diffeomorphisms. The variations of the fields are given by their q-Lie derivative, in analogy with the q=1 case. The algebra of q-Lie derivatives is shown to close with field dependent structure functions. The equations of motion are found, generalizing the Einstein equations and the zero-torsion condition.Comment: 12 pp., LaTeX, DFTT-01/94 (extra blank lines introduced by mailer, corrupting LaTeX syntax, have been hopefully eliminated

Noncommutative gauge fields coupled to noncommutative gravity

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted star product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and twisted gauge transformations. The Seiberg-Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor D_{IJK}.Comment: 18 pages, LaTe

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