Leonardo Castellani

Special Kähler geometry. An intrinsic formulation from N=2 space-time supersymmetry

Abstract N =2, 4D supergravity coupled to vector multiplets (1, 1 2 , 0) is reanalysed in a geometrical setting. By requiring the closure of the supersymmetry transformation laws, we find a coordinate-free characterization of the manifold M spanned by the scalar fields. The geometry of M , called “special geometry”, is relevant to compactified string theories, since it is common to the moduli spaces of Calabi-Yau threefolds and c =9 (2, 2) superconformal field theories.

AN INTRODUCTION TO NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ON QUANTUM GROUPS

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case (q→1 limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan-Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group GLq(2) is given in detail. The softening of a quantum group is considered, and we introduce q curvatures satisfying q Bianchi identities, a basic ingredient for the construction of q gravity and q gauge theories.

The Complete

Abstract The complete N = 3 matter coupling to supergravity is obtained in a geometrical framework. This coupling always exists if the 3 n complex scalars of the n vector multiplets are co-ordinates of the Kahler-grassmannian manifold SU(3, n )/SU(3) × SU( n ) × U(1). Subgroups of SO(3, n ) ⊂ SU(3, n ) of dimension 3 + n can be gauged and give rise to a non-trivial scalar potential. The techniques used in this paper allow for the calculation of scalar potentials of extended supergravities in any dimension without explicit construction of the lagrangian. This opens the possibility of discussing patterns of partial supersymmetry and gauge symmetry breaking on a purely group-theoretical ground.

N=3

We discuss the class of BPS saturated M-branes that are in one-to-one correspondence with the Freund-Rubin compactifications of M-theory on either AdS4 × G/H or AdS7 × G/H, where G/H is any of the seven (or four) dimensional Einstein coset manifolds with Killing spinors classified long ago in the context of Kaluza-Klein supergravity. These G/H M-branes, whose existence was previously pointed out in the literature, are solitons that interpolate between flat space at infinity and the old Kaluza-Klein compactifications at the horizon. They preserve N/2 supersymmetries where N is the number of Killing spinors of the AdS × G/H vacuum. A crucial ingredient in our discussion is the identification of a solvable Lie algebra parametrization of the Lorentzian noncompact coset SO(2, p + 1)/SO(1, p + 1) corresponding to anti-de Sitter space AdSp + 2. The solvable coordinates are those naturally emerging from the near horizon limit of the G/H p-brane and correspond to the Bertotti-Robinson form of the anti-de Sitter metric. The pull-back of anti-de Sitter isometries on the p-brane world-volume contain, in particular, the recently found broken conformal transformations

Matter Coupled Supergravity

We present a general method to deform the inhomogeneous algebras of theBn,Cn,Dn type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromBn+1,Cn+1,Dn+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISOq(N) as a consistent projection of the (bicovariant)q-algebraSOq(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called “minimal” deformations. The case ofISOq(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the “gauging” of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.

G / H M-branes and AdS(p+2) geometries

We present a noncommutative extension of Einstein-Hilbert gravity in the context of twist-deformed space-time, with a -product associated to a quite general triangular Drinfeld twist. In particular the -product can be chosen to be the usual Groenewald-Moyal product. The action is geometric, invariant under diffeomorphisms and centrally extended Lorentz -gauge transformations. In the commutative limit it reduces to ordinary gravity, with local Lorentz invariance and usual real vielbein. This we achieve by imposing a charge conjugation condition on the noncommutative vielbein. The theory is coupled to fermions, by adding the analog of the Dirac action in curved space. A noncommutative Majorana condition can be imposed, consistent with the -gauge transformations. Finally, we discuss the noncommutative version of the Mac-Dowell Mansouri action, quadratic in curvatures.

Differential calculus onISOq(N), quantum Poincaré algebra and q-gravity

This review is based on two lectures given at the 2000 TMR school in Torino (TMR school on contemporary String Theory and Brane Physics, 26 January-2 February 2000, Torino, Italy). We discuss two main themes: (a) Moyal-type deformations of gauge theories, as emerging from M-theory and open string theories, and (b) the non-commutative geometry of finite groups, with the explicit example of Z2, and its application to Kaluza-Klein gauge theories on discrete internal spaces.

Noncommutative D = 4 gravity coupled to fermions

Abstract We find two different q -generalizations of Yang-Mills theories. The corresponding lagrangians are invariant under the q -analogue of infinitesimal gauge transformations. We explicitly give the lagrangian and the transformation rules for the bicovariant q -deformation of SU(2) × U(1). The gauge potentials satisfy q -communications, as one expects from the differential geometry of quantum groups. However, in one of the two schemes we present, the field strengths do commute.

Non-commutative geometry and physics: a review of selected recent results

Abstract The gauging of the q- Poincar e algebra of L. Castellani [Differential calculus on ISOq(N), quantum Poincare algebra and q-gravity, Torino preprint DFTT-70/93, 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.

Gauge theories of quantum groups

A bstractWe use the Seiberg-Witten map (SW map) to expand noncommutative gravity coupled to fermions in terms of ordinary commuting fields. The action is invariant under general coordinate transformations and local Lorentz rotations, and has the same degrees of freedom as the commutative gravity action. The expansion is given up to second order in the noncommutativity parameter θ.A geometric reformulation and generalization of the SW map is presented that applies to any abelian twist. Compatibility of the map with hermiticity and charge conjugation is proven. The action is shown to be real and invariant under charge conjugation at all orders in θ. This implies the bosonic part of the action to be even in θ, while the fermionic part is even in θ for Majorana fermions.