arXiv subjects
Publications and source records attributed to Leonardo Castellani.
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.
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 θ. 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 θ--> 0 is the usual N=1, D=4 AdS supergravity.
In this paper we couple noncommutative (NC) vielbein gravity to scalar fields. Noncommutativity is encoded in a star product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg-Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein-Hilbert and Klein-Gordon actions are organized in successive powers of the noncommutativity parameter θ^{AB}.
We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter θ. We thus express the action at order θ^n+2 in terms of noncommutative fields of order at most θ^n+1 and, iterating the procedure, in terms of noncommutative fields of order at most θ^n. This in particular provides the explicit expression of the action at order θ^2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations.
We 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 conditions 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.
We analyze some aspects of quantum computing with super-qubits (squbits). We propose the analogue of a superfield formalism, and give a physical interpretation for the Grassmann coefficients in the squbit expansion as fermionic creation operators of an auxiliary quantum system. In the simplest case the squbit is a superposition of one Bose X Bose and one Fermi X Fermi state, and its norm is invariant under a U(2) group realized with Clifford-valued matrices. This case can be generalized to a superposition of n_B bosonic and n_F fermionic states, with a norm invariant under U(n_B + n_F). Entanglement between squbits, super quantum gates and teleportation are discussed.
We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented, including the fundamental theorem of noncommutative Riemannian geometry. Inspired by [1, 2], we obtain solutions of noncommutative Einstein equations by considering twists that are compatible with the curved spacetime metric.
We prove the discrete triality invariance of the N=2 NSR superstring moving in a D=2+2 target space. We find that triality holds also in the Siegel-Berkovits formulation of the selfdual superstring. A supersymmetric generalization of Cayley's hyperdeterminant, based on a quartic invariant of the SL(2|1)^3 superalgebra, is presented.
We present a noncommutative D=3, N=1 supergravity, invariant under diffeomorphisms, local U(1,1) noncommutative \star-gauge transformations and local \star-supersymmetry. Its commutative limit is the usual D=3 pure supergravity, without extra fields. A noncommutative deformation of D=4, N=1 supergravity is also obtained, reducing to the usual simple supergravity in the commutative limit. Its action is invariant under diffeomorphisms and local GL(2,C) \star-gauge symmetry. The supersymmetry of the commutative action is broken by noncommutativity. Local \star-supersymmetry invariance can be implemented in a noncommutative D=4, N=1 supergravity with chiral gravitino and complex vierbein.
We present a noncommutative extension of Einstein-Hilbert gravity in the context of twist-deformed space-time, with a $\star$-product associated to a quite general triangular Drinfeld twist. In particular the $\star$-product can be chosen to be the usual Groenewald-Moyal product. The action is geometric, invariant under diffeomorphisms and centrally extended Lorentz $\star$-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 $\star$-gauge transformations. Finally, we discuss the noncommutative version of the Mac-Dowell Mansouri action, quadratic in curvatures.
A \star-product is defined via a set of commuting vector fields X_a = e_a^μ(x) \partial_μ, and used in a phi^*4 theory coupled to the e_a^μ(x) fields. The \star-product is dynamical, and the vacuum solution phi =0, e_a^μ(x)=delta_a^μreproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy-momentum and angular momentum tensors are explicitly derived.
Introducing an extended Lie derivative along the dual of A, the three-form field of d=11 supergravity, the full diffeomorphism algebra of d=11 supergravity is presented. This algebra suggests a new formulation of the theory, where the three-form field A is replaced by bivector B^{ab}, bispinor B^{αβ}, and spinor-vector η^{aβ} one-forms. Only the bivector one-form B^{ab} is propagating, and carries the same degrees of freedom of the three-form in the usual formulation, its curl \Dcal_{[μ} B^{ab}_{ν]} being related to the F_{μνa b} curl of the three-form. The other one-forms are auxiliary, and the transformation rules on all the fields close on the equations of motion of d=11 supergravity.
Free differential algebras (FDA's) provide an algebraic setting for field theories with antisymmetric tensors. The "presentation" of FDA's generalizes the Cartan-Maurer equations of ordinary Lie algebras, by incorporating p-form potentials. An extended Lie derivative along antisymmetric tensor fields can be defined, and used to recover a Lie algebra dual to the FDA, that encodes all the symmetries of the theory including those gauged by the p-forms. The general method is applied to the FDA of D=11 supergravity: the resulting dual Lie superalgebra contains the M-theory supersymmetry anticommutators in presence of 2-branes.
Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations, and under the analogues of Lorentz and general coordinate transformations. The case Z_n \times Z_n \times...\times Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories, and a new discretized action for gravity.
This review is based on two lectures given at the 2000 TMR school in Torino. We discuss two main themes: i) Moyal-type deformations of gauge theories, as emerging from M-theory and open string theories, and ii) the noncommutative geometry of finite groups, with the explicit example of Z_2, and its application to Kaluza-Klein gauge theories on discrete internal spaces.
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.
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.
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.