We find a link between oriented matroid theory and 2d gravity with torsion. Our considerations may be useful in the context of noncommutative phase space in a target spacetime of signature (2+2) and in a possible theory of gravity ramification.

We study a cosmological model in 1+D+d dimensions where D dimensions are associated with the usual Friedman-Robertson-Walker type metric with radio a(t) and d dimensions corresponds to an additional homogeneous space with radio b(t). We make a general analysis of the field equations and then we obtain solutions involving the two cosmological radii, a(t) and b(t). The particular case D=3, d=1 is studied in some detail.

One-dimensional potentials defined by V (S) (x) = S(S + 1)h 2 π 2 /[2ma 2 sin 2 (πx/a)] (for integer S) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalized solutions, ψ (S) n (x), for all V (S) (x) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl-Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behavior of the corresponding momentum-space wave functions φ (S) n (p) for large |p| and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.

We construct a Lagrangian and Hamiltonian formulation for charged black holes in a d-dimensional maximally symmetric spherical space. By considering first new variables that give raise to an interesting dimensional reduction of the problem, we show that the introduction of a charge term is compatible with classical solutions to Einstein equations. In fact, we derive the wellknown solutions for charged black holes, specially in the case of d=4, where the Reissner-Nordström solution holds, without reference to Einstein field equations. We argue that our procedure may be of help for clarifying symmetries and dynamics of black holes, as well as some quantum aspects.

We describe a possible and alternative route to connect gravity with Elko theory. Our approach is based on the possibility to introduce a totally antisymmetric gauge field in the generalized Elko field equation, which is an alternative extension of the Dirac field equation. The corresponding totally antisymmetric field strength ([Formula: see text]-form) is then associated with Grassmann–Plücker coordinates and therefore with a decomposable [Formula: see text]-form. We show that such a totally antisymmetric field strength can be considered as the square root of the Riemann tensor associated to a homogeneous space. Motivated by this result we conjecture that a full connection with Riemann geometry and therefore with general relativity must be possible if such a field strength is not decomposable. We also show how a supergravity version could arise from a generalization of the above ideas.