International Journal of Geometric Methods in Modern Physics

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as in the supergeometric setting. The review is supplemented by some new concepts and examples.

This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as "our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 𝕆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 𝕂 or tensor products 𝕂₁ ⊗ 𝕂₂, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ′, ℍ′, 𝕆′ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 𝕂₁, 𝕂₂, which in all cases are obtained from ℝ, ℂ, ℂ′, ℍ, ℍ′, 𝕆, 𝕆′ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.

Duality symmetries have played a key role in the discovery that the five consistent superstring theories in 10 dimensions emerge as different corners of the moduli space of a single unifying theory, known as M-theory. Focusing on the target space, or T, duality symmetry, we show how it can be formulated in spacetimes with abelian and non-abelian isometries. Finally, we discuss some recent work that realizes non-abelian T-duality as a consistent truncation to seven-dimensional supergravity, thus generalizing to the non-abelian case the realization of abelian T-duality at the supergravity level.

We summarize the results obtained by applying Dirac's gauge fixing formalism to the combinatorial description of the Chern–Simons formulation of (2+1)-gravity and their implications for the symmetries of the quantum theory. While the combinatorial description of the phase space exhibits standard Poisson–Lie symmetries, every gauge fixing condition based on two point particles yields a Poisson structure determined by a dynamical classical r-matrix. By considering transformations between different gauge fixing conditions, it is possible to classify all gauge fixed Poisson structures in terms of two standard solutions of the dynamical classical Yang–Baxter equation. We discuss the conclusions that can be drawn from this about the symmetries of (2+1)-dimensional quantum gravity.

A hypersymplectic structure on a Lie algebroid determines several Poisson–Nijenhuis, ΩN and PΩ structures on that Lie algebroid. We show that these Poisson–Nijenhuis (respectively, ΩN, PΩ) structures on the Lie algebroid, are pairwise compatible.

Integrability of systems in N = 2 dimensions possessing a non-trivial scaling symmetry is reformulated in the context of dynamical symmetries and solvable Lie algebras. A method to construct integrable (Lagrangians) from a dynamical vector field subjected to a scaling symmetry is proposed.

The second-order differential equation on a Lorentzian manifold describes, in particular, the dynamics of particles under the action of an electromagnetic field F and a conservative force -∇V. We provide a first study on the extendability of its solutions, by imposing some natural assumptions.

We find some curvature properties of 3-quasi-Sasakian manifolds which extend certain well-known identities holding in the Sasakian case. As an application, we prove that any 3-quasi-Sasakian manifold of constant horizontal sectional curvature is necessarily either 3-α-Sasakian or 3-cosymplectic.

The concept of entropy was initially defined for systems with thermodynamical equilibrium. We try to extend this notion for quantum non-relativistic decaying states. We use a technique based on path integration on coherent states in order to obtain an approximation to the entropy of a decaying state.

We show that the supermembrane theory compactified on a torus is invariant under T-duality. There are two different topological sectors of the compactified supermembrane (M2) classified according to a vanishing or nonvanishing second cohomology class. We find the explicit T-duality transformation that acts locally on the supermembrane theory and we show that it is an exact symmetry of the theory. We give a global interpretation of the T-duality in terms of bundles. It has a natural description in terms of the cohomology of the base manifold and the homology of the target torus. We show that in the limit when the torus degenerate into a circle and the M2 mass operator restricts to the string-like configurations, the usual closed string T-duality transformation between the type IIA and type IIB mass operators is recovered. Moreover, if we just restrict M2 mass operator to string-like configurations but we perform a generalized T-duality we find the SL(2,Z) nonperturbative multiplet of IIA.

We study complex-pure-and-fullness at the first stage on compact complex nilmanifolds M of complex dimension 3, with special attention to the case of existence of a balanced Hermitian metric on M.

After briefly describing Hamiltonian BRST methods and the multisymplectic approach to field theory, a symmetric geometric Lagrangian is studied by extending the BRST method to the multisymplectic setting. This work uses ideas first introduced by Hrabak [On the multisymplectic origin of the nonabelian deformation algebra of pseudoholomorphic embeddings in strictly almost Kähler ambient manifolds, and the corresponding BRST algebra, preprint (1999), arXiv: math-ph/9904026].

For k-symplectic Hamiltonian field theories, we study infinitesimal transformations generated by some kinds of vector fields which are not Noether symmetries, but which allow us to obtain conservation laws by means of suitable generalizations of Noether's theorem.

We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.

We study the Yang–Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kähler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang–Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin–Kobayashi correspondence we prove that every (poly-)stable principal K_ℂ-bundle has to satisfy this generalized Bogomolov type inequality.

Given a Hamiltonian action of a Lie group (not necessarily compact) on a symplectic manifold M, by means of the integration over M of the coupling class associated with this action, we define an element which is invariant under deformations of the action.