Index of content:
Volume 57, Issue 5, May 2016
- Representation Theory and Algebraic Methods
57(2016); http://dx.doi.org/10.1063/1.4948409View Description Hide Description
We introduce the notion of a universal odd generalized Poisson superalgebra associated with an associative algebra A, by generalizing a construction made in the work of De Sole and Kac [Jpn. J. Math. 8, 1–145 (2013)]. By making use of this notion we give a complete classification of simple linearly compact (generalized) n-Nambu-Poisson algebras over an algebraically closed field of characteristic zero.
- Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory
57(2016); http://dx.doi.org/10.1063/1.4947531View Description Hide Description
We present and study a model of 4–dimensional higher Chern-Simons theory, special Chern–Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2–algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2–connection coupled to a background closed 3–form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2–group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3–form. Finally, SCS theory is related to a 3–dimensional special gauge theory whose 2–connection space has a natural symplectic structure with respect to which the 1–gauge transformation action is Hamiltonian, the 2–curvature map acting as moment map.
- Dynamical Systems
57(2016); http://dx.doi.org/10.1063/1.4947528View Description Hide Description
A relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on Schur-type symmetric functions. A system of orthogonal rational functions is constructed using a multiparameter deformation of the Schur functions. Spectral equations for the orthogonal rational functions are derived by using properties of the Schur-type symmetric functions. The compatibility condition of the spectral equations induces a discrete dressing chain which is a Toda-type discrete integrable system describing dressing transformations for orthogonal rational functions.
57(2016); http://dx.doi.org/10.1063/1.4947530View Description Hide Description
57(2016); http://dx.doi.org/10.1063/1.4948641View Description Hide Description
The higher-order superintegrability of the two-dimensional isotonic oscillator (noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator) with rational ratio of frequencies is directly related with the existence of some complex functions with interesting Poisson bracket properties. First the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi-Hamiltonian complex structure. In the second part several real symplectic structures are obtained and the properties of the recursion operators are studied.
- Methods of Mathematical Physics
57(2016); http://dx.doi.org/10.1063/1.4947529View Description Hide Description
We consider here special Poisson brackets given by the “averaging” of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the “physical meaning” of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of “Hydrodynamic Type” in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a “pseudo-canonical” form under some special (“physical”) requirements on the initial bracket.
Optimal space of linear classical observables for Maxwell k-forms via spacelike and timelike compact de Rham cohomologies57(2016); http://dx.doi.org/10.1063/1.4947563View Description Hide Description
Being motivated by open questions in gauge field theories, we consider non-standard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincaré duality for de Rham cohomology is shown to hold for the case with non-standard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree k of both the vector potential and the Faraday tensor. The term optimal has to be intended in the following sense: The spaces of linear observables we consider distinguish between different configurations; in addition to that, there are no redundant observables. This last point in particular heavily relies on the analog of Poincaré duality for the new cohomology groups.