23 research outputs found
Algebraic theories of brackets and related (co)homologies
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets
in the category of modules over a commutative algebra is described. Some
related structures and (co)homology invariants are discussed, as well as
applications to geometry.Comment: 14 pages; v2: minor correction
On the variational noncommutative Poisson geometry
We outline the notions and concepts of the calculus of variational
multivectors within the Poisson formalism over the spaces of infinite jets of
mappings from commutative (non)graded smooth manifolds to the factors of
noncommutative associative algebras over the equivalence under cyclic
permutations of the letters in the associative words. We state the basic
properties of the variational Schouten bracket and derive an interesting
criterion for (non)commutative differential operators to be Hamiltonian (and
thus determine the (non)commutative Poisson structures). We place the
noncommutative jet-bundle construction at hand in the context of the quantum
string theory.Comment: Proc. Int. workshop SQS'11 `Supersymmetry and Quantum Symmetries'
(July 18-23, 2011; JINR Dubna, Russia), 4 page
Editors’ preface for the topical issue on “The interface between integrability and quantization”
A non-standard Lax formulation of the Harry Dym hierarchy and its supersymmetric extension
For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from
that of Korteweg-de Vries (KdV) equation through a reciprocal transformation.
By supersymmetrizing this Lax operator, a new N=2 supersymmetric extension of
the Harry Dym hierarchy is constructed, and is further shown to be linked to
one of the N=2 supersymmetric KdV equations through superconformal
transformation. The bosonic limit of this new N=2 supersymmetric Harry Dym
equation is related to a coupled system of KdV-MKdV equations.Comment: 9 page
Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry
In the search for vacuum solutions, with or without a cosmological constant,
of the Einstein field equations of Petrov type N with twisting principal null
directions, the CR structures to describe the parameter space for a congruence
of such null vectors provide a very useful tool. A work of Hill, Lewandowski
and Nurowski has given a good foundation for this, reducing the field equations
to a set of differential equations for two functions, one real, one complex, of
three variables. Under the assumption of the existence of one Killing vector,
the (infinite-dimensional) classical symmetries of those equations are
determined and group-invariant solutions are considered. This results in a
single ODE of the third order which may easily be reduced to one of the second
order. A one-parameter class of power series solutions, g(w), of this
second-order equation is realized, holomorphic in a neighborhood of the origin
and behaving asymptotically as a simple quadratic function plus lower-order
terms for large values of w, which constitutes new solutions of the twisting
type N problem. The solution found by Leroy, and also by Nurowski, is shown to
be a special case in this class. Cartan's method for determining equivalence of
CR manifolds is used to show that this class is indeed much more general.
In addition, for a special choice of a parameter, this ODE may be integrated
once, to provide a first-order Abel equation. It can also determine new
solutions to the field equations although no general solution has yet been
found for it.Comment: 28 page
Integrable structures for a generalized Monge-Ampère equation
We consider a 3rd-order generalized Monge-Ampère equa-
tion u yyy − u 2 xxy + u xxx u xyy = 0 (which is closely related to the asso-
ciativity equation in the 2-d topological field theory) and describe all
integrable structures related to it (i.e., Hamiltonian, symplectic, and re-
cursion operators). Infinite hierarchies of symmetries and conservation
laws are constructed as well