Related papers: By Magri's Theorem, Self-Dual Gravity is Completel…
In general terms duality consists of two descriptions of one physical system by using degrees of freedom of different nature. There are different kinds of dualities and they have been extremely useful to uncover the underlying strong…
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom…
The properties of Dirac gamma matrices in a four-dimensional space-time with the $(2,2)$ signature are studied. The basic spinors are classified, and the existence of Majorana-Weyl spinors is noted. Supersymmetry in $2 + 2$ dimensions is…
Static axisymmetric Einstein-Maxwell-Dilaton and stationary axisymmetric Einstein-Maxwell-Dilaton-Axion (EMDA) theories in four space-time dimensions are shown to be integrable by means of the inverse scattering transform method. The proof…
We classify super-symmetric solutions of the minimal $N=2$ gauged Euclidean supergravity in four dimensions. The solutions with anti-self-dual Maxwell field give rise to anti-self-dual Einstein metrics given in terms of solutions to the…
The main results for the two-dimensional quantum gravity, conjectured from the matrix model or integrable approach, are presented in the form to be compared with the world-sheet or Liouville approach. In spherical limit the integrable side…
We develop a classical two-dimensional bi-scalar gravity based on the Kaluza-Klein reduction applied to the four-dimensional Horndeski theory. One of the scalar fields arises from the original four-dimensional theory, while the extra scalar…
The self-dual solution to lattice Euclidean gravity is constructed. In contrast to the well known Eguchi-Hanson solution to continuous Euclidean Gravity, the lattice solution is asymptotically {\it{globally}} Euclidean, i.e., the boundary…
The (2,2) world-sheet supersymmetric string theory is discussed from the viewpoint of string/membrane unification. The effective field theory in the closed string target space is known to be the 2+2 dimensional (integrable) theory of…
First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three…
We study the Hamiltonian formulation of Plebanski theory in both the Euclidean and Lorentzian cases. A careful analysis of the constraints shows that the system is non regular, i.e. the rank of the Dirac matrix is non-constant on the…
We propose that gravity be intrinsically quantum-mechanical, so that in the absence of quantum mechanics the geometry of the universe would be Minkowski. We show that in such a situation gravity does not require any independent quantization…
We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate…
These lectures are devoted to the low energy limit of \N2 SUSY gauge theories, which is described in terms of integrable systems. A special emphasis is on a duality that naturally acts on these integrable systems. The duality turns out to…
An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and…
This paper studies the self-dual Einstein-Dirac theory. A generalization is obtained of the Jacobson-Smolin proof of the equivalence between the self-dual and Palatini purely gravitational actions. Hence one proves equivalence of self-dual…
The Hamiltonian formalism of bigravity and massive gravity is studied here for the general form of the interaction potential of two metrics. In the theories equipped with two spacetime metrics it is natural to use the Kuchar approach,…
We study some symmetry and integrability properties of four-dimensional Einstein-Maxwell gravity with nonvanishing cosmological constant in the presence of Killing vectors. First of all, we consider stationary spacetimes, which lead, after…
We show that various dynamical gauge theories in two, three and four dimensions, obtained as Manin deformations of topological Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) theories, are equivalent to gravitational theories. Since gravity…
We construct a Lagrangian formulation of Hitchin's self-duality equations on a Riemann surface $\Sigma$ using potentials for the connection and Higgs field. This two-dimensional action is then obtained from a four-dimensional Chern-Simons…