Related papers: A New look at the vortex equations and dimensional…
We study the dynamics of supersymmetric theories in five dimensions obtained by compactifications of M-theory on a Calabi-Yau threefold X. For a compact X, this is determined by the geometry of X, in particular the Kahler class dependence…
The maximally complicated arbitrary-dimensional "maximal" Galileon field equations simplify dramatically for symmetric configurations. Thus, spherical symmetry reduces the equations from the D- to the two-dimensional Monge-Ampere equation,…
We introduce generalized dimensional reductions of an integrable 1+1-dimensional dilaton gravity coupled to matter down to one-dimensional static states (black holes in particular), cosmological models and waves. An unusual feature of these…
We find the twisted extensions of the symmetry algebra of the 2D Euler equation in the vorticity form and use them to construct new Lax representation for this equation. Then we generalize this result by considering the transformation…
A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a…
We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X…
A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the D-invariant direct sum of stable subbundles). It has…
We prove a refined contraction inequality for diffusion semigroups with respect to the Wasserstein distance on a compact Riemannian manifold taking account of the dimension. The result generalizes in a Riemannian context, the dimensional…
We discuss no-dimensional (approximate) versions of Carath\'eodory's and Helly's theorems. Our goal is to draw attention to open problems and potential applications related to these results. We survey recent progress and pose several…
Vertex algebras in higher dimensions correspond to models of quantum field theory with global conformal invariance. Any vertex algebra in dimension D admits a restriction to a vertex algebra in any lower dimension and, in particular, to…
In D-dimensional spacetimes which can be foliated by n-dimensional homogeneous subspaces, a quantum field can be decomposed in terms of modes on the subspaces, reducing the system to a collection of (D-n)-dimensional fields. This allows one…
Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systems with homogeneous spatial dimensions,…
A condition of reduction of multidimensional wave equations to the two-dimensional equation is studied, and the necessary conditions of compatibility and exact solutions of the resulting d'Alembert-Hamilton system are obtained.
A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of…
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and…
We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional K\"ahler-Einstein equation with symmetry and the one-sided type-D…
In Kaluza-Klein compactifications, some symmetries of the higher dimensional theory are preserved in lower dimensions, others are broken, and occasionally, there are symmetry enhancements. The symmetries that are enhanced by toroidal…
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the…
This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space CP^2 interacting via a…
I point out how coherence vortices, i.e., topological defects in a correlation function, could help explore new physics if they are created in matter waves. Vortex dynamics could be studied in up to six dimensions, and spin topological…