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Dmitri V. Alekseevsky

Publications and source records attributed to Dmitri V. Alekseevsky.

34 records · Page 2Linked to original sources
arXiv2012

Conification of Kähler and hyper-Kähler manifolds

Given a Kähler manifold $M$ endowed with a Hamiltonian Killing vector field $Z$, we construct a conical Kähler manifold $\hat{M}$ such that $M$ is recovered as a Kähler quotient of $\hat{M}$. Similarly, given a hyper-Kähler manifold $(M,g,J_1,J_2,J_3)$ endowed with a Killing vector field $Z$, Hamiltonian with respect to the Kähler form of $J_1$ and satisfying $\mathcal{L}_ZJ_2= -2J_3$, we construct a hyper-Kähler cone $\hat{M}$ such that $M$ is a certain hyper-Kähler quotient of $\hat{M}$. In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-Kähler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kähler cone, which in turn defines a quaternionic Kähler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.

math.DG↗
arXiv2012

Invariant generalized complex structures on Lie groups

We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (\gk, ω), where \gk is an appropriate regular subalgebra of the complex Lie algebra \gg^{C} associated to G and ωis a closed 2-form on \gk, such that a non-degeneracy condition holds. In the case when G is a semisimple Lie group of inner type (in particular, when G is compact) a classification of regular generalized complex structures on G is given. We show that any invariant generalized complex structure on a compact semisimple Lie group G is regular, provided that an additional natural condition holds. In the case when G is a semisimple Lie group of outer type, we describe the subalgebras \gk in terms of appropriate root subsystems of a root system of \gg^{C} and we construct a large class of admissible pairs (\gk ,ω) (hence, regular generalized complex structures on G).

math.DG↗
arXiv2011

Two-symmetric Lorentzian manifolds

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.

math.DG↗
arXiv2009

On the Geometry of Spaces of Oriented Geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space (other than the octonion hyperbolic plane), and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kaehler structure on L(M).

math.DG↗
arXiv2009

Rolling of Coxeter polyhedra along mirrors

The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.

math.MG↗
arXiv2008

Homogeneous para-Kähler Einstein manifolds

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $ K^2 = \mathrm{Id} $ or, equivalently, as a symplectic manifold $(M,ω)$ with a bi-Lagrangian structure $L^\pm$, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold $M = G/H$ of a semisimple Lie group $G$ admits an invariant para-Kähler structure $(g,K)$ if and only if it is a covering of the adjoint orbit $\mathrm{Ad}_Gh$ of a semisimple element $h.$ We give a description of all invariant para-Kähler structures $(g,K)$ on a such homogeneous manifold. Using a para-complex analogue of basic formulas of Kähler geometry, we prove that any invariant para-complex structure $K$ on $M = G/H$ defines a unique para-Kähler Einstein structure $(g,K)$ with given non-zero scalar curvature. An explicit formula for the Einstein metric $g$ is given. A survey of recent results on para-complex geometry is included.

math.DG↗
arXiv2008

Geometric construction of the r-map: from affine special real to special Kähler manifolds

We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold $M$ as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle $N=TM$ carries a canonical structure of (affine) special Kähler manifold. This gives an intrinsic description of the $r$-map as the map $M\mapsto N=TM$. On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when $M$ is any Hessian manifold.

math.DG↗
arXiv2008

Maximally homogeneous para-CR manifolds of semisimple type

An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in Γ({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras.

math.DG↗
arXiv2004

Polyvector Super-Poincare Algebras

A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W. Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W_0 are submodules of ΛV. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant Λ^k V-valued bilinear forms on the spinor module S.

hep-th↗
arXiv2003

Yang-Mills connections over manifolds with Grassmann structure

Let M be a manifold with Grassmann structure, i.e. with an isomorphism of the cotangent bundle T^*M\cong E\otimes H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection \nabla^W in a vector bundle W\to M as a connection whose curvature F\in S^2E\otimes\wedge^2 H\otimes W \subset\wedge^2 T^*M\otimes W. Under appropriate assumptions, for example, when the Grassmann structure is associated with a quaternionic Kaehler structure on M, half-flatness implies the Yang-Mills equations. Inspired by the harmonic space approach, we develop a local construction of (holomorphic) half-flat connections \nabla^W over a complex manifold with (holomorphic) Grassmann structure equipped with a suitable linear connection. Any such connection \nabla^W can be obtained from a prepotential by solving a system of linear first order ODEs. The construction can be applied, for instance, to the complexification of hyper-Kaehler manifolds or more generally to hyper-Kaehler manifolds with admissible torsion and to their higher-spin analogues. It yields solutions of the Yang-Mills equations.

math.DG↗
arXiv2002

Partially-flat gauge fields on manifolds of dimension greater than four

We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The second is the notion of half-flatness, which is defined for manifolds with certain Grassmann structure T^C M \cong E \otimes H. In some cases, for example for hyper-Kaehler manifolds M, half-flatness implies Omega-self-duality. A construction of half-flat connections inspired by the harmonic space approach is described. Locally, any such connection can be obtained from a free prepotential by solving a system of linear first order ODEs.

hep-th↗
arXiv2001

Flows on quaternionic-Kaehler and very special real manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kaehler manifolds. For the homogeneous quaternionic-Kaehler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p in M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kaehler manifolds we find degenerate local minima.

hep-th↗
arXiv2000

Classification of indefinite hyper-Kaehler symmetric spaces

We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H) over the quaternions on the space (S^4C^n)^τ of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = τS, where τis the real structure induced by the quaternionic structure of C^{2m} = H^m. We define and classify also complex hyper-Kaehler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.

math.DG↗
arXiv1993

Poisson structures on the cotangent bundle of a Lie group or a principal bundle and their reductions

On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $θ$ and the symplectic form $d θ$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on itself), and also in terms of the left Maurer-Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic stuctures on $T\sp*G$ and to Poisson structures by replacing the canonical momenta of the right or left actions of $G$ on itself by arbitrary ones, followed by reduction (to $G$ cross a Weyl-chamber, e.g.). This method also works on principal bundles.

math.SG↗