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A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded variables which can either commute or anticommute, according to their degree. To obtain a consistent global description of graded manifolds, one resorts to sheaves of graded commutative associative algebras on second countable Hausdorff topological spaces, locally isomorphic to a suitable “model space”. This paper aims to build robust mathematical foundations of geometry of graded manifolds. Some known issues in their definition are resolved, especially the case where positively and negatively graded coordinates appear together. The focus is on a detailed exposition of standard geometrical constructions rather than on applications. Necessary excerpts from graded algebra and graded sheaf theory are included.

This is a didactical review on the canonical BV Laplacian on half-densities.

An appropriate definition of the Hodge duality ⋆ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality ⋆ operation on the (2+2)-dimensional supermanifold parametrized by a couple of even (bosonic) space–time variables x^μ(μ = 0, 1) and a couple of odd (fermionic) variables θ and of the Grassmann algebra. The Minkowski space–time manifold, hidden in the supermanifold and parametrized by x^μ(μ = 0, 1), is chosen to be a flat manifold on which a two (1+1)-dimensional (2D) free Abelian gauge theory, taken as a prototype field theoretical model, is defined. We demonstrate the applications of the above definition (and its further generalization) for the discussion of the (anti-)co-BRST symmetries that exist for the field theoretical models of 2D and 4D free Abelian gauge theories considered on the four (2+2)- and six (4+2)-dimensional supermanifolds, respectively.

Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a nondegenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of two different types of odd geometries on supermanifolds.

In this paper, a theory of characteristic classes for super vector bundles over is studied. Some properties of even and odd Chern classes, constructed in this theory, are established. At last, the relevance of the theory to supergeometry is discussed.

We study the relation between the Laplacian associated to an odd metric on a supermanifold and harmonic superfunctions, through the application of the calculus of variations to a supersymmetric sigma model.

A novel supergeometric generalization of common n-sphere, called ν-sphere, is introduced. It is shown that a canonical volume form exists on a ν-sphere. Then a concept of degree is developed for the maps between them. Finally, a semigroup action on ν-spheres is discussed.

ν-classes are supergeometric generalization of the Chern classes. Associated to each superline bundle over supermanifold , ν-class is an element of the second cohomology group of M with coefficients in ℤ[ν] with ν² = 1.

We show that the metric (line element) is the first geometrical object to be associated to a discrete (quantum) structure of the spacetime without necessity of black hole-entropy-area arguments, in sharp contrast with other attempts in the literature. To this end, an emergent metric solution obtained previously in [ Phys. Lett. B661 (2008) 186–191] from a particular non-degenerate Riemannian superspace is introduced. This emergent metric is described by a physical coherent state belonging to the metaplectic group Mp(n) with a Poissonian distribution at lower n (number basis) restoring the classical thermal continuum behavior at large n(n → ∞), or leading to non-classical radiation states, as is conjectured in a quite general basis by means of the Bekenstein–Mukhanov effect. Group-dependent conditions that control the behavior of the macroscopic regime spectrum (thermal or not), as the relationship with the problem of area/entropy of the black hole are presented and discussed.

In this paper, we study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a nontrivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus.

Let V = ℝ^p,q be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cℓ⁰(V). A homogeneous quaternionic manifold (M, Q) is constructed for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map II: ∧²W → V. If the skew symmetric vector valued bilinear form II is nondegenerate then (M, Q) is endowed with a canonical pseudo-Riemannian metric g such that (M, Q, g) is a homogeneous quaternionic pseudo-Kähler manifold. If the metric g is positive definite, i.e. a Riemannian metric, then the quaternionic Kähler manifold (M, Q, g) is shown to admit a simply transitive solvable group of automorphisms. In this special case (p = 3) we recover all the known homogeneous quaternionic Kähler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If p > 3 then M does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kähler manifolds. Then it is shown that for q = 0 the noncompact quaternionic manifold (M, Q) can be endowed with a Riemannian metric h such that (M, Q, h) is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if p > 3.

The twistor bundle Z → M and the canonical SO(3)-principal bundle S → M associated to the quaternionic manifold (M, Q) are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution of complex codimension one, which is a complex contact structure if and only if II is nondegenerate. Moreover, an equivariant open holomorphic immersion into a homogeneous complex manifold of complex algebraic group is constructed.

Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map II : ∨²W → V a homogeneous quaternionic supermanifold (M, Q) is constructed and, moreover, a homogeneous quaternionic pseudo-Kähler supermanifold (M, Q, g) if the symmetric vector valued bilinear form II is nondegenerate.