Supermanifold Research Articles

A coupled time integration algorithm for discontinuous deformation analysis using the numerical manifold method

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.

One-dimensional super Calabi-Yau manifolds and their mirrors

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY’s having reduced manifold equal to {mathrm{mathbb{P}}}^1 , namely the projective super space {mathrm{mathbb{P}}}^{left.1right|2} and the weighted projective super space mathbb{W}{mathrm{mathbb{P}}}_{(2)}^{left.1right|1} . Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces {mathrm{mathbb{P}}}^{left.nright|m} . We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of {mathrm{mathbb{P}}}^{left.1right|2} , whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of {mathrm{mathbb{P}}}^{left.1right|m} , discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that {mathrm{mathbb{P}}}^{left.1right|2} is self-mirror, whereas mathbb{W}{mathrm{mathbb{P}}}_{(2)}^{left.1right|1} has a zero dimensional mirror. Also, the mirror map for {mathrm{mathbb{P}}}^{left.1right|2} naturally endows it with a structure of N = 2 super Riemann surface.

On mixed super quasi-Einstein warped products

In this paper, we study mixed super quasi-Einstein warped product manifolds for arbitrary dimension n ≥ 3 and we give an example of mixed super quasi-Einstein manifold (MS(QE)n) to ensure the existence of such manifold. Also in the last section we also give an example of warped product on mixed super quasi-Einstein manifold.

A Characterization of Warped Product on Mixed Super quasi-Einstein Manifold

In this paper we investigate the expressions of the Ricci tensors and scalar curvatures of warped product manifold with respect to the bases and fibres when warped product manifold is a mixed super quasi-Einstein manifold. In some cases we give some obstructions to the existence of such manifolds and in the last section we give an example of warped product on mixed super quasi-Einstein manifold.

CHARACTERIZATION OF SUPER QUASI-EINSTEIN MANIFOLD

Abstract In this paper we give characterizations of super quasi-Einstein manifold, mixed super quasi-Einstein manifold and mixed generalized quasi-Einstein manifold for both even and odd dimensions.

Some Results on Super Quasi-Einstein Manifolds

This paper deals with the study of super quasi-Einstein manifolds admitting -curvature tensor. The totally umbilical hypersurfaces of are also studied. Among others, the existence of such a manifold is ensured by a nontrivial example.

Lie algebroid fibrations

A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [ Q , Q ] = 1 , namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifolds, that essentially involves a complete Ehresmann connection. As it is the case for Lie algebras, such fibrations turn out not to be just locally trivial products. We also define homotopy groups and prove the expected long exact sequence associated to a fibration. In particular, Crainic and Fernandes's obstruction to the integrability of Lie algebroids is interpreted as the image of a transgression map in this long exact sequence.

On Poisson functions

In this paper, defining Poisson functions on super manifolds, we show that the graphs of Poisson functions are Dirac structures, and find Poisson functions which include as special cases both quasi-Poisson structures and twisted Poisson structures.

NILPOTENT SYMMETRIES FOR A FREE RELATIVISTIC PARTICLE IN AUGMENTED SUPERFIELD FORMALISM

In the framework of the augmented superfield formalism, the local, covariant, continuous and off-shell (as well as on-shell) nilpotent (anti-)BRST symmetry transformations are derived for a (0+1)-dimensional free scalar relativistic particle that provides a prototype physical example for the more general reparametrization invariant string- and gravitational theories. The trajectory (i.e. the world-line) of the free particle, parametrized by a monotonically increasing evolution parameter τ, is embedded in a D-dimensional flat Minkowski target manifold. This one-dimensional system is considered on a (1+2)-dimensional supermanifold parametrized by an even element τ and a couple of odd elements (θ and [Formula: see text]) of a Grassmannian algebra. The horizontality condition and the invariance of the conserved (super)charges on the (super)manifolds play very crucial roles in the above derivations of the nilpotent symmetries. The geometrical interpretations for the nilpotent (anti-)BRST charges are provided in the framework of augmented superfield approach.

Notes on the Super Nambu Bracket

We define a super Nambu-Poisson algebra over a super manifold. A super Nambu bracket does not satisfy the usual skew-symmetric property, and we propose another skew-symmetric property. We show that the divergence of super Nambu-Hamiltonian vector fields leads to a generalization of the Batalin-Vilkovisky algebra.

The N=2 super-Wess–Zumino–Novikov–Witten model valued in superdiffeomorphism (SDIFF) M2 is self-dual supergravity in four dimensions

It is shown that the pure topological term of the N=2 super-Wess–Zumino–Novikov–Witten (SWZNW) model, valued in the area-preserving diffeomorphisms group of a two-dimensional Euclidean surface, is self-dual supergravity (SDSG) in 3+1 dimensions. This is achieved by a suitable slicing and pasting procedure of the (2,0); (0,2) N=2 SWZNW models determined by the embedding of the (2,0); (0,2) superspace supercovariant derivatives into the (3+1) superspace ones. Preliminary steps are provided to show that the N=1 SWZNW model, valued in the area-preserving superdiffeomorphism group of a super Riemann surface (SRS) whose signature is (1+1), is self-dual supergravity in (2+2) dimensions: the so-called Atiyah–Ward space–times.

SUPER KRICHEVER FUNCTOR

A supersymmetric generalization of the Krichever map is proposed. This map assigns injectively a point of an infinite dimensional super Grassmannian to a set of geometric data consisting of an arbitrary algebraic super manifold of dimension 1|1 defined over a field of any characteristic and a line bundle on it. The naturality of this map comes from the fact that it is obtained as the restriction of a contravariant functor to certain special objects. This functor gives an antiequivalence between the category of infinite dimensional super vector spaces satisfying the Fredholm condition together with their stabilizers, and the category of algebraic super curves and certain sheaves on them including all even vector bundles of arbitrary rank.

Some applications of KAC moody algebras defined on a super manifold

Superconformal algebras on the one hand, and area preserving diffeomorphisms on the other hand, are examinated in the framework of the Kac Moody algebras defined on a super manifold.

Differential algebras andD-modules in super toda lattice hierarchy

A new super Toda lattice hierarchy is proposed and formulated in the language of differential algebra. AD-module structure is shown to exist behind this nonlinear system and to play the same role as a similarD-module for the super KP hierarchy. From the structure of thisD-module, one can indeed see a direct connection with a set of affine coordinates on an infinite-dimensional super Grassmannian manifold. These affine coordinates are the basic ingredients of an intrinsic construction of τ functions as well as symmetries.

Symmetries of the super KP hierarchy

Symmetries of the super Kadomtsev-Petviashvili hierarchy are studied. A key role is played by a D-module structure, which connects the nonlinear system with the geometry of an infinite-dimensional super Grassmannian manifold. Infinitesimal action of a Lie superalgebra on the super Grassmannian manifold, via this connection, gives rise to symmetries of the nonlinear system.