Real Grassmann Research Articles

Null fermio-dynamics in superspace: Abelian and non-Abelian 3D massive vector fields

Half-order actions, i.e. actions where each term contains, at most, one spinorial derivative, of all the different known 3D supersymmetric massive vector systems are given. Since the two real Grassmann coordinates ( theta 1, theta 2)= theta of the 3D superspace respectively constitute the two independent (fermionic) light-front projections of theta along the null bosonic directions x+or-=21/2 (x0-or+x1), one of them, theta 1 is regarded as the spinorial time while the other one, theta 2, is seen as a (spinorial) spacelike variable. In terms of the corresponding null spinorial timelike and spinorial spacelike derivatives the field equations contain algebraic as well as differential constraints. The authors solve them for the different known 3D vector systems and obtain their corresponding unconstrained actions in superspace. In each case their evolution is shown to be controlled by a quantity which, by analogy with the standard bosonic case, is called the superenergy of the system. The analysis of the scalar case illustrates the connection between null dynamics in superspace and the standard null dynamics in 3D bosonic spacetime. It also shows how much the superenergy contributes to the null bosonic generator of the 3D dynamics.

On the Stiefel-Whitney classes and the span of real Grassmannians

On the Stiefel-Whitney classes and the span of real Grassmannians

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces

On a symmetric coset space G/H the eigenvalues of the Laplacian and the Lichnerowicz operator acting on arbitrary tensor harmonics are given in terms of the eigenvalues of the quadratic Casimir operators of G and H. Explicit examples for Sn, CPn, and real (complex) Grassmann manifolds are analyzed.

Non-Parallelizability of Grassmann Manifolds

AbstractThe fact that no real Grassmann manifolds Gk(Rn) are parallelizable (or even stably parallelizable) except for the obvious cases G1R2≅S1, G1(R4)≅G3(R4) ≅ RP3, and G1(R8)≅ G7(R8) ≅ RP7 was first noted by Hiller and Stong. Their work in turn depends on induction and the work of Oproiu, who examined detailed calculations of Stiefel-Whitney classes for k = 2, 3. In this note we give a short proof of this result, using elementary results from K-theory, that also covers the complex and quaternionic Grassmann manifolds.

N = 2, three-dimensional massless and topologically massive supersymmetric Yang-Mills theory

The actions for the N = 2 supersymmetric massless and topological massive three-dimensional non-abelian gauge theory are given. The basic potential is a real superscalar field defined on the natural complex superspace induced by the N = 2 extended supersymmetry in the real Grassmann sector.

Bäcklund transformations and soliton-type solutions for σ models with values in real Grassmannian spaces

A Backlund transformation for the two-dimensional σ model with values in oriented real Grassmannian spaces is constructed using the known Backlund transformation for the SO(n) principal models. The construction provides a natural way to linearize the differential system of the Backlund transformation. In the case of the S n≈SO(n+1/SOn) model, the Backlund transformation reduces to that of Pohlmeyer. The one-soliton solution for S 2∼SO(3)/SO(2) is obtained analytically and plots of one-soliton and two-soliton solutions are displayed.

Cup products in Grassmannians

This note calculates the height of the first Stiefel-Whitney class in the cohomology of the real Grassmannians and determines the length of the longest nontrivial cup-product in H ∗(G k(R n+k);Z 2) (k⩽n) with k⩽4.

Immersion dimension for real Grassmannians

Immersion dimension for real Grassmannians

Non-Linear Model of Grassmann Manifold and Non-Abelian Gauge Field with Scalar Coupling

The renormalization group functions of the non-abelian gauge field with scalar coupling are studied near four dimensions. The 1/N expansion is considered for 2≪d≪4. It is shown that this model is related to the non-linear σ model of the real Grassmann manifold O(N)/O(p)×O(N-p) for the Large N case. The case of a complex field is also investigated.

Real vector bundles and spaces with free involutions

The functor KR(X), defined in [4], is a contravariant functor defined in the category of spaces with involutions. It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation. For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold. This fixed point set can be identified with the real Grassmann manifold. It is then shown that, for free involutions, KR(X) is an invariant of the homotopy type of the orbit space X modulo its involution. The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface Q in complex projective space. Q carries a free involution and this classification is again valid for spaces with free involutions.

LIE GROUPS TRANSITIVE ON GRASSMANN AND STIEFEL MANIFOLDS

Lie groups which ate transitive on real Grassmann manifolds Gn,2k and on quaternionic Grassmann manifolds Qn,k are studied. The principal result states that each connected Lie group which acts transitively and effectively on Gn,2k (2 < 2k < n – 2) or on Qn,k (2 < k < n – 2) is similar to the real linear group SL(n, R) or the quaternionic group SU* (2n) or their subgroups SO(n) and Sp(n) respectively. The analogous statement for complex Grassmann manifolds was shown by the author previously (Math. USSR Sb. 4 (1968), 233-240). Also treated are all simple compact Lie groups which are transitive on real, complex or quaternionic Stiefel manifolds (with some exceptions). From this is obtained a classification of all noncompact simple Lie groups which are transitive on these manifolds and whose maximal compact subgroups contain a unique simple normal divisor of rank greater than 1. Bibliography: 15 items.

Symmetric spaces which are real cohomology spheres

This is a survey in which we collate some known results using semi-standard techniques, dropping the condition of simple connectivity in Kostant's work [2] and proving Theorem 1. Let M be a compact connected riemannian symmetric space. Then M is a real cohomology (dim M)-sphere if and only if (1) M is an odd dimensional sphere or real projective space; or (2) M = M/Γ where (a) M = S 2 r i X •. X STM, rt > 0, product of m > 1 even dimensional spheres, and (b) Γ consists of all γ = γλX X γm, where γ{ is the identity map or the antipodal map of S 2 r , and the number of yt which are antipodal maps, is even or (3) M = SU(3)/SO(3) orM = {SU(3)/Z3}/SO(3) or (4) M = O(5)/O(2) X O(3), non-oriented real grassmannian of 2-planes through 0 in R. In (2) we note πx(M) = Γ = {Ίj^ ~ in particular the even dimensional spheres are the case m — 1. In (3) we note that the first case is the universal 3-fold covering of the second case. In (4) we have πλ{M) = Z2. Theorem 1 is based on a series of lemmas which can be pushed, with appropriate modification, to the case of a real cohomology w-sphere of dimension greater than n. Here we make the convention that a 0-sphere is a single point. By using a cohomology theory which satisfies the homotopy axiom (such as singular theory) we can also drop the requirement of compactness. Thus we push the method of proof of Theorem 1 and obtain Theorem 2. Let M be a connected riemannian symmetric space. Then M is a real cohomology n-sphere, 0 0 factors are euclidean spaces and irreducible symmetric spaces of noncompact type, and (β) M is one of the following spaces. (1) W = M/Γ, where M = S r i X X STM is the product of m > 0 spheres of positive even dimensions 2ru Γ=(Z2) m consists of all γλX Xγm such that γi is the identity or antipodal map on S s θ is any one of the 2 characters on Γ, and Γ is the kernel of θ. Express θ = θiχ θi$9 where

Space Forms of Grassmann Manifolds

We shall consider the classification problem for space forms of (Riemannian manifolds which are covered by) real, complex, and quaternionic Grassmann manifolds. In the particular case of the real Grassmann manifold of oriented 1-dimensional subspaces of a real Euclidean space, this is the classical "spherical space form problem" of Clifford and Klein. We shall not consider space forms of the Cayley projective plane because it is easy to see that there are no non-trivial ones.

On the Cohomology of the Real Grassmann Complexes and the Characteristic Classes of n-Plane Bundles

On the Cohomology of the Real Grassmann Complexes and the Characteristic Classes of n-Plane Bundles

On the cohomology of the real Grassmann complexes and the characteristic classes of 𝑛-plane bundles

1. Introduction. This paper deals with two things: first the cohomology of the real Grassmann spaces; and secondly, relations between the various characteristic classes of n-plane bundles. Let Rn be a real n-plane bundle over a paracompact space B. With Pn we associate the Stiefel-Whitney characteristic class