Homogeneous Superspaces Research Articles

Beyond Lorentzian symmetry

This thesis presents a framework in which to explore kinematical symmetries beyond the standard Lorentzian case. This framework consists of an algebraic classification, a geometric classification, and a derivation of the geometric properties required to define physical theories on the classified spacetime geometries. The work completed in substantiating this framework for kinematical, super-kinematical, and super-Bargmann symmetries constitutes the body of this thesis. To this end, the classification of kinematical Lie algebras in spatial dimension $D = 3$, as presented in [3, 4], is reviewed; as is the classification of spatially-isotropic homogeneous spacetimes of [5]. The derivation of geometric properties such as the non-compactness of boosts, soldering forms and vielbeins, and the space of invariant affine connections is then presented. We move on to classify the $\mathcal{N}=1$ kinematical Lie superalgebras in three spatial dimensions, finding 43 isomorphism classes of Lie superalgebras. Once these algebras are determined, we classify the corresponding simply-connected homogeneous (4|4)-dimensional superspaces and show how the resulting 27 homogeneous superspaces may be related to one another via geometric limits. Finally, we turn our attention to generalised Bargmann superalgebras. In the present work, these will be the $\mathcal{N}=1$ and $\mathcal{N}=2$ super-extensions of the Bargmann and Newton-Hooke algebras, as well as the centrally-extended static kinematical Lie algebra, of which the former three all arise as deformations. Focussing solely on three spatial dimensions, we find $9$ isomorphism classes in the $\mathcal{N}=1$ case, and we identify $22$ branches of superalgebras in the $\mathcal{N}=2$ case.

A class of homogeneous superspaces associated to odd involutions

A new generalization of Grassmannians in supergeometry, called $$\nu $$ -Grassmannians, are constructed by gluing $$\nu $$ -domains. By a $$\nu $$ -domain, we mean a superdomain with an odd involution say $$\nu $$ on its structure sheaf, as morphism of modules. Then we show that $$\nu $$ -Grassmannians are homogeneous superspaces. In addition, in the last section, a supergroup associated to the odd involution $$\nu $$ is introduced.

On the construction of supergrassmannians as homogeneous superspaces

In this paper, we give an explicit description of the action of the super Lie group [Formula: see text] on supergrassmannian [Formula: see text] in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the supergrassmannian.

Kinematical superspaces

We classify N =1 d = 4 kinematical and aristotelian Lie superalgebras with spa- tial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quater- nionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of 43 isomorphism classes of Lie superalgebras, some with pa- rameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous (4|4)-dimensional superspaces, resulting in a list of 27 homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.

Quantum homogeneous superspaces and quantum duality principle

Quantum homogeneous superspaces and quantum duality principle

The [formula omitted] Lie superbialgebras

By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all gl(1|1) Lie superbialgebras. Then, by calculating their classical r-matrices, the gl(1|1) coboundary Lie superbialgebras and their types (triangular, quasi-triangular or factorizable) are determined, furthermore in this way super Poisson structures on the GL(1|1) Lie supergroup are obtained. Also, we classify Drinfeld superdoubles based on the gl(1|1) as a theorem. Afterwards, as a physical application of the coboundary Lie superbialgebras, we construct a new integrable system on the homogeneous superspace OSp(1|2)/U(1). Finally, we make use of the Lyakhovsky and Mudrov formalism in order to build up the deformed gl(1|1) Lie superalgebra related to all gl(1|1) coboundary Lie superbialgebras. For one case, the quantization at the supergroup level is also provided, including its quantum R-matrix.

Riemannian supergeometry

Motivated by Zirnbauer in J Math Phys 37(10):4986–5018 (1996), we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g., geodesics, isometry groups and invariant metrics on Lie supergroups and homogeneous superspaces.

Spherical functions on homogeneous superspaces

Homogeneous superspaces arising from the general linear supergroup are studied within a Hopf algebraic framework. Spherical functions on homogeneous superspaces are introduced, and the structures of the superalgebras of the spherical functions on classes of homogeneous superspaces are described explicitly.

EXPLICIT N = 2 SUPERSYMMETRY FOR HIGHER-SPIN FIELDS IN D = 4 AdS SUPERSPACE

We develop two N = 2 superfield formulations of free equations of motion for the joint model of all D = 4 massless higher-superspin fields. The explicit Osp (2|4) supersymmetry is achieved without exploiting the harmonic superspace, and with adding no auxiliary component fields to those of N = 1 superfields. The formulations are developed in two different Osp (2|4) homogeneous superspaces which have a structure of a fibre bundle over the standard D = 4 AdS superspace, with dimensions (7|4) and (7|8). The N = 2 covariant derivatives in these spaces are expressed in terms of N = 1 ones which give simple rules for component analysis.

Geometry and representations of the quantum supergroup OSPq(1|2n)

The quantum supergroup OSPq(1|2n) is studied systematically. A Haar functional is constructed, and an algebraic version of the Peter–Weyl theory is extended to this quantum supergroup. Quantum homogeneous superspaces and quantum homogeneous supervector bundles are defined following the strategy of Connes’ theory. Parabolic induction is developed by employing the quantum homogeneous supervector bundles. Quantum Frobenius reciprocity and a generalized Borel–Weil theorem are established for the induced representations.

On the supersymplectic homogeneous superspace underlying the OSp(1/2) coherent states

In this work Onofri and Perelomov’s coherent states methods are extended to the recently introduced OSp(1/2) coherent states. These latter are shown to be parametrized by points of a supersymplectic supermanifold, namely, the OSp(1/2)/U(1) homogeneous superspace, which is clearly identified with a supercoadjoint orbit of OSp(1/2) by exhibiting the corresponding equivariant supermoment map. Moreover, this supermanifold is shown to be a nontrivial example of Rothstein’s supersymplectic supermanifolds. More precisely, it is shown that its supersymplectic structure is completely determined in terms of SU(1,1)-invariant (but unrelated) Kähler 2-form and Kähler metric on the unit disc. This result leads to the definition of the notions of a super-Kähler supermanifold and a super-Kähler superpotential, the geometric structure of the former being encoded into the latter.

Unified models from gauged supergroups. III. The ‘‘super-2-sphere’’ OSp(3‖2)/OSp(2‖2)

The formalism of coset space dimensional reduction for internal supersymmetry [P. D. Jarvis, J. Math. Phys. 31, 1783 (1990); P. D. Jarvis and D. S. McAnally, J. Math. Phys. 33, 399 (1992)] is extended in the supersymmetric case, and the homogeneous superspace OSp(3‖2)/OSp(2‖2) is studied in detail, with gauge supergroup OSp(m+2‖2).

Superposition formulas for nonlinear superequations

Nonlinear superequations, for which the general solution can be expressed algebraically in terms of a finite number of particular solutions, are obtained. They are based on the orthosymplectic supergroup OSP(m,2n) and its action on a homogeneous superspace. Superposition formulas are discussed for the cases m=1, n arbitrary, and m=2, n=1. For OSP(2,2) the number of particular solutions needed to reconstruct the general solution depends on the dimension of the underlying Grassmann algebra, whereas for OSP(1,2n) it does not.

Characters of super-Kac-Moody algebras

The partition functions for super-Wess-Zumino-Witten models can be expressed in terms of characters of super-Kac-Moody algebras. These characters are examined with the emphasis on maintaining supersymmetry explicitly. It is shown that an analogoue of Borel-Weil theory is at least formally relevant to the representation theory of super-Kac-Moody algebras, and that the characters have an interpretation in terms of fixed points of the action of the corresponding group on a homogeneous superspace. Characters with nontrivial dependence on the supermodular parameters of superconformal and supersymmetric Yang-Mills backgrounds on the torus with (++) spin structure are computed, and for the case of SU(2), they are used to extend the conventional GKO construction for the characters of the discrete series of unitary representations of the superconformal algebra with c< 3 2 to accomodate the odd superconformal supercharacters of Cohn and Friedan. This extension of the GKO construction requires the incorporation of a “spectator” space of free fermions in the standard GKO construction of superconformal characters relevant to the (+−), (−+) and (−−) spin structures.

A homogeneous superspace relevant to superconformal field theory

An infinite-dimensional homogeneous superspace, determined by the choice of annihilation operators in a superconformal field theory, is introduced. Regularization of the field theory induces a central extension for the group which acts transitively on the space, and the cocycle for the central extension is computed. The vacuum states for the theory form a line bundle over the homogeneous superspace, and the curvature of the line bundle is calculated.

Algebraic approach to supergravity in superspace

The methods of algebraic geometry are used to construct supergravity theories in a homogeneous superspace OSp( N|4) with structure group Sl(2C) ⊗ O N and base supermanifold the coset space OSp( N|4)/( Sl(2 C)⊗ O N ). The nature and origin of the supersymmetry transformations is completely elucidated. The equations of motion for O 1 and O 2 supergravity are obtained, as the realization of this symmetry on the space-time components h μ of the “supervierbein”.