Supersymmetric σ-models and graded Lie groups (original) (raw)

The geometry of supersymmetric sigma-models

We review non-linear σ-models with (2,1) and (2,2) supersymmetry. We focus on off-shell closure of the supersymmetry algebra and give a complete list of (2, 2) superfields. We provide evidence to support the conjecture that all N = (2, 2) non-linear σ-models can be described by these fields. This in its turn leads to interesting consequences about the geometry of the target manifolds. One immediate corollary of this conjecture is the existence of a potential for hyper-Kähler manifolds, different from the Kähler potential, which does not only allow for the computation of the metric, but of the three fundamental twoforms as well. Several examples are provided: WZW models on SU (2) × U (1) and SU (2) × SU (2) and four-dimensional special hyper-Kähler manifolds.

Curved Superspaces and Local Supersymmetry in Supermatrix Model

Progress of Theoretical Physics, 2006

In a previous paper, we introduced a new interpretation of matrix models, in which any d-dimensional curved space can be realized in terms of d matrices, and the diffeomorphism and the local Lorentz symmetries are included in the ordinary unitary symmetry of the matrix model. Furthermore, we showed that the Einstein equation is naturally obtained, if we employ the standard form of the action, S = −tr [Aa, A b ][A a , A b ] +• • •. In this paper, we extend this formalism to include supergravity. We show that the supercovariant derivatives on any d-dimensional curved space can be expressed in terms of d supermatrices, and the local supersymmetry can be regarded as a part of the superunitary symmetry. We further show that the Einstein and Rarita-Schwinger equations are compatible with the supermatrix generalization of the standard action. *) Because a covariant derivative introduces a new vector index, it is not an endomorphism. Therefore, it cannot be represented by a set of matrices. *) Ra b and R (a) b represent the same quantity. However, we distinguish them, because a and (a) obey different transformation laws. Specifically, a is transformed by the action of G, while (a) is not. *) Strictly speaking, because Aa is Hermitian, we should introduce the anticommutator { , } in Eq. (2. 10): *) As in §2.2, because Aa and Ψα are Hermitian, we should introduce the commutator and the anticommutator. Here we omit them for simplicity.

Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras

Communications in Mathematical Physics, 2007

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p + 2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz's original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the N = 1 superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the N = 2 supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds.

On the general structure of supersymmetric quantum mechanical models

Nuclear Physics B, 1990

We require total invariance of a lagrangian under supersymmetry transformations and we observe that special variables are singled out. They are identical to those entering the Nicolai mapping. We show that a similarity transformation is connected with the introduction of these new variables. We give a stochastic formulation of this transformation using the Cameron-Martin formula. * Onderzoeksdirecteur NFWO, Belgium. Supported in part by "Fonds zur Förderung der wissenschaftlichen Forschung in Osterreich", project Nr. P 5588. ** Onderzoeker l1KW, Belgium.

On algebraic structures in supersymmetric principal chiral model

The European Physical Journal C, 2008

Using the Poisson current algebra of the supersymmetric principal chiral model, we develop the algebraic canonical structure of the model by evaluating the fundamental Poisson bracket of the Lax matrices that fits into the r − s matrix formalism of nonultralocal integrable models. The fundamental Poisson bracket has been used to compute the Poisson bracket algebra of the monodromy matrix that gives the conserved quantities in involution.

Homogeneous K�hler manifolds: Paving the way towards new supersymmetric sigma models

Communications in Mathematical Physics, 1986

Homogeneous Kahler manifolds give rise to a broad class of supersymmetric sigma models containing, as a rather special subclass, the more familiar supersymmetric sigma models based on Hermitian symmetric spaces. In this article, all homogeneous Kahler manifolds with semisimple symmetry group G are constructed, and are classified in terms of Dynkin diagrams. Explicit expressions for the complex structure and the Kahler structure are given in terms of the Lie algebra cj of G. It is shown that for compact G, one can always find an Einstein-Kahler structure, which is unique up to a constant multiple and for which the Kahler potential takes a simple form. * On leave of absence from Fakultat fur Physik der Universitat Freiburg, FRG 1 The term "homogeneous space" is synonymous for "coset space," and similarly, the term "Hermitian symmetric space" is synonymous for "symmetric Kahler manifold"

Generic supersymmetric hyper-Kähler sigma models in

Physics Letters B, 2007

We analyse the geometry of four-dimensional bosonic manifolds arising within the context of N = 4, D = 1 supersymmetry. We demonstrate that both cases of general hyper-Kähler manifolds, i.e. those with translation or rotational isometries, may be supersymmetrized in the same way. We start from a generic N=4 supersymmetric three-dimensional action and perform dualization of the coupling constant, initially present in the action. As a result, we end up with explicit component actions for N = 4, D = 1 nonlinear sigma-models with hyper-Kähler geometry (with both types of isometries) in the target space. In the case of hyper-Kähler geometry with translational isometry we find that the action possesses an additional hidden N = 4 supersymmetry, and therefore it is N = 8 supersymmetric one.

Supersymmetry in physics: An algebraic overview

Physica D: Nonlinear Phenomena, 1985

We survey the possible uses Graded Lie Algebras (GLA) might have in physics. We first review Kac's list of simple GLA including the hyperclassical algebras. A brief, mostly numerical survey of their representation theory follows. Although all GLA can find nonrelativistic applications, only a few can be applied to relativistic situations in any dimensions. Finally, the massless representations of some of these relativistic GLA are listed.

New supersymmetric σ-models with Wess-Zumino terms

Physics Letters B, 1988

We construct new superfields and a-models with (4, 4) and (2, 2) supersymmetry. These lead to new metrics that include examples of manifolds with two complex structures that neither commute nor anticommute. We also consider the analogous issues in euclidean supersymmetry, and find the curious result that we are naturally led to indefinite signature a-model metrics.

Generalized Kähler Geometry from Supersymmetric Sigma Models

Letters in Mathematical Physics, 2006

We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri [10] regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates-Hull-Roček . When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.