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Special Issue "Supersymmetry"

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A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (30 June 2012)

Special Issue Editor

Guest Editor
Dr. Giuseppe Policastro (Website)

Laboratoire de Physique Theorique, Ecole Normale Superieure, 24, rue Lhomond, 75005 Paris, France

Published Papers (6 papers)

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Research

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Open AccessArticle Supersymmetric Sigma Model Geometry
Symmetry 2012, 4(3), 474-506; doi:10.3390/sym4030474
Received: 6 July 2012 / Revised: 23 July 2012 / Accepted: 2 August 2012 / Published: 23 August 2012
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Abstract
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces. [...] Read more.
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessArticle Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations
Symmetry 2012, 4(3), 441-451; doi:10.3390/sym4030441
Received: 25 May 2012 / Accepted: 27 July 2012 / Published: 8 August 2012
Cited by 1 | PDF Full-text (11001 KB) | HTML Full-text | XML Full-text
Abstract
We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons [...] Read more.
We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessArticle Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection
Symmetry 2012, 4(3), 427-440; doi:10.3390/sym4030427
Received: 18 June 2012 / Revised: 15 July 2012 / Accepted: 26 July 2012 / Published: 7 August 2012
Cited by 6 | PDF Full-text (267 KB) | HTML Full-text | XML Full-text
Abstract
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, [...] Read more.
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom. Full article
(This article belongs to the Special Issue Supersymmetry)

Review

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Open AccessReview Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry
Symmetry 2013, 5(2), 119-214; doi:10.3390/sym5020119
Received: 10 March 2013 / Accepted: 1 April 2013 / Published: 26 April 2013
Cited by 2 | PDF Full-text (1432 KB) | HTML Full-text | XML Full-text
Abstract
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The [...] Read more.
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states. Full article
(This article belongs to the Special Issue Supersymmetry)
Figures

Open AccessReview Supersymmetric Extensions of Non-Relativistic Scaling Algebras
Symmetry 2012, 4(3), 517-536; doi:10.3390/sym4030517
Received: 3 July 2012 / Revised: 7 August 2012 / Accepted: 8 August 2012 / Published: 24 August 2012
Cited by 3 | PDF Full-text (549 KB) | HTML Full-text | XML Full-text
Abstract
An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z [...] Read more.
An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z plays an important role in constructing the gravity duals for non-relativistic field theories. Supersymmetric extensions of symmetry algebras including the anisotropic scaling are very helpful to consider holographic relations accurately. We give a short summary on the classification of superalgebras with the anisotropic scaling as subalgebras of the following Lie superalgebras, psu(2,2|4), osp(8|4) and osp (8*|4), which appear in the study of AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessReview Supersymmetric Quantum Mechanics and Solvable Models
Symmetry 2012, 4(3), 452-473; doi:10.3390/sym4030452
Received: 29 June 2012 / Revised: 20 July 2012 / Accepted: 31 July 2012 / Published: 16 August 2012
Cited by 8 | PDF Full-text (507 KB) | HTML Full-text | XML Full-text
Abstract
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is [...] Read more.
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ. Full article
(This article belongs to the Special Issue Supersymmetry)

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