An N=8 superconformal particle in the half-plane (original) (raw)

https://doi.org/10.48550/ARXIV.1105.4067

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Abstract

By imposing global supersymmetry and scale invariance we construct an N=8 superconformal mechanical system based on the inhomogeneous (2,8,6) linear multiplet. The unique action describes a special K"ahler sigma model with a Calogero-type potential and Fayet-Iliopoulos terms. The classical dynamics of the two propagating bosons is restricted to a (warped) half-plane and bounded. We numerically inspect typical trajectories of this special particle.

03 N = 8 superconformal mechanics

2004

We construct new models of N=8 superconformal mechanics associated with the off-shell N=8, d=1 supermultiplets (3,8,5) and (5,8,3). These two multiplets are derived as N=8 Goldstone superfields and correspond to nonlinear realizations of the N=8, d=1 superconformal group OSp(4⋆|4) in its supercosets OSp(4 |4) U(1)R⊗SO(5) and OSp(4|4) SU(2)R⊗SO(4) , respectively. The irreducibility constraints for these superfields automatically follow from appropriate superconformal covariant conditions on the Cartan superforms. The N=8 superconformal transformations of the superspace coordinates and the Goldstone superfields are explicitly given. Interestingly, each N=8 supermultiplet admits two different off-shell N=4 decompositions, with different N=4 superconformal subgroups SU(1, 1|2) and OSp(4⋆|2) of OSp(4⋆|4) being manifest as superconformal symmetries of the corresponding N=4, d=1 superspaces. We present the actions for all such N=4 splittings of the N=8 multiplets considered.

Superconformal mechanics and nonlinear supersymmetry

Journal of High Energy Physics, 2003

We show that a simple change of the classical boson-fermion coupling constant, 2alphato2alphan2\alpha \to 2\alpha n 2alphato2alphan, ninNn\in \NninN, in the superconformal mechanics model gives rise to a radical change of a symmetry: the modified classical and quantum systems are characterized by the nonlinear superconformal symmetry. It is generated by the four bosonic integrals which form the so(1,2) x u(1) subalgebra, and by the 2(n+1) fermionic integrals constituting the two spin-n/2 so(1,2)-representations and anticommuting for the order n polynomials of the even generators. We find that the modified quantum system with an integer value of the parameter alpha\alphaalpha is described simultaneously by the two nonlinear superconformal symmetries of the orders relatively shifted in odd number. For the original quantum model with ∣alpha∣=p|\alpha|=palpha=p, pinNp\in \NpinN, this means the presence of the order 2p nonlinear superconformal symmetry in addition to the osp(2|2) supersymmetry.

Dual multiplets in N = 4 superconformal mechanics

Journal of Physics A: Mathematical and Theoretical, 2013

We propose Lagrangian formulations of a three-particles translation invariant N = 4 superconformal mechanics based on the standard and twisted (1, 4, 3) supermultiplets. We show that in the appropriate set of coordinates, in which the bosonic kinetic terms for each of the two cases take conformally-flat forms, the corresponding models produce just similar physical systems with the flipped angular part of bosonic potential U (x) → 1/U (x). This flipping looks so simple only being written in the "angle" variable, while in the standard variables it looks more complicated to preserve the superconformal symmetry. We demonstrate at both the superfield and the component level, how familiar N = 4 supersymmetric 3-particles models, including 3 − Calogero, BC2 and G2 ones, can be constructed with twisted supermultiplets. We also present some new explicit examples of 3-dimensional superconformal mechanics.

superconformal n-particle mechanics via superspace

Nuclear Physics B, 2009

We revisit the (untwisted) superfield approach to one-dimensional multi-particle systems with N =4 superconformal invariance. The requirement of a standard (flat) bosonic kinetic energy implies the existence of inertial (super-)coordinates, which is nontrivial beyond three particles. We formulate the corresponding integrability conditions, whose solution directly yields the superpotential, the two prepotentials and the bosonic potential. The structure equations for the two prepotentials, including the WDVV equation, follow automatically. The general solution for translation-invariant three-particle models is presented and illustrated with examples. For the four-particle case, we take advantage of known WDVV solutions to construct a D3 and a B3 model, thus overcoming a previously-found barrier regarding the bosonic potential. The general solution and classification remain a challenge.

OSp(4|2) superconformal mechanics

Journal of High Energy Physics, 2009

A new superconformal mechanics with OSp(4|2) symmetry is obtained by gauging the U(1) isometry of a superfield model. It is the one-particle case of the new N =4 super Calogero model recently proposed in arXiv:0812.4276 [hep-th]. Classical and quantum generators of the osp(4|2) superalgebra are constructed on physical states. As opposed to other realizations of N =4 superconformal algebras, all supertranslation generators are linear in the odd variables, similarly to the N =2 case. The bosonic sector of the component action is standard one-particle (dilatonic) conformal mechanics accompanied by an SU(2)/U(1) Wess-Zumino term, which gives rise to a fuzzy sphere upon quantization. The strength of the conformal potential is quantized.

New super-Calogero models and OSp(4|2) superconformal mechanics

Physics of Atomic Nuclei, 2011

We report on the new approach to constructing superconformal extensions of the Calogero-type systems with an arbitrary number of involved particles. It is based upon the superfield gauging of non-abelian isometries of some supersymmetric matrix models. Among its applications, we focus on the new N =4 superconformal system yielding the U(2) spin Calogero model in the bosonic sector, and the oneparticle case of this system, which is a new OSp(4|2) superconformal mechanics with non-dynamical U(2) spin variables. The characteristic feature of these models is that the strength of the conformal inverse-square potential is quantized.

N=8 non-linear supersymmetric mechanics

Physics Letters B, 2006

We construct a new two-dimensional N = 8 supersymmetric mechanics with nonlinear chiral supermultiplet. Being intrinsically nonlinear this multiplet describes 2 physical bosonic and 8 fermionic degrees of freedom. We construct the most general superfield action of the sigma-model type and propose its simplest extension by a Fayet-Iliopoulos term. The most interesting property of the constructed system is a new type of geometry in the bosonic subsector, which is different from the special Kähler one characterizing the case of the linear chiral N = 8 supermultiplet.

Superconformal SU(1, 1|n) mechanics

Journal of High Energy Physics, 2016

Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1, 1|2) in mechanics. Remarking that SU(1, 1|2) is a particular member of a chain of supergroups SU(1, 1|n) parametrized by an integer n, here we begin a systematic study of SU(1, 1|n) multi-particle mechanics. A representation of the superconformal algebra su(1, 1|n) is constructed on the phase space spanned by m copies of the (1, 2n, 2n−1) supermultiplet. We show that the dynamics is governed by two prepotentials V and F , and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation. All solutions to the latter in terms of root systems reveal decoupled models only. An extension of the dynamical content of the (1, 2n, 2n−1) supermultiplet by angular variables in a way similar to the SU(1, 1|2) case is problematic.

Script N=4 superconformal mechanics as a non linear realization

Journal of High Energy Physics, 2006

An action for a superconformal particle is constructed using the non linear realization method for the group P SU (1, 1|2), without introducing superfields. The connection between P SU (1, 1|2) and black hole physics is discussed. The lagrangian contains six arbitrary constants and describes a non-BPS superconformal particle. The BPS case is obtained if a precise relation between the constants in the lagrangian is verified, which implies that the action becomes

Geometry and integrability in N=8 supersymmetric mechanics

Physical Review D

We construct the N = 8 supersymmetric mechanics with potential term whose configuration space is the special Kähler manifold of rigid type and show that it can be viewed as the Kähler counterpart of N = 4 mechanics related to "curved WDVV equations". Then, we consider the special case of the supersymmetric mechanics with the non-zero potential term defined on the family of U (1)invariant one-(complex)dimensional special Kähler metrics. The bosonic parts of these systems include superintegrable deformations of perturbed two-dimensional oscillator and Coulomb systems.

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References (12)

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Superconformal mechanics

Journal of Physics A: Mathematical and Theoretical, 2012

We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single-and multiparticle systems are covered. ⋆ Invited review by Journal of Physics A: Mathematical and Theoretical × On leave of absence from V.N. Karazin Kharkov National University, Ukraine N =0 , N =2 [61] and N = 4 [9] superconformal mechanics.

N=8 superconformal mechanics

Nuclear Physics B, 2004

We construct new models of N =8 superconformal mechanics associated with the off-shell N =8, d=1 supermultiplets (3, 8, 5) and (5, 8, 3). These two multiplets are derived as N =8 Goldstone superfields and correspond to nonlinear realizations of the N =8, d=1 superconformal group OSp(4 ⋆ |4) in its supercosets OSp(4 ⋆ |4)

New variant ofN= 4 superconformal mechanics

Journal of High Energy Physics, 2003

Proceeding from a nonlinear realization of the most general N =4, d=1 superconformal symmetry, associated with the supergroup D(2, 1; α), we construct a new model of nonrelativistic N =4 superconformal mechanics. In the bosonic sector it combines the worldline dilaton with the fields parametrizing the R-symmetry coset S 2 ∼ SU (2)/U (1). We present invariant off-shell N =4 and N =2 superfield actions for this system and show the existence of an independent N =4 superconformal invariant which extends the dilaton potential. The extended supersymmetry requires this potential to be accompanied by a d=1 WZW term on S 2. We study the classical dynamics of the bosonic action and the geometry of its sigma-model part. It turns out that the relevant target space is a cone over S 2 for any non-zero α = ± 1 2. The constructed model is expected to be related to the 'relativistic' N =4 mechanics of the AdS 2 × S 2 superparticle via a nonlinear transformation of the fields and the time variable.

Many-particle mechanics with D(2, 1; α) superconformal symmetry

Journal of High Energy Physics, 2011

We overcome the barrier of constructing N =4 superconformal models in one space dimension for more than three particles. The D(2, 1; α) superalgebra of our systems is realized on the coordinates and momenta of the particles, their superpartners and one complex pair of harmonic variables. The models are determined by two prepotentials, F and U , which must obey the WDVV and a Killingtype equation plus homogeneity conditions. We investigate permutation-symmetric solutions, with and without translation invariance. Models based on deformed An and BCDn root systems are constructed for any value of α, and exceptional Fn-type and super root systems admit solutions as well. Translation-invariant mechanics occurs for any number of particles at α=− 1 2 (osp(4|2) invariance as a degenerate limit) and for four particles at arbitrary α (three series).

New variant of N = 4 superconformal mechanics

Journal of High Energy Physics, 2003

Proceeding from a nonlinear realization of the most general N =4, d=1 superconformal symmetry, associated with the supergroup D(2, 1; α), we construct a new model of nonrelativistic N =4 superconformal mechanics. In the bosonic sector it combines the worldline dilaton with the fields parametrizing the R-symmetry coset S 2 ∼ SU (2)/U (1). We present invariant off-shell N =4 and N =2 superfield actions for this system and show the existence of an independent N =4 superconformal invariant which extends the dilaton potential. The extended supersymmetry requires this potential to be accompanied by a d=1 WZW term on S 2 . We study the classical dynamics of the bosonic action and the geometry of its sigma-model part. It turns out that the relevant target space is a cone over S 2 for any non-zero α = ± 1 2 . The constructed model is expected to be related to the 'relativistic' N =4 mechanics of the AdS 2 × S 2 superparticle via a nonlinear transformation of the fields and the time variable.

Potentials in N=4 superconformal mechanics

Physical Review D, 2009

Proceeding from nonlinear realizations of (super)conformal symmetries, we explicitly demonstrate that adding the harmonic oscillator potential to the action of conformal mechanics does not break these symmetries but modifies the transformation properties of the (super)fields. We also analyze the possibility to introduce potentials in N = 4 supersymmetric mechanics by coupling it with auxiliary fermionic superfields. The new coupling we considered does not introduce new fermionic degrees of freedom -all our additional fermions are purely auxiliary ones. The new bosonic components have a first order kinetic term and therefore they serve as spin degrees of freedom. The resulting system contains, besides the potential term in the bosonic sector, a non-trivial spin-like interaction in the fermionic sector. The superconformal mechanics we constructed in this paper is invariant under the full D(2, 1; α) superconformal group. This invariance is not evident and is achieved within modified (super)conformal transformations of the superfields.

Deformed N = 8 Supersymmetric Mechanics

Symmetry

We give a brief review of deformed N = 8 supersymmetric mechanics as a generalization of SU(2|1) mechanics. It is based on the worldline realizations of the supergroups SU(2|2) and SU(4|1) in the appropriate N = 8 , d = 1 superspaces. The corresponding models are deformations of the standard N = 8 mechanics models by a mass parameter m.

Two-dimensional N = 8 supersymmetric mechanics in superspace

Physics Letters B, 2005

We construct a two-dimensional N = 8 supersymmetric quantum mechanics which inherits the most interesting properties of N = 2, d = 4 supersymmetric Yang-Mills theory. After dimensional reduction to one dimension in terms of field-strength, we show that only complex scalar fields from the N = 2, d = 4 vector multiplet become physical bosons in d = 1. The rest of the bosonic components are reduced to auxiliary fields, thus giving rise to the (2, 8, 6) supermultiplet in d = 1. We construct the most general superfields action for this supermultiplet and demonstrate that it possesses duality symmetry extended to the fermionic sector of theory. We also explicitly present the Dirac brackets for the canonical variables and construct the supercharges and Hamiltonian which form a N = 8 super Poincarè algebra with central charges. Finally, we discuss the duality transformations which relate the (2, 8, 6) supermultiplet with the (4, 8, 4) one.

𝒩 = 4 superconformal Calogero models

Journal of High Energy Physics, 2007

We continue the research initiated in hep-th/0607215 and apply our method of conformal automorphisms to generate various N =4 superconformal quantum many-body systems on the real line from a set of decoupled particles extended by fermionic degrees of freedom. The su(1, 1|2) invariant models are governed by two scalar potentials obeying a system of nonlinear partial differential equations which generalizes the Witten-Dijkgraaf-Verlinde-Verlinde equations. As an application, the N =4 superconformal extension of the three-particle (A-type) Calogero model generates a unique G 2 -type Hamiltonian featuring three-body interactions. We fully analyze the N =4 superconformal three-and four-particle models based on the root systems of A 1 ⊕ G 2 and F 4 , respectively. Beyond Wyllard's solutions we find a list of new models, whose translational non-invariance of the center-of-mass motion fails to decouple and extends even to the relative particle motion.

non-linear supersymmetric mechanics

Physics Letters B, 2006

We construct a new two-dimensional N = 8 supersymmetric mechanics with nonlinear chiral supermultiplet. Being intrinsically nonlinear this multiplet describes 2 physical bosonic and 8 fermionic degrees of freedom. We construct the most general superfield action of the sigma-model type and propose its simplest extension by a Fayet-Iliopoulos term. The most interesting property of the constructed system is a new type of geometry in the bosonic subsector, which is different from the special Kähler one characterizing the case of the linear chiral N = 8 supermultiplet.