Supersymmetric Mechanics Research Articles

Note on 𝒩 = 8 supersymmetric mechanics with dynamical and semi-dynamical multiplets

Note on 𝒩 = 8 supersymmetric mechanics with dynamical and semi-dynamical multiplets

Supersymmetric field theories and the elliptic index theorem with complex coefficients

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushfoward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $\sigma$-model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and 2-dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai--Quillen Thom form in complexified ${\rm KO}$-theory and a cocycle representative of the $\hat{A}$-class for a family of oriented manifolds.

Global symmetries of Quaternion-K\xe4hler mathcal{N} = 4 supersymmetric mechanics

We analyze the global symmetries of mathcal{N} = 4 supersymmetric mechanics in- volving 4n-dimensional Quaternion-Kähler (QK) 1D sigma models on projective spaces ℍHn and ℍPn as the bosonic core. All Noether charges associated with global worldline symmetries are shown to vanish as a result of equations of motion, which implies that we deal with a severely constrained hamiltonian system. The complete hamiltonian analysis of the bosonic sector is performed.

Geometry and integrability in N=8 supersymmetric mechanics

We construct the $\mathcal{N}=8$ supersymmetric mechanics with potential term whose configuration space is the special K\"ahler manifold of rigid type and show that it can be viewed as the K\"ahler counterpart of $\mathcal{N}=4$ 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 K\"ahler metrics. The bosonic parts of these systems include superintegrable deformations of perturbed two-dimensional oscillator and Coulomb systems.

Symmetries of deformed supersymmetric mechanics on Kähler manifolds

Based on the systematic Hamiltonian and superfield approaches we construct the deformed $\mathcal{N}=4,8$ supersymmetric mechanics on K\"ahler manifolds interacting with constant magnetic field, and study their symmetries. At first we construct the deformed $\mathcal{N}=4,8$ supersymmetric Landau problem via minimal coupling of standard (undeformed) $\mathcal{N}=4,8$ supersymmetric free particle systems on K\"ahler manifold with constant magnetic field. We show that the initial "flat" supersymmetries are necessarily deformed to $SU(2|1)$ and $SU(4|1)$ supersymmetries, with the magnetic field playing the role of deformation parameter, and that the resulting systems inherit all the kinematical symmetries of the initial ones. Then we construct $SU(2|1)$ supersymmetric K\"ahler oscillators and find that they include, as particular cases, the harmonic oscillator models on complex Euclidian and complex projective spaces, as well as superintegrable deformations thereof, viz. $\mathbb{C}^N$-Smorodinsky-Winternitz and $\mathbb{CP}^N$-Rosochatius systems. We show that the supersymmetric extensions proposed inherit all the kinematical symmetries of the initial bosonic models. They also inherit, at least in the case of $\mathbb{C}^N$ systems, hidden (non-kinematical) symmetries. The superfield formulation of these supersymmetric systems is presented, based on the worldline $SU(2|1)$ and $SU(4|1)$ superspace formalisms.

Hidden Supersymmetries of Deformed Supersymmetric Mechanics

We consider quantum models corresponding to superymmetrizations of the two-dimensional harmonic oscillator based on worldline d = 1 realizations of the supergroup SU( N/2 |1), where the number of supersymmetries N is arbitrary even number. Constructed models possess the hidden supersymmetry SU( N/2 |2). Degeneracies of energy levels are spanned by representations of the hidden supersymmetry group.

Superfield approach to higher derivative mathcal{N} = 1 superconformal mechanics

We formulate the equations which determine a potential function in an mathcal{N} = 1 higher derivative supersymmetric mechanics compatible with the osp(2|1)⊕so(d) symmetry and provide a few explicit examples.

New deformed models of and supersymmetric mechanics

We present two types of new deformations of extended supersymmetric mechanics. The first one generalizes SU(2|1) mechanics and encompasses various models of the “curved” SU(2|2) and SU(4|1) mechanics as deformations of the flat mechanics models. Another type is a generalization of the mechanics associated with the multiplets (4, 4, 0) and involving hyper-Kähler d = 1 sigma models in the bosonic sector. This kind of deformation results in models with quaternion-Kähler d = 1 sigma models as the bosonic core.

The curved WDVV equations and superfields

We reproduce the supersymmetric mechanics on curved spaces, constructed earlier within the Hamiltonian formalism, using the superfield methods. We show that for any such mechanics, given by the metric and the third order Codazzi tensor, it is possible to construct a suitable modification of irreducibility conditions of linear multiplets and obtain the superfield Lagrangian by solving a simple differential equation. Also, we prove that the constructed irreducibility conditions are consistent if and only if the metric and Codazzi tensor satisfy the modification of the WDVV equations, which are the conditions of existence of supersymmetry.

Deformed 
 
 
 
 N
 
 
 
 = 8 Supersymmetric Mechanics

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.

Deformed mathcal{N}=8 mechanics of (8, 8, 0) multiplets

We construct new models of “curved” SU(4|1) supersymmetric mechanics based on two versions of the off-shell multiplet (8, 8, 0) which are “mirror” to each other. The worldline realizations of the supergroup SU(4|1) are treated as a deformation of flat mathcal{N}=8 , d = 1 supersymmetry. Using SU(4|1) chiral superfields, we derive invariant actions for the first-type (8, 8, 0) multiplet, which parametrizes special Kähler manifolds. Since we are not aware of a manifestly SU(4|1) covariant superfield formalism for the second-type (8, 8, 0) multiplet, we perform a general construction of SU(4|1) invariant actions for both multiplet types in terms of SU(2|1) superfields. An important class of such actions enjoys superconformal OSp(8|2) invariance. We also build off-shell actions for the SU(4|1) multiplets (6, 8, 2) and (7, 8, 1) through appropriate substitutions for the component fields in the (8, 8, 0) actions. The (6, 8, 2) actions are shown to respect superconformal SU(4|1, 1) invariance.

SU(2|1) supersymmetric mechanics on curved spaces

We present SU(2|1) supersymmetric mechanics on n-dimensional Riemannian manifolds within the Hamiltonian approach. The structure functions including prepotentials entering the supercharges and the Hamiltonian obey extended curved WDVV equations specified by the manifold’s metric and curvature tensor. We consider the most general u(2)-valued prepotential, which contains both types (with and without spin variables), previously considered only separately. For the case of real Kähler manifolds we construct all possible interactions. For isotropic (so(n)-invariant) spaces we provide admissible prepotentials for any solution to the curved WDVV equations. All known one-dimensional SU(2|1) supersymmetric models are reproduced.

N=4 supersymmetric mechanics on curved spaces

We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds constructed within the Hamiltonian approach. The structure functions entering the supercharges and the Hamiltonian obey modified covariant constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde equations specified by the presence of the manifold's curvature tensor. Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In particular, the sum of $n$ copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.

Curved WDVV equation and supersymmetric mechanics

We extend the relation between the Witten-Dijkgraaf-Verlinde-Verlinde equation and N = 4 supersymmetric mechanics to arbitrary curved spaces. The resulting curved WDVV equation is written in terms of the third rank Codazzi tensor. We provide the solutions of the curved WDVV equation for the so(n) symmetric conformally flat metrics. We also explicitly demonstrate how each solution of the flat WDVV equation can be lifted up to the curved WDVV solution on the conformally flat spaces.

New deformations of N = 4 and N = 8 supersymmetric mechanics

This is a review of two different types of the deformed N = 4 and N = 8 supersymmetric mechanics. The first type is associated with the worldline realizations of the supergroups SU(2|1) (four supercharges), as well as of SU(2|2) and SU(4|1) (eight supercharges). The second type is the quaternion- Kähler (QK) deformation of the hyper-Kähler (HK) N = 4 mechanics models. The basic distinguishing feature of the QK models is a local N = 4 supersymmetry realized in d = 1 harmonic superspace.

Deformed supersymmetric quantum mechanics with spin variables

We quantize the one-particle model of the SU(2|1) supersymmetric multiparticle mechanics with the additional semi-dynamical spin degrees of freedom. We find the relevant energy spectrum and the full set of physical states as functions of the mass-dimension deformation parameter m and SU(2) spin qin left({mathrm{mathbb{Z}}}_{>0,}1/2+{mathrm{mathbb{Z}}}_{ge 0}right) . It is found that the states at the fixed energy level form irreducible multiplets of the supergroup SU(2|1). Also, the hidden superconformal symmetry OSp(4|2) of the model is revealed in the classical and quantum cases. We calculate the OSp(4|2) Casimir operators and demonstrate that the full set of the physical states belonging to different energy levels at fixed q are unified into an irreducible OSp(4|2) multiplet.

Quaternion-K\xe4hler mathcal{N} = 4 supersymmetric mechanics

Using the mathcal{N} = 4, 1D harmonic superspace approach, we construct a new type of mathcal{N} = 4 supersymmetric mechanics involving 4n-dimensional Quaternion-Kähler (QK) 1D sigma models as the bosonic core. The basic ingredients of our construction are local mathcal{N} = 4,1D supersymmetry realized by the appropriate transformations in 1D harmonic superspace, the general mathcal{N} = 4, 1D superfield vielbein and a set of 2(n + 1) analytic “matter” superfields representing (n + 1) off-shell supermultiplets (4, 4, 0). Both superfield and component actions are given for the simplest QK models with the manifolds ℍHn = Sp(1, n)/[Sp(1) × Sp(n)] and ℍPn = Sp(1 + n)/[Sp(1) × Sp(n)] as the bosonic targets. For the general case the relevant superfield action and constraints on the (4, 4, 0) “matter” superfields are presented. Further generalizations are briefly discussed.

SU(2|1) supersymmetric mechanics as a deformation of N = 4 mechanics

We give a brief review of SU(2|1) supersymmetric quantum mechanics based on the worldline realizations of the supergroup SU(2|1) in the appropriate N = 4, d = 1 superspaces. The corresponding SU(2|1) models are deformations of standard N = 4, d = 1 models by a mass parameter m.

Long N = 2, 4 multiplets in supersymmetric mechanics

We define SU(2|1) supermultiplets described by chiral superfields having non-zero external spins with respect to SU(2) ⊂ SU(2|1) and show that their splitting into N = 2, d = 1 multiplets contains the so called “long” indecomposable N = 2, d = 1 multiplets (2, 4, 2)l. We give superfield formulation for this type of N = 2 long multiplets and construct their most general superfield action. A simple example of long N = 4, d = 1 multiplet is also considered, both in the superfield and the component formulations.

SU(2|2) supersymmetric mechanics

We introduce a new kind of non-relativistic ${\cal N}{=}\,8$ supersymmetric mechanics, associated with worldline realizations of the supergroup $SU(2|2)$ treated as a deformation of flat ${\cal N}{=}\,8$, $d{=}1$ supersymmetry. Various worldline $SU(2|2)$ superspaces are constructed as coset manifolds of this supergroup, and the corresponding superfield techniques are developed. For the off-shell $SU(2|2)$ multiplets $({\bf 3,8,5})$, $({\bf 4,8,4})$ and $({\bf 5,8,3})$, we construct and analyze the most general superfield and component actions. Common features are mass oscillator-type terms proportional to the deformation parameter and a trigonometric realization of the superconformal group $OSp(4^*|4)$ in the conformal cases. For the simplest $({\bf 5, 8, 3})$ model the quantization is performed.