Supersymmetric Kahler oscillator in a constant magnetic field
We propose the notion of the oscillator on Kähler space and consider its supersymmetrization in the presence of a constant magnetic field. Supersymmetric mechanics attracts permanent interest since its introduction [1]. However, studies focussed mainly on the mechanics with standard N = 2 supersymmetries (see for the review [2] and refs therein). The systems with N = 4 supersymmetries also received much attention: the most general N = 4, D = 1, 3 supersymmetric mechanics described by real superfield actions were studied in Refs. [3, 4] respectively, and those in arbitrary D in Ref.[5]; in [6] N = 4, D = 2 supersymmetric mechanics described by chiral superfield actions were considered. Let us mention also some recent papers on this subject [7]. The study of N = 8 supersymmetric mechanics has been performed recently in Ref.[8]. On the other hand, not enough attention has been paid to systems with non-standard supersymmetry algebra, although they arise in many realistic situations. Some of the systems of that sort were extensively studied by M. Plyushchay [9]. A. Smilga studied the dynamical aspects of “weak supersymmetry” [10] on the simple example of the supersymmetric oscillator. He suggested in this case a nontrivial model of “weak supersymmetric” mechanics, related with quasi-exactly solvable models and the systems with nonlinear supersymmetry. In the present work we consider the supersymmetrization of a specific model of Hamiltonian mechanics on Kähler manifold (M0, gab̄dz dz̄) interacting with constant magnetic field B, viz H = g(πaπ̄b + ω∂aK∂̄bK), Ω0 = dπa ∧ dz + dπ̄a ∧ dz̄ + iBgab̄dza ∧ dz̄, (1) where K(z, z̄) is a Kähler potential of configuration space. Notice, that the Kähler potential is defined up to holomorphic and antiholomorphic terms, K(z, z̄) → K(z, z̄) + U(z) + Ū(z̄) , (2) while the Hamiltonian under consideration is not invariant under these transformations. For example, in the limit ω → 0 it yields the Hamiltonian H = g(πaπ̄b + ∂aU(z)∂̄bŪ(z̄)). (3) This Hamiltonian admits, in the absence of magnetic field, a N = 4 superextension [12], in the spirit of Alvarez-Gaumé-Freedman [11]. The suggested system could be viewed, in many cases, as a generalization of the oscillator on the Kähler manifold. It includes, as special cases, a few interesting exactly• The oscillator on I C = IR, H = ππ̄ + ωzz̄, (4) corresponding to the choice U = zz/2. The constants of motion defining the hidden symmetries of the system, could be represented as follows: I ab = πaπb + ω z̄z̄, I = Ī, Iab̄ = πaπ̄b + ω z̄z. (5) The symmetry algebra of the system is u(2n). Clearly, these constants of motion are functionally-dependent ones. • The oscillator on complex projective space I CP (for n > 1) [14], K = r 0 log(1 + zz̄),⇒ H = gπ̄aπb + ωr 0zz̄. (6) This system is also specified, in the absence of magnetic field, by the hidden symmetry given by the constants of motion Jab̄ = i(z πa − π̄bz̄), Iab̄ = J a J − b r 0 + ωr 0z̄ z , (7) where J a = πa + (z̄π̄)z̄ , J a = J̄ + a are the translation generators. These generators form the nonlinear (quadratic) algebra {Jāb, Jc̄d} = iδādJb̄c − iδc̄bJād, {Iab̄, Jcd̄} = iδcb̄Iad̄ − iδad̄Icb̄ {Iab̄, Icd̄} = iωδcb̄Jad̄ − iωδad̄Jcb̄ + iIcb̄(Jad̄ + J0δad̄)/r 0 − iIad̄(Jcb̄ + J0δcb̄)/r 0 . (8) • The oscillator on I CP could also be extended to the class of Kähler conifolds, defined by the Kähler potential [15] K = r 0 log(1 ± (zz̄) 2 ) , ⇒ H = gπ̄aπb + ωr 0(zz̄)ν 2 , (9) where ν is a numerical parameter. Although the corresponding oscillator systems do not have hidden symmetry for ν 6= 1, i.e. on non-constant curvature spaces, they remain exactly-solvable at both the classical [15] and the quantum [16] level. Moreover, for ν = 2 the conic oscillator reduces to the Higgs oscillator on the threedimensional sphere and pseudosphere interacting with a Dirac monopole field and some specific potential proportional to the squared monopole number. Notice also , that the Hamiltonian system under consideration, yields, in the “large mass limit”, πa → 0, the following one, H0 = ωg∂aK∂̄bK, Ω0 = iBgab̄dza ∧ dz̄, which could be easily extended with N = 2 supersymmetry [13]. The supersymmetrization procedure follows closely the steps we performed in [14] for the oscillator on complex projective space. Next we will show that, although the system under consideration does not possess a standard N = 4 superextension, it admits including, as subalgebras, two copies of N = 2 superalgebras. This nonstandard superextension respects the inclusion of constant magnetic field. We follow the following strategy. At first, we extend the initial phase space to the a (2N.2N) I C-dimensional superspace equipped with the symplectic structure Ω = dπa ∧ dz + dπ̄a ∧ dz̄ + i(Bgab̄ + iRab̄cd̄η αη̄d α)dza ∧ dz̄ + gab̄Dη α ∧ Dη̄ α . (10) Here Dη α = dη a α + Γ a bcη a αdz , α = 1, 2, and Γabc, Rab̄cd̄ are, respectively, the connection and curvature of the Kähler structure. The corresponding Poisson brackets are defined by the following non-zero relations (and their complex-conjugates): {πa, zb} = δ a, {πa, η α} = −Γacη α, {πa, π̄b} = i(Bgab̄ + iRab̄cd̄η αη̄d α), {ηa α, η̄ β} = gδαβ . (11) The symplectic structure (10) becomes canonical in the coordinates (pa, χ ) pa = πa − i 2∂ag, χmi = emb η i : ΩScan = dpa ∧ dz + dp̄ā ∧ dz̄ + iBgab̄dz ∧ dz̄ + dχmα ∧ dχ̄m̄α , (12) where ema are the einbeins of the Kähler structure: e m a δmm̄ē m̄ b̄ = gab̄. So, in order to quantize the system, one chooses