aa r X i v : . [ h e p - t h ] J u l LMP-TPU–5/09ITP-UH–10/09
Harmonic N =2 mechanics Anton Galajinsky a and Olaf Lechtenfeld ba Laboratory of Mathematical Physics, Tomsk Polytechnic University,634050 Tomsk, Lenin Ave. 30, Russian Federation
Email: [email protected] b Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover,Appelstrasse 2, D-30167 Hannover, Germany
Email: [email protected]
Abstract N =2 superconformal many-body quantum mechanics in arbitrary dimensions is gov-erned by a single scalar prepotential which determines the bosonic potential and theboson-fermion couplings. We present a special class of such models, for which thebosonic potential is absent. They are classified by homogeneous harmonic functionssubject to physical symmetry requirements, such as translation, rotation and permuta-tion invariance. The central charge is naturally quantized. We provide some examplesfor systems of identical particles in any dimension. PACS numbers: 03.65.-w, 11.30.Pb, 11.30.-jKeywords: supersymmetric quantum mechanics
Introduction N =4 superconformal many-body quantum mechanics in one dimension is governed by twoscalar prepotentials U and F which obey a coupled set of partial differential equations.While U may vanish, F always takes nonzero values. Recent studies in [1]–[4] (for relateddevelopments see [5]–[10]) revealed an interesting link between N =4 quantum mechanics andthe WDVV equation [11, 12] which plays an important role in d =2 topological field theory [11,12] and N =2 supersymmetric Yang-Mills theory [13]. Because the WDVV equation underliesa potential deformation of a Fr¨obenius algebra [14], it relates N =4 mechanics with Fr¨obeniusmanifolds. All N =4 models with a nontrivial U constructed so far are based on the rootsystems of simple Lie algebras or Coxeter reflection groups.A peculiar feature of N =4 mechanics concerns the center-of-mass coordinate. Although itdecouples from the relative particle motion, its nonzero F prepotential generates an inverse-square potential for the center-of-mass motion, thus breaking translation invariance. If thisis unwanted, one must give up N =4 and soften the model to an N =2 system, which isruled by the prepotential U alone [15]. Our interest in N =2 mechanics is also motivatedby the desire to go beyond d =1 and to construct new exactly solvable many-body modelsin higher dimensions and to explore novel correlations (see e.g. [16] and references therein).It is natural to expect that d> N =2 superconformal many-body models will provide newinsight into the nonrelativistic version of the AdS/CFT correspondence which has currentlysparked substantial interest.A minimal extension of the Galilei algebra by the dilatation and special conformal gener-ators is known in the literature as the Schr¨odinger algebra. A conformal extension obtainedby contracting the relativistic conformal so ( d +1 ,
2) algebra gives an even larger algebrawhich goes under the name of conformal Galilei algebra (for a recent discussion and fur-ther references see e.g. [17]). Because the conformal Galilei algebra requires vanishing mass,the Schr¨odinger algebra has a better prospect for quantum mechanical applications. Sincethe translations are part of the Schr¨odinger algebra, N =2 interacting many-body quantummechanics is likely to be the maximal superextension feasible in higher dimensions.The purpose of this paper is to reconsider the construction of N =2 n -particle quantummechanics in d dimensions and to exhibit a new special class of models determined by asingle harmonic function. These ( n, d ) models are characterized by the absence of bosonicinteractions, yet retain (quantum) boson-fermion couplings. They are classified by the homo-geneous harmonic functions on R nd subject to physical symmetry requirements (Euclideanand permutation invariance) and quantize the central charge of the N =2 algebra.In Section 2 we recall the conventional framework for formulating N =2 many-body mod-els in one dimension and explore the hitherto unexploited possibility of purely boson-fermioncouplings. We show how the Laplace equation arises, explain the central charge quantizationand discover solutions related to Lie-algebra root systems.In Section 3 the analysis is extended beyond one dimension. It is shown that the role of theLaplace equation persists in higher dimensions, but the prepotential is further constrainedby Euclidean invariance in R d , as part of the N =2 Schr¨odinger supersymmetry. We finally1resent a one-parameter family of ( n, d ) models as well as a particular ( n, n −
1) system, bothbeing invariant under particle permutations. Conclusions follow. N =2 mechanics The conventional representation of the d =1, N =2 superconformal algebra on the phase spaceof n identical particles (with unit mass) is provided by a single prepotential U ( x , . . . , x n )which gives rise to the operators [6] H = p i p i + ∂ i U ( x ) ∂ i U ( x ) − ∂ i ∂ j U ( x ) h ψ i ¯ ψ j i , J = h ψ i ¯ ψ i i ,D = tH − ( x i p i + p i x i ) , K = − t H + 2 tD + x i x i ,Q = ψ i ( p i + i ∂ i U ( x )) , ¯ Q = ¯ ψ i ( p i − i ∂ i U ( x )) ,S = x i ψ i − tQ, ¯ S = x i ¯ ψ i − t ¯ Q, (1)where the symbol h . . . i stands for symmetric (or Weyl) ordering of the fermions. The oper-ators H , D and K generate time translations, dilatations and special conformal transforma-tions, respectively, while Q and ¯ Q are supersymmetry generators, and S and ¯ S generate su-perconformal transformations. The U(1) R-symmetry transformation generated by J affectsonly the fermions. Note that the prepotential U ( x ) is defined up to an additive constant.The operators (1) obey the (anti)commutation relations of the d =1, N =2 superconformalalgebra with central charge C (Hermitian conjugates are omitted)[ H, D ] = i H, [ K, D ] = − i K, [ Q, D ] = i2 Q, [ S, D ] = − i2 S, [ Q, J ] = − Q, [ S, J ] = − S, [ H, K ] = 2i D, [ Q, K ] = − i S, [ S, H ] = i Q, { Q, ¯ Q } = 2 H, { S, ¯ S } = 2 K, { Q, ¯ S } = − D − i J − i C, (2)provided the prepotential satisfies the linear partial differential equation x i ∂ i U ( x ) = − C . (3)The general solution to (3) reads U ( x ) = − C ln | x | + Λ( x i x j ) , (4)where Λ( x i x j ) is a function of the coordinate ratios x i x j for i < j . We work in the standard coordinate representation, p i = − i ∂∂x i , [ x i , p j ] = i δ ij , and put ~ =1. Thefermionic operators are mutually conjugate via ( ψ i ) † = ¯ ψ i and obey the anticommutation relations { ψ i , ψ j } =0, { ¯ ψ i , ¯ ψ j } =0, { ψ i , ¯ ψ j } = δ ij . The t -dependent pieces in the generators are kept explicit so as to have a directlink to the classical theory. Throughout the paper summation over repeated indices is understood.
2n order to extract a class of reasonable models from the infinity of N =2 systems encodedin the general solution (4), one can impose additional restrictions like permutation symmetry,translation invariance etc.. Another option is to start with a specific bosonic theory, H B = p i p i + V ( x ) with ( x i ∂ i + 2) V ( x ) = 0 , (5)and then solve the Hamilton-Jacobi equation ∂ i U ( x ) ∂ i U ( x ) = 2 V ( x ) (6)for the given potential − V and zero energy. Each solution U yields an N =2 superconformalextension of the original model (5). In particular, in this way one can treat quantum inte-grable many-body models related to simple Lie algebras, the prominent example being the N =2 Calogero model [15] (see also [6]).Among the many possible bosonic starting points, there exist special bosonic potentials V which can be absorbed into a reordering of the fermions. Since a deviation from Weyl orderingproduces a term proportional to ∂ i ∂ j U in H , this property translates to the condition ∂ i U ( x ) ∂ i U ( x ) + κ ∂ i ∂ i U ( x ) = 0 (7)for some real parameter κ of order ~ . Note that this forces U to be of order ~ as well, so thatthese models are classically free. The value of κ quantifies the deviation from Weyl orderingand takes unit ( ~ ) value for normal ordering. If (7) can be solved, then the Hamiltonianmay be brought to the form H = p i p i − ∂ i ∂ j U ( x ) : ψ i ¯ ψ j : κ (8)for a suitable fermionic ordering prescription, so that the interaction contains only boson-fermion couplings. We now describe a class of solutions to (7) with quantized central charge.The conditions (3) and (7) simplify under the substitution U ( x ) = κ ln G ( x ) to ( x i ∂ i + Cκ ) G ( x ) = 0 and ∂ i ∂ i G ( x ) = 0 , (9)so that G ( x ) is a harmonic homogeneous function of degree ℓ := − Cκ in R n . Such functionsare single-valued only for ℓ ∈ Z and regular at the origin x = x = . . . = x n = 0 for ℓ ≥ κ , C = − ℓ κ with ℓ = 0 , , , . . . , (10)and restrict the prepotential to G ℓ ( x ) = ( x + . . . + x n ) ℓ Y ℓ (angles) , (11) If singular behavior is admitted at coincidence loci x i = x j , more general solutions appear. Y ℓ is a linear combination of S n − spherical harmonics for spin ℓ . Note that linearcombinations of G ℓ are forbidden by the homogeneity condition (9).Each value of ℓ and choice of Y ℓ produces a special N =2 many-body quantum system.The demand for permutation invariance or translation invariance puts restrictions on Y ℓ ,which can be solved. For illustration, we consider a solution related to the positive roots { α } of a simple Lie algebra, G ( x ) = Y α ( αx ) . (12)In this case ℓ equals the number of positive roots. That (12) solves Laplace’s equationis verified with the use of the same root identities which were previously applied in [4]for solving the WDVV equation (see section 6 in [4] for more details). Permutation andtranslation invariance it achieved for the A n root systems, { ( αx ) } = { x i − x j | ≤ i
2. The formulae also work for d =1, but produce G ( x ) ≡ i, j, k in (21) as the vertices of a triangle, this prepotentialappears to be constructed in terms of triangle areas and edge lengths. This suggests toconstruct other prepotentials in terms of generalized volumes. The simplest such situation,specific to d = n − G ( x ) = ǫ α ...α n − r α r α . . . r α n − n . (26)This homogeneous polynomial of degree n − n = d +1 particle locations and is naturally permutation invariant (up to an irrelevantsign). It trivially solves the Laplace equation since each vector x αi occurs at most linearlyin (26). Hence, this example describes a valid ( n, n −
1) particle model.6
Conclusions
We have constructed new interacting N =2 many-body quantum mechanics of a special kind:the bosonic potential is absent, but interaction takes place through boson-fermion couplingsalone. These couplings are governed by a prepotential G = e U/κ which only has to beharmonic and homogeneous. The central charge (in the N =2 superconformal algebra) isgiven by the degree of G and therefore naturally quantized. By changing the fermionicordering prescription, one may generate also a particular bosonic potential which is purelyquantum.In d =1, the admissible prepotentials include models built from the positive roots of simpleLie algebras. The A n root systems yield translation-invariant models of identical particles.In dimensions d>
1, we provided a general framework with N =2 Schr¨odinger supersymmetryand gave two example models, one for generic ( n, d ) with a free parameter and another onefor d = n − R n . The freedom of a deficit angle around the singularity allows for moregeneral harmonic functions and therefore other N =2 models. In the higher-dimensionalsituation, our examples were not the most general ones. A physical classification needsan understanding of all homogeneous harmonic functions on R nd invariant under the n !permutations of the particle labels and under the rigid translations and rotations of R d . Itwould be interesting to learn how the root-system solutions fit into such a scheme. Acknowledgements
This work was supported in part by DFG grant 436 RUS 113/669/0-3, RF Presidential grantMD-2590.2008.2 and RFBR grant 09-02-91349. O.L. thanks D. Fairlie for help with (7).
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