N=2 superconformal many-body quantum mechanics in arbitrary dimensions is
governed by a single scalar prepotential which determines the bosonic potential
and the boson-fermion couplings. We present a special class of such models, for
which the bosonic potential is absent. They are classified by homogeneous
harmonic functions subject to physical symmetry requirements, such as
translation, rotation and permutation invariance. The central charge is
naturally quantized. We provide some examples for systems of identical
particles in any dimension.Comment: 1+8 page

Galajinsky, Anton

Lechtenfeld, Olaf

English

arXiv

N=2 superconformal many-body quantum mechanics in arbitrary dimensions is governed by a single scalar prepotential which determines the bosonic potential and the boson-fermion couplings. We present a special class of such models, for which the bosonic potential is absent. They are classified by homogeneous harmonic functions subject to physical symmetry requirements, such as translation, rotation and permutation invariance. The central charge is naturally quantized. We provide some examples for systems of identical particles in any dimension

Institutional Repository of Leibniz Universität Hannover

HarmonicN ¼ 2mechanicsAnton Galajinsky1,* and Olaf Lechtenfeld2,†1Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Lenin Avenue 30, Russian Federation2Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, D-30167 Hannover, Germany(Received 11 August 2009; published 10 September 2009)N ¼ 2 superconformal many-body quantum mechanics in arbitrary dimensions is governed by asingle scalar prepotential which determines the bosonic potential and the boson-fermion couplings. Wepresent a special class of such models, for which the bosonic potential is absent. They are classified byhomogeneous harmonic functions subject to physical symmetry requirements, such as translation, rotationand permutation invariance. The central charge is naturally quantized. We provide some examples forsystems of identical particles in any dimension.DOI: 10.1103/PhysRevD.80.065012 PACS numbers: 03.65.w, 11.30.j, 11.30.PbI. INTRODUCTIONN ¼ 4 superconformal many-body quantum mechan-ics in one dimension is governed by two scalar prepoten-tials U and F which obey a coupled set of partialdifferential equations. WhileUmay vanish, F always takesnonzero values. Recent studies in [1–4] (for related devel-opments see [5–10]) revealed an interesting link betweenN ¼ 4 quantum mechanics and the Witten, Dijkgraaf,Verlinde, and Verlinde (WDVV) equation [11,12] whichplays an important role in d ¼ 2 topological field theory[11,12] and N ¼ 2 supersymmetric Yang-Mills theory[13]. Because the WDVV equation underlies a potentialdeformation of a Fröbenius algebra [14], it relates N ¼ 4mechanics with Fröbenius manifolds. All N ¼ 4 modelswith a nontrivial U constructed so far are based on the rootsystems of simple Lie algebras or Coxeter reflectiongroups.A peculiar feature of N ¼ 4 mechanics concerns thecenter-of-mass coordinate. Although it decouples from therelative particle motion, its nonzero F prepotential gener-ates an inverse-square potential for the center-of-massmotion, thus breaking translation invariance. If this isunwanted, one must give up N ¼ 4 and soften the modelto an N ¼ 2 system, which is ruled by the prepotential Ualone [15]. Our interest in N ¼ 2 mechanics is alsomotivated by the desire to go beyond d ¼ 1 and to con-struct new exactly solvable many-body models in higherdimensions and to explore novel correlations (see e.g. [16]and references therein). It is natural to expect that d > 1,N ¼ 2 superconformal many-body models will providenew insight into the nonrelativistic version of the AdS/CFTcorrespondence which has currently sparked substantialinterest.A minimal extension of the Galilei algebra by the dila-tation and special conformal generators is known in theliterature as the Schrödinger algebra. A conformal exten-sion obtained by contracting the relativistic conformalsoðdþ 1; 2Þ algebra gives an even larger algebra whichgoes under the name of conformal Galilei algebra (for arecent discussion and further references see e.g. [17]).Because the conformal Galilei algebra requires vanishingmass, the Schrödinger algebra has a better prospect forquantum mechanical applications. Since the translationsare part of the Schrödinger algebra, N ¼ 2 interactingmany-body quantum mechanics is likely to be the maximalsuperextension feasible in higher dimensions.The purpose of this paper is to reconsider the construc-tion of N ¼ 2 n-particle quantum mechanics in d dimen-sions and to exhibit a new special class of modelsdetermined by a single harmonic function. These ðn; dÞmodels are characterized by the absence of bosonic inter-actions, yet retain (quantum) boson-fermion couplings.They are classified by the homogeneous harmonic func-tions on Rnd subject to physical symmetry requirements(Euclidean and permutation invariance) and quantize thecentral charge of the N ¼ 2 algebra.In Sec. II we recall the conventional framework forformulating N ¼ 2 many-body models in one dimensionand explore the hitherto unexploited possibility of purelyboson-fermion couplings. We show how the Laplace equa-tion arises, explain the central charge quantization anddiscover solutions related to Lie-algebra root systems.In Sec. III the analysis is extended beyond one dimen-sion. It is shown that the role of the Laplace equationpersists in higher dimensions, but the prepotential is furtherconstrained by Euclidean invariance in Rd, as part of theN ¼ 2 Schrödinger supersymmetry. We finally present aone-parameter family of ðn; dÞ models as well as a particu-lar ðn; n 1Þ system, both being invariant under particlepermutations. Conclusions follow.II. SPECIAL N ¼ 2 MECHANICSThe conventional representation of the d ¼ 1, N ¼ 2superconformal algebra on the phase space of n identicalparticles (with unit mass) is provided by a single prepo-*galajin@mph.phtd.tpu.ru†lechtenf@itp.uni-hannover.dePHYSICAL REVIEW D 80, 065012 (2009)1550-7998=2009=80(6)=065012(5) 065012-1  2009 The American Physical Societytential Uðx1; . . . ; xnÞ which gives rise to the operators [6]1H ¼ 12pipi þ 12@iUðxÞ@iUðxÞ  @i@jUðxÞhc i c ji;J ¼ hc i c ii;D ¼ tH  14ðxipi þ pixiÞ; K ¼ t2Hþ 2tDþ 12xixi;Q ¼ c iðpi þ i@iUðxÞÞ; Q ¼ c iðpi  i@iUðxÞÞ;S ¼ xic i  tQ; S ¼ xi c i  t Q; (1)where the symbol h. . .i stands for symmetric (or Weyl)ordering of the fermions. The operators H, D and K gen-erate time translations, dilatations and special conformaltransformations, respectively, while Q and Q are super-symmetry generators, and S and S generate superconfor-mal transformations. TheUð1Þ R-symmetry transformationgenerated by J affects only the fermions. Note that theprepotential UðxÞ is defined up to an additive constant.The operators (1) obey the (anti)commutation relationsof the d ¼ 1,N ¼ 2 superconformal algebra with centralcharge C (Hermitian conjugates are omitted)½H;D ¼ iH; ½K;D ¼ iK; ½Q;D ¼ i2Q;½S;D ¼  i2S; ½Q; J ¼ Q; ½S; J ¼ S;½H;K ¼ 2iD; ½Q;K ¼ iS; ½S;H ¼ iQ;fQ; Qg ¼ 2H; fS; Sg ¼ 2K;fQ; Sg ¼ 2D iJ  iC; (2)provided the prepotential satisfies the linear partial differ-ential equationxi@iUðxÞ ¼ C: (3)The general solution to (3) readsUðxÞ ¼ C lnjx1j þxixj; (4)where ðxixjÞ is a function of the coordinate ratiosxixjfor i <j.In order to extract a class of reasonable models from theinfinity ofN ¼ 2 systems encoded in the general solution(4), one can impose additional restrictions like permutationsymmetry, translation invariance, etc. Another option is tostart with a specific bosonic theory,HB ¼ 12pipi þ VðxÞ with ðxi@i þ 2ÞVðxÞ ¼ 0; (5)and then solve the Hamilton-Jacobi equation@iUðxÞ@iUðxÞ ¼ 2VðxÞ (6)for the given potential V and zero energy. Each solutionU yields anN ¼ 2 superconformal extension of the origi-nal model (5). In particular, in this way one can treatquantum integrable many-body models related to simpleLie algebras, the prominent example being the N ¼ 2Calogero model [15] (see also [6]).Among the many possible bosonic starting points, thereexist special bosonic potentials V which can be absorbedinto a reordering of the fermions. Since a deviation fromWeyl ordering produces a term proportional to @i@jU inH,this property translates to the condition@iUðxÞ@iUðxÞ þ @i@iUðxÞ ¼ 0 (7)for some real parameter  of order @. Note that this forcesU to be of order @ as well, so that these models areclassically free. The value of  quantifies the deviationfrom Weyl ordering and takes unit (@) value for normalordering. If (7) can be solved, then the Hamiltonian may bebrought to the formH ¼ 12pipi  @i@jUðxÞ:c i c j: (8)for a suitable fermionic ordering prescription, so that theinteraction contains only boson-fermion couplings. Wenow describe a class of solutions to (7) with quantizedcentral charge.The conditions (3) and (7) simplify under the substitu-tionUðxÞ ¼  lnGðxÞ toxi@i þ CGðxÞ ¼ 0and @i@iGðxÞ ¼ 0; (9)so thatGðxÞ is a harmonic homogeneous function of degree‘ :¼  C in Rn. Such functions are single-valued only for‘ 2 Z and regular at the origin x1 ¼ x2 ¼    ¼ xn ¼ 0for ‘  0. These conditions quantize the central charge inunits of ,C ¼ ‘ with ‘ ¼ 0; 1; 2; . . . ; (10)and restrict the prepotential to2G‘ðxÞ ¼ ðx21 þ    þ x2nÞ‘=2Y‘ ðanglesÞ; (11)where Y‘ is a linear combination of Sn1 spherical har-monics for spin ‘. Note that linear combinations of G‘ areforbidden by the homogeneity condition (9).Each value of ‘ and choice of Y‘ produces a specialN ¼ 2 many-body quantum system. The demand forpermutation invariance or translation invariance puts re-strictions on Y‘, which can be solved. For illustration, weconsider a solution related to the positive roots fg of asimple Lie algebra,1We work in the standard coordinate representation, pi ¼i @@xi , ½xi; pj ¼ iij, and put @ ¼ 1. The fermionic operatorsare mutually conjugate via ðc iÞy ¼ c i and obey the anticom-mutation relations fc i; c jg ¼ 0, f c i; c jg ¼ 0, fc i; c jg ¼ ij.The t-dependent pieces in the generators are kept explicit so as tohave a direct link to the classical theory. Throughout the papersummation over repeated indices is understood.2If singular behavior is admitted at coincidence loci xi ¼ xj,more general solutions appear.ANTON GALAJINSKYAND OLAF LECHTENFELD PHYSICAL REVIEW D 80, 065012 (2009)065012-2GðxÞ ¼ YðxÞ: (12)In this case ‘ equals the number of positive roots. That (12)solves Laplace’s equation is verified with the use of thesame root identities which were previously applied in [4]for solving the WDVVequation (see Sec. 6 in [4] for moredetails). Permutation and translation invariance is achievedfor the An root systems, fðxÞg ¼ fxi  xjj1  i < j nþ 1g. The interaction potential for these models readsVint ¼Xðc Þð c ÞðxÞ2 : (13)III. SPECIAL N ¼ 2 MODELS IN ARBITRARYDIMENSIONWe proceed to the construction of N ¼ 2 models indimensions d greater than 1. At the algebraic level, extradimensions comewith additional generators correspondingto spatial translations P, spatial rotations M, Galileiboosts K and super Galilei transformations L and Lwith ; ¼ 1; . . . ; d. It is assumed that L and L areHermitian conjugates of each other. The set of generatorsfH;D;K; P; K;Mg spans a subalgebra known as theSchrödinger algebra. In what follows, Greek letters arereserved for spatial indices while Latin indices label iden-tical particles of unit mass.Apart from the structure relations (2), which persist inhigher dimensions, the nonvanishing (anti)commutationrelations of the N ¼ 2 Schrödinger superalgebra include(Hermitian conjugates are omitted)½H;K ¼ iP; ½D;K ¼ i2K; ½K;P ¼ iK; ½D;P ¼  i2P;½M; L ¼ iðL  LÞ; ½Q;K ¼ iL; ½K; P ¼ iM;½M; P ¼ iðP  PÞ; fQ; Lg ¼ P; fL; Lg ¼ Z;½M;K ¼ iðK  KÞ; ½S; P ¼ iL; fS; Lg ¼ K;½M;M ¼ iðM þ M  M  MÞ; ½J; L ¼ L; (14)where M and Z are the central charges.In order to build a quantum mechanical representation of this algebra, one introduces bosonic operators xi and pi andfermionic operators c i and ci , which obey the (anti)commutation relations½xi ; pj  ¼ iij and fc i ; c j g ¼ ij: (15)The fermionic operators are related by Hermitian conjugation, i.e. ðc i Þy ¼ c i . A representation of the superalgebra (14)can then be constructed in terms of a single prepotential UðxÞ by analogy with the one-dimensional case,Q ¼ c i ðpi þ i@iUðxÞÞ; Q ¼ c i ðpi  i@iUðxÞÞ; L ¼Xic i ; L ¼ Xic i ;S ¼ xi c i  tQ; S ¼ xi c i  t Q; J ¼ hc i c i i; P ¼Xipi ;M ¼ ðxi pi  xi pi Þ  ihc i c i  c i c i i; K ¼Xixi  tP;C ¼ t2H þ 2tDþ 12xi xi ; D ¼ tH 14ðxi pi þ pi xi Þ;H ¼ 12pi pi þ12@iUðxÞ@iUðxÞ  @i@jUðxÞhc i c j i; (16)where we abbreviated @i ¼ @@xi . This representation fixesthe values of the two central charges to Z ¼ M ¼ n.3 Thecommutation relations of the N ¼ 2 Schrödinger super-algebra (14) constrain the prepotential to obey a set ofpartial differential equations,ðxi @i  xi @iÞUðxÞ ¼ 0;Xi@iUðxÞ ¼ 0;xi @iUðxÞ ¼ C:(17)The first two restrictions in (17) come from rotation andtranslation invariance, while the last one is responsible forconformal symmetry.3For particles of mass m one has Z ¼ M ¼ nm.HARMONIC N ¼ 2 MECHANICS PHYSICAL REVIEW D 80, 065012 (2009)065012-3Like in one dimension, we would like to absorb thebosonic potential V ¼ 12@iU@iU into a reordering ofthe fermions. The condition for this option generalizesour principal equation (7) to@iUðxÞ@iUðxÞ þ @i@iUðxÞ ¼ 0: (18)Introducing GðxÞ as in (9) one getsxi @i þCGðxÞ ¼ 0 and @i@iGðxÞ ¼ 0 (19)besides translation and rotation invariance for GðxÞ.Formally, the n-particle model in d dimensions is just aspecial nd-particle model in one dimension. However, thephysical symmetry requirement is different: We want thepotential to be invariant under permutations of the n par-ticle labels only, and not under permutations of all ndlabels. The translation and rotation invariance, on the otherhand, is more restrictive in d dimensions, but this may bedealt with by passing to a set of SOðdÞ invariants built fromrelative coordinates. We shall see that for d > 1 it ispossible to construct physically acceptable ðn; dÞ modelsfor identical particles.Like in one dimension, we consider prepotentials Gwhich are regular at the origin xi ¼ 0. We do not knowhow to write down the most general rotation andtranslation-invariant harmonic function, but let us presenttwo classes of examples. In order to take into accounttranslation and rotation invariance, we switch to the rela-tive coordinates rij ¼ xi  xj and form SOðdÞ scalarsðrij; rklÞ and 1...dr1i1j1 . . . rdidjd from them. Here, ð ; Þ and1...d denote the Euclidean scalar product and the Levi-Civita tensor, respectively, in Rd. It should be kept in mindthat these building blocks are not independent; e.g. thetriangle rule rij þ rjk þ rki ¼ 0 implies thatðrij; rjkÞ þ ðrjk; rkiÞ þ ðrki; rijÞ¼ 12½ðrij; rijÞ þ ðrjk; rjkÞ þ ðrki; rkiÞ ðno sumÞ: (20)Our first example is a homogeneous and permutationinvariant polynomial of fourth order (thus C ¼ 4),GðxÞ ¼  Xni<j;kðrik; rkjÞ2 þ Xni<j;kðrik; rikÞðrkj; rkjÞþ Xni<jðrij; rijÞ2 (21)with free parameters ,  and . Computing@m@mðrik; rkjÞ2 ¼ 4ðrik; rikÞ þ 4ðrkj; rkjÞ 4ðdþ 1Þðrik; rkjÞ;@m@mðrik; rikÞðrkj; rkjÞ ¼ 4dðrik; rikÞ þ 4dðrkj; rkjÞ 8ðrik; rkjÞ;@m@mðrij; rijÞ2 ¼ 8ðdþ 2Þðrij; rijÞ (22)with sums over m only, and employing the identityXni<j;kðrik; rkjÞ ¼ 2 n2Xni<jðrij; rijÞ (23)following from (20), one arrives at@m@mGðxÞ ¼ Xni<jðrij; rijÞ; (24)with  being a linear expression in ,  and . Therefore,solving (19) amounts to putting  ¼ 0, which isðn 2Þðdþ 5Þþ ðn 2Þð4dþ 2Þþ ð4dþ 8Þ ¼ 0:(25)Since the scale of GðxÞ is irrelevant, this linear relationleaves a one-parameter family (21) of ðn; dÞ prepotentials,for d > 1 and n  2. The formulas also work for d ¼ 1,but produce GðxÞ  0.Viewing the three particle labels i, j, and k in (21) as thevertices of a triangle, this prepotential appears to be con-structed in terms of triangle areas and edge lengths. Thissuggests to construct other prepotentials in terms of gen-eralized volumes. The simplest such situation, specific tod ¼ n 1 dimensions, provides our second example,GðxÞ ¼ 1...n1r112r213 . . . rn11n : (26)This homogeneous polynomial of degree n 1 measuresthe volume of the simplex spanned by the n ¼ dþ 1particle locations and is naturally permutation invariant(up to an irrelevant sign). It trivially solves the Laplaceequation since each vector xi occurs at most linearly in(26). Hence, this example describes a valid (n, n 1)particle model.IV. CONCLUSIONSWe have constructed new interacting N ¼ 2 many-body quantum mechanics of a special kind: The bosonicpotential is absent, but interaction takes place throughboson-fermion couplings alone. These couplings are gov-erned by a prepotential G ¼ eU= which only has to beharmonic and homogeneous. The central charge (in theN ¼ 2 superconformal algebra) is given by the degree ofG and therefore naturally quantized. By changing thefermionic ordering prescription, one may generate also aparticular bosonic potential which is purely quantum.In d ¼ 1, the admissible prepotentials include modelsbuilt from the positive roots of simple Lie algebras. The Anroot systems yield translation-invariant models of identicalparticles. In dimensions d > 1, we provided a generalframework with N ¼ 2 Schrödinger supersymmetry andgave two example models, one for generic ðn; dÞwith a freeparameter and another one for d ¼ n 1.Finally, let us discuss possible further developments ofthis work. The quantization of the central charge may beANTON GALAJINSKYAND OLAF LECHTENFELD PHYSICAL REVIEW D 80, 065012 (2009)065012-4weakened by letting the particle coordinates parametrize acone rather than Rn. The freedom of a deficit angle aroundthe singularity allows for more general harmonic functionsand therefore other N ¼ 2 models. In the higher-dimensional situation, our examples were not the mostgeneral ones. A physical classification needs an under-standing of all homogeneous harmonic functions on Rndinvariant under the n! permutations of the particle labelsand under the rigid translations and rotations of Rd. Itwould be interesting to learn how the root-system solutionsfit into such a scheme.ACKNOWLEDGMENTSThis work was supported in part by DFG Grant No. 436RUS 113/669/0-3, RF Presidential Grant No. MD-2590.2008.2 and RFBR Grant No. 09-02-91349. O. L.thanks D. Fairlie for help with (7).[1] N. Wyllard, J. Math. Phys. (N.Y.) 41, 2826 (2000).[2] S. Bellucci, A. Galajinsky, and E. Latini, Phys. Rev. D 71,044023 (2005).[3] A. Galajinsky, O. Lechtenfeld, and K. Polovnikov, J. HighEnergy Phys. 11 (2007) 008.[4] A. Galajinsky, O. Lechtenfeld, and K. Polovnikov, J. HighEnergy Phys. 03 (2009) 113.[5] S. Bellucci, A. Galajinsky, and S. Krivonos, Phys. Rev. D68, 064010 (2003).[6] A. Galajinsky, O. Lechtenfeld, and K. Polovnikov, Phys.Lett. B 643, 221 (2006).[7] S. Bellucci, S. Krivonos, and A. Sutulin, Nucl. Phys.B805, 24 (2008).[8] S. Fedoruk, E. Ivanov, and O. Lechtenfeld, Phys. Rev. D79, 105015 (2009).[9] S. Krivonos, O. Lechtenfeld, and K. Polovnikov, Nucl.Phys. B817, 265 (2009).[10] S. Fedoruk, E. Ivanov, and O. Lechtenfeld,arXiv:0905.4951.[11] E. Witten, Nucl. Phys. B340, 281 (1990).[12] R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys.B352, 59 (1991).[13] A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B389, 43 (1996).[14] B. Dubrovin, Nucl. Phys. B379, 627 (1992).[15] D. Freedman and P. Mende, Nucl. Phys. B344, 317 (1990).[16] A. Galajinsky and I. Masterov, Phys. Lett. B 675, 116(2009).[17] C. Duval and P. A. Horváthy, arXiv:0904.0531.HARMONIC N ¼ 2 MECHANICS PHYSICAL REVIEW D 80, 065012 (2009)065012-5

Harmonic N=2 Mechanics

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