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A note on N = 4 supersymmetric mechanics on K¨ahler manifolds. Stefano Bellucci1 and Armen Nersessian1,2,3 1 INFN, Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italy 2 JINR, Laboratory of Theoretical Physics, Dubna, 141980 Russia 3 Yerevan State University, A.Manoogian, 1, Yerevan, 375025 Armenia (February 1, 2008) ThegeometricmodelsofN =4supersymmetricmechanicswith(2d.2d)CI-dimensionalphasespace are proposed, which can be viewed as one-dimensional counterparts of two-dimensional N = 2 su- persymmetric sigma-models by Alvarez-Gaum´e and Freedman. The related construction of super- symmetric mechanics whose phase space is a K¨ahler supermanifold is considered. Also, its relation with antisymplectic geometry is discussed. PACS number: 11.30.Pb 1 0 I. INTRODUCTION pa =πa− 2i∂ag, χmi =emb ηib : (3) 0 Ω=dpa∧dza+dp¯a¯∧dz¯a¯+dχmi ∧dχ¯mi¯, 2 Supersymmetric mechanics attracts permanent inter- where em are the einbeins of the K¨ahler structure: an emsatisnilnyceonitsthinetrNodu=cti2onc[a1s]e.,Haonwdevthere, mstuosdtiesimfopcourtsasendt ema δmm¯¯e¯mba¯ =ga¯b. So, to quantize this model, one chooses J 11 cteansetioonf,Ntho=ugh4sommeechiannteicresstdiindgnobotserrveacetiiovnesewneoruegmhaadte- pˆa =−i∂∂za, ˆp¯a¯ =−i∂∂z¯a¯, [χˆmi ,χˆ¯nj¯]+ =δmn¯δij. about this subject: let us mention that the most gen- eral N = 4,D = 1,3 supersymmetric mechanics de- We restrict ourselves by the supersymmetric mechanics 1 v scribed by real superfield actions were studied in Refs. whosesuperchargesarelinearintheGrassmannvariables 5 [2,3] respectively,and those in arbitraryD in Ref. [4]; in ηia,η¯ia¯. Thesesystemscanbeobtainedbydimensionalre- 6 [5] N = 4,D = 2 supersymmetric mechanics described ductionfromN =2supersymmetric(1+1)−dimensional 0 by chiral superfield actions were considered; the general sigma-models by Alvarez-Gaum´e and Freedman [8]; in 1 studyofsupersymmetricmechanicswitharbitraryN was the simplest case of d = 1 and in the absence of central 0 performed recently in Ref. [6]. In the Hamiltonian lan- charge these systems coincide with the N = 4 super- 1 0 guageclassicalsupersymmetric mechanicscanbe formu- symmetric mechanics described by the chiral superfield / lated in terms of superspace equipped with some super- action [5]. The constructed systems are connected also h symplectic structure (and corresponding non-degenerate withtheN =4supersymmetricmechanicsdescribingthe t - Poisson brackets). After quantization the odd coordi- low-energydynamicsofmonopolesanddyonsinN =2,4 p nates are replaced by the generators of Clifford algebra. super-Yang-Mills theories [7]. e h It is easy to verify that the minimal dimension of phase We also propose the relatedconstruction of N =2 su- : superspace, which allows to describe a D−dimensional persymmetric mechanics whose phase superspace is the v supersymmetricmechanicswithnonzeropotentialterms, external algebra of an arbitrary K¨ahler manifold. Un- i X is (2D.2D), while supersymmetry specifies both the ad- der the additional assumption that the base manifold is r missible sets of configuration spaces and potentials. a hyper-K¨ahler one, this system should get the N = 4 a In the present work we propose the N = 4 supersym- supersymmetry. The relation of this system with anti- metric one-dimensional sigma-models (with and with- symplectic geometry is discussed. out central charge) on K¨ahler manifold (M0,ga¯bdzadz¯¯b), with (2d.2d)CI-dimensional phase space equipped with II. SIGMA-MODEL WITH STANDARD N =4 the symplectic structure SUSY. Ω=ω −i∂∂¯g= 0 =dπ ∧dza+dπ¯ ∧dz¯a+ (1) Let us consider a one-dimensional supersymmet- a a +Ra¯bcd¯ηiaη¯ibdza∧dz¯b+ga¯bDηia∧Dη¯ib ric sigma-model on an arbitrary Riemann manifold (M ,g (x)dxµdxν), with (2D.2D)−dimensional phase 0 µν where superspace equipped with a supersymplectic structure g =iga¯bηaσ0η¯b, Dηia =dηia+Γabcηiadza, i=1,2 (2) Ω=d(pµdxµ+θiµgµνDθiν)= =dp ∧dxµ+ 1R θµθνdxλ∧dxρ (4) whileΓabc, Ra¯bcd¯arerespectivelytheconnectionandcur- µ +g D2 θµµν∧λρDiθνi, vature of the K¨ahler structure. The odd coordinates ηa µν i i i belongtotheexternalalgebraΛ(M ),i.e. theytransform 0 where as dza. This symplectic structure becomes canonical in the coordinates (p ,χk) Dθµ =dθµ+Γν θρdxλ, a i i ρλ i 1 and Γµνλ, Rµνλρ are respectively the Cristoffel symbols F =iga¯bηaσ3η¯¯b : {Q±i ,F}=±iQ±i , {H,F}=0. (11) and curvature tensor of underlying metric g dxµdxν. µν On this phase superspace one can formulate the one- PerformingtheLegendretransformationonegetstheLa- dimensional N = 2 supersymmetric sigma-model with grangianof the system supercharges linear on Grassmann variables, viz Q1 =Hp=µθ1µ21g+µνU(,+pµµ(Rxp)νθ+2µ,θUµ,µθQUν1θ,νλ=)θρ+pµ:Uθµ2µ;ν−θ1µUθ,2µν+(x)θ1µ, (5) L=−ggaa¯b¯bUz˙aazU¯˙¯¯b¯b+−+R21iaUη¯bckaad¯g;bηaη1a¯b1aη¯Dηd1bη¯2τbηk¯b2c−η¯+2diU.¯12a¯D;d¯bητη¯ka1a¯gη¯a2¯b¯b+η¯¯b− (12) µνλρ 1 2 1 2 {Qi,Qj}=2δijH, {Qi,H}=0,i=1,2. The supersymmetry transformations of the Lagrangian are of the form One can also introduce a specific constant of motion (“fermionic number”) δ+za =ǫηa, i i F =gµνθ1µθ2ν : {F,Qi}=ǫijQj {F,H}=0. (6) δi+ηja =ǫ(cid:16)iδǫ−ijzU¯a¯bg=¯ba0+, Γabcηibηjc(cid:17); (13) i To get the N = 4 supersymmetric one-dimensional δ−ηa =ǫδ z˙a i j ij sigma-model mechanics, one should require that the target space M0 is a K¨ahler manifold (M0,ga¯bdzadz¯¯b), where ǫ is an odd parameter: p(ǫ)=1. ga¯b = ∂2K(z,z¯)/∂za∂z¯b (this restriction follows also So, we get the action for one-dimensional sigma-model from the considerations of superfield actions: indeed, with four exact real supersymmetries. It can be the N−extended supersymmetric mechanics obtained straightlyobtainedby the dimensionalreductionof N = from the action depending on D real superfields, have 2 supersymmetric (1 + 1) dimensional sigma-model by a (2D.ND) -dimensional symplectic manifold, whereas Alvarez-Gaum´eand Freedman [8] (the mechanical coun- IR thoseobtainedfromthe actiondepending ondchiralsu- terpart of this system without potential term was con- perfields have a (2d.Nd/2)CI−dimensional phase space, structedin[10]). Notice thatthe above-presentedN =4 with the configuration space being a 2d−dimensional SUSY mechanics for the simplest case, i.e. d = 1, was K¨ahlermanifold). In that casethe phase superspace can obtained by Berezovoy and Pashnev [5] from the chiral be equipped by the supersymplectic structure (1). The superfield action correspondingPoissonbracketsaredefinedbythefollow- 1 ing non-zero relations (and their complex-conjugates): S = K(Φ,Φ¯)+2 U(Φ)+2 U¯(Φ¯) (14) 2Z Z Z {π ,zb}=δb, {π ,ηb}=−Γb ηc, a a a i ac i {πa,π¯b}=−Ra¯bcd¯ηkcη¯kd, {ηia,η¯jb}=ga¯bδij. wahseimreilΦarisacchtiiornaldsueppeenrfideinldg. oItnsdeemchsirtaolbsuepoebrvfiieoludsstwhailtl generate the above-presentedN =4 SUSY mechanics. ToconstructonthisphasesuperspacetheHamiltonian mechanics with standard N =4 supersymmetry algebra {Q+,Q−}=δ H, III. N =4 SIGMA-MODEL WITH CENTRAL ± ± i ±j ij (7) CHARGE {Q ,Q }={Q ,H}=0, i=1,2, i j i let us choose the supercharges given by the functions Letusconsiderageneralizationofabovesystem,which possesses N =4 supersymmetry with central charge Q+ =π ηa+iU η¯a¯, Q+ =π ηa−iU η¯a¯. (8) 1 a 1 a¯ 2 2 a 2 a¯ 1 {Θ+,Θ−}=δ H+Zσ3, {Θ±,Θ±}=0, Then,calculatingthe commutators(Poissonbrackets)of i j ij ij ± i j (15) {Z,H}={Z,Θ }=0. these functions, we get that the supercharges (8) belong k to the superalgebra (7) when the functions U ,U¯ are of a a¯ For this purpose one introduces the supercharges the form Θ+ =(π +iG (z,z¯))ηa+iU¯ (z¯)η¯a¯, ∂U(z) ∂U¯(z¯) 1 a ,a 1 ,a¯ 2 (16) Ua(z)= ∂za , U¯a¯(z¯)= ∂z¯a , (9) Θ+2 =(πa−iG,a(z,z¯))η2a−iU¯,a¯(z¯)η¯1a¯, while the Hamiltonian reads where the real function G(z,z¯) obeys the conditions H=ga¯b(πaπ¯b+UaU¯¯b)−iUa;bη1aη2b+iU¯a¯;¯bη¯1a¯η¯2¯b− (10) ∂a∂bG+Γcab∂cG=0, G,a(z,z¯)ga¯b∂¯bU¯(z¯)=0. (17) −Ra¯bcd¯η1aη¯1bη2aη¯2d, Thefirstequationin(17)isnothingbuttheKillingequa- where U ≡∂ ∂ U −Γc ∂ U. tion of the underlying K¨ahler structure (let us remind, a;b a b ab c The constant of motion counting the number of that the isometries of the K¨ahler structure are Hamilto- fermions, reads: nian holomorphic vector fields) given by the vector 2 G=Ga(z)∂ +G¯a(z¯)∂¯ , Ga =iga¯b∂¯ G. (18) IV. THE RELATED CONSTRUCTION a a b ThesecondequationmeansthatthevectorfieldGleaves Let us consider a supersymmetric mechanics whose the holomorphic function invariant: phase superspace is the external algebra of the K¨ahler LGU =0 ⇒ Ga(z)Ua(z)=0. manifoldΛ(M),where(cid:16)M, gAB¯(z,z¯)dzAdz¯B¯(cid:17)playsthe role of the phase space of the underlying Hamiltonian CalculatingthePoissonbracketsofthesesupercharges, mechanics. The phase superspace is a (D.D)CI− di- we get explicit expressions for the Hamiltonian mensional K¨ahler supermanifold equipped by the super- K¨ahler structure [11] H≡ga¯b πaπ¯¯b+G,aG¯b+U,aU¯,¯b − −iUa;bη1aη2b(cid:0)+iU¯a¯;¯bη¯1a¯η¯2¯b+ 21Ga¯b(ηka(cid:1)η¯k¯b)− (19) Ω=i∂∂¯ K(z,z¯)−igAB¯θAθ¯B¯ = −Ra¯bcd¯η1aη¯1bη2cη¯2d =i(gAB¯ −(cid:16)iRAB¯CD¯θCθ¯D¯)dzA∧(cid:17)dz¯B¯ (23) and the central charge +gAB¯DθA∧Dθ¯B¯, Z =i(Gaπa+Ga¯π¯a¯)+ 12∂a∂¯¯bG(ηaσ3η¯¯b). (20) where DθA = dθA +ΓABCθBdzC, and ΓABC, RAB¯CD¯ are respectively the Cristoffel symbols and curvature tensor It can be checked by a straightforward calculation that of the underlying K¨ahler metrics gAB¯ =∂A∂B¯K(z,z¯). the functionZ indeedbelongstothe centerofthe super- The corresponding Poisson bracket can be presented algebra(15). Thescalarpartofeachphasewithstandard in the form N = 2 supersymmetry can be interpreted as a particle mteornvainlgmoangntheeticK¨afihellderwmitahnisfotrldenignththFe p=resieGnac¯bedozfaa∧ndezx¯¯b- { , }=ig˜AB¯∇A∧∇¯B¯ +gAB¯∂θ∂A ∧ ∂θ∂¯B¯ , (24) and in the potential field U,a(z)ga¯bU¯,¯b(z¯). where The Lagrangianof the system is of the form ∂ ∂ ∇ = −ΓC θB L=ga¯b z˙az¯˙b+ 21ηkaDdη¯τk¯b + 12Ddητkaη¯¯b − A ∂zA AB ∂θC (cid:16) (cid:17) and −ga¯b(GaG¯b+UaU¯¯b)+ (21) +iUa;bη1aη2b−iU¯a¯;¯bη¯1a¯η¯2¯b+Ra¯bcd¯η1aη¯1bη2aη¯2d. g˜A−B1¯ =(gAB¯ −iRAB¯CD¯θCθ¯D¯). The supersymmetry transformations read Onthis phasesuperspaceonecanimmediately construct the mechanics with N =2 supersymmetry δ+za =ǫηa, i i δi+ηja =ǫ(cid:16)iδǫ−ijzU¯a¯bg=¯ba0+, Γabcηibηjc(cid:17), (22) {Q+,Q−}=H˜, {Q±,Q±}={Q±,H˜}=0, (25) i given by the supercharges δ−ηa =ǫ(δ z˙a−ǫ Ga). i j ij ij Q+ =∂AH(z,z¯)θA, Q− =∂A¯H(z,z¯)θ¯A¯ (26) Assuming that(M0,ga¯bdzadz¯b)is the hyper-K¨ahlermet- ric and that U(z) + U¯(z¯) is a tri-holomorphic func- where H(z,z¯) is the Killing potential of the underlying tion while the function G(z,z¯) defines a tri-holomorphic K¨ahler structure, Killing vector, one should get the N = 8 supersymmet- ric one-dimensional sigma-model. In that case instead ∂A∂BH −ΓCAB∂CH =0, VA(z)=igAB¯∂B¯H(z,z¯). of the phase with standard N = 2 SUSY arising in the K¨ahlercase,we shallgetthe phase withstandardN =4 The Hamiltonian of the system reads SUSY. The latter system can be viewed as a particu- lar case of N = 4 SUSY mechanics describing the low- H˜ =gAB¯VAV¯B¯ +iV,ACgAB¯V¯,BD¯¯θCθ¯D¯− (27) energy dynamics of monopoles and dyons in N = 2,4 −RAB¯CD¯V,ACV¯;BD¯¯θAθCθ¯B¯θ¯D¯, super-Yang-Mills theory [7]. Notice that, in contrast to theN =4mechanicssuggestedinthementionedpapers, while the supersymmetry transformations are given by inthe above-proposed(hypothetic)constructionalsothe the vector fields δ± ≡{Q±, }, fourhidden supersymmetriescouldbe explicitly written. WewishtoconsiderthisN =8supersymmetricmechan- δ− =−iVA(z)∂θ∂A −iV;ACθCNAD∇D, (28) ics, as well as its application to the solutions of super- Yang-Mills theory, in a forthcoming paper. where N−1 A ≡δA−iRA θCθ¯D¯. B B BCD¯ (cid:0) (cid:1) 3 Requiring that M be a hyper-K¨ahler manifold, we can V. ACKNOWLEDGMENTS double the number of supercharges and get a N = 4 supersymmetric mechanics. In that case the Killing po- The authors are grateful to S. Krivonos for numerous tential should generate a tri-holomorphic vector field. illuminating discussions, and to E. Ivanov and A. Pash- The phase space of the system under consideration nevforusefulcomments. A.N.thanksINFNforfinancial can be equipped, in addition to the Poisson bracket cor- support and kind hospitality during his stay in Frascati, responding to (23), with the antibracket (odd Poisson within the framework of a INFN-JINR agreement. bracket) associated with the odd K¨ahler structure Ω = 1 i∂∂¯K1, where K1 =eiα∂AK(z,z¯)θA+e−iα∂A¯K(z,z¯)θ¯A¯, α=0,π/2, { , }1 =e−iαgA¯B∇A¯∧ ∂θ∂B +c.c. . (29) [1] E. Witten, Nucl. Phys. B188 (1981), 513; ibid. B202 Itiseasytoobservethatthefollowingequalityholds[11] (1982), 253. L≡{Z˜, }={Q, } , (30) [2] E.A.Ivanov,S.O.Krivonos,A.I.Pashnev,Class. Quant. 1 Grav. 8 (1991), 19. where [3] E.A. Ivanov,A.V. Smilga, Phys.Lett. B257 (1991), 79; V.P. Berezovoy, A.I. Pashnev, Class. Quant. Grav. 8 Z˜≡H(z,z¯)+i∂A∂B¯H(z,z¯)θAθ¯B¯ (31) (1991), 2141. Q=eiαQ++e−iαQ−. [4] E.E. Donets, A. Pashnev, J.J. Rosales, M. Tsulaia, Phys. Rev.D61 (2000), 043512. Then, after obvious algebraic manipulation with the Ja- [5] V. Berezovoy, A. Pashnev, Class. Quant. Grav. 13 cobi identities, one gets the following relations: (1996), 1699. [6] C.M. Hull, The Geometry of Supersymmetric Quantum {Z˜,H˜}={Z˜,Q±}=0. (32) Mechanics, hep-th/9910028. [7] D. Bak, K. Lee, P. Yi, Phys. Rev. D61 (2000), 045003; Hence, the function Z˜ is a constant of motion, which ibid. D62 (2000), 025009; belongs to the center of the superalgebra defined by D. Bak, C. Lee, K. Lee, P. Yi, Phys. Rev. D61 (2000), Q±,H,Z˜. One can also introduce another constant of 025001; motion, i.e. the “fermionic number” J. Gauntlett, N. Kim, J. Park, P. Yi, Phys. Rev. D61 (2000), 125012; F˜ =igABθAθ¯B¯ : {Q±,F˜}=±iQ±, {H˜,F˜}=0. (33) J. Gauntlett, C. Kim, K. Lee, P. Yi, hep-th/0008031. [8] L.Alvarez-Gaum´e,D.Freedman,Comm.Math.Phys.91 Noticethatthesupermanifoldsprovidedbytheevenand (1983), 87; for the quantum connection of sigma-models oddsymplectic(andK¨ahler)structures,andparticularly to strings see e.g. [9] and references therein. the equation (30), were studied in the context of the [9] S. Bellucci, Z. Phys. C36 (1987), 229; ibid. Prog. Theor. problem of the description of supersymmetric mechanics Phys. 79 (1988), 1288; ibid. Z. Phys. C41 (1989), 631; in terms of antibrackets [11,12] (this problem was sug- ibid. Phys.Lett.B227 (1989), 61; ibid.Mod. Phys.Lett. gested in [13]). Later this structure was found to be A5 (1990), 2253. useful in equivariant cohomology, e.g. for the construc- [10] H.W. Braden, A.J. Macfarlane, J. Phys. A18 (1985), tionofequivariantcharacteristicclassesandderivationof 2955. localization formulae [14], since the vector field (30) can [11] A.P. Nersessian, Theor. Math. Phys. 96 (1993), 866. beidentifiedwiththeLiederivativealongthevectorfield [12] O.M. Khudaverdian,J. Math. Phys. 32 (1991), 1934. generated by H, while the vector fields {F˜, } , {H, } O.M. Khudaverdian, A.P. Nersessian, J. Math. Phys.32 1 1 correspondstoexternaldifferentialandoperatorofinner (1991), 1938; ibid. 34 (1993), 5533. product. Hence, {H +F˜,H +F˜} = 2Q, and H +F˜ [13] D.V. Volkov, A. I. Pashnev, V.A. Soroka, V.I. Tkach, 1 JETP Lett. 44 (1986), 70. defines an equivariant Chern class; the Lie derivative of [14] A.P.Nersessian,JETPLett.58(1993),66;LectureNotes the even symplectic structure (23) along the vector field in Physics 524, 90 (hep-th/9811110). {H +F˜, } yields the equivariant even pre-symplectic 1 [15] I.A. Batalin, G.A. Vilkovisky, Phys. Lett. B102 (1981), structure,generatingequivariantEulerclassesofthe un- 27; Phys. Rev.D28 (1983), 2567. derlying K¨ahler manifold. [16] I.A. Batalin, P.M. Lavrov, I.V. Tyutin, J. Math. Phys. Notice also, that the antibrackets are the basic ob- 31 (1990), 1487; ibid. 32 (1990), 532; ject of the Batalin-Vilkovisky formalism [15], while the I.A. Batalin, R. Martnelius, A. Semikhatov,Nucl. Phys. pairofantibrackets,correspondingtothe α=0,π/2,to- B 446 (1995), 249; getherwithcorrespondingnilpotentvectorfields{F, }1 A.Nersessian,P.H.Damgaard,Phys.Lett.B355(1995), and the associated∆−operators,form the “triplectic al- 150; gebra” underlying the BRST-antiBRST-invariantexten- I.A. Batalin, R. Marnelius, Nucl. Phys. B465 (1996), sion of Batalin-Vilkovisky formalism [16]. 521. 4

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