Table Of ContentA 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
