Super Landau models on odd cosets.

Super Landau models on odd cosets

M. Goykhman,1,*E. Ivanov,2,†and S. Sidorov2,‡

1_{Institute Lorentz for Theoretical Physics, Leiden University, P. O. Box 9506, Leiden 2300RA, The Netherlands} 2_{Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia}

(Received 2 December 2012; published 24 January 2013)

We constructd¼1sigma models of the Wess-Zumino type on theSUðnj1Þ=UðnÞfermionic cosets. Such models can be regarded as a specific supersymmetric generalization (with a target space supersym-metry) of the classical Landau model, when a charged particle possesses only fermionic coordinates. We consider both classical and quantum models, and we prove the unitarity of the quantum model by introducing the metric operator on the Hilbert space of the quantum states, such that all their norms

become positive definite. It is remarkable that the quantum n¼2 model exhibits hidden SUð2j2Þ

symmetry. We also discuss the planar limit of these models. The Hilbert space in the planar n¼2

case is shown to carrySUð2j2Þsymmetry which is different from that of theSUð2j1Þ=Uð1Þmodel.

DOI:10.1103/PhysRevD.87.025026 PACS numbers: 11.30.Pb, 12.60.Jv, 03.65.w, 03.65.Aa

I. INTRODUCTION

The renowned Landau model [1] describes a charged nonrelativistic particle moving on a two-dimensional Euclidean planeR2 _{ðz;zÞ under the influence of a} uni-form magnetic field which is orthogonal to the plane. Its simplest generalization to a curved manifold is the Haldane model [2], describing a charged particle on the two-sphere S2SUð2Þ=Uð1Þin the field of a Dirac monopole located at the center. These models are exactly solvable, on both the classical and the quantum levels. They can be inter-preted as one-dimensional nonlinear sigma models with the Wess-Zumino (WZ) terms. For instance, the Haldane model is described by thed¼1SUð2Þ=Uð1Þsigma model action with theUð1ÞWZ term.

There are two different approaches to supersymmetrizing this bosonic system. One is based on a worldline supersym-metry (see, e.g., Ref. [3]), while the other deals with a target-space supersymmetry. The latter option corresponds to extending the bosonic manifolds to supermanifolds by add-ing extra fermionic target coordinates. These supermani-folds are identified with cosets of some supergroups, so the relevant invariant actions describe d¼1 WZ sigma models on supergroups. Since supergroups possess a wider set of cosets as compared to their bosonic subgroups, there are several nonequivalent super-Landau models associated with the same supergroup. A minimal superextension of SUð2Þ is the supergroup SUð2j1Þ involving the bosonic subgroupUð2Þ ¼SUð2Þ Uð1Þand a doublet of fermionic generators.1It possesses a few different cosets, each giving rise to some super-Landau d¼1 sigma model. The SUð2Þ=Uð1Þ model can be promoted either to a model on theð2j2Þ-dimensional supersphereSUð2j1Þ=Uð1j1Þ[4,5] or

to a model on the ð2j4Þ-dimensional superflag manifold SUð2j1Þ=½Uð1Þ Uð1Þ [5,6]. Both models are exactly solvable, like their bosonic prototypes, and exhibit further interesting properties [5]. For instance, for some special relations between the coefficients of the corresponding WZ terms they prove to be quantum-mechanically equiva-lent to each other. Another surprising feature of these models is that, contrary to the standard lore, the presence of noncanonical fermionic kinetic terms bilinear in time derivative in their actions does not necessarily lead, upon quantization, to ghost states with negative norms; all norms can be made positive definite by modifying the inner prod-uct on the Hilbert space.

The supergroup SUð2j1Þ also possesses the pure odd supercoset SUð2j1Þ=Uð2Þ of the dimension ð0j4Þ. The Landau-type quantum sigma models on the odd cosets SUðnj1Þ=UðnÞ of the dimension ð0j2nÞ, with the pure WZ term as the action, were studied in Ref. [7] (see also Ref. [4]). The relevant Hilbert spaces studied there involve only single (vacuum) states associated with the lowest Landau levels (LLLs). These LLL states reveal interesting SUðnj1Þrepresentation content.

There remained a problem of finding out the complete sigma model actions on the supercoset SUðnj1Þ=UðnÞ, such that they contain both the WZ term and the term bilinear in the coset Cartan forms [i.e., the one-dimensional pullback of the Killing form on SUðnj1Þ=UðnÞ], and of exploring the relevant quantum mechanics. The basic aim of the present paper is to fill this gap. It is worth noting that sigma models with the supergroup target spaces have attracted a lot of attention over the last few years, in particular because of their intimate relation to superbranes (see, e.g., Refs. [8–10]). This increasing interest is one of the main motivations of our study.

The paper is organized as follows. In Sec. II, we con-struct theSUðnj1Þinvariant action for this model, using the d¼1version of the universal gauge approach which can be traced back to the construction of the two-dimensional

*[email protected]

†_{[email protected]}

‡_{[email protected]}

1_{This is a minimal possibility if one assumes the standard}

bosonicCPn_{sigma model actions in Ref. [11]. The same} general approach works nicely in some other cases elabo-rated as instructive examples in the Appendix [Landau-type models on the bosonic cosetSUðnþ1Þ=UðnÞ CPn and the supersphereSUð2j1Þ=Uð1j1Þ CPð1j1Þ]. In Sec.III we construct the corresponding Hamiltonian andSUðnj1Þ Noether charges, in both the classical and the quantum cases. In Sec.IVwe present the complete set of eigenfunc-tions of the Hamiltonian and compute its spectrum. The salient feature of the quantum case is that the number of Landau level (LL) states is finite and equal tonþ1. We discuss theSUð2j1Þ representation assignment of the full set of states in then¼2case by computing, in particular, the eigenvalues of theSUð2j1ÞCasimir operators. In Sec.V we compute the norms of the LL states and find that for some values of the WZ term strength , there are states with negative and/or zero norms, like in other super-Landau models. In Sec.VIwe give a more detailed treat-ment of then¼2case. We present the explicit form of the metric operator which allows one to make all norms positive definite. At each LL, the quantum states are found to form short multiplets of some hiddenSUð2j2Þsymmetry which is an extension of the original SUð2j1Þ symmetry (the phenomenon of such an enhancement ofSUð2j1Þ to SUð2j2Þ at the quantum level was earlier revealed in the superflag Landau model [5]). In Sec. VII we study the planar limit of the odd-coset Landau models. We show, in particular, that the Hilbert space forn¼2also carries some extendedSUð2j2Þsymmetry. We finish with conclu-sions and outlook in Sec.VIII.

II.SUðnj1Þ=UðnÞACTION FROMUð1ÞGAUGING

The supergroup SUðnj1Þ can be defined as the set of linear transformations of the nþ1-component multiplet ðz; iÞ(i¼1;2;. . .n), such that they preserve its norm

zz¼inv: (2.1)

Here, the componentsz,zare Grassmann even, whilei,i are Grassmann odd. Hereafter,¼ ¼ii. The fermionic transformations are

z¼; i_¼i_{z; (2.2)}

wherei,iare Grassmann-odd parameters. The variables _itransform in the fundamental representation of the group UðnÞ. These UðnÞ transformations are contained in the closure of (2.2).

Now we are going to show how, starting from this linear SUðnj1Þ multiplet, one can construct a nonlinear d¼1 WZ sigma model action associated with the odd ð0j2nÞ-dimensional cosetSUðnj1Þ=UðnÞ.

We start with the followingSUðnj1Þinvariant Lagrangian:

L¼ rzr zþr rþ2A; (2.3)

where

r ¼@tiA; r ¼ @tþiA: (2.4)

The auxiliary gauge fieldAðtÞensures the invariance of the Lagrangian (2.3) under theUð1Þgauge transformations:

z¼iz; i_¼i

i; A¼:_ (2.5) The last term in (2.3) is the Fayet-Iliopoulos (FI) term; it is invariant under (2.5) up to a total time derivative. The gauge fieldAðtÞis aSUðnj1Þsinglet; gauge transformations (2.5) commute with the rigidSUðnj1Þones.

As the next steps, we impose the SUðnj1Þ invariant constraint on the variablesz,i

zz¼1; (2.6)

choose theUð1Þgauge

z¼z (2.7)

and, using (2.6), express in terms of the fermionic variables,

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ

q

: (2.8)

Finally, the auxiliary field A can be eliminated by its algebraic equation of motion:

A¼ 1

2½2þið

__Þ: (2.9)

Upon substituting all of this back into the Lagrangian (2.3), the latter takes the form

L¼__ þ 2þ 4ð1þÞ½ð

_

Þ2_{þ ð}__Þ2

2ð1þÞð _

Þð_Þ þið__Þ:

(2.10)

This Lagrangian is invariant, up to a total time deriva-tive, under the following purely fermionic nonlinear real-ization of the oddSUðnj1Þtransformations:

i_¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ

q

i_þ

i

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ð þÞ: (2.11)

It precisely coincides with the one considered in Ref. [7]. This realization follows from (2.2) by taking into account the gauge (2.7), the constraint (2.6), and the necessity to accompany the original SUðnj1Þ transformations by the compensating gauge transformations (2.5) with

¼ i

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ð þÞ

The number of independent variables was reduced from the (nþ1) complex onesz,i_{to nsuch variables} i _by imposing the constraint (2.6) and choosing the gauge (2.7). In this aspect, the method we applied is quite similar to the gauge approach to the construction of the bosonic d¼2 CPn_{sigma models in Ref. [11]. It is worth pointing out that} ourd¼1gauging procedure automatically yields not only the standard sigma model part of the Lagrangian but also the WZ term with the strength2.2In the ‘‘parent’’ linear sigma model action (2.3), this constant appears as a strength of the FI term. In Ref. [7], only the WZ term in (2.10) was considered. Such truncated Lagrangian can be treated as the largelimit of (2.10).

In what follows, it will be convenient to deal with the coset coordinatesirelated toiby

i_¼

i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ

p (2.12)

and possessing the simple holomorphic transformation law

i_¼i_{þ ðÞ}i_{: (2.13)}

In terms ofi, the Lagrangian (2.10) is rewritten as L¼ _

_ 1þ

ð__Þð__Þ ð1Þ2 i

__

1 : (2.14)

The oddSUðnj1Þ transformations may be expressed in terms of the fermionicSUðnj1ÞgeneratorsQi,Qyj:

¼ ½ðQÞ þ ðQ yÞ; (2.15)

whereQiandQyi¼ ðQiÞysatisfy the relations

fQ_i; Q_jg ¼0; fQyi_{; Q}yj_{g ¼0; (2.16)}

fQi; Qyjg ¼

Jj_ij_i1 nJ

k k

þj_iB: (2.17)

Here J_ij1_nj_iJk

k J~ij are the SUðnÞ generators, and B is the generator of theUð1Þ transformation. They consti-tute the bosonic ‘‘body’’UðnÞof the supergroupSUðnj1Þ. In the realization on the variablesi_,

k, these generators are

Jj_i ¼i @ @j

j @

@i; B¼

1 n1

Jk_k: (2.18)

The explicit expressions for the conserved Noether supercharges corresponding to the odd SUðnj1Þ transfor-mations are

Qi¼ iþiðÞ ii; (2.19)

Qyi_¼i_þi_{ðÞ þi}i_{: (2.20)}

Here, _i¼@L=@_i _and i_¼@L=@_

i are the momenta canonically conjugate to the fieldsi_and

i. The correspond-ing conservedUðnÞgeneratorsJi

jandBare expressed as Jj_i ¼ijji;

B¼

₁

n1

ð

iiiiÞ þconstant;

(2.21)

where a constant in B(the central charge) will be fixed in the quantum model by requiring the generators to close on thesuð2j1Þalgebra as in (2.16) and (2.17).

III. QUANTIZATION

A. Hamiltonian formulation

To quantize the classical SUðnj1Þ=UðnÞ coset model constructed in the previous section, we have to build its Hamiltonian formulation. First of all, it is convenient to rewrite the Lagrangian (2.14) in a geometric way, in terms of the metric on the coset space SUðnj1Þ=UðnÞ and the external gauge potentials given on this manifold. The corresponding Hamiltonian will have the form convenient for quantization, and it will be easy to find its spectrum.

We can write down the metric on the SUðnj1Þ=UðnÞ coset parametrized by i coordinates, using the Ka¨hler potential

K ¼lnð1Þ: (3.1)

The metric is given by

gi j¼@j@

i_{K ¼} i j 1

i j

ð1Þ2: (3.2)

Also, we define the gauge connections

A_i¼ i@_iK¼i i 1;

Ai_¼i@i _K¼i i

1: (3.3)

Note thatAi¼ ðAiÞ. The inverse metric is given by ðg1Þi_j¼ ð1Þði_jþijÞ: (3.4)

In terms of these quantities, the Lagrangian (2.14) can be written as

L¼gi

j_i_jþð_iAiþ_iAiÞ: (3.5) The momenta canonically conjugate to the variablesi andiare

i¼ @L @_i¼g

k

i_kþAi; i¼ @L

@__i¼g i k_

k_þAi_{: (3.6)}

Then the Hamiltonian is given by

H¼ ðg1_Þi

jðjAjÞðiAiÞ: (3.7) Using the anticommutativity of the Grassmann varia-bles, we rewrite the Hamiltonian in the form

H¼1 2ðg

1_Þi

j½jAj; iAi; (3.8)

and perform canonical quantization in the coordinate representation by replacing

i! i@i; i! i@

i_{: (3.9)}

As a result, we obtain the quantum Hamiltonian

H¼1 2ðg

1_Þi j½r

ðÞ

i ;r

ðÞj_{; (3.10)}

where we have introduced

rðÞ i ¼@iþ

_i 1; r

ðÞj_¼@j_þ j

1: (3.11)

These ‘‘semicovariant’’ derivatives satisfy the following anticommutation relations:3

frðÞ i ;r

ðÞ j g ¼0; frðÞi_;rðÞj_{g ¼0;} frðÞ

i ;r

ð0Þj_{g ¼ ðþ}0_Þgj i:

(3.12)

Using them, we can rewrite the Hamiltonian (3.10) in the convenient equivalent form:

H¼ ðg1Þi_jrð_iÞrðÞjnH0n: (3.13)

For what follows, it will be useful to writeH0explicitly:

H0¼ ð1Þð@i@ i_þi

j@i@

j_Þþð1Þð i@

ij_@

jÞ

2þn: (3.14)

The Hamiltonian is Hermitian with respect to the appro-priate inner product (see below).

B. QuantumSUðnj1Þgenerators

The quantumSUðnj1Þgenerators can be obtained from the classical expressions (2.19), (2.20), and (2.21):

Qi¼ @iþij@ j_þ

i;

Qyi_{¼ @}iij_@

jþi;

(3.15)

Jj_i ¼i @ @_j

j @ @iJ~

j iþ

1 n

j iJkk;

F¼

1 n1

Jk k 2;

(3.16)

where we properly fixed the ordering ambiguities, based on the same reasonings as in Ref. [7]. A constant in the

expression for the Noether chargeB[see (2.21)] was fixed to be 2 and the resulting operator was denoted F in order to have thesuðnj1Þalgebra (2.16) and (2.17) in the quantum case:

fQi; Qjg ¼0; fQyi; Qyjg ¼0; (3.17)

fQi; Qyjg ¼J~ j iþ

j

iF: (3.18)

Using the explicit form (3.14) ofH0, it is straightforward to check that theSUðnj1Þgenerators (3.15) and (3.16) indeed commute with the Hamiltonian.

For further use, we explicitly present the generators in the n¼2 case. The corresponding Landau model pos-sesses the eight-parameter symmetry supergroupSUð2j1Þ.

We define

¼Q1; Q¼Q2: (3.19)

Then the full set of theSUð2j1Þgenerators as differential operators acting on the manifold with the odd coordinates ði;iÞ,i¼1, 2, is given by the expressions

Q¼ @2þ21@1þ2; (3.20)

¼ @₁þ₁₂@2þ₁; (3.21)

Jþ¼i2@1i1@2; (3.22)

J¼i2@1i1@2; (3.23)

J₃ ¼1 2ð

1_@

12@21@1þ2@2Þ; (3.24)

F¼1 2ð

1_@

1þ2@21@12@2Þ 2: (3.25)

The corresponding nonvanishing (anti)commutation rela-tions read

½Jþ; J ¼2J3; ½J3; J ¼ J; (3.26)

½Jþ; ¼iQ; ½J; Q ¼ i;

½J3; ¼ 1

2; ½J3; Q ¼ 1 2Q;

½F; ¼ 1

2; ½F; Q ¼ 1 2Q;

(3.27)

½J;y ¼iQy; ½Jþ; Qy ¼ iy;

½J₃;y ¼1 2

y_{; ½J}

3; Qy ¼ 1 2Q

y_;

½F;y ¼1 2

y_{; ½F; Q}y _¼1

2Q

y_;

(3.28)

3_{The semicovariant derivatives are essentially complex, in}

accordance with the fact that the wave superfunctions are com-plex. Under the complex conjugation, they are transformed as

ðrðÞ i ;rðÞ

j_{Þ ) ðr}ðÞi_;rðÞ

j Þ, when acting, respectively, on

f;yg ¼ J₃þF; fQ; Qyg ¼J₃þF;

f; Qyg ¼iJ; fy; Qg ¼ iJþ: (3.29)

The quadratic Casimir operator of the superalgebra suð2j1Þ,

C2 ¼ 1

2fJþ; Jg J 2

3þF2þ 1 2½Q; Q

y _þ1

2½;

y_;

(3.30)

is related to the Hamiltonian of then¼2model as

C2¼Hþ42¼H02þ42: (3.31)

One can also define a third-order Casimir operator:

C₃ ¼i 2Jþ½Q

y_; i

2½

y_{; QJ}

þ1

2J3ð½Q; Q

y

½;yÞ 1

2Fð½;

y _{þ ½Q; Q}y_{Þ þ2C}

2F

yQQy: (3.32)

In then¼2model it can be represented as

C3 ¼6H0þ2ð21Þð41Þ: (3.33)

IV. THE ENERGY SPECTRUM AND WAVE FUNCTIONS

In this section we turn to the study of the quantum SUðnj1Þ=UðnÞ model. We construct the complete set of the wave superfunctions and find the corresponding energy levels. We also obtain the realization of the SUðnj1Þ symmetry group on the wave superfunctions.

A. The spectrum

In this subsection we construct the complete set of wave superfunctions for theSUðnÞsinglet sector of the full space of quantum states. The corresponding superfunctions carry no externalSUðnÞindices, but possess the fixedBcharge 2Þ0, in accord with the explicit structure of the quantum generators (3.15) and (3.16). Possible wave func-tions with nonzero externalSUðnÞ spins form a subspace orthogonal to the SUðnÞ singlet one with respect to the inner product to be defined below. A similar situation occurs in the case of super-Landau models associated with the supersphere SUð2j1Þ=Uð1j1Þ [5], where one can consider only those wave functions which are singlets of the semisimple partSUð1j1Þ Uð1j1Þ.

To proceed, we will need two important properties:

ðg1_Þi j½r

ðÞ i ; g

j k ¼

ð1nÞ_k

1 (4.1)

and

rðÞ

½i ðg j

kÞ ¼g j

½kr

ðkþ2Þ

i ; (4.2)

where is an arbitrary superfunction, ¼ð;Þ, and square brackets denote antisymmetrization of indices (with the factor1=n!). These relations can be proved using the definitions (3.2) and (3.11).

We will deal with the shifted Hamiltonian H0 defined in (3.13). The lowest (vacuum) Landau level wave function 0corresponds to the zero eigenvalue ofH0and is defined by the same chirality condition as in Ref. [7]:

rðÞj

0¼0!0 ¼ 1

ð1Þ0ðÞ; (4.3)

where the analytic wave function is defined by the expansion

0ðÞ ¼c0þic0i þ þi1 inc0i1in: (4.4)

The wave superfunctions corresponding to the excited Landau levels are constructed by the covariant derivatives rð0Þ_{acting on the chiral superfunctions. The latter should} carry the appropriate external SUðnÞ indices in order to ensure the full wave functions to beSUðnÞsinglets.

The first LL wave superfunction is defined as

₁¼ rð_kþ1nÞk_{; (4.5)}

where k _{is the chiral superfunction in the fundamental} representation ofUðnÞ:

rðÞjk_¼0!k_¼ 1 ð1Þ

k_{ðÞ: (4.6)}

Using (4.1), it is easy to check that₁is the eigenfunction ofH0,

H01¼ ð2þ1nÞ1: (4.7)

The second LL wave superfunction is defined by

₂ ¼ rð_iþ1nÞr_kðþ3nÞ½ik_{; (4.8)}

where ½ik is a chiral superfield [it is expressed through the holomorphic reduced wave superfunction ½ik_{ðÞ in} the same way as in (4.3) and (4.6)]. The reason why the chiral superfunctions should belong to the irreducibleUðnÞ representations constructed by antisymmetrizing the indi-ces in the fundamental representation will be explained in the next subsection.

Using (4.1) and (4.2), one may verify that

H0₂¼2ð2þ2nÞ₂: (4.9)

In then¼2case, when the indices take values 1 and 2, it is the last level in the spectrum, because no nonzero higher-rank antisymmetric tensors can be defined.

‘¼ r

ðþ1nÞ

m1 r

ðþ3nÞ

m2 r

ðþ2‘1nÞ

m‘

½m1m2m‘_;

(4.10)

where the reduced wave superfunction½m1m2m‘_{is chiral:}

½m1m2m‘¼ 1

ð1Þ

½m1m2m‘ðÞ_{: (4.11)}

The corresponding energy eigenvalue is

E_‘ ¼‘ð‘nþ2Þ: (4.12)

Sometimes it is convenient to use the equivalent representa-tion for‘:

‘:¼ 1

ð1Þ^‘; ^

_‘¼ rðm21þ1nÞr

ð2þ3nÞ m2 r

ð2þ2‘1nÞ

m‘

½m1m2m‘_:

(4.13)

It is natural to require that the energies of the excited Landau levels are not negative and exceed (or at least are not less than) the energy of LLL. Therefore, in what follows we will consider only the options when the WZ term strength is restricted to the values

ðn1Þ=2: (4.14)

The obtained set ofnþ1chiral superfunctions captures the whole spectrum of the model in theSUðnÞsinglet sector. To prove this, we should check that any SUðnÞ singlet superfunction can be expressed as a linear superposition of the wave superfunctions ofnþ1Landau levels. The total number of independent functions ofncomplex Grassmann variables is equal to22n_{. The total number of independent} coefficients in the i _{expansion of a holomorphic} super-function corresponding to the level ‘ is 2n_ðn

lÞ. Different levels possess independent wave superfunctions. So the total number of independent coefficients is

2nX n

‘¼0 n

l

!

¼22n_{: (4.15)}

Thus, the constructed set of wave superfunctions indeed spans the fullSUðnÞsinglet Hilbert space.

In the remainder of this subsection we will briefly discuss the case with <0. Consider, instead of the H0 eigenvalues (4.12), those of the full Hamiltonian H¼H0n:

E‘ )E‘ ¼‘ð‘nþ2Þ n: (4.16)

Now we assume that ¼ jj and redefine the level number‘as

‘¼ ‘0þn: (4.17)

In terms of‘0, the spectrum (4.16) for <0becomes

Eð_‘<00 Þ¼‘0ð‘0nþ2jjÞ jjn: (4.18)

This formula coincides with that for >0,

Eð_‘>0Þ¼‘ð‘nþ2jjÞ jjn; (4.19)

modulo the substitution‘!‘0. Thus, for <0, the tower of the LL states is reversed: the highest LL with‘¼nbecomes the LLL with‘0 ¼0, while the LLL with‘¼0becomes the highest LL with‘0¼n. In order to have, in both cases, the excited LL energies not less than the LLL energy, one needs to impose the following general condition:

jj ðn1Þ=2: (4.20)

The <0LLL wave superfunction~_‘0_¼0 :¼_‘¼ncan be checked to satisfy the antichirality condition4

rðÞ

j ~‘0¼0 ¼0!~‘0¼0¼ ð1Þjj0ðÞ; (4.21) whereas all other ones, up to~_‘0_¼n :¼_‘¼0, are obtained through the successive action of the proper antiholomor-phic covariant derivatives on the antichiral superfunctions ð1Þjj ½i1...i‘0ðÞ. Passing to the

complex-conjugate set of the wave superfunctions,~‘0 !~?‘0 takes us back to the holomorphic representation, i.e., to the same picture as in the >0case (with replacing! jj every-where). Thus, without loss of generality, we can basically limit our study to the >0option.

B. Transformation properties

TheSUðnj1Þtransformation law of the wave function for any LL,

ð;Þ ¼ ðQþQyÞð;Þ; (4.22)

is fully specified by the form of the quantum supercharges:

Q_i¼ @_iþ_ij@j_þ

i; (4.23)

Qyi¼ @iij@jþi: (4.24)

The bosonic transformations are contained in the closure of these odd ones.

Sometimes it is more convenient to deal with the equiva-lentpassiveform of the sameSUðnj1Þtransformation:

ð;Þ ’0ð0;0Þ ð;Þ

¼ðþÞð;Þ: (4.25)

4_{The proof of this property is rather tricky. One rewritesr}ðÞ

Note that this transformation law indicates that the wave superfunctions cannot be real unless¼0.

Given the transformation law of the full wave super-function for the level‘, one can restore the transformation rules of the reduced chiral superfunctions defined in the previous subsection. To this end, one should take into account the ‘‘passive’’ transformation properties of the semicovariant derivatives

rð_j0Þ¼ ðÞr_jð0Þþ_ji_rð0Þ i þ

0

j;

rð0Þj¼ ðÞrð0Þjjirð

0_Þi_þ0j_:

(4.26)

Then we consider the transformation law of some chiral wave function i1i‘ _{with ‘ 2 and require that the}

corresponding transformation of‘is given by the passive form of (4.22), i.e., by (4.25). We find, first, that i1i‘

should necessarily be fully antisymmetric in its SUðnÞ indices, i1i‘ ¼½i1i‘_{, and second, that the}

transfor-mation law of½i1i‘_{should be}

½i1i‘¼ðÞ½i1i‘þ ðþ_‘ÞðÞ½i1i‘

þi1

j½ji‘þ þi‘j½i1i‘1j:

(4.27)

The chirality conditions are automatically covariant. For the first LL functioni_{, we have the same transformation} law; for the SUðnÞ singlet LLL function 0, the UðnÞ rotation terms are obviously absent and the transformation law coincides with (4.25). For the holomorphic wave superfunction

½i1i‘ðÞ ¼ ð₁Þ½i1i‘ _(4.28)

the weight factors are properly combined in such a way that the holomorphy property is preserved:

½i1i‘ðÞ ¼ ð₂þ_‘ÞðÞ½i1i‘ðÞ þ _{; (4.29)}

where ‘‘dots’’ stand for the holomorphic UðnÞ rotations, which are the same as in (4.27). This transformation law is valid for any‘ 0.

C.n¼2

Here we consider in more detail then¼2case, which corresponds to the SUð2j1Þ=Uð2Þ model. In this case we can lower and raise the SUð2Þ indices with the help of skew-symmetric symbols,

"ik; "ik; "ik"kj¼ij; "12¼"21¼1: (4.30)

The holomorphic LLL superfunction0ðÞhas the follow-ingexpansion:

₀ðÞ ¼c0_þi_c0 iþ

1 2

i1i2"

i1i2c1; (4.31)

wherec0 andc1areSUð2Þsinglets. From the transforma-tion law

₀ ¼2ðÞ₀ ½i_{þ ðÞ}i_@

i0; (4.32) we derive the following transformations of the component wave functions:

c0_¼ i_c0

i; c0i ¼ 2ic0ic1;

c₁ ¼ ð21Þk_c0 k:

(4.33)

Here, the complex conjugation rule foriis ði_Þ¼

i; ðiÞ¼ i: (4.34) Next, we consider the first-level wave superfunction:

½kðÞ ¼c½kþjc½_jkþ1 2

j1j2_" j1j2c

½k

2 : (4.35)

It transforms according to the rule

½k_{¼ ð2þ1ÞðÞ}½k_þk i½i ½i_{þ ðÞ}i_@

i½k; (4.36) which implies the following component transformation laws:

c½k_¼ i_c½k i ;

c½_jk¼ jc

½k

2 ð2þ1Þjc½kþkjic½i;

c½₂k¼"kj ic

½i

j þ2jc

½k j :

(4.37)

Finally, consider the second-level wave superfunction:

½k1k2ðÞ ¼_c½k1k2þj_c½k1k2

j þj1j2c

½k1k2

j1j2 : (4.38) Introducing

A¼"i1i2c

½i1i2_{; B}¼_"k1k2_" i1i2c

½i1i2

k1k2 ; Fk¼"i1i2c

½i1i2 k ;

(4.39)

we find

A¼ k_F

k; B¼ 2kFk;

Fk ¼kB ð1þ2ÞkA:

(4.40)

Note that these transformations can be brought precisely into the form (4.33) after redefinition B! B, 2! 21. This means that the‘¼0and‘¼2wave super-functions constitute isomorphicSUð2j1Þmultiplets.

It is appropriate here to give the relevant values of the suð2j1ÞCasimirsC2 andC3 defined in (3.31) and (3.33).

The eigenvalues of the Casimir operators on the LLL state are

C2¼2ð21Þ; C3 ¼2ð21Þð41Þ: (4.41)

On the first LL, the Casimir operators become

C2¼ ð2þ1Þð21Þ; C3 ¼4ð21Þð2þ1Þ:

(4.42)

C₂¼2ð2þ1Þ; C₃¼2ð2þ1Þð4þ1Þ: (4.43)

Note that the replacement 2!21 in (4.43) yields just (4.41), in accord with the remark after Eqs. (4.40).

The spectrum of Casimir operators for the finite-dimensional representations of SUð2j1Þ was studied in Refs. [12,13]. These representations are characterized by some positive number(‘‘highest weight’’) which can be a half-integer or an integer and an arbitrary additional real number which is related to the eigenvalues of the gen-eratorF(‘‘baryon charge’’). The values (4.41), (4.42), and (4.43) can be uniformly written in the generic form given in Ref. [12] as

C2¼ ð 22Þ; C3¼2 ð 22Þ ¼2 C2; (4.44)

with

LLL: ¼1 2; ¼

41

2 ; (4.45)

LLL: ¼1; ¼2; (4.46)

2nd LL: ¼1 2; ¼

4þ1

2 : (4.47)

Note that ourC₂andC₃were defined to have the opposite sign of those in Refs. [12,13] andC₃ also differs by the factor 2. We also took into account that 1=2 in our case. The isospins andFcharges of the component wave functions are expressed through the appropriate quantum numbers and in full agreement with the general for-mulas of Ref. [12].

At >1=2, we deal with what is called ‘‘typical’’ SUð2j1Þ representations (both Casimirs are nonzero); at the special value¼1=2, both Casimirs are zero for the LLL and first LL multiplets. So in this case the latter belong to the so-called ‘‘atypical’’ SUð2j1Þ representations. In accord with the consideration in Ref. [13], they are not completely irreducible: they contain invariant subspaces, the quotients over which, in turn, yield some further irre-ducible representations. As is seen from the transformation properties (4.33) and (4.37), at2¼1the componentc₁ of ₀ isSUð2j1Þsinglet and the subsetðc_k½k; c½₂kÞin₁ also forms a closedSUð2j1Þmultiplet.

In the alternative <0 case, the eigenvalues of the quadratic Casimir are

C2¼2jjð2jj þ1Þ; ‘¼0ð‘0¼2Þ;

C2¼ ð2jj 1Þð2jj þ1Þ; ‘¼1ð‘0¼1Þ;

C2¼2jjð2jj 1Þ; ‘¼2ð‘0¼0Þ:

(4.48)

These values are not negative forjj 1=2, in accord with the general condition (4.20).

V.SUðnj1ÞINVARIANT NORMS

A. General case

TheSUðnj1Þinvariant Berezin integral is defined as [7]

Z

d¼Z d0½1 ðÞn1; (5.1)

where

Z

d0 ¼

Y

@i@

i_{: (5.2)}

Using this integration measure, we can define the inner product on the Hilbert space of wave superfunctions:

hji ¼Z d?_{: (5.3)}

It is manifestly SUðnj1Þ invariant, taking into account the invariance of the measure d and the fact that the weight factor in the general transformation law (4.25) is imaginary.

To express the norms in terms of the reduced chiral superfunctions ½k1kn_{, defined in (4.10), we will need}

the following rule of integration by parts for two super-functions,and:

Z

dðrð_kÞÞ?

¼ ð1ÞPðÞþPðÞZ _dðrðþn1Þk_Þ?_{; (5.4)}

wherePðÞandPðÞare Grassmann parities of the super-functions. We will also employ the identity

rðþ2Þ½i_gk j ¼g

½k jrðÞ

j_{: (5.5)}

Using these rules, together with the anticommutation relations (3.12), and the antichirality condition

rðÞ

j ð½k1knÞ¼0;

one can express the norm of the full wave superfunction‘ for the‘th LL in terms of the chiral wave functions as

jj‘jj2¼

‘!ð2nþ2‘1Þ! ð2nþ‘1Þ!

Z dgm1 i1 g

m‘

i‘ ð

½m1m‘Þ?½i1i‘_{: (5.6)}

It is also straightforward to show that the wave superfunc-tions associated with different Landau levels are mutually orthogonal.

Expressing ½k1kn through holomorphic wave

func-tions by Eq. (4.28),

½k1kn¼ ð₁Þ½k1knðÞ_;

expansion of the½k1knðÞ_{. As an illustration, we present}

this final form of the norm of the LLL wave superfunction, with theexpansion defined in (4.4):

jj0jj2¼ð1Þn

Xn

k¼0

n21

k

!

k!ðnkÞ! c0i1ink_c0

i1ink:

(5.7)

In the next subsection, as an instructive example, we will give the explicit expressions for norms of all three LLL states of then¼2model.

As should be clear from the expression (5.7), there are values offor which the squared norms are negative; the same is true for the norms of higher LLs. This situation is typical for quantum-mechanical systems with Grassmann-odd target-space coordinates [7]. In the next section, we analyze this issue in some detail on the n¼2example. The way to make all norms positive definite is to modify the inner product by introducing some metric operator on the Hilbert space (like in all other known examples of super-Landau models). This operator proves to be especially simple in the planar limit (Sec.VII).

B. Norms for theSUð2j1Þ=Uð2Þmodel

Here we specialize to the n¼2 case and present the explicit expressions for the ‘‘naive’’ norms, using the general formulas of the previous subsection.

The norm of the vacuum wave superfunction is given by

jj0jj2 ¼c1c1þ ð12Þc0ic0iþ2ð21Þc0c0; (5.8)

where the component wave functions were defined in (4.31). The first excited level is described by the wave superfunc-tion with the norm

jj1jj2 ¼ ð21Þ½ð2þ1Þð21Þc½kc½k þ ð12Þc_½i_kc½_ikck_½kc_i½iþc2½kc

½k

2 : (5.9)

The second-level wave superfunction has the norm

jj2jj2 ¼2ð2þ1Þ½2ð2þ1ÞAA þBB 2FkFk:

(5.10)

It is straightforward to check that these norms are in-variant under the transformations (4.33), (4.34), (4.35), (4.36), (4.37), (4.38), (4.39), and (4.40). Also, we observe that these norms are not positive definite. In the next section, we will see in detail how this unwanted property can be cured. At2¼1there are zero norms for the LLL and the first LL wave functions. In this case, it is natural to define the physical Hilbert space as a quotient over the subspace of zero-norm states, so it is spanned by the LLL SUð2j1Þ singlet ‘‘wave function’’c1 and four wave func-tionsðA; B; FkÞof the second LL.

VI. UNITARY NORMS FORn¼2AND HIDDEN

SUð2j2ÞSYMMETRY

A. Redefining the inner product

All norms for the case n¼2 can be made positive by modifying the inner product in the Hilbert space, like in the cases worked out in [5]:

hhjii ¼Z d?G; (6.1)

where G is a metric operator on Hilbert space. As was already mentioned in Sec.VI A, we will consider only the casejj 1=2, because only in this case are the energies of the excited Landau levels non-negative. For the case 1=2, we choose the metric operator to be

G¼14ð2Fþ4þ‘Þ þ2ð2Fþ4þ‘Þ2_{: (6.2)}

It satisfies the conditions

G2 ¼1; ½H; G ¼0; (6.3)

which mean that the metric operator just alters the sign in front of all negative terms in the expressions for the norms of the wave superfunctions. Accordingly, the positive norms of the wave superfunctions are given by

jj0jj2¼c1c1þ ð21Þc0ic0i þ2ð21Þc0c0; jj1jj2¼ ð21Þ½ð2þ1Þð21Þc½kc½k

þ ð21Þc_½i_kc½_ikþck_½kc_i½iþc2½kc

½k 2 ; jj2jj2¼2ð2þ1Þ½2ð2þ1ÞAA þBB þ2FkFk:

(6.4)

With¼1=2, the norms become jj0jj2¼c1c1; jj1jj2 ¼0;

jj₂jj2_{¼2ð2þ1Þ½2ð2þ1ÞAA þBB þ2F}k_F k:

(6.5)

As was already mentioned, the Hilbert space in this special case, obtained as a quotient over the zero-norm states, involves only two physical states, one corresponding to LLL with unbroken SUð2j1Þ symmetry and another one corresponding to the second LL.

Now, let O be some operator that commutes with the Hamiltonian, and hence, generates some symmetry of the model, and let Oy be its Hermitian conjugated operator with respect to the naive inner product (5.3). Then its Hermitian conjugate with respect to the ‘‘improved’’ prod-uct (6.1) is given by

Oz_GOy_G¼Oy_þG½Oy_{; G: (6.6)}

¼ @₁;

y¼ 12@2þ1

2þ‘ 2B^3

2B^þ;

Q¼ @2;

Qy¼ 21_@ 1þ2

2þ‘ 2þB^3

1_B^

;

Jþ¼ i1@2þiB^þ; J¼i2@1iB^;

J₃ ¼1 2ð

1_@

12@2Þ B^3;

F¼1 2ð

1_@

1þ2@24‘Þ:

(6.7)

The matrix partsB^of theSUð2Þgenerators satisfy, on their own, thesuð2Þcommutation relations,

½_B^_þ;B^ _{¼ 2 ^B3; ½B}^_3;B^ _{¼ B}^_{: (6.8)}

They can take nonvanishing values only when applied to the first-level analytic wave functions½k _{which form a} doublet with respect to the external index:

^

Bþ½2¼½1; B^þ½1¼0; B^½1¼½2;

^

B½2¼0; B^3½1¼ 1 2

½1_{; B^}

3½2¼ 1 2

½2_:

(6.9)

In the holomorphic realization, the differential ‘‘metric’’ operatorGdefined in (6.2) takes the form

G_an ¼12i_@

iþ412@2@1 ¼ ð121_@

1Þð122@2Þ: (6.10) One can check that Gan anticommutes with the super-charges and commutes with the bosonic SUð2Þ Uð1Þ generators. Owing to these properties, we find

Qz ¼ Qy; z¼ y: (6.11)

In Ref. [5], there was given a general definition of supercharge that commutes withG:

~

O¼Oþ1

2½G; OG: (6.12) However, in the model under consideration, it identically vanishes. So one cannot define a modifiedsuð2j1Þ super-algebra which would commute with the operator G. Therefore, the suð2j1Þ transformation properties of the component wave functions are slightly changed after pass-ing to the new Hermitian conjugation.

The oddSUð2j1Þtransformations are now generated by

ðQþQ zÞ; (6.13)

giving rise to the following modified transformation prop-erties of the component wave functions:

c0_¼ i_c0 i;

c0

i ¼2ic0ic1;

c₁¼ ð21Þk_c0 k;

(6.14)

c½k_¼ i_c½k i ;

c½_jk¼ jc

½k

2 þ ð2þ1Þjc½kkjic½i;

c½₂k¼ "kjic

½i

j 2jc

½k j ;

(6.15)

A¼ kFk; B¼ 2kFk;

F_k ¼_kB ð1þ2Þ_kA: (6.16)

These transformations are similar to (4.33), (4.37), and (4.40), the difference being the opposite sign before the terms with_i. The modified norms (6.4) are invariant just under these transformations. The set of the group gener-ators is now Q, , Qz, z, F, J, J3. The quadratic Casimir operator for them is

C2¼ 1

2fJþ; Jg J 2

3 þF2 1 2½Q; Q

z 1

2½;

z_:

(6.17)

It is related to the Hamiltonian by the same Eq. (3.31) and so takes the same values (4.41), (4.42), and (4.43) on the LL wave superfunctions [the same is true for the 3d order Casimir (3.33)]. In particular, for¼1=2, Casimir opera-tors vanish for the LLL and the first LL superfunctions, implying that these levels carry atypical representations of SUð2j1Þ. It is worthwhile to note that, although thesuð2j1Þ algebra (3.26), (3.27), (3.28), and (3.29) changes its form after the replacement Qy, y! Qz, z, the original form can be restored by passing [in the SUð2Þ covariant notation (3.17) and (3.18)] to the new generators

~

Q_i¼ _ikQzk_,Q~zi_¼ik_Q

k,F~¼ F.

B.SUð2j2Þsymmetry

It was shown in Ref. [5] that the quantum Hilbert space of the superflag Landau model carries hidden SUð2j2Þ sym-metry which is, in a sense, an analog of the hidden worldline N ¼2 supersymmetry of the superplane Landau model [14,15]. It turns out that this phenomenon of enhancing the originalSUð2j1Þsymmetry toSUð2j2Þat the quantum level persists as well in the considered odd-coset super-Landau model. Below we assume that >1=2, i.e., that the Casimir operators are nonvanishing for all three LLs.

We define

_G 1

2½; G ¼G;

z

G

z_G;

QG 1

2½Q; G ¼QG; Q

z

G QzG:

(6.18)

These operators can be used to generate the second SUð2Þalgebra5as

J 1

2 ffiffiffiffiffiffiC2

p fQ;Gg ¼ 1 2 ffiffiffiffiffiffiC2

p f; QGg ¼ 1

ffiffiffiffiffiffi

C2

p QG;

Jþ 1

2 ffiffiffiffiffiffiC2

p fz_G; Qzg ¼ 1 2 ffiffiffiffiffiffiC2

p fQz_G;zg

¼ 1ffiffiffiffiffiffi

C2

p zQzG; J3 ¼

1

2½Jþ;J: (6.19)

The Casimir operatorC2 was defined in (3.31). Thus, we have two distinctSUð2Þalgebras

½Jþ; J ¼2J3; ½J3; J ¼ J; ½Jþ;J ¼2J3; ½J3;J ¼ J;

(6.20)

which can be checked to commute with each other. Explicitly, in the holomorphic realization, these generators read

Jþ¼J21¼1@2B^þ; J¼J21 ¼2@1B^;

J3¼J11¼ J22 ¼ 1 2ð

1_@

12@2Þ B^3; (6.21)

Jþ¼J21 ¼

ffiffiffiffiffiffi

C2

p

12_{; J}

¼J12¼ 1

ffiffiffiffiffiffi

C2 p @2@1;

J3 ¼J11 ¼ J22¼ 1 2ð

1_@

1þ2@21Þ: (6.22)

Remember that the SUð2Þ matrix operators B^ are nonvanishing only at the first LL (‘¼1) and are zero for other levels. Then we can define the supercharges Sa

iði¼1;2;a¼1;2Þ which are doublets with respect to either of twoSUð2Þgroups:

S1₁ ¼; S2₁ ½;Jþ;

S1

2 ¼Q; S22 ½Q;Jþ;

(6.23)

Si

a:¼ ðSaiÞz: (6.24) Explicitly, these supercharges are

S1

1¼ @1; S21 ¼

ffiffiffiffiffiffi

C2

p

2_;

S1

2¼ @2; S22 ¼ þ

ffiffiffiffiffiffi

C2

p

1_; (6.25)

S1₁ ¼12@21

2þ‘ 2B^3

þ2B^þ;

S2

1 ¼21@12

2þ‘ 2þB^3

þ1_B^

;

ffiffiffiffiffiffi

C2

p

S1₂ ¼ þð11@1Þ@2

2þ‘ 2B^3

@2B^þ@1;

ffiffiffiffiffiffi

C2

p

S2

2 ¼ ð12@2Þ@1þ

2þ‘ 2þB^3

@1þB^@2:

(6.26)

It is straightforward to check that they satisfy the (anti) commutation relations of the superalgebra suð2j2Þ with three central charges:

fSa_i;Sj_bg ¼a_bJ_ijj_iJa_bþa_bj_i

2þ‘ 2

1 2

;

fSa

i; Sbjg ¼"ij"ab

ffiffiffiffiffiffi

C2

p

;

f_Si

a;Sjbg ¼"ij"ab

ffiffiffiffiffiffi

C2

p

;

½Sa

i;Jcb ¼abSci 1 2

c bSai;

½Sa i; J

j k ¼

j iSak

1 2

j kSai:

(6.27)

As an example, let us give how thisSUð2j2Þsymmetry is realized on the LLL (i.e., ‘¼0) wave functions. We denote i_,

i as the parameters associated with the extra pair of fermionic generators, i.e., with S2

i, Si2. Then the additional transformations are

c0_¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21 p

ffiffiffiffiffiffi

2 p i_c0

i; c1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þð2Þ

p

k_c0 k; c0 i ¼ ffiffiffiffiffiffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21

p

ic1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þð2Þ

p

ic0: (6.28)

It is easy to check that they leave invariant the normjj₀jj in (6.4). Their bracket with the SUð2j1Þ transformations (6.14) produces the secondSUð2Þ, with respect to whichc0 andc1 form a doublet. The remaining componentsc0i are singlets of the second SUð2Þ. It is also easy to find the realization of thetransformations on the‘¼1and‘¼2 wave function multiplets.

According to Ref. [16] (see also Ref. [17]), the central charges can be combined into the three-vectorC~ as

~

Cð‘Þ ¼ ð2þ‘=21=2;

ffiffiffiffiffiffiffiffiffiffiffiffi

C₂ð‘Þ

q

;

ffiffiffiffiffiffiffiffiffiffiffiffi

C₂ð‘Þ

q

Þ ðC; P; KÞ:

(6.29)

5_{Due to the anticommutativity of the metric operatorGwith}

and Q (and its commutativity with bosonic generators) the

operatorsG,

z

G andQG,Q

z

G form the samesuð2j1Þalgebra

as,z andQ,Qz themselves, but they do not produce any

The norm ofC~defined asC~2¼C2_{PKis invariant under} thesoð1;2Þouter automorphisms of thesuð2j2Þ superalge-bra (6.27) [16]. Exploiting thissoð1;2Þfreedom, one can castC~ in the form

~

C¼ ðZ;0;0Þ; (6.30) where

Z¼1

2 for‘¼0;2; and Z¼1 for ‘¼1: (6.31) Explicitly, the soð1;2Þ rotated supercharges for all three levels are

~

Sa_i ¼pffiffiffiffiffiffi2Sa i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21 p

"ab_" ijS

j

b; ‘¼0; ~

Sa i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ1 p

Sa i

ffiffiffiffiffiffi

2 p

"ab_" ijS

j

b; ‘¼2;

~ Sa_i ¼ 1ffiffiffi

2

p ½pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ1Sa i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21 p

"ab_" ijS

j

b; ‘¼1:

(6.32)

In the new frame (6.30) and (6.32) the (anti)commutation relations ofsuð2j2Þbecome

f_S~a i;S~

j

bg ¼abJ j i

j

iJabþZab j

i; fS~ai;S~bjg ¼0;

f_S~i a;S~

j

bg ¼0; ½S~ai;Jcb ¼abS~ci 1 2

c bS~ai;

½_S~a i; J

j k ¼

j iS~ak

1 2

j kS~

a i:

(6.33)

With respect to the redefined suð2j2Þ generators, the transformations of the LLL supermultiplet take the form

c0 _¼ i_c0

i

ffiffiffiffiffiffi

2

p ;

c0_i ¼pffiffiffiffiffiffi2ic0 !ic1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21

p ;

c1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21 p

!kc0_k;

(6.34)

where!i_,!

iandi,iare the parameters associated with the superchargesS~1_i,S~i₁ andS~2_i,~Si₂.

As the final topic of this section, we indicate which SUð2j2Þ multiplets the wave functions form for different values of‘.

In general, irreps of SUð2j2Þ are characterized by the triple [16]

hm; n;C~i; (6.35)

where the non-negative integersm,nare Dynkin labels of subalgebrasuð2Þ suð2Þ(in the considered case, they are twice the external isospins of the wave superfunctions) and

~

Crepresents three central charges. The special case is the ‘‘short’’ irreps, with

hm; n;C~i; C~2¼1

4ðmþnþ1Þ

2_{: (6.36)}

It turns out that our wave superfunction multiplets belong just to this restricted class of thesuð2j2Þrepresentations.

At the levels ‘¼0and‘¼2, thesuð2j2Þ algebra has the central charge Z¼1=2. Hence, the relevant wave superfunctions encompass the2j2multiplets characterized by the triple

h0;0;C~i; C~2 ¼Z2_¼1

4: (6.37)

This option corresponds to the fundamental representa-tions ofsuð2j2Þ.

At the level ‘¼1, the wave superfunction multiplet comprises four bosonic and four fermionic components, with the central charge Z¼1. It is characterized by the triple

h1;0;C~i; C~2 ¼Z2 ¼1: (6.38)

With respect to the bosonic subalgebrasuð2Þ suð2Þ, the bosonic components of the ‘¼1 supermultiplet are in ð0;0Þ ð1;0Þ, while the fermionic ones are inð1=2;1=2Þ. This means that, with respect to the firstSUð2Þ, the bosonic fields are split into the singlet cþ and the tripletc½12,c

½1

2 c, where

cþ¼1

2ðc

½1 1 þc

½2

2 Þ; c¼ 1 2ðc

½1 1 c

½2 2 Þ:

With respect to the secondSUð2Þ, all of these components are just singlets. The fermionic components constitute doublets with respect to both SUð2Þ groups. To be more precise, either ofc½k_andc½k

2 is a doublet of the firstSUð2Þ acting on the index ½k, while another SUð2Þ combines them both into its doublet (irrespective of the value of½k). Let us briefly address the case <0. With the condition <1=2taken into account, the norms (5.8) and (5.10) are positive, while the first-level norm (5.9) is negative (we leave aside the degenerate case jj ¼1=2). The metric operator chosen in the form

G¼1þ8

2_2H2

142 (6.39)

has the eigenvalues

G¼ ð1Þ‘_{; (6.40)}

The corresponding generators are obtained by making the substitutions

¼ jj; ‘¼n‘0; Sa

i¼ai; Sia¼ ia (6.41) in the definitions (6.25) and (6.26). Thesuð2j2Þsuperalgebra which these generators form is slightly different from the one defined by the relations (6.27) pertinent to the >0 case. In the basic anticommutator, the two sets of the bosonic SUð2Þ generators, JJ

i and Jab, switch their places, and in the central charge one should replace! jj,‘!‘0. The three-vectorC~ defined in (6.29) preserves its form modulo these substitutions.

A different treatment of the <0case is based on the consideration in the end of Sec. IVA. The LLL wave superfunction is defined as ~‘0¼0:¼‘¼2. It satisfies the antichirality condition

rðÞ

j ~‘0¼0¼ r

ðÞ j r

ð1Þ i r

ðþ1Þ k

½ik_¼0!~ ‘0¼0

¼ ð1Þjj~

0ðÞ: (6.42)

Passing to the complex conjugate set of the wave super-functions, i.e.,~‘0 !~?‘0,~

?

‘0¼0 ¼ ð1Þ

jj~ 0ðÞ, reduces the <0case to the already studied >0one. The relevantSUð2j1Þgenerators are obtained just through the replacement! jjin the >0expressions, and the passive form of the supertransformation of the wave super-functions mimics (4.25)

~?ð;Þ ¼ jjðþÞ~?ð;Þ: (6.43)

Thus, the structure of the quantum Hilbert space in the <0 case is the same as for >0. For the ‘‘physical’’ values <1=2, the metric operatorGmaking all norms positive definite is of the same form as in (6.2), up to the substitutions ! jj and ‘!‘0. The hidden SUð2j2Þ symmetry generators leaving invariant the unitary norms are obtained from (6.25) and (6.26) through the same substitutions.

VII. THE PLANAR LIMIT

In this section, we consider the planar limit of the SUðnj1Þ=UðnÞcoset model.

We introduce a scale parameterr, rescale the odd vari-ables and the time coordinate in the action corresponding to the Lagrangian (2.14) as

!=r; t!t=r2_{; !r}2_{; (7.1)}

and then sendr! 1. The Lagrangian (2.14) goes over to L¼ _i_

iþið_iiþ_iiÞ: (7.2) This is equivalent to the following redefinition of the Ka¨hler superpotential:

K¼r2log

1 r2

: (7.3)

In the planar limitKbecomes, and the target metric and gauge connections read

gi_j¼@j@ i_K¼i

j; (7.4)

Ai¼ i@iK¼ii; Ai¼i@

i_K¼ii_{: (7.5)}

The Hamiltonian is rescaled as

Hð;; Þ !1=r2Hð=r;=r; r 2Þ (7.6)

and in the planar limit becomes

H¼ rð_iÞrðÞin:¼H0n; (7.7)

where now

rðÞ

i ¼@iþi; rðÞ

i_¼@i_þi_{: (7.8)}

The covariant derivatives obey the anticommutation relation

frðÞ i ;r

ðÞj_{g ¼2}j

i: (7.9)

The Lagrangian (7.2) enjoys the symmetry under odd magnetic translations with the generators

Qi¼ @iþi; Qyi¼ @ i_þi_;

fQi; Qykg ¼ 2ik; fQi; Qkg ¼0; (7.10)

as well as a symmetry under UðnÞrotations of the coordi-natesi_{, which define ‘‘R symmetry’’ of the superalgebra} (7.10). The full symmetry structure is an obvious contraction of the superalgebrasuðnj1Þ. The spinor generators form just an extendedN ¼n,d¼1Poincare´ superalgebra, with the central charge2in place of the Hamiltonian appearing in the standard extended supersymmetric mechanics models (see, e.g., Ref. [18]). The Hamiltonian (7.7) admits the Sugawara-type representation

H¼QiQyi2Fþn; (7.11)

whereF¼i_@ ii@

i_{is theUð1Þgenerator. After putting} i^_{¼0, i^¼1;. . .n1, and replacing ! , for the} remaining fermionic variable n_{, the Lagrangian (7.2) is} reduced to that of the ‘‘toy’’ fermionic Landau model of Refs. [14,19]. Also note that (7.2), up to some redefinitions, in the n¼2case, coincides with the pure fermionic trun-cation of the Lagrangian of N ¼4 super-Landau model recently constructed in Ref. [20].

Let us sketch some salient features of the quantum theory. For simplicity, we will mainly concentrate on the case with >0.

The ground state wave superfunction is defined by the chirality conditionrðÞi0¼0, which amounts to express-ing this superfunction through the holomorphic function

0 ¼e

The higher LL wave superfunctions are obtained as

‘¼ r

ðÞ

i1 r

ðÞ

i‘ ½i1i‘_;

½i1i‘¼_e½i1i‘ðÞ_;

(7.13)

with the energy of the‘th Landau level beingE‘ ¼2‘. This formula forE‘can be directly reproduced from (4.12), making rescaling (7.6) in (4.12) and taking the planar limit r! 1. Note that the exponential prefactors in (7.12) and (7.13) can be regarded as ther! 1limit of the prefactor in (4.3), (4.6), and (4.11):ð1r2Þr2!e.

The invariant norm is defined by

jjjj2 ¼Z d0?; (7.14)

where Rd₀ ¼Q@_i@i_{. Using the anticommutation} rela-tion (7.9), it is straightforward to compute the norm of the ‘th-level wave superfunction:

jj_‘jj2 _¼‘!ð2Þ‘Z _d

0?½i1i‘ ½i1i‘

¼‘!ð2Þ‘Z _d 0e2

?

½i1i‘

½i1i‘_{: (7.15)}

This integral is obviously not positive definite, for both >0 and <0. To make all norms positive, we should redefine the inner product:

hhjii ¼Z d?_{G: (7.16)}

The metric operatorGis defined as

ð1Þ G¼ ð12 ^m1Þ. . .ð12 ^mnÞ; for >0; (7.17)

ð2ÞG¼ ð12 ^n1Þ. . .ð12 ^nnÞ; for <0: (7.18)

Here

^ mi¼

rðÞ i r

ðÞi

2 þ ð i_@

ii@ i_{Þ; n^}

i¼ rðÞ

i r

ðÞi

2jj ; (7.19) and no summation over the indexiis assumed.

The operatorGanticommutes with the supercharges for >0,

fQ_i; Gg ¼0; fQyi_{; Gg ¼0; (7.20)}

and commutes with them for <0. Note that, within our conventions, just the latter option corresponds to the toy SUð1j1Þ invariant fermionic Landau model considered in Refs. [14,19]. The relevant metric operator is the n¼1 case of (7.18). The alternative choice of, with the metric operator being then¼1case of (7.17), was not addressed in these papers.

Though in the notation (7.17) and (7.18) covariance with respect to theUðnÞ Rsymmetry is not manifest, one can check that the operatorsGcommute withUðnÞgenerators. For instance, in then¼2case and for <0, the metric

operator Gcan be rewritten in the manifestlyUð2Þ cova-riant form asG¼1þ1

2jj

2_H02_þ2jj1_H0_{. This} opera-tor can be obtained as the planarr! 1limit of the metric operator (6.39), in whichiandare rescaled according to (7.1). Upon truncation to n¼1, we find that H02! 2jjH0 and G!1þ jj1H0. This G is the metric operator of the fermionic model of Refs. [14,19].

Finally, let us illustrate the general consideration by the example of the positive norm for the LLL wave function in then¼2case with >0:

jj0jj2 ¼c1c1þ2c0ic0iþ ð2Þ2c0c0: (7.21)

The coefficients in thei_{expansion of the holomorphic LLL} superfunction0ðÞwere defined precisely as in (4.31).

A. Symmetries of the quantum planarn¼2model

The superplane Landau models obtained as a large radius limit of the SUð2j1Þ=Uð1j1Þ supersphere or the SUð2j1Þ=½Uð1Þ Uð1Þ superflag Landau models respect the worldlineN ¼2supersymmetry which emerges as a contraction of the appropriate extensions ofSUð2j1Þ, real-ized in the Hilbert space of the quantum curved models [5]. In the planar limit, no worldline supersymmetry appears in then¼2odd-coset model. To see what kind of algebraic structure is recovered in this limit from the suð2j2Þ gener-ators defined in Sec. VI B, we should make, in all odd generators (6.25) and (6.26), the rescaling of,, accord-ing to (7.1) and multiply them by the additional factor1=r, while no such factor is needed in the case of the bosonic generators (6.21) and (6.22) (it is enough to make the rescaling of the odd variables and in them). Keeping in mind that, to the leading order,pffiffiffiffiffiffiC₂¼2r2_{þ for any} level‘, it is easy to check that in the limitr! 1, all odd SUð2j2Þ generators become either @_i or 2i _(modulo signs), which are just the generators Qi or Qyi in the holomorphic representation, with the only nonvanishing anticommutatorfQ; Qyg, as is given in (7.10) (forn¼2). Thus, no new fermionic generators appear in this limit. The bosonic generators (6.21) retain their form and define the SUð2Þ R-symmetry group of the flat n¼2 superalgebra (7.10). The extraSUð2Þ generators (6.22) also retain their form, modulo the substitution ffiffiffiffiffiffiC2

p

!2. They generate the second R-symmetry SUð2Þ group of the flat n¼2 superalgebra, such that the generatorsQ_iandQyi_{are mixed} under thisSUð2Þ.

Surprisingly, it is still possible, at least for the choice of >0, to show that the space of quantum states of the n¼2 fermionic planar Landau model exhibits SUð2j2Þ symmetry.

For >0, the Hermitian conjugates of the supercharges Qiare given by

Qzi_GQyi_{G¼ Q}yi_{; i¼1;2: (7.22)}

ðQGÞi 1

2½Qi; G ¼QiG¼ GQi: (7.23) These two pairs can be combined into the complex quartet superchargesS^a_i as

^ S1₁ ¼1

2ð2Þ

1

2ð1þGÞQ 1;

^ S21 ¼

1 2ð2Þ

1

2ð1þGÞQz2;

^ S1₂ ¼1

2ð2Þ

1

2ð1þGÞQ 2;

^ S2₂ ¼1

2ð2Þ

1

2ð1þGÞQz1:

(7.24)

These supercharges together with their conjugates S^z_ai satisfy just the (anti)commutation relations (6.33) of the superalgebrasuð2j2Þ, with Z¼1=2and theSUð2Þ gener-atorsðJ^;J^3Þ,ðJ^;J^3Þdefined as

^

Jþ¼ ð2Þ1Qz1Q2; J^ ¼ ð2Þ1Qz2Q1;

^ J3 ¼

1

2ðm^1m^2Þ; J^þ ¼ ð2Þ

1_Qz1_Qz2_;

^

J¼ ð2Þ1Q2Q1; J^3¼ 1

2ð1m^1m^2Þ: (7.25)

It is instructive to rewrite the first set of the SUð2Þ generators in the covariant form

^ Ji_j¼ 1

4

½Qzi_{; Q}

j 1 2

i

j½Qzk; Qk

; J^þ¼J^12;

^

J ¼J^21; J^3 ¼J^11¼ J^22: (7.26)

We see that they do not coincide with the generators of the R-symmetry SUð2Þ: in the holomorphic basis, with Qi¼ @i,Qzi¼ 2i, one has

^

Jij¼i@j 1 2

i

jk@k; (7.27)

which should be compared with the R-symmetry SUð2Þ generators (6.21) in the same basis (their form is preserved upon passing to the planar limit). The difference is the absence of the matrix parts B^ in (7.27) for any level‘ as compared to the generators (6.21) which necessarily include theB^ parts for‘¼1. Nevertheless, the positive norms are invariant under both of theseSUð2Þsymmetries separately. The R-symmetry SUð2Þ can be interpreted as one of the outer automorphisms of the consideredsuð2j2Þ superalge-bra: it uniformly rotates the doublet indices i, j, k of the fermionic generatorsS^a_i and the bosonic generatorsJ^i_j.

As for the second set ofSUð2Þgenerators in (7.25), they coincide with those of the second R-symmetry group SUð2Þand have, in the holomorphic basis, the same form as in (6.22), up to the substitution ffiffiffiffiffiffiC2

p

!2.

Note that the pair of generatorsðQGÞi¼QiG,ðQGÞzi¼ Qzi_{Gform the same flat algebrafQ}

G; QzGg ¼2as the pair Qi, Qzi. Their crossing anticommutators produce

the suð2Þ generators (7.25). The fermionic generators Q, Q_G and their conjugates just fix another basis in the odd sector (7.24) of the suð2j2Þ superalgebra (this basis was employed, e.g., in Ref. [20]).

The LLL and the second LL wave superfunctions belong to the short representation of thissuð2j2Þ

h0;0;C~i; (7.28)

with C~¼ ð1=2;0;0Þ. The first level is described by the direct sum of such representations:

h0;0;C~i h0;0;C~i: (7.29)

VIII. SUMMARY AND OUTLOOK

In this paper, we continued the study of super-Landau models associated with the supergroup SUðnj1Þ. This study was initiated in Refs. [4,7] for an arbitrary nand further performed in more detail for n¼2in Ref. [5,6]. We constructed the model on the pure odd coset SUðnj1Þ=UðnÞ as a generalization of the consideration of Ref. [4], which dealt only with the lowest Landau level sector of such a model. An important peculiarity of this model is the finite numbernþ1of Landau levels in the sector spanned by the wave superfunctions with the van-ishing externalSUðnÞ‘‘spin.’’ We presented the action of the model, as well as its quantum Hamiltonian, for an arbitrary n, found the energy spectrum, and defined the relevant wave superfunctions. For the particular case of n¼2, we showed that the space of quantum states of the model reveals hiddenSUð2j2Þsymmetry: at each Landau level ‘¼0, 1, 2, the relevant wave superfunctions con-stitute ‘‘short’’ SUð2j2Þ multiplets. Like in other super-Landau models, the Hilbert space for anynincludes the states the norms of which are not positive definite; for n¼2, we gave the explicit form of the metric operator which modifies the inner product in such a way that the norms of all states become non-negative, thus demon-strating that the model is unitary. Such operator is expected to exist for any n. We also studied the planar limit of theSUðnj1Þ=UðnÞmodels. In this limit, the super-algebra suðnj1Þ converts into a superalgebra which is isomorphic to n-extended d¼1 Poincare´ superalgebra, with a central charge playing the role of the d¼1 Hamiltonian. We presented, for an arbitraryn, the explicit form of the metric operator ensuring the norms of all quantum states to be non-negative. For n¼2, we find out that the Hilbert space for all three levels carries a dynamical hiddenSUð2j2Þsymmetry, like in theSUð2j1Þ model, though these twoSUð2j2Þsymmetries are realized in entirely different ways.