Physics Letters B 642 (2006) 535–539

www.elsevier.com/locate/physletb

A note on N = (2,2) superfields in two dimensions

Joris Maes, Alexander Sevrin ∗

Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay Institutes, Pleinlaan 2, B-1050 Brussel, Belgium

Received 18 July 2006; accepted 3 October 2006

Available online 11 October 2006

Editor: L. Alvarez-Gaumé

Abstract

Motivated by the results in hep-th/0508228, we perform a careful analysis of the allowed linear constraints on N = (2,2) scalar superfields. Weshow that only chiral, twisted-chiral and semi-chiral superfields are possible. Various subtleties are discussed.© 2006 Elsevier B.V. All rights reserved.

Non-linear σ -models in two dimensions with an N = (2,2)

supersymmetry play a central role in the description of typeII superstrings in the absence of R–R fluxes. The interest inthese models was recently rekindled as well in the physics as inthe mathematics community. For physicists, these models allowfor the study of compactifications in the presence of non-trivialNS–NS fluxes while for mathematicians the models provide aconcrete realization of generalized complex geometries. Theideal setting for studying them is provided by N = (2,2) su-perspace where the whole (local) geometry gets encoded in asingle scalar function, the Lagrange density. The N = (2,2)

superfields and as a direct consequence the geometry of theresulting σ -model as well, are fully characterized by their con-straints. The analysis of auxiliary field configurations made in[1] raised the suspicion that more general N = (2,2) superfieldsthan those known up till now might exist. This point will be ex-plored in the present Letter.

The target space geometry of a bosonic non-linear σ -modelin two dimensions1 is characterized by the metric gab and aclosed 3-form (the torsion) Tabc on it. We denote the local co-ordinates on the target manifold by Xa . The indices a, b, . . .

run from 1 to D, with D the dimension of the target manifold.Such a model can be lifted to an N = (1,1) supersymmetricσ -model without any further restrictions on the geometry. How-

* Corresponding author.E-mail address: alexandre.sevrin@vub.ac.be (A. Sevrin).

1 We consider here only σ -models without boundaries.

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.10.002

ever, passing from N = (1,1) to N = (2,2) supersymmetryintroduces additional geometric structure [2–5].

Indeed N = (2,2) requires the existence of two (1,1) ten-sors, J a+b(X) and J a−b(X), on the target manifold. On-shellclosure of the N = (2,2) supersymmetry algebra is realizedprovided both J+ and J− are complex structures. I.e. theysquare to −1, J a±cJ

c±b = −δab and their Nijenhuis tensors2 van-

ish, N [J±, J±]abc = 0. One finds that the off-shell non-closingterms in the algebra are proportional to the commutator of thetwo complex structures, [J+, J−]. Decomposing the tangenttarget space as ker[J+, J−] ⊕ coker[J+, J−], one anticipatesthat the description of ker[J+, J−] will be possible with thefields at hand while the description of coker[J+, J−] will re-quire the introduction of additional auxiliary fields.

Realizing the N = (2,2) supersymmetry algebra is obvi-ously not sufficient, the N = (1,1) supersymmetric non-linearσ -model has to be invariant under the additional supersymme-try transformations as well. One finds that this is indeed soprovided the metric is hermitean with respect to both complexstructures3

2 Out of two (1,1) tensors Rab and Sa

b , one constructs a (1,2) ten-

sor N [R,S]abc , the Nijenhuis tensor, as N [R,S]abc = RadSd [b,c] +

Rd [bSac],d + R ↔ S.

3 This implies the existence of two two-forms ω±ab

= −ω±ba

= gacJc±b .

In general they are not closed. Using Eq. (2), one shows that ω±[ab,c] =

∓2Jd±[aTbc]d = ∓(2/3)J d±aJ e±bJf± cTdef, where for the last step we used the

fact that the Nijenhuis tensors vanish.

536 J. Maes, A. Sevrin / Physics Letters B 642 (2006) 535–539

(1)J c±aJd±bgcd = gab.

Furthermore, both complex structures have to be covariantlyconstant

(2)0 = ∇±c J a±b ≡ ∂cJ

a±b + Γ a±dcJd±b − Γ d±bcJ

a±d ,

with the connections Γ± given by

(3)Γ a±bc ≡{

a

bc

}± T a

bc,

where the first term at the right-hand side is the standardChristoffel symbol and T a

bc = gadTdbc .Finding an off-shell description of a general N = (2,2) su-

persymmetric non-linear σ -model remained for more than twodecades an open problem. Recently this has been solved in[6] (papers preparing the road to this result are e.g. [7–10]).There it was shown that chiral, twisted-chiral [3] and semi-chiral [7] multiplets are sufficient to describe any N = (2,2)

non-linear σ -model. Roughly speaking one gets that when writ-ing ker[J+, J−] = ker(J+ −J−)⊕ker(J+ +J−), ker(J+ −J−)

and ker(J+ + J−) respectively can be integrated to chiral andtwisted chiral multiplets respectively [8]. Semi-chiral multipletsallow then for a description of coker[J+, J−] [6,9,10].

However, chiral, twisted-chiral and semi-chiral multipletsare by no means the only representations of d = 2, N = (2,2)

supersymmetry. Other representations are known such as lin-ear [11–13] and twisted linear multiplets [14]. Having othermultiplets at hand allow e.g. for dual formulations of a model(see e.g. [15] and references therein). In [1], a detailed analysisof potential auxiliary field configurations in d = 2, N = (2,2)

σ -models was performed. The resulting expressions were veryinvolved and raise the pertinent question whether other solu-tions besides semi-chiral multiplets exist. This motivates thepresent Letter. We analyze constraints linear in derivatives onN = (2,2) superfields in order to clarify this.

The d = 2, N = (1,1) superspace has two bosonic (light-cone) coordinates σ=| ≡ τ + σ , σ= ≡ τ − σ and two (chiral,real) fermionic coordinates θ+ and θ−. Passing to N = (2,2)

superspace requires the introduction of two additional (chiral,real) fermionic coordinates θ+ and θ−. We introduce the fermi-onic derivatives w.r.t. θ±, D±, and those w.r.t. θ±, D±. They aredefined by

(4)D2+ = D2+ = − i

2∂=| , D2− = D2− = − i

2∂=,

and all other anti-commutators vanish.The action S in N = (2,2) superspace is given by

(5)S =∫

d2σ d2θ d2θ V .

As the measure has dimension zero, the Lagrange density Vcan only be some function of scalar N = (2,2) superfields. Itis clear that in order to generate dynamics, one will have tojudiciously constrain the N = (2,2) superfields.

Consider a set of bosonic, scalar N = (2,2) superfields Xa ,a ∈ {1, . . . ,D}. Expanding them in powers of θ+ and θ−, onefinds that each general N = (2,2) superfield consists of 4 N =(1,1) superfields. As a warming up exercise, let us first study

those constraints linear in the derivatives that reduce the numberof N = (1,1) components in a general N = (2,2) superfield toone. Fixing both D+Xa and D−Xa simultaneously does thejob. The most general constraints consistent with dimensionsand Lorentz covariance are then given by

(6)D±Xa = J a±b(X)D±Xb,

where J±(X) are at this point two arbitrary (1,1) tensors. Fromthis, one gets immediately

D2+Xa = + i

2

(J 2+

)ab∂=| Xb + 1

2N [J+, J+]abcD+XbD+Xc,

(7)

D2−Xa = + i

2

(J 2+

)ab∂=Xb + 1

2N [J−, J−]abcD−XbD−Xc,

and4

{D+, D−}Xa = [J+, J−]abD−D+Xb

(8)+ 2M[J−, J+]abcD+XbD−Xc.

Requiring that these constraints are consistent with D2+ =−(i/2)∂=| , D2− = −(i/2)∂= and {D+, D−} = 0, we get the fol-lowing conditions,

J 2± = −1, [J+, J−] = 0,

(9)N [J±, J±] =M[J+, J−] = 0.

All conditions in Eq. (9) are necessary. Indeed, in the literatureit is often erroneously stated that two complex structures aresimultaneously integrable if they commute.5

In other words, J 2± = −1, N [J±, J±] = 0 and [J+, J−] = 0would imply that N [J+, J−] = 0 holds. However this is wrong!A very nice and explicit counter example was provided in [16].Consider a six-dimensional target manifold with the followingtwo complex structures,

(10)J+ =

⎛⎜⎜⎜⎜⎜⎝

0 −1 0 0 0 01 0 0 0 0 00 0 0 1 0 00 0 −1 0 0 0

−X1 −X1 X3 −X3 0 −1−X1 X1 X3 X3 1 0

⎞⎟⎟⎟⎟⎟⎠ ,

and

(11)J− =

⎛⎜⎜⎜⎜⎜⎝

0 −1 0 0 0 01 0 0 0 0 00 0 0 1 0 00 0 −1 0 0 0

−X1 X1 X3 X3 0 1−X1 −X1 X3 −X3 −1 0

⎞⎟⎟⎟⎟⎟⎠ .

4 Out of two commuting (1,1) tensors Rab and Sa

b , one constructs a (1,2)

tensor, M[R,S]abc = (RadSd

b,c − SadRd

c,b + SdbRa

c,d − RdcS

ab,d )/2.

One has that M[R,S]abc = −M[S,R]acb and N [R,S]abc =M[R,S]abc +M[S,R]abc .

5 While this statement is made in numerous papers, we just mention one paperfor which one of the present authors is responsible [9]. In that paper a multi-plicative factor of two is lacking in front of the last term in Eq. (A.6), whichinvalidates the proof of Lemma 1.

J. Maes, A. Sevrin / Physics Letters B 642 (2006) 535–539 537

One readily verifies that they are complex structures, i.e. J 2± =−1 and N [J±, J±] = 0. They commute, [J+, J−] = 0, as well.However, when calculating M[J+, J−], one finds that all com-ponents vanish except for

M[J+, J−]511 =M[J+, J−]5

21 =M[J+, J−]522

=M[J+, J−]543 =M[J+, J−]6

21

=M[J+, J−]633 =M[J+, J−]6

43

=M[J+, J−]644 = +1,

M[J+, J−]512 =M[J+, J−]5

33 =M[J+, J−]534

=M[J+, J−]544 =M[J+, J−]6

11

=M[J+, J−]612 =M[J+, J−]6

22

(12)=M[J+, J−]634 = −1.

Consequently, the mixed Nijenhuis tensor N [J+, J−] =M[J+, J−] + M[J−, J+] does not vanish! In other words, allfour conditions in Eq. (9) have to be imposed independently.The non-linear σ -model constructed out of superfields con-strained as in Eq. (6) so that Eq. (9) holds, will be such thatEqs. (1) and (2) are automatically satisfied.6

Let us now come back to Eq. (9) which does imply that bothJ+ and J− are simultaneously integrable. This means that wecan make a coordinate transformation such that both J+ andJ− are diagonal with eigenvalues ±i. If the eigenvalue of J+and J− have the same (opposite) sign, we are dealing with a(twisted) chiral field. Indeed, a chiral field Z, and its hermiteanconjugate Z, satisfy

(13)D±Z = +iD±Z, D±Z = −iD±Z,

while for a twisted chiral field Y (and its hermitean conju-gate Y ) we get

D+Y = +iD+Y, D−Y = −iD−Y,

(14)D+Y = −iD+Y , D−Y = +iD−Y .

The first explicit example of a non-linear σ -model which re-quires both chiral and twisted chiral superfields—the S3 × S1

WZW model—was given in [17].We now generalize the constraints by allowing for additional

auxiliary fields. We start from a set of m general N = (2,2)

superfields which we combine in an m × 1 matrix X. The mostgeneral right-handed constraint linear in the derivatives whichwe can impose is

(15)D+X = J+D+X + K+Ψ+,

6 One can also look at things from the point of view of the N = (1,1)

non-linear σ -model without referring to N = (2,2) superspace. Using the co-variantly constancy of the complex structures, one shows 2Ma

bc[J+, J−] =[J+, J−]adΓ d

(−)bc . Put differently: the fact that two complex structures com-mute and that they are covariantly constant does imply that they are simul-taneously integrable. However, we reiterate that in an N = (2,2) superspacetreatment one only deals with Eq. (9) which will imply—at least for the case ofcommuting complex structures—Eqs. (1) and (2).

where Ψ+ is a m × 1 column matrix of fermionic N = (2,2)

superfields and J+ and K+ are constant m × m matrices.7

Through a linear transformation we can bring K+ in its Jor-dan normal form. Making an appropriate linear combination ofthe components of Ψ+ allows one to reduce K+ to the form ofa projection operator

(16)K+ =(

0 00 1k×k

),

where 1k×k is the k × k (k � m) unit matrix. It is clear thatk determines the number of fermionic (auxiliary) superfieldswhich will appear in Eq. (15). We now split X into a k × 1matrix X and an (m−k)×1 matrix X of superfields. Appropri-ately redefining the fermionic superfields in Ψ+, we can rewriteEq. (15) without any loss of generality as

D+X = J+D+X + K+D+X,

(17)D+X = ψ+,

where J+ and K+ respectively are (m − k) × (m − k) and(m − k) × k constant matrices respectively and ψ+ is a k × 1column matrix of fermionic (auxiliary) superfields. Implement-ing D2+ = −(i/2)∂=| in Eq. (17) yields

(18)K+ = 0, J+2 = −1(m−k)×(m−k).

This implies that m − k should be even and we put 2n = m − k

with n ∈ N. At this point we can, through an appropriate lin-ear transformation on X, diagonalize J+ with eigenvalues ±i.Summarizing, we get—without any loss of generality—the fol-lowing right-handed constraints,

D+X = iPD+X,

(19)D+X = ψ+,

where P is given by

(20)P =(

1n×n 00 −1n×n

).

At this point we can still make arbitrary linear transformationson X. On X, linear transformations which commute with P areallowed as well. This freedom will be used later on.

The most general left-handed constraints are of a form simi-lar to the one in Eq. (15),

(21)D−X = J−D−X + K−Ψ−.

This can be further simplified by looking at the non-linearσ -model we ultimately want to describe. We start with a La-grange density V(X) which is some scalar function of the su-perfields X. Integrating over θ±, we get schematically the fol-lowing dependence on Ψ+ and Ψ− of the action in N = (1,1)

7 In order to keep the analysis feasible, we only consider constraints linearin the superfields. This is a reasonable simplification as non-linear constraintswould significantly complicate—if not make it impossible—the quantization ofthe resulting non-linear σ -model.

538 J. Maes, A. Sevrin / Physics Letters B 642 (2006) 535–539

superspace,

S =∫

d2σ d2θ d2θ V(X)

(22)=∫

d2σ d2θ(A1 + Ψ T+ A2 + A3Ψ− + Ψ T+ A4Ψ−

),

where Aα , α ∈ {1, . . . ,4} are matrices which depend on theremainder of the N = (1,1) superfields. It is clear from this ex-pression that the components of Ψ± appear as auxiliary fields.In order to solve for them, we have to require that the number ofcomponents of Ψ+ which appear in Eq. (15) equals the numberof components of Ψ− appearing in Eq. (21).

A further exploration of the structure of the N = (1,1) ac-tion will show that the right-hand side of D−X cannot containany components of Ψ−. Indeed, assume that some componentsof Ψ− do appear at the right-hand side of one or more compo-nents of D−X. Call these components Xr , r ∈ {1, . . . , l � k}.Making an appropriate field redefinition on those componentsof Ψ−, we would get the following schematical structure for theconstraints,

(23)D+Xr = ψr+, D−Xr = ψr−.

Because of {D+, D−} = 0, we get,

(24)D+ψr− = −D−ψr+ ≡ F r+−,

where when reducing to N = (1,1) superspace, F r+− will sur-vive as new N = (1,1) superfields. They will appear in theaction as,

S =∫

d2σ d2θ d2θ V(X)

(25)=∫

d2σ d2θ

(l∑

r=1

∂rV(X)F r+− + · · ·)

.

The equations of motion for Fr+− force the potential V to be in-dependent of Xr . Consequently we can impose that D−X doesnot depend on any of the components of Ψ−.

These considerations lead us to the conclusion that 2n � k.Decomposing X as a k × 1 column matrix X and a 2n − k col-umn matrix X, we arrive—still without any loss of generality(but using input from the goal, the σ -model, we aim for)—atthe following form for the constraints,

D+X = iPD+X, D−X = ψ−,

D+X = ψ+, D−X = J−D−X + K−D−X + L−D−X,

D+X = iPD+X,

(26)D−X = J−D−X + K−D−X + L−D−X + M−ψ−,

where ψ− is a k × 1 column matrix of fermionic superfieldswhich are related to the non-vanishing components of Ψ−through an appropriate coordinate transformation. Furthermore,we used the notation

P =(

1k/2×k/2 00 −1k/2×k/2

),

(27)P =(

1(n−k/2)×(n−k/2) 00 −1

),

(n−k/2)×(n−k/2)

where the reality properties of the fields force us to take k ∈ 2N.Rests us to impose the integrability conditions. One ver-

ifies that 0 = {D+, D−}X, 0 = {D+, D−}X and D2−X =−(i/2)∂=X respectively give us expressions for D−ψ+, D+ψ−and D−ψ−.

Implementing the integrability conditions which followfrom 0 = {D+, D−}X, D2−X = −(i/2)∂=X and D2−X =−(i/2)∂=X reduce Eq. (26) to

D+X = iPD+X, D−X = ψ−,

D+X = ψ+,

D−X = J−D−X + K−D−X − K−J−L−D−X,

D+X = iPD+X,

(28)D−X = J−D−X + L−D−X + J−L−ψ−,

with

J 2− = −1(2n−k)×(2n−k), J 2− = −1k×k,

[J−, P] = 0,

L−P = PL−,

(29)J−K− = −K−J−.

Combining the second of these equations with the freedom tomake an arbitrary linear transformation on X (making simulta-neously the same transformation on ψ+) allows one to put

(30)J− = iP,

where P was defined in Eq. (27). A final simplification isachieved by making the following field redefinitions,

X → X′ = X − J−L−X,

X → X′ = X + 1

2K−J−X + 1

2K−L−X,

(31)ψ+ → ψ ′+ = ψ+ + i

2K−J−PD+X′.

This reduces Eq. (28) to

D+X = iPD+X, D−X = ψ−,

D+X′ = ψ ′+, D−X′ = iPD−X′,(32)D+X′ = iPD+X′, D−X′ = J−D−X′,

where

(33)J 2− = −1, [J−, P] = 0.

We still have the freedom to make a linear transformation onX′ provided it commutes with P. We use this freedom to diag-onalize J− with eigenvalues ±i. Pending upon the sign of theeigenvalues, X′ constitutes of chiral and twisted chiral super-fields [3]. The remaining superfields (X, X′) are recognized assemi-chiral superfields [7].

While the analysis of [1] raised the hope that other auxil-iary field structures beyond semi-chiral superfields might exist,we showed in the present Letter that this is not so. We madetwo important restrictions: we limited ourselves to constraintswhich were both linear in the derivatives as well as linear in the

J. Maes, A. Sevrin / Physics Letters B 642 (2006) 535–539 539

superfields (and it might very well be that using an appropriatecoordinate transformation any constraint non-linear in the su-perfields can be brought to a constraint linear in the fields). Ifone wants to quantize the model, the latter restriction is essen-tially unavoidable. This note strengthens the quite unique roleof semi-chiral superfields. When studying d = 2, N = (2,2)

non-linear σ -models in the presence of NS–NS fluxes, semi-chiral superfields will be very generic. Indeed, consider e.g. avery large but quite simple class of integrable σ -models, theWZW models. In [18] it was shown that any even dimensionalWZW model allowed for an N = (2,2) supersymmetry. How-ever, as argued in [17], only the SU(2) × U(1) model canbe described solely in terms of chiral and twisted chiral su-perfields. All other WZW models on even dimensional groupmanifolds will require semi-chiral superfields as well (some ex-plicit examples can be found in [8,9]). Up till now, the literatureon semi-chiral superfields is rather limited. We hope that thepresent Letter will raise the interest in them.

Acknowledgements

We thank Jim Gates, Arjan Keurentjes, Ulf Lindström,Dario Martelli, Martin Rocek and Jan Troost for useful com-ments, discussions and suggestions. This work was supportedin part by the Belgian Federal Science Policy Office throughthe Interuniversity Attraction Pole P5/27, in part by the Eu-ropean Commission FP6 RTN programme MRTN-CT-2004-005104 and in part by the “FWO-Vlaanderen” through projectG.0428.06.

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