VUB/TENA/99/03
[3mm] hepth/9905141
Properties of SemiChiral Superfields
J. Bogaerts^{1}^{1}1Aspirant FWO, A. Sevrin, S. van der Loo and S. Van Gils
Theoretische Natuurkunde, Vrije Universiteit Brussel
Pleinlaan 2, B1050 Brussel, Belgium
bogaerts, asevrin,
Abstract
Whenever the supersymmetry algebra of nonlinear models in two dimensions does not close offshell, a holomorphic twoform can be defined. The only known superfields providing candidate auxiliary fields to achieve an offshell formulation are semichiral fields. Such a semichiral description is only possible when the twoform is constant. Using an explicit example, hyperKähler manifolds, we show that this is not always the case. Finally, we give a concrete construction of semichiral potentials for a class of hyperKähler manifolds using the duality exchanging a pair consisting of a chiral and a twistedchiral superfield for one semichiral multiplet.
May 1999
1. Introduction
Starting with Zumino’s discovery that the scalar fields of an nonlinear model in four dimensions should be viewed as coordinates on a Kähler manifold [1], ample evidence for the interplay between supersymmetry and complex geometry was found.
Requiring more supersymmetry or raising the dimension puts further restrictions on the geometry, while lowering the dimensions relaxes the requirements. Examples of the former statement are well known in 4 dimensions where passing from to supersymmetry restricts the geometry of the scalars in vector multiplets to socalled special Kähler manifolds [2]. Similarly, the scalars in hypermultiplets describe special hyperKähler manifolds [3].
Particularly interesting examples of supersymmetric models in lower dimensions are twodimensional nonlinear models which are used e.g. for the worldsheet description of stringtheory. The closest analog in two dimensions of supersymmetry in four dimensions is supersymmetry. As long as torsion is absent, the target manifold is indeed Kähler [4] and an offshell description is known. However, once torsion is present, the geometry becomes much richer [5], [6] and an offshell description is much harder to achieve. Finding an offshell description has been the subject of numerous studies, [5], [7], [8], [9] and [10], which culminated in [11]. There strong evidence was put forward to support the conjecture that chiral, twistedchiral and semichiral superfields are sufficient to give a manifest offshell description of these models. These are the only superfields which can be defined by constraints on a general superfield which are linear in the fermionic derivatives. Several explicit examples are known. All Kählermanifolds can be described by chiral superfields, the WesszuminoWitten model is described either by a chiral and a twistedchiral field [8] or by a semichiral multiplet [11]. Finally, the WessZuminoWitten model requires one semichiral and one chiral multiplet [11]. Various models dual to the above mentioned models were constructed in [9], [8], [12].
In the present paper we focus on the case where the does not close offshell in all directions and investigate under which conditions semichiral superfields provide an offshell formulation. Nonclosure implies the existence of a holomorphic twoform. A necessary condition for the semichiral description to be possible is that there exists a complex coordinate system in which this twoform is constant. Using a particular example, hyperKähler manifolds, we are able to show that semichiral fields alone are not able to give a full offshell description thus falsifying the conjecture given in [11]. Semichiral potentials do describe hyperKähler manifolds provided the potential satisfies a nonlinear differential equation.
We end the paper with a short study of duality transformations involving semichiral fields. An interesting duality brings a model described by one chiral and one twistedchiral superfield to a model formulated in terms of one semichiral multiplet [12], [9]. If the original model has an supersymmetry, which is true when the potential satisfies the Laplace equation [5], then the dual model is a hyperKähler manifold. In this way we generate a class of solutions to the nonlinear differential equation using solutions of a linear differential equation. This construction is similar in spirit to the construction in [13], where hyperKähler potentials were constructed using the duality between a real linear and a chiral superfield.
2 supersymmetric nonlinear models
A bosonic nonlinear model in two dimensions is characterized by a manifold, the target manifold, endowed with a metric and a closed 3form . Locally, the torsion can be written as the exterior derivative of the torsion potential ,
(2.1) 
Such a model can be promoted to an supersymmetric model without any additional conditions on the geometry. However, passing from to supersymmetry requires further structure. Two (1,1) tensors and are needed which satisfy
(2.2)  
(2.3)  
(2.4)  
(2.5) 
where and denote covariant differentiation^{1}^{1}1Covariant derivatives are taken as and . using the and connections resp. The first two conditions arise from requiring that the supersymmetry algebra is satisfied onshell and the last two conditions follow from the invariance of the action. Eqs. (2.2) and (2.3) imply that both and are complex structures. Eq. (2.4) imposes hermiticity of the metric with respect to both complex structures and eq. (2.5) states that both complex structures are covariantly constant but, when torsion is present, with respect to different connections.
The models characterized by eqs. (2.22.5) realize the supersymmetry algebra onshell only. One can show that the offshell nonclosing terms in the algebra, are proportional to the commutator of the two complex structures, [11]. The construction of a manifest offshell supersymmetric version of these model was the subject of intense investigations [5], [7], [8], [9], [11]. Locally, the cotangent space can be decomposed as
(2.6) 
In [9], it was shown that and are integrable to chiral and twistedchiral superfields resp. These superfields count as many components as superfields. Indeed, as the algebra closes offshell in these directions, one does not expect that any new auxiliary fields are needed.
Chiral and twisted chiral superfields separately describe Kähler manifolds. However when both of them are simultanously present, the resulting manifold exhibits a product structure which projects on two Kähler subspaces [5]. The complete manifold is not Kähler, which can be seen from the fact that it has torsion.
In [11], it was shown that the dimension of is a multiple of four (see also further in this section). Furthermore, offshell closure of the algebra requires additional auxiliary fields compared to the manifestly formulation of the model. Only one class of superfields defined by constraints linear in the derivatives satisfies these requirements: the semichiral superfields[7]. This, combined with several nontrivial examples, led to the conjecture [11] that semichiral superfields are sufficient to describe . In the present paper, we will give a class of explicit examples disproving the conjecture.
From now on we focus our attention on the case where . Denoting , we construct a nondegenerate twoform or in local coordinates
(2.7) 
The inverse of the commutator can be written as the formal power series
(2.8) 
The twoform satisfies .
Introducing complex coordinates and , we diagonalize : and . In complex coordinates, only and its complex conjugate are nonvanishing. Eqs. (2.3) and (2.4) for imply that and resp. Finally eq. (2.5) for yields . Combining this with eq. (2.5) for gives
(2.9) 
Finally, let us give a very short proof that implies that with . We view as the components of an antisymmetric matrix. From combined with the nondegeneracy of the metric, we get that its determinant is nonvanishing implying that should be even.
3 Semichiral parametrization
We denote the semichiral coordinates by , , and , , , , , and we introduce a real function , the semichiral potential. It is determined modulo the transformation with and arbitrary holomorphic functions of and resp. The potential is the Lagrange density in superspace. Passing to superspace and eliminating the auxiliary fields through their equations of motion yields explicit expressions for the metric, torsion potential and the complex structures [11].
In order to facilitate the notation, we introduce the matrices , , , and .
(3.1) 
(3.2) 
where e.g. stands for . Finally we also need the matrix , defined by
(3.3) 
In terms of these matrices, the complex structures are given by^{2}^{2}2Rows and columns are labeled as and we rescaled by a factor .
(3.4) 
The metric and torsion potential have simple expressions in terms of the complex structures,
(3.5)  
(3.6) 
Eq. (3.5) clearly shows that the vanishing of is necessary and sufficient for the existence of a nondegenerate metric. Furthermore, the potential should be such that . Eq. (3.5) gives the explicit form for the twoform ,
(3.7) 
Quite remarkable is the existence of simple coordinate transformations which diagonalize either or . Consider
(3.8) 
then with
(3.9) 
and and where
(3.10) 
Rows and columns are labeled as , , and . Note that there is also a simple coordinate transformation which diagonalizes which is obtained by reversing the roles of and in the previous expressions.
Eq. (3.10) now gives the twoform in complex coordinates,
(3.11) 
So we reach the conclusion that a necessary condition for a semichiral parametrization to be possible is the existence of a complex coordinate system in which the twoform is constant! Note that for , we can always find a holomorphic coordinate transformation which makes the twoform constant. For this is not the case anymore.
Finally, for the sake of completeness, let us mention that these models are 1loop UV finite provided they are Ricci flat where the Ricci tensor is computed using the connection with torsion. In [16] the oneloop function was directly computed in superspace. Its vanishing yields a constraint on the potential which requires the existence of a holomorphic function of , and a holomorphic function of , , such that
(3.12) 
where is the number of semichiral multiplets and and are matrices given by
(3.13) 
4 HyperKähler manifolds
As is clear from eq. (3.5), the semichiral parametrization is well defined, provided . The most familiar class of complex manifolds satisfying this are the hyperKähler manifolds. A hyperKähler manifold has three complex structures , which satisfy
(4.1) 
and which are such that the manifold is Kähler with respect to all three of them. It is easy to see that a hyperKähler manifold has a twosphere worth of complex structures. Indeed is a complex structure provided that . Choosing e.g. , and requiring that , we find that can be any element of the twosphere, except for the north and southpole, . Choosing for the northpole, we obtain the description of the manifold in terms of chiral superfields, while choosing the southpole we get the parametrization in terms of twistedchiral superfields. Clearly, these are the only two choices where and commute. In other words, there is a cylinder worth of choices for where the algebra does not close offshell and which can potentially be described by semichiral coordinates. In order for this to work, we need at least that the torsion vanishes, . Indeed, choosing one element of the cylinder , we get . From eq. (3.6), one obtains that for a generic element of the cylinder, the torsion potential differs from zero, but the torsion still vanishes, . Turning to the twoform introduced in section 2, one easily shows that, for the present choice for and , it is given by
(4.2) 
where are the fundamental twoforms of the hyperKähler manifold, defined by , . As was shown in the previous section, a semichiral parametrization is only possible, if complex coordinates exist, diagonalizing , where is constant. In this case is given as a linear combination of the the two fundamental twoforms and . Several explicit examples are known of higher dimensional hyperKähler manifolds where this is not the case [13].
As was already mentioned, there is still the simplest case , where both and can be made constant through a holomorphic coordinate transformation. However we find, as we will see in next section, that the metric satisfies the MongeAmpere equation after passing from semichiral to complex coordinates. To achieve this on a hyperKähler manifold, one already needs a holomorphic coordinate transformation. The residual holomorphic coordinate transformations can now only turn the twoform to a constant provided the twoform was originally a phase, which as far as we know, is not necessarily true. Nonetheless, many interesting examples are of this kind. In particular, the four dimensional hyperKähler manifolds constructed in [13] are of this form. This includes familiar examples such as the multiEguchiHanson [14] and TaubNUT [15] selfdual instantons.
5 The fourdimensional case
The simplest hyperKähler manifolds are the four dimensional ones. We will choose and . Requiring we find using eq. (3.4) that the semichiral potential should satisfy
(5.1) 
Performing the coordinate transformation eq. (3.8), we find that the metric eq. (3.10) satisfies the MongeAmpère equation,
(5.2) 
iff. eq. (5.1) holds. Indeed, in four dimensions one finds that eq. (3.5) yields the following nonvanishing components of the metric (and their complex conjugates),
(5.3) 
where . After the coordinate transformation eq. (3.8), the components of the metric are given by
(5.4) 
As a result we get that eq. (5.1) is indeed a necessary and sufficient condition on the potential so that it describes a hyperKähler potential. Note that, as expected, eq. (3.12) is satisfied with .
In [13], a powerful method was developed to construct solutions to eq. (5.2), the Legendre transformation construction. In the remainder of this section we will discuss various duality transformations which involve semichiral fields and we will give a construction of solutions to eq. (5.1) analogous to the Legendre transform construction in [13].
In [12] (see also [9]) various duality transformations in superspace were catalogued. The simplest ones are those which do not need any isometries. They arise by passing to a first order action in superspace. I.e. the superfield constraints are imposed through Lagrange multipliers. The best known example is the duality between chiral and complex linear superfields (for a recent account see e.g. [17]). Both describe Kähler geometry and the former gives the minimal description while the latter is a nonminimal description. The two potentials are related through a simple Legendre transformation.
Similar duality transformations exist for semichiral superfields. One can perform a Legendre transformation either with respect to or with respect to or with respect to both of them. Given a potential , we can construct three potentials
(5.5) 
where in the first case , in the second case and in the last case and hold. One verifies immediately that if satisfies eq. (5.1) then so do , and . These three duality transformations simply shuffle around the auxiliary field content of the system and act as mere coordinate transformations on the physical fields.
In case isometries are present more interesting duality transformations become possible. The most typical semichiral example is the one which interchanges one semichiral multiplet for one chiral and one twisted chiral multiplet. The geometry obviously changes now. In the present case the semichiral coordinates describe a hyperKähler manifold, while the chiral/twisted chiral combination describes a manifold with a product structure which has e.g. a nontrivial torsion. At the chiral/twistedchiral side the model shows a simple enhancement of the supersymmetry to , provided the potential satisfies the Laplace equation. The dual potential turns out to describe a hyperKähler manifold. By this construction, one obtains immediately the semichiral parametrization of the hyperKähler manifold. The advantage of this construction is that the nonlinear differential equation (5.1) gets replaced by a linear differential equation, the Laplace equation.
The starting point is a real prepotential , where and , which satisfies the Laplace equation
(5.6) 
This combines two requirements: the chiral/twistedchiral potential, which is precisely the prepotential under consideration, simultanously exihibits an Abelian isometry and it has supersymmetry. Full details can be found in the appendix. The present coordinates and are related to the chiral and twistedchiral coordinates and by , and .
The semichiral potential is obtained from the prepotential through a Legendre transformation,
(5.7)  
where
(5.8) 
Using eqs. (5.6), (5.7) and (5.8), we get
(5.9) 
where
(5.10) 
Using this, one immediately checks that the resulting semichiral potential satisfies eq. (5.1). In other words, dualizing a chiral/twistedchiral potential having an supersymmetry and an Abelian isometry yields a semichiral potential which describes a hyperKähler manifold! The resulting potential obviously still has an Abelian isometry, and with .
This construction is strikingly similar to the Legendre transform construction in [13], which follows from the duality between an model described by a real linear and a chiral superfield and an model described by two chiral superfields. Again this duality requires an Abelian isometry. There too, eq. (5.6) was the starting point for the construction of hyperKähler potentials directly in local complex coordinates. From a real prepotential , satisfying , one obtains the Kähler potential, , through the Legendre transformation
(5.11) 
where
(5.12) 
This allows for explicit expressions for the metric in terms of the prepotential,
(5.13) 
and and follows from the MongeAmpère equation. The extra fundamental twoforms are constant.
Comparing both constructions is possible if we pass from semichiral to complex coordinates. For simplicity, we use the coordinate transformation which diagonalizes ,
(5.14) 
and from eq. (5.7), one gets the identification . The metric is then given by eq. (5.4) in which and are interchanged. Combining this with eq. (5.9) yields the expression for the metric in complex coordinates,
(5.15) 
where once more we obtain from the MongeAmpère equation. Comparing eq. (5.13) to eq. (5.15) we get, after identifying , a relation between and ,
(5.16) 
with a real constant. Given either or , this allows the construction of and resp. The requirement that the resulting prepotential satisfies the Laplace equation fully determines the prepotential , once is given. However, given , is only determined modulo a function which satisfies and . This has no influence on both eqs. (5.16) and (5.6) but it might be needed in order to have a welldefined legendre transformation in eq. (5.8) which is equivalent to requiring that in eq. (5.10) does not vanish.
We give a few examples where each time both the complex and the semichiral prepotential are given. Each time it is straightforward to check that both satisfy the Laplace equation and that they are related by eq. (5.16).
The fourdimensional special hyperKähler manifolds [3] are descibed by
(5.17) 
where is a holomorphic function of and its complex conjugate and is an arbitrary real constant.
In [13], two different representations of flat space were given
(5.18) 
and
(5.19)  
where
(5.20) 
Other prepotentials for hyperKähler manifolds can now be constructed by superimposing and or equivalently and . In this way one obtains the multiEguchiHanson manifolds [14, 15, 13] by superimposing or about different points ,
(5.21) 
or the TaubNUT manifolds [15, 13] by adding an to this
(5.22) 
and similar expressions where the complex prepotentials are replaced by semichiral prepotentials.
6 Conclusions
In [11], it was conjectured that chiral, twistedchiral and semichiral superfields are sufficient to give a full offshell, manifest supersymmetric description of supersymmetric nonlinear models in two dimensions. The chiral and twistedchiral superfields do give a complete description of the directions along which the supersymmetry closes [9] while the semichiral superfields were expected to introduce the necessary auxiliary fields for those directions where no offshell closure was achieved.
In the present paper we showed that this is not true. Nonclosure of the supersymmetry implies the existence of a holomorphic twoform. Moving from semichiral coordinates to complex coordinates, one gets that this twoform is constant. HyperKähler manifolds provide particularly interesting examples. Choosing left and rightcomplex structures to be anticommuting, we do get full nonclosure of the algebra. The above mentioned twoform is the fundamental twoform associated with the “third” complex structure which is the product of the left with the right complex structure. A necessary condition for the semichiral parametrization to be possible is that this fundamental twoform is a constant, which is not the case for an arbitrary hyperKähler manifold!
As a result, the problem of finding a manifest supersymmetric description of nonlinear models is once more open. Chiral, twistedchiral and semichiral superfields exhaust the superfields which can be defined by constraints linear in the superderivatives. What remains are constraints which are higher order in the derivatives. These have not been systematically studied, but past experience shows that most often they give nonminimal descriptions of known supermultiplets. Finally, there is a last possibility which would certainly work but which involves harmonic superspace. A drawback of this approach is that it is extremely hard to extract explicit expressions for metric and torsion.
Finally, we presented a systematic way to construct hyperKähler manifolds starting from an intriguing duality transformation between an model described by one chiral and one twisted chiral superfield and a hyperKähler manifold. In particular, this implies that wellknown hyperKähler manifolds such as multiEguchiHanson and TaubNUT have a dual which is a complex manifold with a product structure. The consequences of this duality transformation and the relation, if any, with the nonabelian Tduals of these models given in [18], certainly merit further study.
Acknowledgements: We thank Martin Roček, Kostas Sfetsos and Jan Troost for useful discussions. This work was supported in part by the European Commission TMR programme ERBFMRXCT960045 in which all authors are associated to K.U. Leuven.
Appendix: superspace
In this appendix we summarize some properties of superspace and superfields, together with some aspects of duality transformations. The fermionic coordinates which parametrize the superspace are denoted by , , and and the bosonic coordinates by and . The fermionic derivatives , , and satisfy
(A.1) 
with all other (anti)commutators between derivatives vanishing. The superderivatives are given by the real part of the fermionic derivatives,
(A.2) 
while the extra supersymmetry generators are then proportional to the imaginary part of the superderivatives,
(A.3) 
Consider a set of general superfields, , . The most general constraints linear in the derivatives are
(A.4) 
A detailed analysis of the integrability conditions following from eq. (A.4) yields that both and are complex structures which mutually commute [11]. Through a suitable coordinate transformation they can be diagonalized resulting in two classes of superfields, chiral and twistedchiral [5] superfields.

Chiral superfields, and ,
or and . 
Twisted chiral superfields, and ,
or and .
On a chiral superfield , both and have eigenvalue . On a twisted chiral superfield , one finds that has eigenvalue while has eigenvalue . Chiral and twisted chiral fields have the same number of components as a general superfield, consistent with the fact that the algebra closes in the directions where the complex structures commute. A weaker set of constraints is still possible where only one chirality gets constrained. A detailed analysis shows that, in order to get nontrivial dynamics, they should occur in pairs, the members of which have constraints of opposite chirality. This results in semichiral superfields [7].

Semichiral superfields, , , and ,
or and .
Semichiral superfields contain twice as many components as an superfield, however, half of them turn out to be auxiliary. On , is diagonal with eigenvalue , while on , is diagonal with eigenvalue . The precise action of on and on is model dependent and can only be obained after elimination of the auxiliary fields.
Other constraints, linear in the derivatives, are still possible, but they imply restrictions on the dependence of the superfields on the bosonic coordinates (see e.g. [19]). We do not consider this here as it does not seem relevant to the present case.
Finally, we comment on duality transformations involving semichiral superfields. It is well known that in the presence of an abelian isometry a chiral field can be dualized to a twistedchiral superfield and viceversa. Similarly [9, 11], when a specific abelian isometry is present, a pair consisting of a chiral and a twistedchiral superfield can be dualized to a semichiral field and viceversa. Consider a model described by a chiral/twisted chiral potential , which is such that an abelian isometry exists,
(A.5) 
with a real constant. We introduce prepotentials and which are complex unconstrained superfields and one set of semichiral superfields and . We consider the first order action
(A.6)  
If we first integrate over the semichiral fields, we recover the original model in terms of a chiral and a twistedchiral field. Indeed, the equations of motion for and ,
(A.7) 
imply that
(A.8) 
which are solved by putting and . The dual model is obtained by first integrating over the prepotentials which yields
(A.9) 
which can be solved for the prepotentials and . The semichiral potential, , is simply the Legendre transform of ,