First-order supersymmetric sigma models and target space geometry
ABSTRACT We study the conditions under which N=(1,1) generalized sigma models support an extension to N=(2,2). The enhanced supersymmetry is related to the target space complex geometry. Concentrating on a simple situation, related to Poisson sigma models, we develop a language that may help us analyze more complicated models in the future. In particular, we uncover a geometrical framework which contains generalized complex geometry as a special case. Comment: 1+19 pages, JHEP style, published version
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ABSTRACT: This is a review of the relation between supersymmetric non-linear sigma models and target space geometry. In particular, we report on the derivation of generalized K\"ahler geometry from sigma models with additional spinorial superfields. Some of the results reviewed are: Generalized complex geometry from sigma models in the Lagrangian formulation; Coordinatization of generalized K\"ahler geometry in terms of chiral, twisted chiral and semi-chiral superfields; Generalized K\"ahler geometry from sigma models in the Hamiltonian formulation.04/2006;
arXiv:hep-th/0508228v3 15 Mar 2006
Preprint typeset in JHEP style - HYPER VERSION
First-order supersymmetric sigma models
and target space geometry
Andreas Bredthauer∗, Ulf Lindstr¨ om∗,§Jonas Persson∗
∗Department of Theoretical Physics
Box 803, SE-75108 Uppsala
§Helsinki Institute of Physics
P.O. Box 64
FIN-00014 University of Helsinki
Andreas.Bredthauer, Ulf.Lindstrom, Jonas.Persson@teorfys.uu.se
Abstract: We study the conditions under which N = (1,1) generalized sigma
models support an extension to N = (2,2). The enhanced supersymmetry is related
to the target space complex geometry. Concentrating on a simple situation, related
to Poisson sigma models we develop a language that may help us analyze more
complicated models in the future. In particular, we uncover a geometrical framework
which contains generalized complex geometry as a special case.
Keywords: sigma model, supersymmetry, generalized complex geometry.
2. N = (2,2) sigma models, preliminaries3
3. Auxiliary fields and supersymmetry algebra5
4. Almost complex structures on TM ⊕ (T∗M+⊕ T∗M−)7
5. Towards a more general solution11
6. Symplectic sigma model and B-transformation12
7. Manifest supersymmetry and left-/right-chiral superfields14
Supersymmetry has a number of interesting relations to geometry. The analogue of
Minkowski space is superspace, whose geometry is nontrivial even in the ‘flat’ case
. Curved superspace is the setting for supergravity and has a wealth of interesting
geometrical aspects [1, 2, 3]. Superembeddings in curved superspace constrains the
geometry very stringently and in many cases even determines the dynamics of the
embedded super p-branes . Extended supersymmetry is covariantly described in
various extended superspaces with auxiliary degrees of freedom [5, 6, 7]. The target
space of supersymmetric nonlinear sigma models, finally, has to be of a certain type
depending on the dimension and on the number of supersymmetries. It is this latter
situation which concerns us in this paper, more precicely the geometry of twodimen-
sional N = (2,2) supersymmetric nonlinear sigma models with an antisymmetric
– 1 –
In a classic paper  it was shown that the target space of such a sigma model has
to be bi-hermitean, i.e. there are two complex structures preserving the metric and
they are covariantly constant with respect to connections whose torsions are ±dB.
Recently this geometry has been reinterpreted in terms of a generalized complex
geometry, which arose in the context of generalized Calabi-Yau manifolds with B-
field fluxes . In  many aspects of this geometry are investigated and described.
In particular, it is shown that a subclass called generalized K¨ ahler geometry precicely
describes the bi-hermitean geometry.
A natural question to ask is then how generalized K¨ ahler geometry can be directly
realized in a sigma model. Since generalized complex geometry is defined on the sum
of the tangent and cotangent bundles, TM ⊕ T∗M, and the usual sigma model is
defined only on TM, the first task is to find an appropriate extension of the sigma
model to include fields on T∗M. This was done in  where auxiliary spinorial T∗M-
fields were introduced in the N = (1,1) model and the conditions for non-manifest
N = (2,2) supersymmetry investigated under certain assumptions. This investiga-
tion was repeated in , for the case when the metric is absent. Relaxing these
assumptions and limiting the study mainly to extending N = (1,0) to N = (2,0),
a direct relation to generalized complex geometry was found in in most cases .
However, in that investigation it seemed that the geometry in the N = (2,0) case
might be even more general, although the study was incomplete.
To further investigate the geometry, a manifest N = (2,2) model in terms of left
and right (anti-)chiral superfields  was reduced to N = (1,1) superfields and the
generalized complex structures identified in . An interesting aspect of this model
is that the reduction automatically provides the auxiliary spinorial N = (1,1) fields.
In a separate line of investigation [16, 17], it has been shown that generalized com-
plex geometry bears a close relation to the Batalin-Vilkovisky (BV) treatment of
the Poisson sigma model, or more precisely to the Hitchin sigma model. Namely,
the generalized complex geometry implies that the BV-master equation is satisfied.
Also in this case the implication seems to go only in one direction. Generalized com-
plex geometry has also appeared in the sigma model context, e.g. in a hamiltonian
discussion  and for topological strings .
The reason that the investigation of the conditions for N = (2,0) supersymmetry
(and for N = (2,2) supersymmetry) was not carried out  was mainly the technical
complications of having to find solutions to a large number of algebraic and differen-
tial constraints. In the present paper we show how an appropriate field-redefinition
can be used to put the sigma model action in a form where invariance under su-
persymmetry restricts many of the tensors in the supersymmetry transformations
of the fields to vanish. This allows us to completely determine the target space ge-
ometry, at least for the case of vanishing metric, i.e. with only a B-field present.
– 2 –
In doing this we unravel a target space structure where the natural objects live on
TM⊕(T∗M+⊕T∗M−), i.e. the geometry involves two copies of the cotangent bundle
rather than one. Correspondingly all the fundamental geometric objects such as al-
most complex structures, metric and connections have a natural formulation in terms
of 3d×3d matrices. In some respects this structure resembles the bi-hermitean geom-
etry of the second order action (auxiliary fields removed) more than the generalized
K¨ ahler geometry. In particular, the Courant integrability condition of the gener-
alized complex geometry is replaced by covariantly constancy of the matrix-valued
almost complex structures. Now, one of the nice features of generalized complex
geometry is that it naturally puts the so-called b-transform on the same footing as
the diffeomorphisms since they are both automorphisms of the Courant-bracket. It
is thus gratifying that we find that the b-transform can be extended to act on our
matrix-objects, and that this extended b-transform is indeed a gauge transformation
of our basic bundle which preserves the covariantly constancy condition. Finally,
under certain conditions the 3d × 3d matrices collapse to 2d × 2d matrices recov-
ering generalized complex geometry. In other words, the latter is contained in the
structure we have found.
The paper is organized as follows: After a short recapitulation of the basic facts
about N = (2,2) supersymmetric sigma models in section 2, we turn towards a toy
model which we extend to a first order formalism in section 3. For this model, we
give a huge family of solutions for the additional supersymmetry that all close off-
shell. Section 4 is devoted to the development of a proper language that collects the
results in a way similar to the notion of generalized complex geometry. Based on
these results, we discuss in section 5 how to find more general solutions. In section 6,
we show how this relates to the geometry of N = (2,2) symplectic sigma models in
a way that extends the b-transformation. In section 7 we speculate about the role of
manifest N = (2,2) supersymmetry before ending with a short discussion and open
questions in section 8.
2. N = (2,2) sigma models, preliminaries
The action for a N = (1,1) supersymmetric non-linear sigma model under the pres-
ence of a background metric Gµνand an antisymmetric field Bµν
possesses N = (2,2) supersymmetry  provided that the target space geometry is
bi-hermitian. Here, D±are the spinorial derivatives, D2
The additional, non-manifest supersymmetry is given by
±= i∂+ +
=, and Eµν= Gµν+Bµν.
– 3 –
where J(±)are complex structures. The metric is hermitian with respect to both of
them and the complex structures are covariantly constant, i.e.
J(±)2= − ?
N(J(±)) = 0
Here, N(J(±)) is the Nijenhuis torsion for J(±),
The covariant derivatives ∇(±)are given by the connections
that H = dB is related to the complex structures in a certain way.
βγis the metric connection and Tα
2HβγκGκαis the torsion. This implies
The above conditions ensure that the additional supersymmetry commutes with the
first manifest supersymmetry and that its algebra closes on-shell. Off-shell closure is
achieved provided that the two complex structures commute,
[J(+),J(−)] = 0.
This and (2.3) imply that the Magri-Morosi concomitant [20, 21]
vanishes and that both complex structures and the product structure π = J(+)J(−)
are integrable and simultaneously diagonalizable.
While in the previous discussion the metric Gµνplayed a crucial role, we now repeat
the analysis in the case of an antisymmetric background field Bµν only, i.e. we set
Eµν≡ Bµνin the action (2.1) and obtain
Requiring off-shell supersymmetry, we learn that the set of constraints on the trans-
formations (2.2) reduces to
[J(+),J(−)] = 0
N(J(±)) = 0
M(J(+),J(−)) = 0.
H = 0
Thus, the target-space geometry is bicomplex. The condition H = 0 implies that
the model is topological. This is a perfect toy model for our purpose, as we see
it as a first step towards understanding more general sigma models with extended
– 4 –
3. Auxiliary fields and supersymmetry algebra
First order sigma model actions have recently come into the focus of research due
to their relation to generalized complex geometry on the target manifold. While it
is straightforward but lengthy to work out the on-shell supersymmetry transforma-
tions , off-shell supersymmetry is still not really understood in geometrical terms,
partly due to the lack of notation. Several attempts were made to identify those mod-
els that admit or require generalized complex geometry [12, 13, 16, 17, 18, 22, 23].
Here, we follow a different approach to investigate the question of off-shell supersym-
metry. We focus on the action (2.8) and introduce spinorial auxiliary fields S±on
T∗M. They are combined into an auxiliary term added to the action
To keep things simple, we assume that Π is a Poisson tensor of full rank, i.e. it is
symplectic and hence satisfies the Jacobi identity Π[αβρΠρ|γ]= 0.
By dimensional arguments, see e.g. , the most general form of the second super-
symmetry is given by
The action (3.1) is invariant under these transformations provided that
Tαβ= 0 (3.3)
and that a set of differential equations hold. One of these is H = 0. For the time
being, we make the assumption that P(+)and P(−)are invertible. It turns out that
things simplify drastically under this assumption. Indeed, already the commutators
of the second supersymmetry with itself provide 113 conditions to be satisfied. We
comment on the situation for more general P(±)in section 5. Off-shell closure of
the additional supersymmetry algebra is guaranteed if J(±)are commuting complex
structures that are covariantly constant with respect to certain torsionfree connec-
J(±)2= −1[J(+),J(−)] = 0∇(J(±))J(±)= 0
– 5 –
The transformations (3.2) are determined by the composite tensors:
The second rank tensors are ‘covariantly constant’ according to
∇ρP(±)αβ≡ P(±)αβ,ρ+ P(±)ανΓ(K(±))β
The connections are related as
The corresponding Riemann tensors R(·)κλµν= Γ(·)κ
R(R(±))= R(K(±))= R(J(±))= 0.
From the non-derivative parts of the algebra, one constraint remains:
– 6 –
We observe that, except for being covariantly constant, there is no constraint on
P(±). Equations (3.5) imply
The relation (3.12) shows that it is possible, at least in certain situations, to choose
P(±)in such a way that both Z(±)vanish. This requires both complex structures to
commute with ωαβ≡ (P(−)Π−1P(+)t)αβ= −(P(+)Π−1P(−)t)βα. In other words, ω has
to be antihermitian with respect to both complex structures. If, on the other hand,
ω is antisymmetric and in additions satisfies the Jacobi identity then we may identify
its inverse with the two-form of a symplectic manifold. Clearly, G(±)= J(±)ω are
then candidates for effective metrics. One such example is the case P(−)αβ= P(+)αβ.
It follows that R(±)= −K(±), [K(+),K(−)] = 0 and Π is antihermitian with respect
to K(±). However, this alternative is only possible if Π is covariantly constant.
Παβ= Παβ,ρ+ Γ(K)[α
where Γ(K)≡ Γ(K(+))= Γ(K(−)).
This covers the discussion of the second supersymmetry transformations under the
assumptions (3.4) for the particular model we study. Equation (3.9) is sufficient for
off-shell closure. It might not be necessary though we find this quite unlikely due to
the way (3.9) contributes to the solution.
4. Almost complex structures on TM ⊕ (T∗M+⊕ T∗M−)
In the previous section we found that the complete data identifying the solution is
encoded in the objects B, Π, J(±), P(±)and Γ(J). We want a formulation as closely
related as possible to generalized complex geometry [9, 10] and shall try to find a
role for the components of (3.2) in that context. We start with a recapitulation of
the notion of generalized complex geometry.
An almost complex structure is a linear map J : TM → TM that squares to −1. If
we define projection operators π±=1
2(1 ± iJ), then J is integrable if
π∓[π±X,π±Y ] = 0(4.1)
for any X,Y ∈ TM, where [·,·] is the Lie bracket on TM. Hitchin  proposed and
later Gualtieri  investigated a generalization of this notion, where TM is replaced
by TM ⊕ T∗M and the Lie bracket is replaced by the so-called Courant bracket. A
generalized complex structure is defined as a map J : TM ⊕ T∗M → TM ⊕ T∗M,
such that J2= −1 and it leaves the natural symmetric inner product
?X + ξ,Y + η? =1
2(iXη + iYξ)
X + ξ,Y + η ∈ TM ⊕ T∗M
– 7 –
invariant. In a coordinate basis (∂µ,dxµ), the metric
is hermitian with respect to J. Furthermore, the +i eigenbundle of J is closed under
the Courant bracket , which is defined as
[X + ξ,Y + η]C= [X,Y ] + LXη − LYξ −1
2d(iXη − iYξ).
This bracket allows to define Courant integrability as a straightforward generalization
of (4.1). In a coordinate basis, generalized complex structures can be written in terms
of 2d × 2d matrices
An important feature of the Courant bracket is the existence of non-trivial automor-
phisms defined by closed two-forms b ∈ Ω2
complex structure J, we can define a new such structure by the b-transformation
closed(M). Consequently, given a generalized
The automorphism of the Courant bracket guarantees this structure to be integrable.
For a detailed discussion, we refer to the original works [9, 10].
In , the authors constructed examples of sigma models admitting generalized
complex geometry in the target space. Mainly as a curiosity, they found that the
algebraic conditions for closure of the algebra could be combined into a single 3d×3d
matrix squaring to −1. This object seems like a natural extension of the concept of
generalized complex structures. Here, we elaborate this idea in detail and use it as a
basis for the description of the target space geometry. We thus combine the tensors
into two 3d × 3d matrices
The components of these matrices are the linear maps
J(+): TM → TM
L(+): TM → T∗M+
T(+): TM → T∗M− Z(+): T∗M+→ T∗M−
P(+): T∗M+→ TM
K(+): T∗M+→ T∗M+
R(+): T∗M−→ T∗M−.
– 8 –
The components of J(−)are defined analogously. Here, T∗M+and T∗M−are two
copies of the cotangent bundle. They are associated with the two Grassmann direc-
tions on the worldsheet. Thus, J(±)map the bundle E = TM ⊕ (T∗M+⊕ T∗M−)
onto itself. Guided by the action (3.1) we introduce a (degenerate) symmetric inner
product on E, an equivalent to the metric for the ordinary sigma model:
G = Gt=1
We note that G is degenerate because we set E(µν) = 0 in (2.8) and that G is
antisymmetric in the fermionic components. The algebraic conditions arising from
the invariance of the action, eqns. (3.3), and the non-differential part of the algebra
(3.5) can be written in a compact way:
J(±)tGJ(±)= GJ(±)2= −1[J(+),J(−)] = 0.
This allows us to regard J(±)as (almost) complex structures on E. Eqns. (3.8) tell
us that these structures are covariantly constant,
∇(±)J(±)≡ ∂J(±)− J(±)· Γ(±)+ Γ(±)· J(±)= 0(4.11)
with respect to certain connection matrices
and a partial derivative ∂ = 1∂. Equation (3.9) translates into
Γ ≡ Γ(+)= Γ(−).
The components of Γ are torsionfree, Γt= Γ, where the transposition is acting on
the two lower indices, and its Riemann tensor is
R = [∇,∇] = dΓ − Γ ◦ Γ(4.14)
where d = 1d is the generalized exterior derivative. According to (3.10), this matrix
R = 0.
In K¨ ahler geometry, the Nijenhuis torsion and the Levi-Civita connection are related
N(J)(X,Y ) = (∇JXJ)Y − (∇JYJ)X + (∇XJ)JY − (∇YJ)JX,
– 9 –
with X,Y ∈ TM. Clearly, if J is covariantly constant with respect to the Levi-Civita
connection, then N(J) = 0. The generalization to a matrix-valued Nijenhuis torsion
N(J(±)) would make use of ∇ and Γ and hence vanishes if ∇J(±)= 0. Thus, (4.11)
is an integrability condition ensuring the integrability of J(±), K(±)and R(±).
We find that the above description completely covers closure of the supersymmetry
algebra and most of the conditions that arise from the invariance of the action. In
fact, the only condition left is H = 0. We define an antisymmetric tensor by
and define its field strength in the usual way,
H = dB =1
Here, HΠ= d(Π−1) which vanishes in our case, since Π is symplectic. With this, we
H = 0.
There are actually four different possibilities for choosing the two almost complex
structure matrices describing one and the same situation. They are obtained from
(4.7) by acting on J(±)with C(±)= diag(1,∓1,±1) and S = C(+)C(−):
The covariant derivative is changed accordingly, e.g.
This symmetry is reminicent of the discrete symmetries of the first order sigma model
action discussed in, e.g. . The whole discussion may equally well be formulated
in terms of any of these choices.
This completes the discussion of the model (3.1) in this language. However, it is
worth noticing that the geometry of the ordinary second order sigma model (2.1) is
embedded in this framework in a natural way. It corresponds to
G = diag(G,0,0)
B = diag(B,0,0).
Of course, then Γ(+)and Γ(−)are no longer related in the same way, since B generates
torsion in the tangent space directions.
– 10 –
5. Towards a more general solution
One of the main ingredients of the solution given in section 3 is the invertibility
of P(±). This assumption was made because the conditions for the supersymmetry
algebra to close simplified drastically. This helped us to introduce the compact
notation in the previous section. However, the spacetime geometry turned out to
be completely empty, since there is neither a metric nor a three-form field strength.
Here, we elaborate the case where P(±)may have degeneracies. This implies that the
tangent bundle complex structures J(±)are no longer related to the cotangent bundle
ones K(±), R(±)in a unique way. The non-differential conditions for invariance of
the action and closure of the algebra are still ensured by (4.10)
J(±)2= −1[J(+),J(−)] = 0
We observe that the higher order tensors of the solution (3.6) do not depend on
Γ(J(±))but rather on the connections for K(±)and R(±). This allows us to go beyond
flat space in the following way: To stick as close as possible to the solution given in
the previous sections, we start with the assumption that there are two connections
Γ(R(±))such that R(±)are two covariantly constant complex structures. With this,
the solution on the two copies of the cotangent bundle T∗M+⊕ T∗M−remains the
same as before, since
ρν Πσµ+ Πνµ,ρ
and since closure of the algebra requires R(R(±))= R(K(±))= 0. In order for the
higher order tensors to remain defined as in (3.6), we need the further assumption
that there exists A(±)α
Together with the equation
we read off the connection for the tangent bundle Γ(J(±))α
J(±)only has to be covariantly constant in the directions where P(±)is invertible.
On ker(P), we do no longer get any differential conditions and thus, locally, the
tangent space geometry becomes bicomplex. This fits to the original second order
sigma model with a B-field only where we obtained a bicomplex geometry and no
differential conditions for J(±). Especially for P(±)µν= 0, we recover this situation,
as expected, since the supersymmetry transformation for φµdecouples from the aux-
iliary fields S±µ. Since R(J(±))now in general is non-vanishing, we obtain a more
involved geometry of the tangent bundle, while the cotangent bundle does not carry
any additional geometric structure.
ρνand learn that
– 11 –
6. Symplectic sigma model and B-transformation
The N = (2,2) supersymmetric symplectic sigma model action [11, 25]
is obtained from (3.1) by the transformation
S±µ→ˆS±µ= S±µ− ΠµνD±φν
and by identifying Bµν≡ Πµν. To be a bit more general, however, we consider the
We notice that if we take Πµν to be a globally defined two-form, this is precisely
the action used to discuss the WZW term in  with the metric set to zero. By
rewriting the transformations (3.2) in terms of φ andˆS±, we obtain the contributions
to the new tensors, which we denote by a hat to distinguish them from the previous
results. We omit the (±) for a better legibility.
ˆ Mµ[νρ]= ΠµκˆJκ
ˆ Xµνρ= −ΠµσˆJσ
µν− ΠµκPκρν+ N[κρ]
µν+ ΠµσPσρν+ Πµν,σPσρ+ Yρσ
µ = Yνρ
– 12 –
The transformation of J(±)with components (6.4) can be written in a compact way:
ˆ J(±)= UJ(±)U−1
This implies that
ˆ G = (U−1)tGU−1=1
ˆ B = (U−1)tBU−1=1
2(B − Π−1) −1 −1
ˆG is hermitian with respect toˆ J(±),ˆ H = 0 and U is unitary. If we regard (6.2) as a
gauge transformation, that is, an automorphism of the bundle E, then Γ transforms
as a connection and (4.11) is invariant,ˆ∇ˆ J(±)= U∇J(±)U−1= 0. Equations
(6.6) and (6.2) extend the b-transform (4.6) of generalized complex geometry to our
formulation. Hence, it is suggestive to regard (4.11) as an integrability condition. It
is puzzling how to fit in (6.2) in a proper way. Obviously,
ˆΛ ?= UΛΛ = (φ,S+,S−)t
due to the derivatives on φ. In generalized complex geometry, this problem does not
occur, since the fermionic derivative can be included in the definition of Λ. Here,
there are two of them, D±, which complicates the situation. To inspect this in more
detail, we promote the matrices to operators in the following way:
Even if B is antisymmetric, the inner product ?Λ1,Λ2? ≡ Λt
metric in Λ1,Λ2. Accordingly, the almost complex structure matrices become
?Λ2is actually sym-
?(+)= C(+)= diag(1,−1,1).
– 13 –
We introduce the following object:
is given by
?(−)defined correspondingly. With this notation the transformation
? = (?−1)t
? = (?−1)t
The action (3.1) can be written as
?. The supersymmetry transformations become
δ(±)Λ = ǫ±
?(±)Λ + ǫ±Λt
Thus, the matrices C(±)arise here as well. Closure of the algebra reduces to the
[δ1,δ2]Λ = 2ǫ+
2∂+ +Λ + 2ǫ−
It is not difficult to check that this operator formulation works also for (ordinary)
second order sigma models and in the context of generalized complex geometry on
TM ⊕ T∗M.
7. Manifest supersymmetry and left-/right-chiral superfields
There are several ways to construct manifest N = (2,2) sigma models by using
constrained N = (2,2) superfields. The different possibilities are chiral, twisted chiral
and left-/right-chiral ones together with their antichiral partners [8, 15, 26, 27, 28]. To
understand how the latter, originally called semichiral fields, fit into our description
in terms of N = (1,1) manifest supersymmetry, we start with the simple toy model
S = −
– 14 –