Imperial-TP-2024-CH-03

UUITP-05/24

$N=(2,2)$ superfields and geometry revisited
Chris Hulla and Maxim Zabzineb,c

a The Blackett Laboratory, Imperial College London, Prince Consort Road,
London SW7 2AZ, United Kingdom
bDepartment of Physics and Astronomy, Uppsala University,
Box 516, SE-75120 Uppsala, Sweden
c Centre for Geometry and Physics, Uppsala University,
Box 480, SE-75106 Uppsala, Sweden

###### Abstract

We take a fresh look at the relation between generalised Kähler geometry and
$N=(2,2)$ supersymmetric sigma models in two dimensions formulated in terms of
$(2,2)$ superfields. Dual formulations in terms of different kinds of
superfield are combined to give a formulation with a doubled target space and
both the original superfield and the dual superfield. For Kähler geometry, we
show that this doubled geometry is Donaldson’s deformation of the holomorphic
cotangent bundle of the original Kähler manifold. This doubled formulation
gives an elegant geometric reformulation of the equations of motion. We
interpret the equations of motion as the intersection of two Lagrangian
submanifolds (or of a Lagrangian submanifold with an isotropic one) in the
infinite dimensional symplectic supermanifold which is the analogue of phase
space. We then consider further extensions of this formalism, including one in
which the geometry is quadrupled, and discuss their geometry.

###### Contents

  1. 1 Introduction
  2. 2 Kähler Geometry and the $N=2$ Supersymmetric Sigma Model
    1. 2.1 The Kähler Sigma Model and its Dual Formulation
    2. 2.2  Lagrangian System for the Kähler Sigma Model
    3. 2.3 Global Structure
    4. 2.4 Interpretation
    5. 2.5 Solving the linear constraints and extended space
  3. 3 Generalised Kähler geometry and sigma models
    1. 3.1 The Generalised Kähler Sigma Model and its Dual Formulation
    2. 3.2 Lagrangian System for the Generalised Kähler Sigma Model
    3. 3.3 Local symplectic interpretation
  4. 4 Global issues
    1. 4.1 Sigma models with semi-chiral fields only
    2. 4.2 Sigma models with semi-chiral fields and chiral fields
    3. 4.3 Chiral and twisted chiral case
    4. 4.4 Geometric Structure of General Case
  5. 5 Extended geometry
    1. 5.1 Dual superfield presentation
    2. 5.2 Geometric Structure
    3. 5.3 An Extended Formulation
  6. 6 Doubly extended space and superfields
    1. 6.1 Model with semi-chiral superfields only
    2. 6.2 General case
  7. 7 Summary and outlook
  8. A Appendix: $N=(1,1)$ and $N=(2,2)$ superspace conventions
  9. B Appendix: Generalised Kähler geometry, gerbes and glueings

##  1 Introduction

> Superfields are smarter than we are!
>
> Martin Roček told us this many times

Two dimensional $N=(2,2)$ supersymmetric non-linear sigma models play a
prominent role in string theory and in mathematical physics, with a beautiful
relation between supersymmetry and the geometry of the target space. Zumino
[1] was the first to realise, in 1979, that $N=(2,2)$ supersymmetry constrains
the target space of a non-linear sigma-model to be Kähler. Chiral superfields
played a prominent role and the $N=(2,2)$ Lagrangian density was identified
with the Kähler potential. In [2], this was generalised to the case of sigma-
models with Wess-Zumino (WZ) term and it was found that the general $N=(2,2)$
models were described in terms of a bihermitian geometry. This geometry is now
called generalised Kähler geometry. In [2], a $N=(2,2)$ superfield formulation
was found for a special class of generalised Kähler geometries (those with
commuting complex structures that were simultaneously integrable) and it
involved both chiral and twisted-chiral superfields. A further kind of
superfield – the semichiral superfield – was introduced in [3] and in [4] the
full $N=(2,2)$ superfield description of generalised Kähler geometry was found
(subject to a certain regularity assumption) and this superfield description
was given a geometric realisation in terms of local coordinates (which are
associated with the various superfields). For this general model, the
$N=(2,2)$ Lagrangian density is given in terms of single function, the
generalised Kähler potential, which encodes all local geometry.

In parallel, there were interesting developments in mathematics, starting with
the work of Hitchin [5] who introduced the notion of generalised complex
structure. Gualtieri, in his PhD thesis [6], introduced the notion of
generalised Kähler geometry and showed that this is the same as the
bihermitian geometry of [2] (see also [7]). Later, a global interpretation of
the generalised Kähler potential was given in [8] for the case of a
generalised Kähler structure of symplectic type.

In this paper, we return to the superfield description of the $N=(2,2)$ sigma
model and find new geometrical structure. Strikingly, it is the superfield
structure that suggests the new geometry.

We start with the Kähler geometry of the $N=(2,2)$ sigma-model without WZ
term. As well as the usual formulation in terms of chiral superfields, there
is a dual formulation in terms of complex linear superfields.111As we will
discuss later, there is, strictly speaking, only a full global dual
formulation for a special class of target spaces. In one formulation, the
coordinates appear as the lowest component of chiral superfields, in the other
as the lowest component of complex linear superfields. We reformulate the
general Kähler sigma-model in a ‘doubled target space’ with both chiral
superfields and complex linear superfields. We then impose a relation between
the two types of superfield such that the superspace constraint on either one
implies the field equation of the other. This gives an elegant set of
equations that captures the classical structure of the model. Moroever, for
those target spaces for which there are two dual formulations, this makes the
duality manifest. We show that the geometry of the doubled space is that of
Donaldson’s deformation of the holomorphic cotangent bundle of the original
Kähler manifold, while the relation between the two types of superfield
constructs the original space as a Lagrangian submanifold of the deformed
cotangent bundle.

Next, we lift this structure to the space of unconstrained maps from $(2,2)$
superspace to the deformed cotangent bundle, which might be thought of as a
kind of phase space. We show that this has a holomorphic symplectic structure.
We show that the superspace constraints correspond to restricting to a
holomorphic isotropic submanifold, so that the sigma model is specified by the
intersection of an isotropic submanifold and a Lagrangian submanifold. We also
give an extended formulation in which thesubmanifold specified by the
superfield constraints is also Lagrangian.

We then extend this formulation to the case of $N=(2,2)$ sigma-models with
Wess-Zumino term and their generalised Kähler geometry. We find an enlarged
formulation in which, as well as the chiral, twisted-chiral and semichiral
superfields, we include the dual superfields. The model is specified by a
constraint that selects a submanifold of this enlarged space and implies the
field equations. We discuss the symplectic geometry of this and show that the
model can be formulated as the intersection of two submanifolds of the space
of supermaps to this enlarged phase space. (These submanifolds are either both
Lagrangian, or one is Lagrangian and one is isotropic, depending on the choice
of extended space.) We discuss several different enlarged spaces, each of
which gives an extended formulation of the original sigma models.

In our discussion, we combine the superfield description from [4] with ideas
coming from dualities [9] and global considerations from [10]. The superfield
formalism imposes a very rigid structure on field transformations both at
local and global levels and as a result constrains the kind of transition
functions that are permitted. The gerbe structure found in [10] underpins the
geometry, and we find that in some cases it leads to interesting spaces that
might be considered as generalisations of fibre bundles. It will be
interesting to compare all this with the global discussion in [8]. We think
that the ideas we set out here can be further developed and generalised beyond
the present paper.

The paper is organised as follows. In section 2 we review Kähler geometry and
the formulation of the corresponding sigma model in terms of chiral
superfields and the dual formulation in terms of complex linear superfields.
We then present a doubled formulation in terms of both kinds of superfields
and show that the doubled target space is Donaldson’s deformation of the
holomorphic contangent bundle. In section 3 we find a similar doubled
formulation of the general $N=(2,2)$ sigma model with WZ term; in this
section, we present only a local analysis. In section 4 we discuss the global
issues for this construction. Sections 5 and 6 extend the phase space further
in constructions dictated by the superfield formalism. In Section 5 we
consider a construction in which we quadruple some fields and double others,
while in Section 6 we offer a democratic formulation in which all fields are
quadrupled. Finally, in section 7 we summarise our results and outline open
problems. In two appendices we collect some basic properties of the $N=(2,2)$
superfield formalism and review some features of generalised Kähler geometry
and its gerbe structure.

##  2 Kähler Geometry and the $N=2$ Supersymmetric Sigma Model

In this section we consider the well known case of Kähler geometry and the
corresponding $N=(2,2)$ supersymmetric sigma models. We first review the local
structure of the sigma model and its dual formulation. We then provide a
system of equations combining the equations of motion and the superspace
constraints in a unified way. This system has a manifest duality that
interchanges field equations with constraints. The equations govern
superfields taking values in a ‘doubled space’, with the original space
arising as a Lagrangian submanifold. We show that this doubled space is
Donaldson’s deformed cotangent bundle and this facilitates the discussion of
global issues. Finally, we propose a new geometric interpretation of the
system in an infinite-dimensional setting.

###  2.1 The Kähler Sigma Model and its Dual Formulation

The $N=(2,2)$ sigma model with Kähler target $M$ is a theory of maps

${\cal N}=\\{\phi:\mathbb{R}^{2|4}\longrightarrow{M}\\}$ (2.1)

from $N=(2,2)$ superspace $\mathbb{R}^{2|4}$ with coordinates
$(x^{m},\theta^{\alpha})$ (where $\alpha=+,-$ are spinor indices and
$\theta^{\alpha}$ are complex) to the manifold $M$ with local complex
coordinates $(\phi^{a},\bar{\phi}^{\bar{a}})$. The maps are locally specified
by coordinate maps $\phi^{a}(x^{m},\theta^{\alpha})$ which are constrained to
be chiral

$\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},~{}~{}~{}~{}~{}~{}\mathbb{D}_{\pm}\bar{\phi}^{\bar{a}}=0$
(2.2)

and the action is

$S=\int d^{2}x~{}d^{4}\theta~{}K(\phi^{a},\bar{\phi}^{\bar{a}})~{},$ (2.3)

where the real function $K(\phi^{a},\bar{\phi}^{\bar{a}})$ is the Kähler
potential. See Appendix A for details of our superspace conventions and
notation.

The sigma-model has a remarkable dual formulation in terms of complex linear
superfields, given in the superspace textbook [9] for the four-dimensional
sigma model (see the discussion around eqs (4.5.10a,b)) and for the two-
dimensional sigma model in [11]. First, note that the action (2.3) can be
rewritten in a first-order form as

$S_{\rm dual}=\int
d^{2}x~{}d^{4}\theta~{}\Big{[}K(\Phi^{b},\bar{\Phi}^{\bar{b}})-\Sigma_{a}\Phi^{a}-\bar{\Sigma}_{\bar{a}}\bar{\Phi}^{\bar{a}}\Big{]}~{},$
(2.4)

where $\Phi^{b}$ are unconstrained complex superfields and $\Sigma_{a}$ are
complex linear superfields satisfying the constraint

$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0$ (2.5)

($\bar{\Sigma}_{\bar{a}}$ are the complex conjugate superfields). The field
equation for $\Sigma_{a}$ imposes the chirality constraint
$\bar{\mathbb{D}}_{\pm}\Phi^{a}=0$ so that integrating out the $\Sigma_{a}$’s
recovers the original action (2.3) (on identifying the constrained $\Phi^{a}$
with the chiral superfields $\phi^{a}$). On the other hand, the field equation
for $\Phi^{a}$ is

$\Sigma_{a}=\frac{\partial K}{\partial\Phi^{a}}$ (2.6)

giving $\Sigma_{a}$ as a function of $\Phi^{a},\bar{\Phi}^{\bar{a}}$. If this
can be inverted to give $\Phi^{a}$ as a function of
$\Sigma_{a},\bar{\Sigma}_{\bar{a}}$, then integrating out
$\Phi^{a},\bar{\Phi}^{\bar{a}}$ sets $\Phi^{a}$ to be the function
$\Phi^{a}(\Sigma_{a},\bar{\Sigma}_{\bar{a}})$ giving the dual formulation

$S=\int
d^{2}x~{}d^{4}\theta~{}\widetilde{K}(\Sigma_{a},\bar{\Sigma}_{\bar{a}})~{},$
(2.7)

where $\widetilde{K}$ is the Legendre transform of $K$:

$\widetilde{K}=K(\Phi^{b},\bar{\Phi}^{\bar{b}})-\Sigma_{a}\Phi^{a}-\bar{\Sigma}_{\bar{a}}\bar{\Phi}^{\bar{a}}~{},$
(2.8)

and (2.6) is inverted and used to write the RHS as a function of
$\Sigma_{a},\bar{\Sigma}_{\bar{a}}$. For this to be a good duality requires
that the function $\Sigma_{a}(\Phi^{a},\bar{\Phi}^{\bar{a}})$ is invertible
and that the Legendre function is well-defined (not multivalued) and so in
particular needs $K$ to be a convex function. The general case in which it is
not convex is interesting and will be discussed elsewhere.

Now the dual action (2.7) can itself be written in a first-order form as

$\widetilde{S}_{\rm dual}=\int
d^{2}x~{}d^{4}\theta~{}\Big{[}\widetilde{K}(\Phi_{b},\bar{\Phi}_{\bar{b}})-\phi^{a}\Phi_{a}-\bar{\phi}^{\bar{a}}\bar{\Phi}_{\bar{a}}\Big{]}~{},$
(2.9)

where the $\Phi_{b}$ are complex unconstrained superfields and the $\phi^{a}$
are chiral superfields. The $\phi$ equation of motion imposes the constraint
that $\Phi_{a}$ is a complex linear superfield,
$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Phi_{a}=0$, so that integrating out
$\phi$ recovers the action (2.7) (on identifying the constrained $\Phi_{a}$
with the complex linear superfield $\Sigma_{a}$). On the other hand, the
$\Phi$ field equation gives

$\phi^{a}=\frac{\partial\widetilde{K}}{\partial\Sigma_{a}}~{}.$ (2.10)

Integrating out the fields $\Phi_{a}$ requires solving this to determine
$\Sigma$ as a function of $\phi^{a},\bar{\phi}^{\bar{a}}$ and substituting in
the action (2.9) to arrive back at the original action (2.3) (for this to be
well-defined, $\widetilde{K}$ should be convex).

###  2.2  Lagrangian System for the Kähler Sigma Model

The equations of motion following from the action (2.4) together with the
superfield constraints can be recast as the following system of equations

$\boxed{\Sigma_{a}=\frac{\partial
K}{\partial\phi^{a}}~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},}$
(2.11)

where $K(\phi,\bar{\phi})$ is the Kähler potential. Indeed, the first two
equations combine to give

${\bar{\mathbb{D}}}^{2}\frac{\partial K}{\partial\phi^{a}}=0~{},$ (2.12)

which are precisely the equations of motion following from (2.3).
Alternatively, the equations of motion derived from the dual action (2.9) can
be recast as the following system

$\boxed{\phi^{a}=\frac{\partial\widetilde{K}}{\partial\Sigma_{a}}~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},}$
(2.13)

where $\widetilde{K}(\Sigma,\bar{\Sigma})$ is the dual potential. Then the
first and last equation combine to give the field equation for $\Sigma$.

For both systems (2.11) and (2.13) we have maps from superspace to a space
with complex coordinates $(\phi^{a},\Sigma_{a})$ satisfying the constraints
$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0$,
$\bar{\mathbb{D}}_{\pm}\phi^{a}=0$ and the model is defined by restricting to
a subspace defined by the $d$ complex equations (for a manifold of complex
dimension $d$) $\Sigma_{a}=\frac{\partial K}{\partial\phi^{a}}$ or
$\phi^{a}=\frac{\partial\widetilde{K}}{\partial\Sigma_{a}}$.

The two systems of equations (2.11) and (2.13) are equivalent provided that
the Legendre transform between $K$ and $\widetilde{K}$ is well-defined. This
duality between the two systems interchanges constraints with field equations.
In the first system (2.11), we can regard $\phi$ as the fundamental field with
$\bar{\mathbb{D}}_{\pm}\phi^{a}=0$ viewed as a constraint and the other two
equations combining to give the field equation for $\phi$ that follows from
the action (2.3). For the dual system (2.13) we can regard $\Sigma$ as the
fundamental field with $\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0$
viewed as a constraint and the other two equations combining to give the field
equation for $\Sigma$ that follows from the action (2.7).

For example, in the simple case of a flat target space, the Kähler potential
is quadratic, $K=\frac{1}{2}\sum_{a}\phi^{a}\bar{\phi}^{\bar{a}}$. Then
$\Sigma_{a}=\frac{\partial K}{\partial\phi^{a}}$ gives
$\Sigma_{a}=\bar{\phi}^{\bar{a}}$ so that the complex linear constraint on
$\Sigma$, which is $\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0$,
gives the field equation for a free chiral superfield,
$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\bar{\phi}^{\bar{a}}=0$. The dual
potential is
$\widetilde{K}=\frac{1}{2}\sum_{a}\Sigma_{a}\bar{\Sigma}_{\bar{a}}$ so that we
again obtain $\Sigma_{a}=\bar{\phi}^{\bar{a}}$ and now the chiral constraint
on $\phi$ implies the complex linear constraint on $\Sigma$, i.e.
$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0$.

This duality becomes rather subtle if $K$ is not convex as then the Legendre
transform $\widetilde{K}$ becomes multivalued and may even be singular. We
will not discuss this duality further here and will return to this subject
elsewhere. Instead, we will focus here on the system (2.11) which is defined
(locally) for any Kähler manifold and not assume the existence of a dual
potential $\widetilde{K}$. We now turn to the global formulation of this
system for any Kähler manifold.

###  2.3 Global Structure

We now look at the global structure of the system (2.11). First we discuss the
symmetries of the equations (2.11). The chirality conditions allow the field
redefinitions $\phi^{a}\rightarrow{\phi^{\prime}}^{a}(\phi)$ corresponding to
a holomorphic change of complex coordinates on $M$. Then
$\Sigma_{a}=\frac{\partial K}{\partial\phi^{a}}$ requires that the linear
complex superfield transforms as

$\Sigma_{a}\rightarrow\Sigma^{\prime}_{a}=\frac{\partial\phi^{b}}{\partial{\phi^{\prime}}^{a}}~{}\Sigma_{b}~{}.$
(2.14)

Note that this is compatible with the superfield structure (see Appendix): if
$\Sigma_{a}$ satisfies the complex linear constraint
$\bar{\mathbb{D}}^{2}\Sigma_{a}=0$, then so does $\Sigma^{\prime}_{a}$. Then
$\Sigma_{a}$ transforms like a holomorphic 1-form and $(\phi,\Sigma)$
transform under changes of coordinates in the same way as the holomorphic
coordinates of the cotangent bundle. However, although it is tempting to
suppose that $(\phi,\Sigma)$ correspond to complex coordinates on the
cotangent bundle $T^{*}M$, there is a subtlety related to the way the
coordinates $\Sigma$ are glued.

The superspace action is unchanged if the Kähler potential is transformed by

$K\to K^{\prime}=K+f(\phi)+\bar{f}(\bar{\phi})$ (2.15)

for any holomorphic function $f(\phi)$. This is sometimes referred to as a
Kähler gauge transformation. For this to be consistent with
$\Sigma_{a}=\frac{\partial K}{\partial\phi^{a}}$, it is necessary that
$\Sigma_{a}$ transforms as

$\Sigma_{a}\to\Sigma^{\prime}_{a}=\Sigma_{a}+\frac{\partial}{\partial\phi^{a}}f(\phi)$
(2.16)

so that $\Sigma$ can be viewed as a connection one-form for these
transformations. These transformations are important for the global structure,
as the glueing between patches is through a Kähler gauge transformation
composed with a holomorphic diffeomorphism.

For an open cover of the Kähler manifold with (contractible) open sets
$U_{\alpha}$ the Kähler structure leads to a locally-defined Kähler potential
$K_{\alpha}$ on each patch $U_{\alpha}$. On the intersection of two patches
$U_{\alpha}\cap U_{\beta}$ they are glued by a Kähler gauge transformation

$K_{\alpha}(\phi,\bar{\phi})-K_{\beta}(\phi,\bar{\phi})=f_{\alpha\beta}(\phi)+\bar{f}_{\alpha\beta}(\bar{\phi})~{}$
(2.17)

for some holomorphic function $f_{\alpha\beta}$ on $U_{\alpha}\cap U_{\beta}$.
This leads to a well-defined superspace action as the right hand side vanishes
identically under the superspace integral. To preserve the condition
$\Sigma_{a}=\frac{\partial K}{\partial\phi^{a}}$ we postulate the following
glueing of $\Sigma$’s on the intersections

$(\Sigma_{a})_{\beta}~{}=~{}(\Sigma_{a})_{\alpha}+\frac{\partial}{\partial\phi^{a}}f_{\alpha\beta}(\phi)~{}~{}~{}~{}~{}{\rm
on}~{}U_{\alpha}\cap U_{\beta}~{},$ (2.18)

which is again compatible with superfield constraints. The full glueing
conditions combine these with a holomorphic diffeomorphism from the
coordinates in $U_{\alpha}$ to those in $U_{\beta}$. In (2.17),(2.18) we
suppress the diffeomorphism part of the transition function; this can be
thought of as expressing all quantities in the relation in terms of the same
coordinate system.222We will similarly the suppress the diffeomorphism part of
glueing conditions throughout this paper.

The structure here is a deformation of the holomorphic cotangent bundle and we
follow Donaldson’s construction of this [12] (see also [8]). With local
complex coordinates $(z^{a}_{\alpha},p_{a\alpha})$ on $T^{*}U_{\alpha}$, we
define a 1-form $p_{\alpha}=p_{a\alpha}dz_{\alpha}^{a}$ on each patch
$U_{\alpha}$ with the holomorphic affine glueing relation for the fibres over
$U_{\alpha}\cap U_{\beta}$

$p_{\beta}=p_{\alpha}+\partial f_{\alpha\beta}~{},$ (2.19)

which satisfy the cocycle condition. Expressing all terms in the same
coordinate system, this can be written as

$(p_{a})_{\beta}~{}=~{}(p_{a})_{\alpha}+\partial_{a}f_{\alpha\beta}(z)~{}.$
(2.20)

The total space is constructed from patches of the form $T^{*}U_{\alpha}$
glued together using these transition functions. That is, we can define the
holomorphic symplectic affine bundle $({\cal Z},\omega^{(2,0)})$ as

${\cal Z}=\Big{(}\coprod\limits_{\alpha}T^{*}U_{\alpha}\Big{)}/\sim$ (2.21)

with the equivalence relation $p_{\alpha}\sim p_{\beta}$ defined by (2.20).
The bundle ${\cal Z}$ inherits a canonical holomorphic symplectic structure
from $T^{*}M$

$\omega^{(2,0)}=dp_{a}\wedge dz^{a}~{},$ (2.22)

which is well-defined under the identification. In other words, the
transformations between patches (2.20) are symplectomorphisms and so the
symplectic structure $\omega^{(2,0)}$ is globally defined on ${\cal Z}$. The
choice of Kähler potential is encoded in the choice of a global (non-
holomorphic) section ${\cal L}:M\rightarrow{\cal Z}$ that is given by

$p_{a}dz^{a}=\partial K~{}.$ (2.23)

This equation specifies a globally defined submanifold ${\cal L}$ of ${\cal
Z}$ that is Lagrangian with respect to ${\rm Re}(\omega^{(2,0)})$ and
symplectic with respect to ${\rm Im}(\omega^{(2,0)})$. For a more detailed
discussion we refer the reader to [12] and [8].333Here and in the rest of the
paper we follow conventions which allow us minimise the appearance of factors
of $i$ in our formulae. In some other works different conventions are used in
which the roles of ${\rm Re}(\omega^{(2,0)})$ and ${\rm Im}(\omega^{(2,0)})$
are interchanged. We use the same letter for the section and the Lagrangian
submanifold since they are two ways of viewing the same submanifold.

###  2.4 Interpretation

The system (2.11) is then one of supermaps
$\mathbb{R}^{2|4}\longrightarrow{\cal L}$ with $p(x,\theta,\bar{\theta})$
satisfying the complex linear constraints and $z(x,\theta,\bar{\theta})$
satisfying the chiral constraints. Let us reformulate this observation in
geometrical terms.

We introduce the infinite dimensional space of unconstrained supermaps from
$\mathbb{R}^{2|4}$ to the holomorphic symplectic affine bundle

${\cal M}=\\{\Phi:\mathbb{R}^{2|4}\longrightarrow{\cal Z}\\}$ (2.24)

with local holomorphic Darboux coordinates $\Phi^{a},\Phi_{a}$ on ${\cal Z}$
with $\Phi^{a}$ corresponding to complex coordinates on $M$ (referred to as
$z^{a}$ in the last subsection) while $\Phi_{a}$ are complex coordinates along
the fibre (referred to as $p_{a}$ in the last subsection). On the space of
unconstrained supermaps given locally by
$\Phi^{a}(x,\theta,\bar{\theta}),\Phi_{a}(x,\theta,\bar{\theta})$, we denote
the exterior derivative by $\delta$ so that the basic 1-forms are
$\delta\Phi^{a},\delta\Phi_{a}$ and use $\wedge$ for the wedge product. Then
this infinite dimensional space is equipped with the homolorphic symplectic
structure

$\Omega=\int d^{2}x~{}d^{4}\theta~{}\delta\Phi_{a}\wedge\delta\Phi^{a}~{},$
(2.25)

which arises from the holomorphic symplectic structure $\omega^{(2,0)}$ (2.22)
on ${\cal Z}$.

The main idea here is that the system of equations (2.11) can be interpreted
as the intersection of two infinite dimensional Lagrangian submanifolds of
${\cal M}$. A Lagrangian submanifold is isotropic (i.e. the symplectic
structure restricted to the submanifold vanishes) and satisfies a maximality
condition. In the finite dimensional setting, a Lagrangian submanifold of a
manifold of dimension $2d$ is an isotropic submanifold with dimension $d$. In
the infinite dimensional setting, a Lagrangian subspace of a symplectic vector
space is an isotropic subspace whose symplectic compliment that is also
isotropic. The extension of this to infinite dimensional manifolds is subtle,
but intuitively the maximality of a Lagrangian submanifold means that if one
tries to enlarge the space then the property of being isotropic is lost.

The first equation (and its complex conjugate) in (2.11) corresponds to

$\Phi_{a}=\frac{\partial K}{\partial\Phi^{a}}~{},$ (2.26)

which defines a real Lagrangian submanifold of ${\cal M}$ with respect to
${\rm Re}(\Omega)$ (here $K$ is a real function of
$(\Phi^{b},\bar{\Phi}^{\bar{b}})$). This infinite dimensional Lagrangian
submanifold is induced by a finite dimensional Lagrangian submanifold ${\cal
L}$ of ${\cal Z}$. $K$ can be promoted to become the generating function of an
infinite dimensional space, as we show in next subsection.

On the other hand, the superfield constraints in (2.11) specify a holomorphic
isotropic submanifold with respect to $\Omega$. To see that this submanifold
is isotropic we evaluate the holomorphic symplectic form on the subspace in
which the superfields are constrained to obtain

$\Omega|_{\bar{\mathbb{D}}_{\pm}\Phi^{a}=0,~{}\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Phi_{a}=0}=\int
d^{2}x~{}d^{4}\theta~{}\delta\Phi_{a}\wedge\delta\Phi^{a}=0~{},$ (2.27)

where we have used the representation (A.16) for these constrained
superfields. The integral vanishes identically (possibly up to boundary terms,
but we assume boundary conditions that eliminate these). For example, (A.16)
implies that $\delta\Phi^{a}$, which is chiral on the constrained submanifold,
can be written as $\delta\Phi^{a}=\bar{\mathbb{D}}^{2}W^{a}$ for some $W^{a}$,
so that on integrating by parts the integrand in (2.27) becomes
$W^{a}\wedge\bar{\mathbb{D}}^{2}\delta\Phi_{a}$ which vanishes on the
submanifold as $\delta\Phi_{a}$ satisfies the complex linear constraints
there. Thus we see that these superfield constraints correspond to restricting
to a holomorphic isotropic submanifold of ${\cal M}$. In the discussion below
we show that the corresponding submanifold in an extended space is actually
holomorphic Lagrangian.

###  2.5 Solving the linear constraints and extended space

The complex linear constraint

$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0$ (2.28)

is solved by

$\Sigma_{a}=\bar{\mathbb{D}}_{\alpha}\Psi^{\alpha}_{a}$ (2.29)

for some unconstrained spinor superfields
$\Psi^{\alpha}_{a}=(\Psi^{+}_{a},\Psi^{-}_{a})$. Then the first-order action
(2.4) can be rewritten as

$S_{\rm dual}=\int
d^{2}x~{}d^{4}\theta~{}\Big{[}K(\Phi^{b},\bar{\Phi}^{\bar{b}})+\Psi^{\alpha}_{a}\bar{\mathbb{D}}_{\alpha}\Phi^{a}+\bar{\Psi}^{\alpha}_{\bar{a}}{\mathbb{D}}_{\alpha}\bar{\Phi}^{\bar{a}}\Big{]}~{}.$
(2.30)

The superfields $\Psi^{\alpha}_{a}$ are lagrange multipliers imposing the
chirality constraint $\bar{\mathbb{D}}_{\pm}\Phi^{a}=0$. This can be written
as

$S_{\rm dual}={\cal K}+2{\rm Re}({\cal H})~{},$ (2.31)

where

${\cal K}=\int d^{2}x~{}d^{4}\theta~{}K(\Phi^{a},\bar{\Phi}^{\bar{a}})$ (2.32)

and

${\cal H}=\int
d^{2}x~{}d^{4}\theta~{}\Psi^{\alpha}_{a}\bar{\mathbb{D}}_{\alpha}\Phi^{a}=\int
d^{2}x~{}d^{4}\theta~{}(-i\Psi_{-a}\bar{\mathbb{D}}_{+}\Phi^{a}+i\Psi_{+a}\bar{\mathbb{D}}_{-}\Phi^{a})~{}.$
(2.33)

Note that the fields $\Psi^{\alpha}_{a}$ transform as (1,0) forms under
holomorphic coordinate transformations for $M$ so that this action is
covariant.

We now return to the geometrical setup. We can enlarge the Kähler target space
$M$ with the coordinates $(z^{a},\bar{z}^{\bar{a}})$ by introducing odd
additional coordinates $(\psi_{+a},\bar{\psi}_{+\bar{a}})$ and
$(\psi_{-a},\bar{\psi}_{-\bar{a}})$ for two copies of the fibre of the odd
cotangent bundle, so that we have $(\Pi T^{*}\oplus\Pi T^{*})M$. (Here we
follow the standard notation in which $\Pi$ indicates the parity-reversed
fibre.) Next we introduce the cotangent bundle to this supermanifold

$T^{*}\Big{(}(\Pi T^{*}\oplus\Pi T^{*})M\Big{)}$ (2.34)

and denote new fibre coordinates by $(p_{a},\psi_{+}^{a},\psi_{-}^{a})$
(together with their complex conjugates). This supermanifold is equipped with
the canonical holomorphic symplectic structure

$\omega^{\prime(2,0)}=dp_{a}\wedge dz^{a}-id\psi^{a}_{+}\wedge
d\psi_{-a}+id\psi^{a}_{-}\wedge d\psi_{+a}~{}.$ (2.35)

Here the odd coordinates $\psi_{\pm}^{a}$ transform as fibre coordinates for
the odd tangent bundle. The transformations of $p_{a}$ are a bit more
complicated, the $p_{a}$ transform as section of $T^{*}M$ plus an additional
term which is quadratic in $\psi$. This supermanifold has been studied
explicitly and the detailed formulae for the transformation of all
coordinates, together with other properties, can be found in [13] (e.g. see
Remark 3.3.2 in [13] for explicit formulae in the real case). Next we deform
this contangent bundle in the same way as described in subsection 2.3 using
the identification (2.20) for the even coordinate $p_{a}$. Thus we obtain the
deformed cotangent bundle which is now the supermanifold ${\cal Z}^{\prime}$
and is equipped with the holomorphic symplectic structure given again by
formula (2.35) (in Darboux coordinates).

We now extend the discussion of the previous subsection. Instead of using the
target space ${\cal Z}$ we work with the supermanifold ${\cal Z}^{\prime}$ .
We introduce the infinite dimensional space of supermaps from
$\mathbb{R}^{2|4}$ to ${\cal Z}^{\prime}$

${\cal M}^{\prime}=\\{\mathbb{R}^{2|4}\longrightarrow{\cal Z}^{\prime}\\}$
(2.36)

with superfields $(\Phi^{a},\Phi_{a},\Psi^{a}_{\pm},\Psi_{\pm a})$ (plus their
complex conjugates). Here the superfields $\Phi^{a}$ are associated with
coordinates $z^{a}$, the superfields $\Phi_{a}$ are associated with $p_{a}$,
the superfields $\Psi_{\pm}^{a}$ are associated with $\psi_{\pm}^{a}$ and
finally the superfields $\Psi_{\pm a}$ are associated with $\psi_{\pm a}$. The
transformation rules for the superfields follow from those of the coordinates
of the supermanifold ${\cal Z^{\prime}}$.

The space ${\cal M}^{\prime}$ is equipped with the holomorphic symplectic form

$\Omega_{\rm large}=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}\delta\Phi_{a}\wedge\delta\Phi^{a}-i\delta\Psi^{a}_{+}\wedge\delta\Psi_{-a}+i\delta\Psi^{a}_{-}\wedge\delta\Psi_{+a}\Big{)}~{}.$
(2.37)

(Note that this formalism with an enlarged phase space has some analogies to
the BV/BRST formalism.) We now use generating functions to define Lagrangian
submanifolds. We define a holomorphic Lagrangian submanifold of ${\cal
Z}^{\prime}$ through the holomorphic generating function of
$(\Phi^{a},\Psi_{+a},\Psi_{-a})$ given by ${\cal H}$ defined in (2.33) which
gives

$\Phi_{a}=\frac{\delta{\cal
H}}{\delta\Phi^{a}}=i\bar{\mathbb{D}}_{+}\Psi_{-a}-i\bar{\mathbb{D}}_{-}\Psi_{+a}~{},~{}~{}~{}~{}\Psi_{+}^{a}=\frac{\delta{\cal
H}}{\delta\Psi_{-a}}=\bar{\mathbb{D}}_{+}\Phi^{a}~{},~{}~{}~{}~{}\Psi_{-}^{a}=\frac{\delta{\cal
H}}{\delta\Psi_{+a}}=\bar{\mathbb{D}}_{-}\Phi^{a}~{}.$ (2.38)

A real Lagrangian submanifold with respect to ${\rm Re}(\Omega_{\rm large})$
can be defined using the real generating function of
$(\Phi^{a},\Psi_{+a},\Psi_{-a})$ given by ${\cal K}$ in (2.32) so that we have

$\Phi_{a}=\frac{\delta{\cal K}}{\delta\Phi^{a}}=\frac{\partial
K}{\partial\Phi^{a}}~{},~{}~{}~{}\Psi_{+}^{a}=\frac{\delta{\cal
K}}{\delta\Psi_{-a}}=0~{},~{}~{}~{}~{}\Psi_{-}^{a}=\frac{\delta{\cal
K}}{\delta\Psi_{+a}}=0~{}.$ (2.39)

Remarkably, combining (2.38) and (2.39) and using (2.29) we arrive at the
system of equations (2.11). We interpret the equations of motion as the
intersection of a real Lagrangian submanifold with a holomorphic Lagrangian
submanifold. By enlarging the space from ${\cal M}$ to ${\cal M}^{\prime}$ we
are able to encode both of these submanifolds in terms of generating
functions.

##  3 Generalised Kähler geometry and sigma models

Next we generalise the discussion of the last section to the case of the
general $N=(2,2)$ non-linear sigma model with target space a generalised
Kähler manifold. These are manifolds equipped with two complex structures,
$I_{+}$ and $I_{-}$, and a metric that is hermitian with respect to both of
them. They are also equipped with a closed 3-form $H$; see [7] for details of
the geometry. In this section we will focus on the local structure and we
postpone the discussion of global geometrical issues until the next section.

###  3.1 The Generalised Kähler Sigma Model and its Dual Formulation

The general $N=(2,2)$ non-linear sigma model can be written in terms of four
types of $N=(2,2)$ superfields [4]. These are the chiral and anti-chiral
fields

$\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},~{}~{}~{}~{}~{}~{}\mathbb{D}_{\pm}\bar{\phi}^{\bar{a}}=0~{},$
(3.1)

the twisted chiral and twisted anti-chiral fields

$\bar{\mathbb{D}}_{+}\chi^{a^{\prime}}=0~{},~{}~{}~{}\mathbb{D}_{-}\chi^{a^{\prime}}=0~{},~{}~{}~{}\mathbb{D}_{+}\bar{\chi}^{\bar{a}^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{-}\bar{\chi}^{\bar{a}^{\prime}}=0~{},$
(3.2)

the left semichiral fields

$\bar{\mathbb{D}}_{+}X^{n}_{L}=0~{},~{}~{}~{}\mathbb{D}_{+}\bar{X}_{L}^{\bar{n}^{\prime}}=0~{},$
(3.3)

and finally the right semichiral fields

$\bar{\mathbb{D}}_{-}X_{R}^{n^{\prime}}=0~{},~{}~{}~{}\mathbb{D}_{-}\bar{X}_{R}^{\bar{n^{\prime}}}=0~{}.$
(3.4)

Each superfield has a bosonic component that is interpreted as one of the
coordinates of the target space, so the split into four kinds of superfields
corresponds to a split of the coordinates into four different kinds. This is
always possible locally, and will be possible globally if we assume that all
relevant Poisson structures are regular [4]; we shall assume that this is the
case for the rest of the paper. Then the coordinates are
$\varphi^{A}=(\phi^{a},\chi^{a^{\prime}},X^{n}_{L},X_{R}^{n^{\prime}})$ plus
their complex conjugates and the indices split as
$A=(a,a^{\prime},n,n^{\prime})$,
$\bar{A}=(\bar{a},\bar{a}^{\prime},\bar{n},\bar{n}^{\prime})$. There are equal
numbers of left semi-chirals $X_{L}$ and right semi-chirals $X_{R}$. Then the
general $N=(2,2)$ sigma model action is

$S=\int
d^{2}x~{}d^{4}\theta~{}K(\phi,\bar{\phi},\chi,\bar{\chi},X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})~{},$
(3.5)

where $K$ is real function of the superfields; see [4] for further
explanation.

In analogy with the Kähler case (2.4) we can rewrite the action in first-order
form. We replace the constrained superfields
$\varphi^{A}=(\phi^{a},\chi^{a^{\prime}},X^{n}_{L},X_{R}^{n^{\prime}})$ with
unconstrained superfields
$\Phi^{A}=(\Phi^{a},\Phi^{a^{\prime}},\Phi^{n}_{L},\Phi_{R}^{n^{\prime}})$ and
introduce constrained lagrange multiplier fields
$\varphi_{A}=(\Sigma_{a},\Lambda_{a^{\prime}},Y_{Ln},\bar{Y}_{R\bar{n}^{\prime}})$
(plus their complex conjugates) whose field equations impose the appropriate
constraints on the fields $\Phi^{A}$. The action is

$S_{\rm dual}=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}K(\Phi,\bar{\Phi})-\varphi_{A}\Phi^{A}-\bar{\varphi}_{\bar{A}}\bar{\Phi}^{\bar{A}}\Big{)}~{},$
(3.6)

which can be expanded to give

$\displaystyle S_{\rm dual}=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}K(\Phi,\bar{\Phi})-\Sigma_{a}\Phi^{a}-\bar{\Sigma}_{\bar{a}}\bar{\Phi}^{\bar{a}}-\Lambda_{a^{\prime}}\Phi^{a^{\prime}}-\bar{\Lambda}_{\bar{a}^{\prime}}\bar{\Phi}^{\bar{a}^{\prime}}$
$\displaystyle-
Y_{Ln}\Phi^{n}-\bar{Y}_{L\bar{n}}\bar{\Phi}^{\bar{n}}-Y_{Rn^{\prime}}\Phi^{n^{\prime}}-\bar{Y}_{R\bar{n}^{\prime}}\bar{\Phi}^{\bar{n}^{\prime}}\Big{)}~{}.$
(3.7)

As seen in the last section, the $\Sigma_{a}$ are complex linear superfields.
The constraints on the other fields are as follows: the $\Lambda_{a^{\prime}}$
are twisted complex linear superfields, the $Y_{Ln}$ are left semichiral
superfields and the $Y_{Rn^{\prime}}$ are right semichiral superfields (see
the appendix for the definitions). The field equations for $\Sigma$,
$\Lambda$, $Y_{L}$ and $Y_{R}$ impose the appropriate chirality constraints on
the $\Phi$’s: they impose that $\Phi^{a}$ is chiral, $\Phi^{a^{\prime}}$ is
twisted chiral, $\Phi^{n}_{L}$ is left semi-chiral and $\Phi_{R}^{n^{\prime}}$
is right semi-chiral. Then integrating out the fields $\varphi_{A}$ we recover
the original action (3.5) (after renaming the fields
$\Phi^{A}=(\Phi^{a},\Phi^{a^{\prime}},\Phi^{n}_{L},\Phi_{R}^{n^{\prime}})$ as
$\varphi_{A}=(\Sigma_{a},\Lambda_{a^{\prime}},Y_{Ln}\Phi^{n},\bar{Y}_{R\bar{n}^{\prime}})$).

Alternatively we can integrate out the $\Phi$’s to arrive at the dual action

$S_{\rm dual}=\int
d^{2}x~{}d^{4}\theta~{}\widetilde{K}(\Sigma,\bar{\Sigma},\Lambda,\bar{\Lambda},Y_{L},\bar{Y}_{L},Y_{R},\bar{Y}_{R})~{},$
(3.8)

where
$\widetilde{K}(\Sigma,\bar{\Sigma},\Lambda,\bar{\Lambda},Y_{L},\bar{Y}_{L},Y_{R},\bar{Y}_{R})$
is a generalised Legendre transform of $K$.444This duality and the action
(3.1) have also been considered by U. Lindström, in unpublished work. As in
the last section, the duality needs the Legendre transform to be well-defined,
which requires $K$ to be convex.

###  3.2 Lagrangian System for the Generalised Kähler Sigma Model

The field equations that follow from (3.1) can be re-expressed as the
following system of equations for constrained superfields
$\varphi^{A}=(\phi^{a},\chi^{a^{\prime}},X^{n}_{L},X_{R}^{n^{\prime}})$ and
$\varphi_{A}=(\Sigma_{a},\Lambda_{a^{\prime}},Y_{Ln},\bar{Y}_{R\bar{n}^{\prime}})$:

$\varphi_{A}=\frac{\partial K}{\partial\varphi^{A}}~{},$ (3.9)

where $K(\varphi^{A},\bar{\varphi}^{\bar{A}})$ is the generalised Kähler
potential. Explicitly, this gives the set of equations

$\displaystyle\Sigma_{a}=\frac{\partial
K}{\partial\phi^{a}}~{},~{}~{}~{}~{}\Lambda_{a^{\prime}}=\frac{\partial
K}{\partial\chi^{a^{\prime}}}~{},~{}~{}~{}~{}Y_{Ln}=\frac{\partial K}{\partial
X^{n}_{L}}~{},~{}~{}~{}~{}Y_{Rn^{\prime}}=\frac{\partial K}{\partial
X^{n^{\prime}}_{R}}$
$\displaystyle\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}\mathbb{D}_{-}\Lambda_{a^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}Y_{Ln}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}Y_{Rn^{\prime}}=0$
(3.10)
$\displaystyle\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}\chi^{a^{\prime}}=\mathbb{D}_{-}\chi^{a^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}X^{n}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}X^{n^{\prime}}_{R}=0$
together with the complex conjugate equations. Regarding these as equations
for the constrained superfields $\varphi^{A}$, the constraints on
$\varphi_{A}$ applied to $\frac{\partial K}{\partial\varphi^{A}}$ give the
field equations for $\varphi^{A}$ that follow from the action (3.5).

If a Legendre transform $\widetilde{K}$ exists, then the equations can also be
cast in the dual form

$\varphi^{A}=\frac{\partial\widetilde{K}}{\partial\varphi_{A}}~{},$ (3.11)

where $\widetilde{K}(\varphi_{A},\bar{\varphi}_{\bar{A}})$ is the Legendre
transform of the generalised Kähler potential. Explicitly, this gives the set
of equations

$\displaystyle\phi^{a}=\frac{\partial\widetilde{K}}{\partial\Sigma_{a}}~{},~{}~{}~{}~{}\chi^{a^{\prime}}=\frac{\partial\widetilde{K}}{\partial\Lambda_{a^{\prime}}}~{},~{}~{}~{}~{}X^{n}_{L}=\frac{\partial\widetilde{K}}{\partial
Y_{Ln}}~{},~{}~{}~{}~{}X^{n^{\prime}}_{R}=\frac{\partial\widetilde{K}}{\partial
Y_{Rn^{\prime}}}$
$\displaystyle\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}\mathbb{D}_{-}\Lambda_{a^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}Y_{Ln}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}Y_{Rn^{\prime}}=0$
(3.12)
$\displaystyle\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}\chi^{a^{\prime}}=\mathbb{D}_{-}\chi^{a^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}X^{n}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}X^{n^{\prime}}_{R}=0$
together with the complex conjugate equations. We leave this dual system for
future discussion and concentrate here on the equations (3.2).

###  3.3 Local symplectic interpretation

We now take a first look at the symplectic geometry associated with the system
(3.2). We will investigate the local geometry in this subsection and postpone
discussion of the global structure to the following section. In particular, we
restrict ourselves to a single coordinate patch $U$ with coordinates
$\varphi^{A},\varphi_{A}$. This coordinate patch has a complex structure with
respect to which
$(\varphi^{A},\varphi_{A})=(\phi^{a},\Sigma_{a},\chi^{a^{\prime}},\Lambda_{a^{\prime}},X_{L}^{n},Y_{Ln},X_{R}^{n^{\prime}},Y_{Rn^{\prime}})$
are holomorphic coordinates and their complex conjugates are anti-holomorphic
coordinates.

Following our discussion of the Kähler case, we re-cast the theory in terms of
unconstrained superfields, replacing the constrained superfields
$\varphi^{A},\varphi_{A}$ with unrestricted fields $\Phi^{A},\Phi_{A}$
(together with their complex conjugates
$\bar{\Phi}^{\bar{A}},\bar{\Phi}_{\bar{A}}$). First, we regard
$\Phi^{A},\Phi_{A}$ as coordinates on $U$. Then $U$ has a holomorphic
symplectic structure

$\omega^{(2,0)}=d\Phi_{A}\wedge d\Phi^{A}~{}.$ (3.13)

Following our discussion of the Kähler case, we now consider unrestricted
superfields $\Phi^{A}(x,\theta,\bar{\theta}),\Phi_{A}(x,\theta,\bar{\theta})$
which map superspace to the set $U$. Of course, we will be interested in the
generalisation to superfields mapping to the whole space rather than to the
subset $U$, but for now we work with this restricted case. Then the symplectic
form $\omega^{(2,0)}$ lifts to

$\Omega=\int d^{2}x~{}d^{4}\theta~{}\delta\Phi_{A}\wedge\delta\Phi^{A}~{}.$
(3.14)

As in the Kähler case, the first line in equation (3.2) corresponds to a
submanifold of the space of supermaps to $U$

$\Phi_{A}=\frac{\partial K}{\partial\Phi^{A}}$ (3.15)

that is real Lagrangian with respect to ${\rm Re}(\Omega)$. The superfield
constraints in (3.2) define a holomorphic isotropic submanifold, as $\Omega$
vanishes when restricted to superfields satisfying the constraints:

$\Omega\Big{|}_{\rm constraints}=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}\delta\Phi_{a}\wedge\delta\Phi^{a}+\delta\Phi_{a^{\prime}}\wedge\delta\Phi^{a^{\prime}}+\delta\Phi_{n}\wedge\delta\Phi^{n}+\delta\Phi_{n^{\prime}}\wedge\delta\Phi^{n^{\prime}}\Big{)}=0~{},$
(3.16)

where we have used the superfield representations (A.16). Thus we see that the
corresponding submanifold is holomorphic isotropic with respect to the complex
structure on $U$. Using the language of generating functions we now show that
the corresponding manifold arises as a Lagrangian submanifold of an enlarged
space.

As in the Kähler case, we enlarge our space by adding additional fermionic
fields $\Psi_{+A},\Psi_{-A}$ and take the symplectic form to be

$\Omega_{\rm large}=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}\delta\Phi_{A}\wedge\delta\Phi^{A}-i\delta\Psi^{A}_{+}\wedge\delta\Psi_{-A}+i\delta\Psi^{A}_{-}\wedge\delta\Psi_{+A}\Big{)}~{}.$
(3.17)

Next we choose two generating functions which depend on
$(\Phi^{A},\Psi_{+A},\Psi_{-A})$. We define a real generating function

${\cal K}=\int d^{2}x~{}d^{4}\theta~{}K(\Phi^{A},\bar{\Phi}^{\bar{A}})~{},$
(3.18)

which gives rise to a real Lagrangian submanifold with respect to ${\rm
Re}(\Omega)$. We also define the holomorphic generating function

${\cal H}=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}i\Psi_{-a}\bar{\mathbb{D}}_{+}\Phi^{a}-i\Psi_{+a}\bar{\mathbb{D}}_{-}\Phi^{a}+i\Psi_{-a^{\prime}}\bar{\mathbb{D}}_{+}\Phi^{a^{\prime}}-i\Psi_{+a^{\prime}}\mathbb{D}_{-}\Phi^{a^{\prime}}+i\Psi_{-n}\bar{\mathbb{D}}_{+}\Phi^{n}-i\Psi_{+n^{\prime}}\bar{\mathbb{D}}_{-}\Phi^{n^{\prime}}\Big{)}~{},$
(3.19)

which defines a holomorphic Lagrangian submanifold. The intersection of these
two Lagrangian submanifolds give rise to the equations (3.2).

However, this local discussion, attractive though it is, does not in general
extend to the full space as the local complex structure on $U$ used here does
not in general extend to a complex structure on the whole space, because the
transition functions mix coordinates that are holomorphic with respect to the
local complex structure with ones that are anti-holomorphic.

##  4 Global issues

In this section, we seek a global structure that is consistent with the
equations (3.2). Recall that in the Kähler case we considered the sigma
model’s symmetries – the holomorphic diffeopmorphisms and the Kähler gauge
transformations – and saw in section 2 how requiring these to be symmetries of
the equations (2.11) fixed the transformation of $\Sigma$ under these
transformations. This then led to a global structure in which these symmetries
were used in the glueing relations. In this section, we will study the
symmetries of the equations (3.2) and attempt to define a global structure by
using these symmetries as glueing relations in the overlaps of patches.

There are two kinds of symmetry here. The first consists of those
diffeomorphisms $\Phi^{A}\to{\Phi^{\prime}}^{A}(\Phi^{A},\bar{\Phi}^{A})$ that
are compatible with the choice of coordinates for symplectic foliations and
with the various superfield constraints; these transformations are given in
(A.17). Requiring these to extend to symmetries of (3.2) then determines the
transformations of the fields $\varphi_{A}$, and these are then used in the
glueing relations. The second kind of symmetry arises from the fact that the
generalised Kähler potential is not uniquely defined, but is only defined up
to transformations that generalise (2.15). On the intersection of two patches
$U_{\alpha}\cap U_{\beta}$ the generalised Kähler potentials are related by
the transformations [10]

$K_{\alpha}-K_{\beta}=F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+\bar{F}^{+}(\bar{\phi},\bar{\chi},\bar{X}_{L})+F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})+\bar{F}_{\alpha\beta}^{-}(\bar{\phi},\chi,\bar{X}_{R})~{},$
(4.1)

since the combinations on the right hand side vanish identically after
superintegration. For this to be a symmetry of the system of equations (3.2)
we require that the fields $\varphi_{A}$ are glued on the intersection of two
patches $U_{\alpha}\cap U_{\beta}$ as follows

$\displaystyle(\Sigma_{a})_{\beta}=(\Sigma_{a})_{\alpha}+\frac{\partial}{\partial\phi^{a}}\Big{(}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})\Big{)}~{},$
(4.2)
$\displaystyle(\Lambda_{a^{\prime}})_{\beta}=(\Lambda_{a^{\prime}})_{\alpha}+\frac{\partial}{\partial\chi^{a^{\prime}}}\Big{(}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+\bar{F}_{\alpha\beta}^{-}(\bar{\phi},\chi,\bar{X}_{R})\Big{)}~{},$
(4.3)
$\displaystyle(Y_{Ln})_{\beta}=(Y_{Ln})_{\alpha}+\frac{\partial}{\partial
X^{n}_{L}}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})~{},$ (4.4)
$\displaystyle(Y_{Rn^{\prime}})_{\beta}=(Y_{Rn^{\prime}})_{\alpha}+\frac{\partial}{\partial
X^{n^{\prime}}_{R}}F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})~{},$ (4.5)

(These glueing relations are to be composed with the transition functions
expressing the change of coordinates between the two patches.) As explained in
Appendix B, the above shifts satisfy the cocycle conditions on the triple
intersections $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$.

Both kinds of symmetry should be compatible with the superfield redefinitions
(A.17) allowed by the chirality constraints. It will be useful to state the
most general superfield redefinitions that are compatible with the superfield
constraints (we suppress all indices for the sake of clarity)

$\displaystyle\phi~{}\longrightarrow~{}f_{1}(\phi)~{},$
$\displaystyle\Sigma~{}\longrightarrow~{}f_{2}(\phi)\Sigma+f_{3}(\phi,\chi,X_{L},Y_{L})+f_{4}(\phi,\bar{\chi},X_{R},Y_{R})~{},$
$\displaystyle\chi~{}\longrightarrow~{}f_{5}(\chi)~{},$ (4.6)
$\displaystyle\Lambda~{}\longrightarrow~{}f_{6}(\chi)\Lambda+f_{7}(\phi,\chi,X_{L},Y_{L})+f_{8}(\bar{\phi},\chi,\bar{X}_{R},\bar{Y}_{R})~{},$
$\displaystyle X_{L}~{}\longrightarrow~{}f_{9}(\phi,\chi,X_{L},Y_{L})~{},$
$\displaystyle
X_{R}~{}\longrightarrow~{}f_{10}(\phi,\bar{\chi},X_{R},Y_{R})~{},$
$\displaystyle Y_{L}~{}\longrightarrow~{}f_{9}(\phi,\chi,X_{L},Y_{L})~{},$
$\displaystyle
Y_{R}~{}\longrightarrow~{}f_{10}(\phi,\bar{\chi},X_{R},Y_{R})~{},$

where $f_{i}$ are arbitrary functions of their arguments. We will use these
field redefinitions as a guiding principle in our discussion of the global
issues.

###  4.1 Sigma models with semi-chiral fields only

We consider first the special case in which the generalised Kähler geometry is
described in terms of semi-chiral fields only. The system of equations (3.2)
for the case with only semi-chiral fields reduces to

$\displaystyle Y_{Ln}=\frac{\partial K}{\partial
X^{n}_{L}}~{},~{}~{}~{}~{}Y_{Rn^{\prime}}=\frac{\partial K}{\partial
X^{n^{\prime}}_{R}}$
$\displaystyle\bar{\mathbb{D}}_{+}Y_{Ln}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}Y_{Rn^{\prime}}=0$
(4.7)
$\displaystyle\bar{\mathbb{D}}_{+}X^{n}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}X^{n^{\prime}}_{R}=0$
We now look at the symmetries of this system. The constraints on the
superfields allow the following transformations

$\displaystyle X_{L}^{n}\rightarrow
f^{n}(X_{L},Y_{L})~{},~{}~{}~{}Y_{Ln}\rightarrow g_{n}(X_{L},Y_{L})~{},$ (4.8)
$\displaystyle X_{R}^{n^{\prime}}\rightarrow
h^{n^{\prime}}(X_{R},Y_{R})~{},~{}~{}~{}Y_{Rn^{\prime}}\rightarrow
k_{n^{\prime}}(X_{R},Y_{R})~{},$ (4.9)

for arbitrary functions $f,g,h,k$ and the shifts (4.4)-(4.5) are just a
special form of these transformations. This suggests that there is no linear
structure associated with the $Y$ directions so that, unlike the Kähler case,
the conjugate coordinates are not fibre coordinates for some vector bundle.

The correct global interpretation of the system (4.1) follows from [4]. We now
briefly review the relevant geometry. We introduce local coordinates
$Z^{A}=(\zeta_{L}^{n},\zeta^{n^{\prime}}_{R})$ on the manifold $M$
corresponding to the lowest components of the superfields
$\varphi^{A}=(X_{L}^{n},X_{R}^{n^{\prime}})$ together with variables
$P_{A}=(\pi_{Ln},\pi_{Rn^{\prime}})$ corresponding to the lowest components of
the superfields $\varphi_{A}=(Y_{Ln},Y_{Rn^{\prime}})$. The space $M$ has two
complex structures $I_{\pm}$ and it was shown in [4] that
$(\zeta_{L}^{n},\pi_{Ln})$ can be taken as coordinates for $M$ and that
moreover they are holomorphic coordinates with respect to $I_{+}$. The
symplectic form $d\pi_{Ln}\wedge d\zeta^{n}_{L}$ is then holomorphic with
respect to $I_{+}$. Similarly, $(\zeta^{n^{\prime}}_{R},\pi_{Rn^{\prime}})$
can be also taken as coordinates for $M$ and these are holomorphic coordinates
with respect to $I_{-}$. The symplectic structure $d\pi_{Rn^{\prime}}\wedge
d\zeta^{n^{\prime}}_{R}$ is holomorphic with respect to $I_{-}$. The
transformation from the coordinates $(\zeta_{L},\pi_{L})$ to the coordinates
$(\zeta_{R},\pi_{R})$ is a canonical transformation that is generated by the
generalised Kähler potential $K(\zeta_{L},\zeta_{R})$.

This extends to a global formulation. In each patch $U$ for $M$, there are
$I_{+}$-holomorphic coordinates $(\zeta_{L},\pi_{L})$ and in each overlap
$U\cap U^{\prime}$ the coordinates $(\zeta_{L},\pi_{L})$ in $U$ and the
coordinates $(\zeta^{\prime}_{L},\pi^{\prime}_{L})$ in $U^{\prime}$ are
related by $I_{+}$-holomorphic reparameterisations,
$\zeta^{\prime}_{L}=\zeta^{\prime}_{L}(\zeta_{L},\pi_{L})$,
$\pi_{L}^{\prime}=\pi^{\prime}_{L}(\zeta_{L},\pi_{L})$. Similarly,
$I_{-}$-holomorphic coordinates
$(\zeta^{n^{\prime}}_{R},\widetilde{\pi}_{Rn^{\prime}})$ can also be
introduced for each patch and the transition functions for the coordinates
$(\zeta_{R},\pi_{R})$ are $I_{-}$-holomorphic reparameterisations
$\zeta^{\prime}_{R}=\zeta_{R}^{\prime}(\zeta_{R},\pi_{R})$,
$\pi^{\prime}_{R}=\pi_{R}^{\prime}(\zeta_{R},\pi_{R})$. These transition
functions give glueing relations for the corresponding superfields that are
consistent with (4.8), so that they preserve the constraints. Then in each
patch, there is a generalised Kähler potential $K(\zeta_{L},\zeta_{R})$
generating the transformation between the $I_{+}$-holomorphic coordinates
$(\zeta_{L}^{n},\pi_{Ln})$ and the $I_{-}$-holomorphic coordinates
$(\zeta^{n^{\prime}}_{R},\pi_{Rn^{\prime}})$ and the potentials in overlapping
patches have the glueing relations (4.1).

This can be elegantly reformulated in terms of a double space ${\cal
Z}=M\times M$ with coordinates $(\zeta_{L},\pi_{L},\zeta_{R},\pi_{R})$. The
first $M$ is taken to have coordinates $(\zeta_{L},\pi_{L})$ which are
holomorphic with respect to $I_{+}$. It has the symplectic form

$\omega^{(2,0)}_{+}=d\pi_{Ln}\wedge d\zeta^{n}_{L}$ (4.10)

which is holomorphic with respect to $I_{+}$. The second $M$ is taken to have
coordinates $(\zeta_{R},\pi_{R})$ that are holomorphic with respect to $I_{-}$
and an $I_{-}$-holomorphic symplectic form

$\omega^{(2,0)}_{-}=d\pi_{Rn^{\prime}}\wedge d\zeta^{n^{\prime}}_{R}~{}.$
(4.11)

The space ${\cal Z}=M\times M$ is itself equipped with the complex structure

$I=I_{+}\times I_{-}$ (4.12)

and the symplectic structure

$\omega^{(2,0)}=dP_{A}\wedge dZ^{A}=d\pi_{Ln}\wedge
d\zeta^{n}_{L}+d\pi_{Rn^{\prime}}\wedge d\zeta^{n^{\prime}}_{R}~{},$ (4.13)

which is holomorphic with respect to $I$. The splitting of the coordinates
into $P$ and $Z$ is a choice of polarization and $P,Z$ are Darboux
coordinates. It will be useful to write this as

${\cal Z}=M_{+}\times M_{-}~{},$ (4.14)

where $M_{+}=(M,I_{+})$ and $M_{-}=(M,I_{-})$.

Then the Lagrangian submanifold of ${\cal Z}$ defined by the equation with a
real function
$K(\zeta_{L}^{n},\bar{\zeta}_{L}^{\bar{n}},\zeta^{n^{\prime}}_{R},\bar{\zeta}_{R}^{\bar{n}^{\prime}})$

$P_{A}=\frac{\partial K}{\partial Z^{A}}$ (4.15)

gives a diagonally embedded submanifold $M$, with coordinates
$(\zeta_{L}^{n},\zeta^{n^{\prime}}_{R})$, and this is the sigma model target
space.

The space ${\cal Z}=M_{+}\times M_{-}$ is then the setting for the equations
(4.1). Then $X_{L},Y_{L}$ are supermaps to the first factor $M_{+}$ and
$X_{R},Y_{R}$ are supermaps to the second factor $M_{-}$ while $X_{L},X_{R}$
are supermaps to the Lagrangian submanifold $M$.

We now reformulate this in terms of _unconstrained_ superfields
$\Phi_{A},\Phi^{A}$ where
$\Phi^{A}=(\zeta_{L}(x,\theta,\bar{\theta}),\zeta_{R}(x,\theta,\bar{\theta}))$
and
$\Phi_{A}=(\pi_{L}(x,\theta,\bar{\theta}),\pi_{R}(x,\theta,\bar{\theta}))$.
The space of supermaps

${\cal M}=\\{\Phi:\mathbb{R}^{2|4}\longrightarrow M_{+}\times M_{-}\\}$ (4.16)

is equipped with the holomorphic symplectic structure

$\Omega=\int d^{2}x~{}d^{4}\theta~{}\delta\Phi_{A}\wedge\delta\Phi^{A}~{}.$
(4.17)

The equations of motion (4.1) then specify the intersection of two
submanifolds of ${\cal M}$. The submanifold determined by

$\Phi_{A}=\frac{\partial K}{\partial\Phi^{A}}$ (4.18)

is a real Lagrangian submanifold with respect to ${\rm Re}(\Omega)$, while the
submanifold specified by the semi-chiral superfield constraints is a
holomorphic isotropic submanifold with respect to $\Omega$. As before,
enlarging the space of supermaps by introducing auxiliary fermionic
superfields gives a formulation in which the submanifold (of the enlarged
space) specified by the semi-chiral superfield constraints is Lagrangian.

###  4.2 Sigma models with semi-chiral fields and chiral fields

For our next example, we consider the case in which there are chiral and
semichiral superfields but there are no $(\chi,\Lambda)$-fields in the
equations (3.2). As we shall see, this combines features of the Kähler case
(with only chiral fields) with features of the purely semi-chiral case from
the previous subsection. The most general field redefinitions (A.17)
compatible with the constraints can be written in this case as follows

$\displaystyle\phi~{}\longrightarrow~{}f_{1}(\phi)~{},$
$\displaystyle\Sigma~{}\longrightarrow~{}f_{2}(\phi)\Sigma+f_{3}(\phi,X_{L},Y_{L})+f_{4}(\phi,X_{R},Y_{R})~{},$
$\displaystyle
X_{L}~{}\longrightarrow~{}f_{9}(\phi,X_{L},Y_{L})~{},~{}~{}~{}Y_{L}~{}\longrightarrow~{}\widetilde{f}_{9}(\phi,X_{L},Y_{L})~{},$
$\displaystyle
X_{R}~{}\longrightarrow~{}f_{10}(\phi,X_{R},Y_{R})~{},~{}~{}~{}Y_{R}~{}\longrightarrow~{}\widetilde{f}_{10}(\phi,X_{R},Y_{R})~{},$
(4.19)

(together with their complex conjugates) for some functions $f_{i}$.
Importantly, there is no mixing between barred and unbarred fields in this
case. Thus the holomorphic superfields $(\phi,\Sigma,X_{L},Y_{L},X_{R},Y_{R})$
are glued to each other and the transition functions do not depend on the
conjugate fields and thus we have a complex structure on the space of
superfields. Here the glueing combines two models, the deformed holomorphic
cotangent bundle from the Kähler case for the $(\phi,\Sigma)$ coordinates and
the product $M\times M$ for the semi-chiral coordinates.

We now explain the construction in more detail. We introduce local coordinates
$Z^{A}=(z^{a},\zeta_{L}^{n},\zeta^{n^{\prime}}_{R})$ on the manifold $M$
corresponding to the lowest components of the superfields
$\varphi^{A}=(\phi^{a},X_{L}^{n},X_{R}^{n^{\prime}})$ together with variables
$P_{A}=(p_{a},\pi_{Ln},\pi_{Rn^{\prime}})$ corresponding to the lowest
components of the superfields
$\varphi_{A}=(\Sigma_{a},Y_{Ln},Y_{Rn^{\prime}})$. The glueing relations (4.1)
for the generalised Kähler potential
$K(z,\bar{z},\zeta_{L},\zeta_{R},\bar{\zeta}_{L},\bar{\zeta}_{R})$ in an
overlap $U\cap U^{\prime}$ become in this case

$K^{\prime}-K=F^{+}(z,\zeta_{L})+F^{-}(z,\zeta_{R})+{\rm complex~{}conjugate}$
(4.20)

where $K$ is the potential on $U$ and $K^{\prime}$ is the potential on
$U^{\prime}$. With $Z^{A}$ the coordinates of $M$, the $P_{A}$ can be taken to
be defined by

$P_{A}=\frac{\partial K}{\partial Z^{A}}$ (4.21)

Then (4.20) implies that the glueing conditions for $P_{A}$ in an overlap
$U\cap U^{\prime}$ must be

$p^{\prime}_{a}=p_{a}+\frac{\partial}{\partial
z^{a}}\left[F^{+}(z,\zeta_{L})+F^{-}(z,\zeta_{R})\right]~{},$ (4.22)
$\displaystyle\pi^{\prime}_{Ln}=\pi_{Ln}+\frac{\partial}{\partial\zeta_{L}^{n}}F^{+}(z,\zeta_{L})~{},$
$\displaystyle\pi^{\prime}_{Rn^{\prime}}=\pi_{Rn^{\prime}}+\frac{\partial}{\partial\zeta_{R}^{n^{\prime}}}F^{-}(z,\zeta_{R})~{},$
(4.23)

where $F^{\pm}$ are the functions appearing in (4.20) and these glueing
relations satisfy the cocycle condition in triple overlaps (see Appendix B).
Note that (4.23) are a special case of the transformations (4.26). Then the
variables $(Z^{A},P_{A})$ can be regarded as coordinates on a manifold, which
we refer to as the ‘phase space’. We construct this phase space ${\cal Z}$
explicitly below.

On $M$ we can use the coordinates
$Z^{A}=(z^{a},\zeta_{L}^{n},\zeta^{n^{\prime}}_{R})$, or the
$I_{+}$-holomorphic coordinates $(z,\zeta_{L},\pi_{L})$ or the
$I_{-}$-holomorphic coordinates $(z,\zeta_{R},\pi_{R})$. The manifold $M$ is
foliated by subspaces of constant $z$. The coordinates on each leaf can be
taken to be $\zeta_{L},\zeta_{R}$ or $\rho_{L}=(\zeta_{L},\pi_{L})$ or
$\rho_{R}=(\zeta_{R},\pi_{R})$. Choosing the leaf coordinates to be
$\rho_{L}=(\zeta_{L},\pi_{L})$, the transition functions in an overlap $U\cap
U^{\prime}$ are $I_{+}$-holomorphic and of the form

$z^{\prime}=z^{\prime}(z)~{},\qquad\rho_{L}^{\prime}=\rho_{L}^{\prime}(z,\rho_{L})~{},$
(4.24)

which is consistent with (4.19). Alternatively, choosing the coordinates on
each leaf to be $\rho_{R}=(\zeta_{R},\pi_{R})$, the transition functions in an
overlap $U\cap U^{\prime}$ are $I_{-}$-holomorphic of the form

$z^{\prime}=z^{\prime}(z)~{},\qquad\rho_{R}^{\prime}=\rho_{R}^{\prime}(z,\rho_{R})~{}.$
(4.25)

We now construct a space $\widehat{M}$ in which the leaves of this foliation
are ‘doubled’. For each open set $U$ of an atlas for $M$ we can choose
coordinates $(z,\rho_{L})$ or $(z,\rho_{R})$ adapted to the foliation. Then
for each such $U$ we introduce a space $\widehat{U}$ with coordinates
$z,\rho_{L},\rho_{R}$. Then $\widehat{M}$ is constructed by glueing together
the patches $\widehat{U}$ with transition functions

$z^{\prime}=z^{\prime}(z)~{},\qquad\rho_{L}^{\prime}=\rho_{L}^{\prime}(z,\rho_{L})~{},\qquad\rho_{R}^{\prime}=\rho_{R}^{\prime}(z,\rho_{R})~{}.$
(4.26)

Note that the foliated structure of $M$ is essential for this construction of
$\widehat{M}$. This space $\widehat{M}$ can be regarded as a quotient of
$M_{+}\times M_{-}$. With coordinates $(z,\rho_{L})$ for $M_{+}$ and
$(z^{\prime},\rho_{R})$ for $M_{-}$, taking the quotient by the relation
$z\sim z^{\prime}$ gives $\widehat{M}$.

The next step is to construct a bundle ${\cal Z}$ over $\widehat{M}$ with
fibre coordinates $p_{a}$ and glueing conditions (4.22). It is important that
these satisfy the cocycle condition in triple overlaps. Then ${\cal Z}$ has
coordinates $Z^{A},P_{A}$ and the transition functions are all holomorphic in
$Z^{A},P_{A}$ and so this endows ${\cal Z}$ with a complex structure.
Moreover, ${\cal Z}$ has a holomorphic symplectic structure

$\omega^{(2,0)}=dP_{A}\wedge dZ^{A}=dp_{a}\wedge dz^{a}+d\pi_{Ln}\wedge
d\zeta_{L}^{n}+d\pi_{Rn^{\prime}}\wedge d\zeta^{n^{\prime}}_{R}~{},$ (4.27)

which is invariant under the glueing (4.22) supplemented by the
diffeormorphisms (4.23). The equation

$P_{A}=\frac{\partial K}{\partial Z^{A}}$ (4.28)

specifies a real Lagrangian submanifold of ${\cal Z}$ with respect to ${\rm
Re}(\omega^{(2,0)})$, which is the original manifold $M$.

We now consider the space of unrestricted supermaps
$\Phi^{A}=Z^{A}(x,\theta,\bar{\theta})$,
$\Phi_{A}=P_{A}(x,\theta,\bar{\theta})$

${\cal M}=\\{\Phi:\mathbb{R}^{2|4}\longrightarrow{\cal Z}\\}~{},$ (4.29)

which has the holomorphic symplectic structure

$\Omega=\int d^{2}x~{}d^{4}\theta~{}\delta\Phi_{A}\wedge\delta\Phi^{A}~{}.$
(4.30)

The submanifold determined by

$\Phi_{A}=\frac{\partial K}{\partial\Phi^{A}}$ (4.31)

is a real Lagrangian submanifold with respect to ${\rm Re}(\Omega)$ and is the
space of unconstrained supermaps to $M$. The superfield constraints that
$z^{a}(x,\theta,\bar{\theta})=\phi^{a}$ is chiral,
$p_{a}(x,\theta,\bar{\theta})=\Sigma_{a}$ is complex linear,
$\zeta^{n}_{L}(x,\theta,\bar{\theta})=X_{L}^{n}$,
$\pi_{Ln}(x,\theta,\bar{\theta})=Y_{Ln}$ are left-semi-chiral and
$\zeta_{R}^{n^{\prime}}(x,\theta,\bar{\theta})=X^{n^{\prime}}_{R}$,
$\pi^{Rn^{\prime}}(x,\theta,\bar{\theta})=Y_{Rn^{\prime}}$ are right-semi-
chiral specify a submanifold of ${\cal M}$ that is a holomorphic isotropic
submanifold with respect to $\Omega$.

As in the previous cases, enlarging the space of supermaps by introducing
auxiliary fermionic superfields $\Psi$ should give a formulation in which the
submanifold (of the enlarged space) specified by the superfield constraints is
Lagrangian. This construction is straightforward locally. Globally, a problem
which may arise with the $\Psi$ fields is that of defining them globally since
we need to understand the global structures associated with our construction
and this may require extra input from the geometry, in particular a better
understanding of the global issues related to symplectic realisations.

There is another interesting case to consider consisting of the system (3.2)
with no $(\phi,\Sigma)$ superfields. If we write the general field
redefinitions (A.17) for this specific case then we have

$\displaystyle\chi~{}\longrightarrow~{}f_{5}(\chi)~{},$
$\displaystyle\Lambda~{}\longrightarrow~{}f_{6}(\chi)\Lambda+f_{7}(\chi,X_{L})+f_{8}(\chi,\bar{X}_{R})~{},$
$\displaystyle
X_{L}~{}\longrightarrow~{}f_{9}(\chi,X_{L},Y_{L})~{},~{}~{}~{}Y_{L}~{}\longrightarrow~{}\widetilde{f}_{9}(\chi,X_{L},Y_{L})~{},$
$\displaystyle\bar{X}_{R}~{}\longrightarrow~{}{f}_{10}(\chi,\bar{X}_{R},\bar{Y}_{R})~{},~{}~{}~{}\bar{Y}_{R}~{}\longrightarrow~{}{\widetilde{f}}_{10}(\chi,\bar{X}_{R},\bar{Y}_{R})~{}.$

In these transformations the superfields
$(\chi,\Lambda,X_{L},Y_{L},\bar{X}_{R},\bar{Y}_{R})$ do not mix with
superfields $(\bar{\chi},\bar{\Lambda},\bar{X}_{L},\bar{Y}_{L},X_{R},Y_{R})$.
Thus we can claim that the set
$(\chi,\Lambda,X_{L},Y_{L},\bar{X}_{R},\bar{Y}_{R})$ corresponds to
holomorphic coordinates and we again have a well-defined holomorphic
symplectic structure

$\Omega=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}\delta\Phi_{a^{\prime}}\wedge\delta\Phi^{a^{\prime}}+\delta\Phi_{n}\wedge\delta\Phi^{n}+\delta\bar{\Phi}_{\bar{n}^{\prime}}\wedge\delta\bar{\Phi}^{\bar{n}^{\prime}}\Big{)}$
(4.32)

and the discussions from the previous case go through similarly, after
interchanging $n^{\prime}\leftrightarrow{\bar{n}}^{\prime}$. Thus the system
(3.2) without $(\phi,\Sigma)$ can again be interpreted as the intersection of
real Lagrangian and holomorphic isotropic (or Lagrangian) submanifolds but
with a modified holomorphic symplectic structure.

These two cases, either without $(\chi,\Lambda)$ or without $(\phi,\Sigma)$,
have target spaces which are generalised Kähler manifolds of symplectic type
and the corresponding geometry has been studied in [8]. For any generalised
Kähler manifold of symplectic type a ‘doubled space’ was constructed using a
holomorphic Morita equivalence in [8] that is a holomorphic symplectic
manifold. Here we have also constructed a ‘doubled’ holomorphic symplectic
manifold $({\cal Z},\omega)$ and we conjecture that the two constructions in
fact agree. As far as we can see, the present discussion is consistent with
the global considerations in [8].

###  4.3 Chiral and twisted chiral case

We now consider the case of generalised Kähler geometry for which the
corresponding sigma model is formulated only in terms of chiral and twisted
chiral fields. We again investigate the global structure of the system (3.2)
without $X$ and $Y$ fields by examining the symmetries of these equations. The
chirality conditions allow the field redefinitions
$\phi^{a}\rightarrow{\phi^{\prime}}^{a}(\phi)$,
$\chi^{a^{\prime}}\rightarrow\chi^{\prime a^{\prime}}(\chi)$ and the linear
complex superfields $\Sigma_{a}$ and twisted linear complex superfields
$\Lambda_{a^{\prime}}$ transform under these coordinate transformations as

$\Sigma_{a}\rightarrow\Sigma^{\prime}_{a}=\frac{\partial\phi^{b}}{\partial{\phi^{\prime}}^{a}}~{}\Sigma_{b}~{},~{}~{}~{}~{}~{}\Lambda_{a^{\prime}}\rightarrow\Lambda^{\prime}_{a^{\prime}}=\frac{\partial\chi^{b^{\prime}}}{\partial{\chi^{\prime}}^{a^{\prime}}}~{}\Lambda_{b^{\prime}}~{}.$
(4.33)

Note that this is compatible with the superfield structure (see Appendix A).
In addition to these symmetries we have the following glueing of $\Sigma$’s
and $\Lambda$’s

$\displaystyle(\Sigma_{a})_{\beta}=(\Sigma_{a})_{\alpha}+\frac{\partial}{\partial\phi^{a}}\Big{(}F_{\alpha\beta}^{+}(\phi,\chi)+F_{\alpha\beta}^{-}(\phi,\bar{\chi})\Big{)}~{},$
(4.34)
$\displaystyle(\Lambda_{a^{\prime}})_{\beta}=(\Lambda_{a^{\prime}})_{\alpha}+\frac{\partial}{\partial\chi^{a^{\prime}}}\Big{(}F_{\alpha\beta}^{+}(\phi,\chi)+\bar{F}_{\alpha\beta}^{-}(\bar{\phi},\chi)\Big{)}~{}.$
(4.35)

The structure here is again a deformation ${\cal Z}$ of the cotangent bundle
of the generalised Kähler manifold $M$, generalising that considered in the
previous section. However there is an important difference as we do not have a
holomorphic structure on ${\cal Z}$ since the above transformations mix barred
and unbarred fields. For a patch $U_{\alpha}$ of $M$ (with $2d$ the real
dimension of $M$), we introduce $d$ local complex coordinates
$Z^{A}_{\alpha}=(z^{a}_{\alpha},w^{a^{\prime}}_{\alpha})$ (corresponding to
the lowest components of the superfields
$\varphi^{A}=(\phi^{a},\chi^{a^{\prime}})$). Then for the cotangent bundle
$T^{*}(U_{\alpha})=U_{\alpha}\times\mathbb{R}^{2d}$ we inroduce $d$ complex
fibre coordinates
$(P_{A})_{\alpha}=((p_{a}))_{\alpha},(r_{a^{\prime}})_{\alpha})$
(corresponding to the lowest components of the superfields
$\varphi_{A}=(\Sigma_{a},\Lambda_{a^{\prime}})$). We then construct the
deformed cotangent bundle ${\cal Z}$ by glueing together the patches
$T^{*}(U_{\alpha})$ with the following transition conditions in
$T^{*}(U_{\alpha}\cap U_{\beta})$:

$\displaystyle(p_{a})_{\beta}=(p_{a})_{\alpha}+\frac{\partial}{\partial
z^{a}}\Big{(}F_{\alpha\beta}^{+}+F_{\alpha\beta}^{-}\Big{)}~{},$ (4.36)
$\displaystyle(r_{a^{\prime}})_{\beta}=(r_{a^{\prime}})_{\alpha}+\frac{\partial}{\partial
w^{a^{\prime}}}\Big{(}F_{\alpha\beta}^{+}+\bar{F}_{\alpha\beta}^{-}\Big{)}~{},$
(4.37)

where the functions $F^{\pm}$ have the following dependence on the
coordinates:

$F_{\alpha\beta}^{+}=F_{\alpha\beta}^{+}(z^{a},w^{a^{\prime}})~{},\qquad
F_{\alpha\beta}^{-}=F_{\alpha\beta}^{-}(z^{a},\bar{w}^{{\bar{a}}^{\prime}})~{}.$
(4.38)

The properties of the glueing in (4.1) were studied in [10] where it was shown
that they are ultimately related to gerbes. One consequence of [10] is that
the above glueing relations (4.36)-(4.37) satisfy the coycle conditions in
triple overlaps and thus they are consistent glueing rules for a bundle (see
Appendix B).

In each patch $T^{*}(U_{\alpha})$ there is, as the notation suggests, a
complex structure that acts as $+i$ on $(dZ^{A},dP_{A})$, together with a
holomorphic symplectic structure

$\omega^{(2,0)}=dP_{A}\wedge dZ^{A}~{}.$ (4.39)

However, the transition functions (4.36)-(4.37) mix the holomoprhic
coordinates $(Z^{A},P_{A})$ with the antiholomorphic ones
$(\bar{Z}^{\bar{A}},\bar{P}_{\bar{A}})$ so the local complex structure and
holomorphic symplectic structure do not extend to holomorphic structures on
${\cal Z}$. Thus there is no _manifest_ complex structure on ${\cal Z}$ in
general, although of course the generalised Kähler manifold has two complex
structures $I_{\pm}$. The bundle is then a real bundle with fibres
$\mathbb{R}^{2d}$ and can’t be viewed as a holomorphic bundle using the
complex structure on each patch. However, there is a well-defined real
symplectic form

$2~{}{\rm Re}(\omega^{(2,0)})=dP_{A}\wedge dZ^{A}+d\bar{P}_{\bar{A}}\wedge
d\bar{Z}^{\bar{A}}~{}.$ (4.40)

Explicit calculation shows that this symplectic form is invariant under the
transformations in the transition functions (4.36)-(4.37), so that they are
symplectomorphisms. Then ${\rm Re}(\omega)$ is globally well-defined on the
deformed cotangent bundle ${\cal Z}$. In this way we obtain an affine
symplectic bundle ${\cal Z}$. The transition functions here are closely
related to gerbes, see [10]. Moreover, it is not clear how the space ${\cal
Z}$ encodes generalised Kähler geometry of $M$ (this is known for the previous
examples).

Next we define the infinite dimensional space of supermaps from
$\mathbb{R}^{2|4}$ to the symplectic affine bundle constructed above

${\cal M}=\\{\Phi:\mathbb{R}^{2|4}\longrightarrow{\cal Z}\\}~{},$ (4.41)

which is equipped with the real symplectic form

$2{\rm Re}(\Omega)=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}\delta\Phi_{a}\wedge\delta\Phi^{a}+\delta\Phi_{a^{\prime}}\wedge\delta\Phi^{a^{\prime}}+\delta\bar{\Phi}_{\bar{a}}\wedge\delta\bar{\Phi}^{\bar{a}}+\delta\bar{\Phi}_{\bar{a}^{\prime}}\wedge\delta\bar{\Phi}^{\bar{a}^{\prime}}\Big{)}~{}.$
(4.42)

The condition

$\Phi_{A}=\frac{\partial K}{\partial\Phi^{A}}$ (4.43)

defines a real Lagrangian submanifold with respect to ${\rm Re}(\Omega)$. The
superfields constraints with their complex conjugate define a real isotropic
submanifold. In analogy with Kähler case we can introduce introduce fermionic
fields and define the corresponding generating functions so that the subspace
defined by the superfield constraints is a real Lagrangian submanifold.

###  4.4 Geometric Structure of General Case

We now consider the target space of the of general sigma model with chiral,
twisted chiral and semi-chiral superfields. We introduce local coordinates
$Z^{A}=(z^{a},w^{a^{\prime}},\zeta_{L}^{n},\zeta_{R}^{n^{\prime}})$ on the
manifold $M$ corresponding to the lowest components of the superfields
$\varphi^{A}=(\phi^{a},\chi^{a^{\prime}},X_{L}^{n},X_{R}^{n^{\prime}})$
together with dual coordinates $\pi_{Ln},\pi_{Rn^{\prime}}$, corresponding to
the lowest components of the superfields $Y_{Ln},Y_{Rn^{\prime}}$, that are
defined by

$\pi_{L}=\frac{\partial K}{\partial\zeta_{L}}~{},\qquad\pi_{R}=\frac{\partial
K}{\partial\zeta_{R}}~{},$ (4.44)

where $K(Z,\bar{Z})$ is the generalised Kähler potential. Then
$z,w,\zeta_{L},\pi_{L}$ are $I_{+}$-holomorphic coordinates on $M$ and
$z,\bar{w},\zeta_{R},\pi_{R}$ are $I_{-}$-holomorphic coordinates on $M$. On
the overlap $U_{\alpha}\cap U_{\beta}$ of two patches on $M$, (4.1) gives the
glueing conditions for the generalised Kähler potential in the two patches to
be

$K_{\alpha}-K_{\beta}=F_{\alpha\beta}^{+}(z,w,\zeta_{L})+\bar{F}^{+}(\bar{z},\bar{w},\bar{\zeta}_{L})+F_{\alpha\beta}^{-}(z,\bar{w},\zeta_{R})+\bar{F}_{\alpha\beta}^{-}(\bar{z},w,\bar{\zeta}_{R})~{}.$
(4.45)

Note that $F^{+}$ is $I_{+}$-holomorphic and $F^{-}$ is $I_{-}$-holomorphic.

This can be recast in terms of a product space of the kind considered in the
previous subsections:

${\cal N}=M_{+}\times M_{-}~{},$ (4.46)

where $M_{+}=(M,I_{+})$ and $M_{-}=(M,I_{-})$ and ${\cal N}$ has complex
structure $I=I_{+}\times I_{-}$. Taking $z,w,\zeta_{L},\pi_{L}$ as
$I_{+}$-holomorphic coordinates on $M_{+}$ and
$z^{\prime},w^{\prime},\zeta_{R},\pi_{R}$ as $I_{-}$-holomorphic coordinates
on $M_{-}$, the manifold $M$ with coordinates
$z^{a},w^{a^{\prime}},\zeta^{n}_{R},\zeta^{n^{\prime}}_{R}$ is obtained as the
submanifold specified by

$z^{\prime}=z~{},\qquad w^{\prime}=\bar{w}~{},\qquad\pi_{L}=\frac{\partial
K}{\partial\zeta_{L}}~{},\qquad\pi_{R}=\frac{\partial
K}{\partial\zeta_{R}}~{}.$ (4.47)

Note the twist: $z^{\prime}$ is identified with $z$ but $w^{\prime}$ is
identified with the _complex conjugate_ of $w$.

The space $M_{+}$ is foliated by leaves on which $z$ and $w$ are constant,
with coordinates $\zeta_{L},\pi_{L}$ on each leaf. Similarly, $M_{-}$ is
foliated by leaves on which $z^{\prime}$ and $w^{\prime}$ are constant, with
coordinates $\zeta_{R},\pi_{R}$ on each leaf. Introducing a relation $z\sim
z^{\prime}$, $w\sim\bar{w}^{\prime}$, then taking the quotient

${\widehat{M}}={\cal N}/\sim$ (4.48)

gives a space with coordinates $z,w,\zeta_{L},\pi_{L},\zeta_{R},\pi_{R}$,
which can be thought of as the original space $M$ with ‘doubled’ leaves, which
agrees with the space $\widehat{M}$ from subsection 4.2 in the case in which
there are no twisted chiral fields and hence no coordinates $w$. Much of the
discussion from subsection 4.2 extends to this case.

We now construct a bundle ${\cal Z}$ over $\widehat{M}$ with fibre coordinates
$p_{a},r_{a^{\prime}}$ and glueing conditions

$\displaystyle p^{\prime}_{a}$ $\displaystyle=$ $\displaystyle
p_{a}+\frac{\partial}{\partial
z^{a}}\left[F^{+}(z,w,\zeta_{L})+F^{-}(z,\bar{w},\zeta_{R})\right]~{},$ (4.49)
$\displaystyle r^{\prime}_{a^{\prime}}$ $\displaystyle=$ $\displaystyle
r_{a^{\prime}}+\frac{\partial}{\partial
w^{a^{\prime}}}\left[F^{+}(z,w,\zeta_{L})+\bar{F}^{-}(\bar{z},w,\zeta_{R})\right]~{}.$
(4.50)

It is important that these satisfy the cocycle condition in triple overlaps
(see Appendix B) so that this is indeed a fibre bundle. With these glueing
rules, the section of ${\cal Z}$ defined by

$p_{a}=\frac{\partial}{\partial z^{a}}K,\qquad
r_{a^{\prime}}=\frac{\partial}{\partial w^{a^{\prime}}}K$ (4.51)

is a global section defining a submanifold.

This is then the geometric setting for the equations (3.2). Globally we are
able to define only a real symplectic structure and there is no natural
complex structure. Thus at best we can interpret the equations (3.2) as the
defining intersection of two real Lagrangian submanifolds in the space of
unconstrained superfields. In the next section we present an alternative
formulation which does have a holomorphic structure.

##  5 Extended geometry

In the previous sections we discussed the symplectic interpretation of the
equations (3.2). In certain special cases we have a superfield phase space
equipped with a complex structure and a holomorphic symplectic form. However,
this is not the case in the general situation. We seek here an extension of
the target space that carries a holomorphic symplectic structure. We find a
surprising reformulation that has an interesting geometry and is dictated by
the superfield structure.

###  5.1 Dual superfield presentation

The (twisted) complex linear superfields can be re-expressed in terms of semi-
chiral superfields. The complex linear constraints

$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma_{a}=0~{},\qquad\bar{\mathbb{D}}_{+}\mathbb{D}_{-}\Lambda_{a^{\prime}}=0$
(5.1)

are solved by

$\Sigma_{a}=\bar{\mathbb{D}}_{+}\Psi^{+}_{a}+\bar{\mathbb{D}}_{-}\Psi^{+}_{a}~{},~{}~{}~{}~{}~{}\Lambda_{a^{\prime}}=\bar{\mathbb{D}}_{+}\Upsilon_{a^{\prime}}^{+}+\mathbb{D}_{-}\Upsilon_{a^{\prime}}^{-}$
(5.2)

for some unconstrained spinor superfields
$\Psi^{\pm}_{a},\Upsilon_{a^{\prime}}^{\pm}$. Then (twisted) complex linear
superfields can be rewritten as

$\Sigma_{a}=U_{La}+U_{Ra}~{},\qquad\Lambda_{a^{\prime}}=V_{La^{\prime}}+\bar{V}_{Ra^{\prime}}~{},$
(5.3)

where

$U_{La}=\bar{\mathbb{D}}_{+}\Psi^{+}_{a},\qquad
U_{Ra}=\bar{\mathbb{D}}_{-}\Psi^{+}_{a},$ (5.4)
$V_{La^{\prime}}=\bar{\mathbb{D}}_{+}\Upsilon_{a^{\prime}}^{+},\qquad\bar{V}_{Ra^{\prime}}=\mathbb{D}_{-}\Upsilon_{a^{\prime}}^{-}$
(5.5)

are semi-chiral superfields. As we shall see, formulating the theory in terms
of the semi-chirals $U_{L},U_{R},V_{L},V_{R}$ instead of the linear
superfields $\Sigma,\Lambda$ is very helpful. Using this we can rewrite the
equations (3.2) as follows

$\displaystyle U_{La}+U_{Ra}=\frac{\partial
K}{\partial\phi^{a}}~{},~{}~{}~{}~{}V_{La^{\prime}}+\bar{V}_{Ra^{\prime}}=\frac{\partial
K}{\partial\chi^{a^{\prime}}}~{},~{}~{}~{}~{}Y_{Ln}=\frac{\partial K}{\partial
X^{n}_{L}}~{},~{}~{}~{}~{}Y_{Rn^{\prime}}=\frac{\partial K}{\partial
X^{n^{\prime}}_{R}}$
$\displaystyle\bar{\mathbb{D}}_{+}U_{La}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{-}U_{Ra}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}V_{La^{\prime}}=0~{},~{}~{}~{}\mathbb{D}_{-}\bar{V}_{Ra^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}Y_{Ln}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}Y_{Rn^{\prime}}=0$
$\displaystyle\bar{\mathbb{D}}_{\pm}\phi^{a}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}\chi^{a^{\prime}}=\mathbb{D}_{-}\chi^{a^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}X^{n}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}X^{n^{\prime}}_{R}=0$
(5.6)

Let us discuss the symmetries of these equations. From the glueing conditions
(4.2),(4.3) we get the following glueing in terms of the new variables

$\displaystyle(U_{La})_{\beta}=(U_{La})_{\alpha}+\frac{\partial}{\partial\phi^{a}}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})~{},$
(5.7)
$\displaystyle(U_{Ra})_{\beta}=(U_{Ra})_{\alpha}+\frac{\partial}{\partial\phi^{a}}F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})~{},$
(5.8)
$\displaystyle(V_{La^{\prime}})_{\beta}=(V_{La^{\prime}})_{\alpha}+\frac{\partial}{\partial\chi^{a^{\prime}}}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})~{},$
(5.9)
$\displaystyle(\bar{V}_{Ra^{\prime}})_{\beta}=(\bar{V}_{Ra^{\prime}})_{\alpha}+\frac{\partial}{\partial\chi^{a^{\prime}}}\bar{F}_{\alpha\beta}^{-}(\bar{\phi},\chi,\bar{X}_{R})~{},$
(5.10)

whereas the transformations (4.6) for $\Sigma$ and $\Lambda$ become

$\displaystyle
U_{L}~{}\longrightarrow~{}f_{2}(\phi)U_{L}+f_{3}(\phi,\chi,X_{L},Y_{L})~{},$
$\displaystyle
U_{R}~{}\longrightarrow~{}f_{2}(\phi)U_{R}+f_{4}(\phi,\bar{\chi},X_{R},Y_{R})~{},$
$\displaystyle
V_{L}~{}\longrightarrow~{}f_{6}(\chi)V_{L}+f_{7}(\phi,\chi,X_{L},Y_{L})~{},$
$\displaystyle\bar{V}_{R}~{}\longrightarrow~{}f_{6}(\chi)\bar{V}_{R}+f_{8}(\bar{\phi},\chi,\bar{X}_{R},\bar{Y}_{R})~{}.$
(5.11)

The glueing conditions and transformations for $U_{L},V_{L}$ are
$I_{+}$-holomorphic and those for $U_{R},V_{R}$ are $I_{-}$-holomorphic. This
would be consistent with $(U_{L},V_{L})$ being fibre coordinates for some
holomorphic bundle over $M_{+}$ and $(U_{R},V_{R})$ being fibre coordinates
for some holomorphic bundle over $M_{-}$. Moreover, the transformations (5.11)
are not the most general ones consistent with the chirality constraints and
their special form is indicative of a role as fibre coordinates. However, this
is not the correct picture. The crucial point is that (5.7)-(5.10) do not
satisfy the cocycle conditions on triple intersections (see Appendix B) so
that there is no interpretation of $(U_{L},V_{L})$ or $(U_{R},V_{R})$ as extra
coordinates for an enlarged manifold. Instead, adding the extra variables
gives rise to a more general structure. We discuss this further below.

###  5.2 Geometric Structure

In subsection 4.4, we considered the manifold ${\cal N}=M_{+}\times M_{-}$
with complex structure $I=I_{+}\times I_{-}$. The manifold $M$ was recovered
as the submanifold specified by (4.47). The space ${\widehat{M}}$ with
coordinates $z,w,\zeta_{L},\pi_{L},\zeta_{R},\pi_{R}$ (obtained from the space
$M$ by doubling the leaves) was constructed as the quotient
${\widehat{M}}={\cal N}/\sim$ of $\widehat{M}$ by the relation $z\sim
z^{\prime}$, $w\sim\bar{w}^{\prime}$. We then constructed a bundle ${\cal Z}$
over $\widehat{M}$ with fibre coordinates $p_{a},r_{a^{\prime}}$ and glueing
conditions (4.49), but this did not have a natural complex structure.

We now introduce variables $p_{La},p_{Ra},r_{La^{\prime}},r_{Ra^{\prime}}$
corresponding to the lowest components of the superfields
$U_{La},U_{Ra},V_{La^{\prime}},V_{Ra^{\prime}}$. The relations (5.3) give

$p=p_{L}+p_{R}~{},\qquad r=r_{L}+\bar{r}_{R}~{}.$ (5.12)

It would be natural to attempt to generalise the previous constructions by
interpreting $p_{L},r_{L}$ as fibre coordinates for some bundle $E_{+}$ over
$M_{+}$ and $p_{R},r_{R}$ as fibre coordinates for some bundle $E_{-}$ over
$M_{-}$ so that $E_{+}\times E_{-}$ has a natural complex structure and the
bundle ${\cal Z}$ can be constructed from it by taking the quotient by $\sim$
and using (5.12). We shall develop this picture below, finding a natural
holomorphic setting for our equations. However, as we shall see, the $E_{\pm}$
that arise in this way are _not_ fibre bundles and are instead generalised
spaces.

For $M_{+}$ we use the $I_{+}$-holomorphic coordinates $z,w,\zeta_{L},\pi_{L}$
and for each patch $U_{\alpha}$ we introduce additional variables
$p_{L},r_{L}$ parameterising local fibres of a bundle $V_{\alpha}$ over
$U_{\alpha}$. Over intersections $U_{\alpha}\cap U_{\beta}$, we have, from
(5.7),(5.9), the transition functions

$\displaystyle(p_{La})_{\beta}=(p_{La})_{\alpha}+\frac{\partial}{\partial
z^{a}}F_{\alpha\beta}^{+}(z,w,\zeta_{L})~{},$ (5.13)
$\displaystyle(r_{La^{\prime}})_{\beta}=(r_{La^{\prime}})_{\alpha}+\frac{\partial}{\partial
w^{a^{\prime}}}F_{\alpha\beta}^{+}(z,w,\zeta_{L})~{},$ (5.14)
$\displaystyle(\pi_{Ln})_{\beta}=(\pi_{Ln})_{\alpha}+\frac{\partial}{\partial\zeta_{L}^{n}}F_{\alpha\beta}^{+}(z,w,\zeta_{L})~{}$
(5.15)

The $V_{\alpha}$ are then glued using the $I_{+}$-holomorphic glueing
relations (5.13),(5.14) to construct some ‘generalised space’ $E_{+}$.
Locally, $p_{La},r_{La^{\prime}}$ are fibre coordinates for a bundle
$V_{\alpha}$ over a patch $U_{\alpha}$. However, globally this does not extend
to a bundle over $M_{+}$, or indeed any picture in which these variables are
coordinates of some manifold. This is because the transition functions for
$p_{La},r_{La^{\prime}}$ do not satisfy the cocycle condition in triple
overlaps (see Appendix B) and so these variables cannot be interpreted as
coordinates on any bundle over $M_{+}$. Then globally, $E_{+}$ is not a bundle
and not a manifold.

We proceed formally and note that we can introduce a complex structure and
symplectic structure on each $V_{\alpha}$. These are invariant under the
glueing relations and so extend to structures on $E_{+}$. Then $E_{+}$ has a
complex structure $I_{+}$ for which $z,w,p_{L},r_{L},\zeta_{L},\pi_{L}$ are
holomorphic variables, and an $I_{+}$-holomorphic symplectic structure

$\omega^{(2,0)}_{+}=d\pi_{Ln}\wedge d\zeta^{n}_{L}+dp_{La}\wedge
dz^{a}+dr_{La^{\prime}}\wedge dw^{a^{\prime}}~{},$ (5.16)

which is invariant under the glueing relations.

Similarly, we take $p_{R},r_{R}$ to be additional variables for a generalised
space $E_{-}$ over $M_{-}$ From (5.8),(5.10) we have the transition functions

$\displaystyle(p_{Ra})_{\beta}=(p_{Ra})_{\alpha}+\frac{\partial}{\partial{z^{\prime}}^{a}}F_{\alpha\beta}^{-}(z^{\prime},w^{\prime},\zeta_{R})~{},$
(5.17)
$\displaystyle(r_{Ra^{\prime}})_{\beta}=(r_{Ra^{\prime}})_{\alpha}+\frac{\partial}{\partial
w^{\prime
a^{\prime}}}F_{\alpha\beta}^{-}(z^{\prime},w^{\prime},\zeta_{R})~{},$ (5.18)
$\displaystyle(\pi_{Rn^{\prime}})_{\beta}=(\pi_{Rn^{\prime}})_{\alpha}+\frac{\partial}{\partial\zeta_{R}^{n^{\prime}}}F_{\alpha\beta}^{-}(z^{\prime},w^{\prime},\zeta_{R})~{}.$
(5.19)

This has simliar properties to $E_{+}$, and for the same reasons $E_{-}$ does
not define a manifold. However, $E_{-}$ has a complex structure $I_{-}$, for
which $z^{\prime},w^{\prime},p_{R},r_{R},\zeta_{R},\pi_{R}$ are holomorphic,
and it has a $I_{-}$-holomorphic symplectic structure

$\omega^{(2,0)}_{-}=d\pi_{Rn^{\prime}}\wedge
d\zeta^{n^{\prime}}_{R}+dp_{Ra}\wedge d{z^{\prime}}^{a}+dr_{Ra^{\prime}}\wedge
dw^{\prime a^{\prime}}~{},$ (5.20)

which is invariant under the glueing.

To understand the geometry of the construction, recall that for the general
(2,2) sigma models the 3-form $H$ represents an integral cohomology class in
the quantum theory and is the curvature for a gerbe on $M$. The interplay
between the gerbe strucure and the complex structures was explored in [10]
where it was found that a rich system of holomorphic gerbes underpins the
theory. In particular, the transition functions $F^{+}_{\alpha\beta}$ and
$F^{-}_{\alpha\beta}$ defined in intersections do not satisfy the cocycle
condition that $\delta F^{\pm}\equiv
F_{\alpha\beta}^{\pm}+F_{\beta\gamma}^{\pm}+F_{\gamma\alpha}^{\pm}$ vanishes
(or vanishes modulo $2\pi$ when appropriately normalised) but instead $\delta
F^{+}$ and $\delta F^{-}$ each provides a non-trivial map from triple
intersections $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$ to $U(1)$ that defines
a gerbe structure. Moreover, as $F^{+}$ is $I_{+}$-holomorphic and $F^{-}$ is
$I_{-}$-holomorphic, these are holomorphic gerbes. See Appendix B for more
details. In our construction, 1-forms $p_{La},r_{La^{\prime}}$ are introduced
on each patch of $M_{+}$ and glued together using the gerbe transition
functions through (5.13),(5.14), while $p_{Ra},r_{Ra^{\prime}}$ are introduced
on each patch of $M_{-}$ and glued using (5.17),(5.18). In fact, they are
glued in the same way as certain prepotentials for holmorphic gerbe
connections (see Appendix B). This can be viewed as a generalisation of the
Donaldson construction, where (2.19) implies that $p$ has the glueing
relations of a holomorphic connection.

The advantage of this construction is its manifest holomorphic structure and
its holomorphic symplectic structure. We introduce the space

${\cal X}=E_{+}\times E_{-}~{},$ (5.21)

which has a complex structure $I=I_{+}\times I_{-}$ and a $I$-holomorphic
symplectic structure $\omega^{(2,0)}=\omega^{(2,0)}_{+}+\omega^{(2,0)}_{-}$.
The $I$-holomorphic coordinates $\widehat{Z}^{A}$ and conjugate variables
$\widehat{P}_{A}$ are

$\widehat{Z}^{A}=(z,w,z^{\prime},w^{\prime},\zeta_{L},\zeta_{R})~{},\qquad\widehat{P}_{A}=(p_{L},r_{L},p_{R},r_{R},\pi_{L},\pi_{R})~{}.$
(5.22)

Then the $I$-holomorphic symplectic structure can be written as

$\omega^{(2,0)}=d\widehat{P}_{A}\wedge d\widehat{Z}^{A}~{}.$ (5.23)

This space is an extended arena for our equations. In particular, our
equations lead to the subspace of this given by (4.47) together with

$p_{L}+p_{R}=\frac{\partial K}{\partial z}~{},\qquad
r_{L}+\bar{r}_{R}=\frac{\partial K}{\partial w}~{}.$ (5.24)

Recall that the manifold $M$ is obtained as the submanifold of $M_{+}\times
M_{-}$ specified by (4.47). Then from ${\cal X}$ we obtain a subspace in which
over each point in this submanifold $M$ we have fibre coordinates
$p_{L},r_{L},p_{R},r_{R}$. We now define

$p=p_{L}+p_{R}~{},\qquad r=r_{L}+\bar{r}_{R}~{}.$ (5.25)

and from (5.13),(5.14),(5.17),(5.18) the transition functions for $p,r$ are

$\displaystyle(p_{a})_{\beta}=(p_{a})_{\alpha}+\frac{\partial}{\partial
z^{a}}\Big{(}F_{\alpha\beta}^{+}+F_{\alpha\beta}^{-}\Big{)}~{},$ (5.26)
$\displaystyle(r_{a^{\prime}})_{\beta}=(r_{a^{\prime}})_{\alpha}+\frac{\partial}{\partial
w^{a^{\prime}}}\Big{(}F_{\alpha\beta}^{+}+\bar{F}_{\alpha\beta}^{-}\Big{)}~{},$
(5.27)

where the functions $F^{\pm}$ have the following dependence on the
coordinates:

$F_{\alpha\beta}^{+}=F_{\alpha\beta}^{+}(z,w,\zeta_{L})~{},\qquad
F_{\alpha\beta}^{-}=F_{\alpha\beta}^{-}(z,\bar{w},\zeta_{R})~{}.$ (5.28)

These glueing relations do satisfy the cocycle condition in triple overlaps
(see Appendix B) and so define a bundle over $M$, with coordinates
$z,w,\zeta_{L},\zeta_{R}$ on $M$ and fibre coordinates $p,r$. As the functions
$F^{\pm}$ are independent of $\pi_{L},\pi_{R}$, this bundle in fact extends to
a bundle $\widehat{\cal Z}$ over $\widehat{M}$ with fibre coordinates $p,r$
and coordinates $z,w,\zeta_{L},\pi_{L},\zeta_{R},\pi_{R}$ on $\widehat{M}$.

The space $\widehat{\cal Z}$ inherits a real symplectic structure from
$\omega^{(2,0)}$ given by

$\omega=dp_{a}\wedge dz^{a}+dr_{a^{\prime}}\wedge
d\bar{w}^{a^{\prime}}+d\pi_{Ln}\wedge d\zeta_{L}^{n}+\pi_{Rn^{\prime}}\wedge
d\zeta^{n^{\prime}}_{R}+{\rm complex~{}conjugate}$ (5.29)

With the notation

$Z^{A}=(z,w,\zeta_{L},\zeta_{R}),\qquad P_{A}=(p,r,\pi_{L},\pi_{R})$ (5.30)

this symplectic structure can be written as

$\omega=dP_{A}\wedge dZ^{A}+d\bar{P}^{\bar{A}}\wedge d\bar{Z}^{\bar{A}}~{}.$
(5.31)

However, ${\cal Z}$ does not inherit a complex structure in general. In the
special case in which there are no coordinates $w,r$ this does inherit a
complex structure from $I$ for which $z,p,\zeta_{L},\pi_{L},\zeta_{R},\pi_{R}$
are holomorphic coordinates; this is the case discussed in section 4.2, and
similarly for the case with no coordinates $z,p$. In general, the symplectic
structure $\omega$ of ${\cal Z}$ is not holomorphic but descends from a
holomorphic symplectic structure on ${\cal N}$.

The field equations specify the subspace of ${\cal Z}$

$p_{a}=\frac{\partial K}{\partial z^{a}}~{},\qquad
r_{a^{\prime}}=\frac{\partial K}{\partial
w^{a^{\prime}}}~{},\qquad\pi_{L}=\frac{\partial
K}{\partial\zeta_{L}}~{},\qquad\pi_{R}=\frac{\partial
K}{\partial\zeta_{R}}~{},$ (5.32)

which is Lagrangian with respect to $\omega$.

###  5.3 An Extended Formulation

We now consider supermaps to the extended space ${\cal X}$

${\cal Q}=\\{\Phi:\mathbb{R}^{2|4}\longrightarrow{\cal X}\\}~{},$ (5.33)

in order to interpret the superfield constraints as defining an isotropic
submanifold. The coordinates on ${\cal X}$ are
$\widehat{Z}^{A},\widehat{P}_{A}$ given by (5.22) with holomorphic symplectic
structure $\omega^{(2,0)}=d\widehat{P}_{A}\wedge d\widehat{Z}^{A}$. The
supermaps are unconstrained superfields
$\widehat{\Phi}^{A}=\widehat{Z}^{A}(x,\theta,\bar{\theta})$,
$\widehat{\Phi}_{A}=\widehat{P}_{A}(x,\theta,\bar{\theta})$ and this space has
a holomorphic symplectic structure derived from (5.23), which is

$\widehat{\Omega}=\int
d^{2}x~{}d^{4}\theta~{}\delta\widehat{\Phi}_{A}\wedge\delta\widehat{\Phi}^{A}~{}.$
(5.34)

We now wish to formulate the superfield constraints as defining a submanifold
of ${\cal Q}$. For
$\zeta_{L},\zeta_{R},\pi_{L},\pi_{R},p_{L},r_{L},p_{R},r_{R}$ we choose the
constraints that correspond to identifying these supermaps with
$X_{L},X_{R},Y_{L},Y_{R},U_{L},V_{L},U_{R},V_{R}$ respectively. For
$z,z^{\prime},w,w^{\prime}$ we want constraints such that, on setting
$z=z^{\prime},w=\bar{w}^{\prime}$, the supermap $z=z^{\prime}$ corresponds to
the chiral superfield $\phi$ and the supermap $w=\bar{w}^{\prime}$ corresponds
to the twisted chiral superfield $\chi$. One way of doing this is to take
$z,z^{\prime}$ to be chiral and $w,w^{\prime}$ to be twisted chiral. However,
there is another possibility involving semi-chirals instead. We choose
constraints that identify $z,z^{\prime},w,w^{\prime}$ with semi-chiral fields
$Z,W$:

$z\sim Z_{L}~{},\qquad z^{\prime}\sim Z_{R}~{},\qquad w\sim W_{L}~{},\qquad
w^{\prime}\sim W_{R}~{}.$ (5.35)

Then $z=z^{\prime}$ corresponds to

$Z_{L}=Z_{R}$ (5.36)

and the constraints on $Z_{L},Z_{R}$ imply that $\phi\equiv Z_{L}=Z_{R}$ is a
chiral superfield. Similarly, $w=\bar{w}^{\prime}$ corresponds to

$W_{L}=\bar{W}_{R}$ (5.37)

and the constraints on $W_{L},W_{R}$ imply that $\chi\equiv W_{L}=\bar{W}_{R}$
is a twisted chiral superfield.

This gives a formalism in which _all_ superfields are semi-chiral. The
superfields are

$X_{L},X_{R},Y_{L},Y_{R},U_{L},V_{L},U_{R},V_{R},Z_{L},Z_{R},W_{L},W_{R}$
(5.38)

and the constraints are

$\displaystyle U_{La}=\frac{\partial\widehat{K}}{\partial
Z_{L}^{a}}~{},~{}~{}Z_{R}^{a}=\frac{\partial\widehat{K}}{\partial
U_{Ra}}~{},~{}~{}V_{Ra^{\prime}}=\frac{\partial\widehat{K}}{\partial
W_{R}^{a^{\prime}}}~{},~{}~{}W_{L}^{a^{\prime}}=\frac{\partial\widehat{K}}{\partial
V_{La^{\prime}}}~{},~{}~{}Y_{Ln}=\frac{\partial\widehat{K}}{\partial
X^{n}_{L}}~{},~{}~{}Y_{Rn^{\prime}}=\frac{\partial\widehat{K}}{\partial
X^{n^{\prime}}_{R}}$
$\displaystyle\bar{\mathbb{D}}_{+}U_{La}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{-}U_{Ra}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}V_{La^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{-}V_{Ra^{\prime}}=0~{},~{}~{}~{}\bar{\mathbb{D}}_{+}Y_{Ln}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}Y_{Rn^{\prime}}=0$
$\displaystyle\bar{\mathbb{D}}_{+}X^{n}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}X^{n^{\prime}}_{R}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{+}Z^{a}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}Z^{a}_{R}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{+}W^{a^{\prime}}_{L}=0~{},~{}~{}~{}~{}\bar{\mathbb{D}}_{-}W^{a^{\prime}}_{R}=0$
where we have defined

$\widehat{K}=K(Z_{L},\bar{Z}_{L},\bar{W}_{R},W_{R},X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})-U_{Ra}Z^{a}_{L}-\bar{U}_{R\bar{a}}\bar{Z}^{\bar{a}}_{L}-V_{La^{\prime}}\bar{W}^{a^{\prime}}_{R}-\bar{V}_{L\bar{a}^{\prime}}W_{R}^{\bar{a}^{\prime}}$
(5.40)

in terms of the generalised Kähler potential
$K(\phi,\bar{\phi},\chi,\bar{\chi},X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})$. One
can easily check that the above equations are equivalent to the equations
(5.1). The two last lines in (5.3) are the semi-chirality constraints and can
be interpreted as defining a holomorphic isotropic submanifold of ${\cal Q}$
with respect to $\widehat{\Omega}$. The first line in (5.3) defines a real
Lagrangian submanifold of ${\cal Q}$ with respect to ${\rm
Re}(\widehat{\Omega})$. The interesting new feature of this extended
construction is that the chirality constraints for chiral and twisted chiral
superfields appear as emergent constraints from the intersection of a
Lagrangian submanifold with an isotropic one.

The ambiguities (4.1) in the definition of $K$ are now realised as holomorphic
diffeomorphisms

$\displaystyle
U_{L}~{}\rightarrow~{}U_{L}+\partial_{Z_{L}}F^{+}(Z_{L},W_{L},X_{L})~{},$
$\displaystyle
V_{L}~{}\rightarrow~{}V_{L}+\partial_{W_{L}}F^{+}(Z_{L},W_{L},X_{L})~{},$
$\displaystyle
Y_{L}~{}\rightarrow~{}Y_{R}+\partial_{X_{L}}F^{+}(Z_{L},W_{L},X_{L})~{},$
(5.41) $\displaystyle
V_{R}~{}\rightarrow~{}V_{R}+\partial_{W_{R}}F^{-}(Z_{R},W_{R},X_{R})~{},$
$\displaystyle
U_{R}~{}\rightarrow~{}U_{R}+\partial_{Z_{R}}F^{-}(Z_{R},W_{R},X_{R})~{},$
$\displaystyle
Y_{R}~{}\rightarrow~{}Y_{R}+\partial_{X_{R}}F^{-}(Z_{R},W_{R},X_{R})~{},$

which obviously respect the corresponding chirality constraints.

##  6 Doubly extended space and superfields

In the previous section we constructed an extension of the original superfield
phase space that is equipped with a holomorphic symplectic form. The equations
of motion and constraints are equivalent to the intersection of a real
Lagrangian submanifold with a holomorphic isotropic submanifold. In our
construction of ${\cal X}$, the set of coordinates $z,w$ were ‘quadrupled’ to
the set $z,z^{\prime},w,w,p_{L},p_{R},r_{L},r_{R}$ while the leaf coordinates
$\zeta_{L},\zeta_{R}$ were doubled to $\zeta_{L},\zeta_{R},\pi_{L},\pi_{R}$.
In this section, we present an alternative formulation in which all
coordinates are quadrupled. Here we present the superfield formulation and
proceed formally. The present construction could be related to the double
groupoid picture outlined in [14].

###  6.1 Model with semi-chiral superfields only

Let us start with the discussion of the case in which there are only semi-
chiral fields. We follow the discussion and notation from subsection 4.1. We
double the construction given there and define a complex manifold

${\cal Z}\times{\cal Z}=M_{+}\times M_{-}\times M_{+}\times M_{-}$ (6.1)

with the holomorphic symplectic structure

$\omega^{(2,0)}=dP_{A}\wedge dZ^{A}+dP^{*}_{A}\wedge dZ^{*A}=d\pi_{Ln}\wedge
d\zeta^{n}_{L}+d\pi_{Rn^{\prime}}\wedge
d\zeta^{n^{\prime}}_{R}+d\pi^{*}_{Ln}\wedge
d\zeta^{*n}_{L}+d\pi^{*}_{Rn^{\prime}}\wedge d\zeta^{*n^{\prime}}_{R}~{},$
(6.2)

where the coordinates with a star are the coordinates for the second ${\cal
Z}$. Doubling the coordinates also doubles the corresponding superfields. We
use the same notation, adding a star for the superfields taking values in the
second ${\cal Z}$. For a given generalized Kähler potential
$K(X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})$ we define the following real
generating function

$\widehat{K}=K(X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})-X_{L}^{n}Y^{*}_{Ln}-\bar{X}_{L}^{\bar{n}}\bar{Y}^{*}_{L\bar{n}}-X_{R}^{n^{\prime}}Y^{*}_{Rn^{\prime}}-\bar{X}_{R}^{\bar{n^{\prime}}}\bar{Y}^{*}_{R\bar{n^{\prime}}}~{},$
(6.3)

which depends on exactly half of the fields. We observe that $\widehat{K}$
defines the submanifold specified by the following equations

$\displaystyle Y_{Ln}=\frac{\partial\widehat{K}}{\partial
X_{L}^{n}}=\frac{\partial K}{\partial X^{n}_{L}}-Y_{Ln}^{*}~{},$
$\displaystyle X_{L}^{*n}=\frac{\partial\widehat{K}}{\partial
Y^{*}_{Ln}}=X^{n}_{L}~{},$ (6.4) $\displaystyle
Y_{Rn^{\prime}}=\frac{\partial\widehat{K}}{\partial
X^{n^{\prime}}_{R}}=\frac{\partial K}{\partial
X^{n^{\prime}}_{R}}-Y_{Rn^{\prime}}^{*}~{},$ $\displaystyle
X_{R}^{*n^{\prime}}=\frac{\partial\widehat{K}}{\partial
Y_{Rn^{\prime}}}=X^{n^{\prime}}_{R}~{}.$

If we make the shifts $Y_{L}\to Y_{L}+Y_{L}^{*}$ and $Y_{R}\to
Y_{R}+Y_{R}^{*}$ we recover the equations (4.1).

The space of unconstrained superfields

$\\{\Phi:\mathbb{R}^{2|4}\longrightarrow{\cal Z}\times{\cal Z}\\}~{},$ (6.5)

is equipped with the holomorphic symplectic form

$\Omega=\int
d^{2}x~{}d^{4}\theta~{}\Big{(}\delta\Phi_{A}\wedge\delta\Phi^{A}+\delta\Phi^{*}_{A}\wedge\delta\Phi^{*A}\Big{)}~{}.$
(6.6)

The semi-chiral constraints on the superfields correspond to a holomorphic
isotropic submanifold with respect to $\Omega$. On the other hand, the
equations (6.1) correspond to a real Lagrangian submanifold with respect to
${\rm Re}(\Omega)$. Thus we conclude that the original equations of motion and
constraints can be interpreted as the intersection of these two submanifolds.

###  6.2 General case

Formally, the general case can be viewed as a combination of the constructions
from subsections 5.3 and 6.1. For a generalised Kähler potential
$K(\phi,\bar{\phi},\chi,\bar{\chi},X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})$ we
define

$\displaystyle\widehat{K}=K(Z_{L},\bar{Z}_{L},\bar{W}_{R},W_{R},X_{L},\bar{X}_{L},X_{R},\bar{X}_{R})-U_{Ra}Z^{a}_{L}-\bar{U}_{R\bar{a}}\bar{Z}^{\bar{a}}_{L}-V_{La^{\prime}}\bar{W}^{a^{\prime}}_{R}-\bar{V}_{L\bar{a}^{\prime}}W_{R}^{\bar{a}^{\prime}}$
$\displaystyle-
X_{L}^{n}Y^{*}_{Ln}-\bar{X}_{L}^{\bar{n}}\bar{Y}^{*}_{L\bar{n}}-X_{R}^{n^{\prime}}Y^{*}_{Rn^{\prime}}-\bar{X}_{R}^{\bar{n^{\prime}}}\bar{Y}^{*}_{R\bar{n^{\prime}}}~{}.$
(6.7)

Then the equations of motion can be written as follows

$\displaystyle U_{La}=\frac{\partial\widehat{K}}{\partial
Z_{L}^{a}}~{},~{}~{}~{}Z_{R}^{a}=\frac{\partial\widehat{K}}{\partial
U_{Ra}}~{},~{}~{}~{}V_{Ra^{\prime}}=\frac{\partial\widehat{K}}{\partial
W_{R}^{a^{\prime}}}~{},~{}~{}~{}W_{L}^{a^{\prime}}=\frac{\partial\widehat{K}}{\partial
V_{La^{\prime}}}~{},$ $\displaystyle
X_{L}^{*n}=\frac{\partial\widehat{K}}{\partial
Y^{*}_{Ln}}~{},~{}~{}~{}X_{R}^{*n^{\prime}}=\frac{\partial\widehat{K}}{\partial
Y_{Rn^{\prime}}}~{},~{}~{}~{}Y_{Ln}=\frac{\partial\widehat{K}}{\partial
X^{n}_{L}}~{},~{}~{}~{}Y_{Rn^{\prime}}=\frac{\partial\widehat{K}}{\partial
X^{n^{\prime}}_{R}}~{},$
provided that we impose the semi-chiral conditions on all fields. All local
geometry and holomorphic symplectic structures follow from those in
subsections 5.3 and 6.1. In this bigger space we can interpret the equations
of motion as the intersection of a holomorphic isotropic submanifold (imposing
the semi-chiral conditions on the superfields) and the real Lagrangian
submanifold defined by $\widehat{K}$. As previously discussed, the ambiguities
in $K$ are now realized as diffeomorphisms in this bigger space.

##  7 Summary and outlook

We have taken a fresh look at generalised Kähler geometry motivated by the
$N=(2,2)$ superfield formulation of the corresponding sigma model. Instead of
looking at the action, we have developed a formulation with a doubled target
space, and constraining to a Lagrangian submanifold implies the field
equations. This might be viewed as a kind of superspace Hamiltonian formalism.

Somewhat surprisingly, alternative superfield formulations leads us to
quadruple some or all of the superfields so as to elegantly embed the original
equations of motion in a bigger framework. We saw that in some situations our
glueing conditions do not satisfy the cocycle condition on triple
intersections so that the space is not properly a manifold, but formally these
spaces appear to have well-defined complex and symplectic structures.

One of the motivations of this work was to try find a physicists’ explanation
of the construction of [8]. It is interesting to also compare with the PhD
thesis by Yucong Jiang [14] where bigger spaces associated with generalised
Kähler manifold are constructed. There are interesting formal similarities and
it would be interesting to understand the relation between these constructions
in more detail.

One drawback of our discussion is that it assumes the regularity of the
Poisson foliations appearing in the generalised Kähler geometry. We need to
impose this requirement in order to use the original superfield formalism
involving different kind of fields (chiral, twisted chiral and semi-chiral)
globally. However, in general this is not the case: generalised Kähler
geometry can exhibit type change so that different regions of the space will
have different foliation structures [6]. In Section 6 we reformulated the
equations of motion in terms of semi-chiral fields only with a quadrupled
target space. In this formulation the chirality conditions are emergent,
arising from imposing a real Lagrangian condition via the real generating
function $\widehat{K}$ defined on the bigger space. We can imagine a situation
in which the changes in $\widehat{K}$ lead to changes in the superspace
constraints (corresponding to type change). That is, a formulation in such an
extended space could provide a global description even for target spaces in
which there is type change. We find this observation exciting and plan to
study it further elsewhere.

Acknowledgement:

We are grateful to Francesco Bonechi and Marco Gualtieri for illuminating
discussions. We thank the Galileo Galilei Institute for Theoretical Physics,
Florence for providing the stimulating atmosphere where this work was
initiated. The research of CH is supported by the STFC Consolidated Grant
ST/T000791/1. The research of MZ is supported by the VR excellence center
grant “Geometry and Physics” 2022-06593.

##  Appendix A Appendix: $N=(1,1)$ and $N=(2,2)$ superspace conventions

Here we sumarise briefly our superspace conventions. We follow the conventions
used in [15] and in [4]. For a detailed introduction to superfield formalism,
see [9].

We work in 2-dimensional Minkowski space and use two-component spinors
$\psi^{\alpha}$ which carry a spinor index $\alpha=(+,-)$. The spinor indices
are raised and lowered using the antisymmetric charge conjugation matrix
$C_{\alpha\beta}$ with

$C_{+-}=i~{},\qquad C_{\alpha\beta}=-C_{\beta\alpha}=-C^{\alpha\beta}~{},$
(A.1)

so that

$\psi^{\alpha}=C^{\alpha\beta}\psi_{\beta}~{},\qquad\psi_{\alpha}=\psi^{\beta}C_{\beta\alpha}~{},\qquad\psi_{\pm}=\mp
i\psi^{\mp}~{}.$ (A.2)

The Lorentz-invariant spinor inner product is

$C_{\alpha\beta}\psi^{\alpha}\chi^{\beta}=\psi_{\alpha}\chi^{\alpha}=i\psi^{+}\chi^{-}-i\psi^{-}\chi^{+}$
(A.3)

and its complex conjugate is

$(C_{\alpha\beta}\psi^{\alpha}\chi^{\beta})^{*}=-i(\psi^{+})^{*}(\chi^{-})^{*}+i(\psi^{-})^{*}(\chi^{+})^{*}=C_{\alpha\beta}(\chi^{\alpha})^{*}(\psi^{\beta})^{*}$
(A.4)

(where we have assumed that both $\psi$ and $\chi$ are Grassman odd).

The $N=(1,1)$ superspace $\mathbb{R}^{2|2}$ has an even sector which is two-
dimensional Minkowski space with coordinates $x^{\alpha\beta}$ which we
represent as (symmetric traceless) bispinors, and we use the notation
$x^{{+\\!\\!\\!+}}=x^{++},x^{=}=x^{--}$. The corresponding even derivative is
written as $\partial_{\alpha\beta}=\frac{\partial}{\partial x^{\alpha\beta}}$.
The Grassmann (odd) coordinates $\theta^{\alpha}$ are real (Majorana) two
component spinors. On $\mathbb{R}^{2|2}$ we define two odd first order
differential operators $D_{\alpha}$ and $Q_{\alpha}$ as follows

$\displaystyle
D_{+}=\partial_{+}+i\theta^{+}\partial_{{+\\!\\!\\!+}}~{},~{}~{}~{}~{}~{}D_{-}=\partial_{-}+i\theta^{-}\partial_{=}~{},$
(A.5) $\displaystyle
Q_{+}=i\partial_{+}+\theta^{+}\partial_{{+\\!\\!\\!+}}~{},~{}~{}~{}~{}~{}Q_{-}=i\partial_{-}+\theta^{-}\partial_{=}~{}.$
(A.6)

They satisfy the algebra

$\displaystyle
D_{+}^{2}=i\partial_{+\\!\\!\\!+}~{},~{}~{}~{}~{}D_{-}^{2}=i\partial_{=}~{},~{}~{}~{}~{}\\{D_{+},D_{-}\\}=0~{},$
(A.7) $\displaystyle
Q_{+}^{2}=i\partial_{+\\!\\!\\!+}~{},~{}~{}~{}~{}Q_{-}^{2}=i\partial_{=}~{},~{}~{}~{}~{}\\{Q_{+},Q_{-}\\}=0~{},$
(A.8)

with

$\\{Q_{\alpha},D_{\beta}\\}=0~{}.$ (A.9)

In the supersymmetry literature, the $D_{\alpha}$ are referred to as spinorial
covariant derivatives and the $Q_{\alpha}$ are referred to as the
supersymmetry generators. On the space of maps

$\\{\mathbb{R}^{2|2}~{}\longrightarrow~{}M\\}$ (A.10)

$Q_{\alpha}$ and $D_{\alpha}$ can be viewed as odd vector fields satisfying
the algebra presented above.

Next we discuss the $N=(2,2)$ superspace $\mathbb{R}^{2|4}$. The bosonic
coordinates are $x^{\alpha\beta}$ as before, but now the grassmann variables
are _complex_ spinors $\theta^{\alpha}$ with complex conjugates
$\bar{\theta}^{\alpha}$ and odd derivatives
$(\partial_{\alpha},\bar{\partial}_{\alpha})$. We define two sets of first
order odd differential operators
$(\mathbb{D}_{\alpha},\bar{\mathbb{D}}_{\alpha})$ and
$(\mathbb{Q}_{\alpha},\bar{\mathbb{Q}}_{\alpha})$ by

$\displaystyle\mathbb{D}_{+}=\partial_{+}+\frac{i}{2}\bar{\theta}^{+}\partial_{{+\\!\\!\\!+}}~{},~{}~{}~{}~{}~{}\bar{\mathbb{D}}_{+}=\bar{\partial}_{+}+\frac{i}{2}\theta^{+}\partial_{{+\\!\\!\\!+}}~{},$
(A.11)
$\displaystyle\mathbb{Q}_{+}=i\partial_{+}+\frac{1}{2}\bar{\theta}^{+}\partial_{{+\\!\\!\\!+}}~{},~{}~{}~{}~{}~{}\bar{\mathbb{Q}}_{+}=i\bar{\partial}_{+}+\frac{1}{2}\theta^{+}\partial_{{+\\!\\!\\!+}}~{},$
(A.12)

for the $+$ sector, with analogous formulae for the $-$ sector. The algebra is
given by the following relations

$\displaystyle\\{\mathbb{D}_{+},\bar{\mathbb{D}}_{+}\\}=i\partial_{{+\\!\\!\\!+}}~{},~{}~{}~{}~{}\\{\mathbb{D}_{-},\bar{\mathbb{D}}_{-}\\}=i\partial_{=}~{},~{}~{}~{}~{}\\{\mathbb{D}_{\alpha},\mathbb{D}_{\beta}\\}=0~{},~{}~{}~{}~{}\\{\bar{\mathbb{D}}_{\alpha},\bar{\mathbb{D}}_{\beta}\\}=0~{},$
(A.13)
$\displaystyle\\{\mathbb{Q}_{+},\bar{\mathbb{Q}}_{+}\\}=i\partial_{{+\\!\\!\\!+}}~{},~{}~{}~{}~{}\\{\mathbb{Q}_{-},\bar{\mathbb{Q}}_{-}\\}=i\partial_{=}~{},~{}~{}~{}~{}\\{\mathbb{Q}_{\alpha},\mathbb{Q}_{\beta}\\}=0~{},~{}~{}~{}~{}\\{\bar{\mathbb{Q}}_{\alpha},\bar{\mathbb{Q}}_{\beta}\\}=0~{},$
(A.14)

where all $\mathbb{Q}/\bar{\mathbb{Q}}$-operators anti commute with all
$\mathbb{D}/\bar{\mathbb{D}}$-operators. Again, in the supersymmetry
literature, $\mathbb{D}_{\alpha},\bar{\mathbb{D}}_{\alpha}$ are referred to as
spinorial covariant derivatives and
$\mathbb{Q}_{\alpha},\bar{\mathbb{Q}}_{\alpha}$ are referred to as
supersymmetry generators. On the space of maps

$\\{\mathbb{R}^{2|4}~{}\longrightarrow~{}M\\}$ (A.15)

$\mathbb{D}_{\alpha},\bar{\mathbb{D}}_{\alpha}$ and
$\mathbb{Q}_{\alpha},\bar{\mathbb{Q}}_{\alpha}$ can be interpreted as odd
vector fields with the algebra given above.

Here we have a new feature, we can consider maps which are annihilated by
specific nilpotent combinations of $\mathbb{D}$’s and $\bar{\mathbb{D}}$’s so
that the supersymmetry algebra generated by $\mathbb{Q}$, $\bar{\mathbb{Q}}$
acts on the subspace that is constrained in this way. That is, the superfields
constrained in this way furnish a representation of the supersymmetry algebra
generated by $\mathbb{Q},\bar{\mathbb{Q}}$ and $\frac{\partial}{\partial
x^{\alpha\beta}}$. Below we summarise the constraints, that we use throughout
the paper

$\bar{\mathbb{D}}_{\pm}\phi=0$ | chiral superfield
---|---
$\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Sigma=0$ | complex linear superfield
$\bar{\mathbb{D}}_{+}\chi=0~{},~{}~{}~{}\mathbb{D}_{-}\chi=0$ | twisted chiral fields
$\bar{\mathbb{D}}_{+}\mathbb{D}_{-}\Lambda=0$ | twisted complex linear superfield
$\bar{\mathbb{D}}_{+}\bar{X}_{L}=0$ (or $\bar{\mathbb{D}}_{+}\bar{Y}_{L}=0$) | left semichiral superfield
$\bar{\mathbb{D}}_{-}X_{R}=0$ (or $\bar{\mathbb{D}}_{-}Y_{R}=0$) | right semichiral superfield

Each of these constraints has a general solution in terms of unconstrained
complex superfields. These are as follows:

$\displaystyle\phi=\bar{\mathbb{D}}_{+}\bar{\mathbb{D}}_{-}\Phi$
$\displaystyle\Sigma=\bar{\mathbb{D}}_{+}\Phi_{1}+\bar{\mathbb{D}}_{-}\Phi_{2}$
$\displaystyle\chi=\bar{\mathbb{D}}_{+}\mathbb{D}_{-}\Phi$ (A.16)
$\displaystyle\Lambda=\bar{\mathbb{D}}_{+}\Phi_{1}+\mathbb{D}_{-}\Phi_{2}$
$\displaystyle X_{L}=\bar{\mathbb{D}}_{+}\Phi$ $\displaystyle
X_{R}=\bar{\mathbb{D}}_{-}\Phi$

Here the $\Phi$’s are unrestricted complex superfields.

The constrains on the superfields allow the following list of field
redefinitions which are compatible with the constraints

$\displaystyle\phi~{}\longrightarrow~{}f_{1}(\phi)~{},$
$\displaystyle\Sigma~{}\longrightarrow~{}f_{2}(\phi)\Sigma+f_{3}(\phi,\chi,X_{L})+f_{4}(\phi,\bar{\chi},X_{R})~{},$
$\displaystyle\chi~{}\longrightarrow~{}f_{5}(\chi)~{},$ (A.17)
$\displaystyle\Lambda~{}\longrightarrow~{}f_{6}(\chi)\Lambda+f_{7}(\phi,\chi,X_{L})+f_{8}(\bar{\phi},\chi,\bar{X}_{R})~{},$
$\displaystyle X_{L}~{}\longrightarrow~{}f_{9}(\phi,\chi,X_{L})~{},$
$\displaystyle X_{R}~{}\longrightarrow~{}f_{10}(\phi,\bar{\chi},X_{R})~{},$

(together with their complex conjugates) and here $f_{i}$ are arbitrary
functions. In the context of non-linear sigma models the lowerest components
of these superfields are interpreted as coordinates on the target space $M$
and thus we get the restrictions on the diffeomorphisms that are compatible
with the constraints, and these in turn are related to geometric structures on
$M$.

##  Appendix B Appendix: Generalised Kähler geometry, gerbes and glueings

In this appendix we review the relevant facts about generalised Kähler
geometry and gerbes. We follow closely [10] and point out some properties
which were not discussed in [10] but which are important for this paper. We
consider a smooth manifold $M$ with an open cover $\\{U_{\alpha}\\}$ such that
all open sets and finite intersections thereof are contractible.

A set of transition functions $f_{\alpha\beta}:U_{\alpha}\cap
U_{\beta}\to\mathbb{R}$ with ($f_{\alpha\beta}=-f_{\beta\alpha}$) from the
intersection of any two patches $U_{\alpha}\cap U_{\beta}$ to $\mathbb{R}$
define a real line bundle if they satisfy the cocycle condition on triple
overlaps $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$ that

$f_{\alpha\beta}+f_{\beta\gamma}+f_{\gamma\alpha}=0$ (B.1)

while if they satisfy this modulo $2\pi$ then the maps
$g_{\alpha\beta}:U_{\alpha}\cap U_{\beta}\to U(1)$ with
$g_{\alpha\beta}=e^{if_{\alpha\beta}}$ satisfy the cocycle condition

$g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=1$ (B.2)

on $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$ and so define a complex line
bundle. A gerbe is a set of maps on triple intersections
$h_{\alpha\beta\gamma}:U_{\alpha}\cap U_{\beta}\cap U_{\gamma}\to\mathbb{R}$
satisfying

$h_{\beta\gamma\delta}+h_{\delta\gamma\alpha}+h_{\alpha\beta\delta}+h_{\beta\alpha\gamma}=0$
(B.3)

modulo $2\pi$ so that $k_{\alpha\beta\gamma}=e^{ih_{\alpha\beta\gamma}}$ are
maps $k_{\alpha\beta\gamma}:U_{\alpha}\cap U_{\beta}\cap U_{\gamma}\to U(1)$
satisfying

$k_{\beta\gamma\delta}k_{\delta\gamma\alpha}k_{\alpha\beta\delta}k_{\beta\alpha\gamma}=1$
(B.4)

We follow the setup and the notation of [10]. On the intersection of two
patches $U_{\alpha}\cap U_{\beta}$ the generalized real Kähler potential has
the glueing

$K_{\alpha}-K_{\beta}=F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+\bar{F}^{+}(\bar{\phi},\bar{\chi},\bar{X}_{L})+F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})+\bar{F}_{\alpha\beta}^{-}(\bar{\phi},\chi,\bar{X}_{R})~{}.$
(B.5)

The functions $F^{\pm}$ are defined up to the following ambiguities

$\displaystyle
F_{\alpha\beta}^{+}(\phi,\chi,X_{L})~{}\rightarrow~{}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+\rho_{\alpha\beta}(\phi)+\sigma_{\alpha\beta}(\chi)~{},$
(B.6) $\displaystyle
F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})~{}\rightarrow~{}F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})-\rho_{\alpha\beta}(\phi)-\bar{\sigma}_{\alpha\beta}(\bar{\chi})~{}$
(B.7)

and they can be chosen such that
$F_{\alpha\beta}^{\pm}=-F_{\beta\alpha}^{\pm}$. Moreover they satisfy the
following conditions on the triple intersections $U_{\alpha}\cap U_{\beta}\cap
U_{\gamma}$

$\displaystyle
F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+F_{\beta\gamma}^{+}(\phi,\chi,X_{L})+F_{\gamma\alpha}^{+}(\phi,\chi,X_{L})=ic_{\alpha\beta\gamma}(\phi)-ib_{\alpha\beta\gamma}(\chi)~{},$
(B.8) $\displaystyle
F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})+F_{\beta\gamma}^{-}(\phi,\bar{\chi},X_{R})+F_{\gamma\alpha}^{-}(\phi,\bar{\chi},X_{R})=-ic_{\alpha\beta\gamma}(\phi)-i\bar{b}_{\alpha\beta\gamma}(\bar{\chi})~{}$
(B.9)

for some functions
$c_{\alpha\beta\gamma}(\phi),{b}_{\alpha\beta\gamma}({\chi})$. On quadruple
intersections $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}\cap U_{\delta}$

$\displaystyle
c_{\beta\gamma\delta}(\phi)+c_{\delta\gamma\alpha}(\phi)+c_{\alpha\beta\delta}(\phi)+c_{\beta\alpha\gamma}(\phi)=\frac{i}{4}d_{\alpha\beta\gamma\delta}~{},$
(B.10) $\displaystyle
b_{\beta\gamma\delta}(\chi)+b_{\delta\gamma\alpha}(\chi)+b_{\alpha\beta\delta}(\chi)+b_{\beta\alpha\gamma}(\chi)=\frac{i}{4}d_{\alpha\beta\gamma\delta}~{},\
$ (B.11)

where $d_{\alpha\beta\gamma\delta}$ is constant. If the 3-form $H/2\pi$ is
quantised, i.e. if it represents an integral cohomology class, then
$d_{\alpha\beta\gamma\delta}\in 2\pi\mathbb{Z}$ so that
$e^{4c_{\alpha\beta\gamma}}$ and $e^{4b_{\alpha\beta\gamma}}$ each satisfy
(B.4) and so each defines a gerbe, and since these are holomorphic functions,
they are holomorphic gerbes. In our discussion here of the classical theory,
the quantisation condition on $H$ does not play a role.

Using the above properties we can construct different objects which satisfy
the cocycle condition on the triple intersection. For example we have the
following relation on $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$

$\displaystyle\frac{\partial}{\partial\phi^{a}}\Big{(}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})\Big{)}+\frac{\partial}{\partial\phi^{a}}\Big{(}F_{\beta\gamma}^{+}(\phi,\chi,X_{L})+F_{\beta\gamma}^{-}(\phi,\bar{\chi},X_{R})\Big{)}$
$\displaystyle+\frac{\partial}{\partial\phi^{a}}\Big{(}F_{\gamma\alpha}^{+}(\phi,\chi,X_{L})+F_{\gamma\alpha}^{-}(\phi,\bar{\chi},X_{R})\Big{)}=0~{}.$
(B.12)

Analogously, the following objects satisfy the cocycle conditions on triple
intersections $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$

$\displaystyle\frac{\partial}{\partial\chi^{a^{\prime}}}\Big{(}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})+\bar{F}_{\alpha\beta}^{-}(\bar{\phi},\chi,\bar{X}_{R})\Big{)}~{},$
(B.13) $\displaystyle\frac{\partial}{\partial
X^{n}_{L}}F_{\alpha\beta}^{+}(\phi,\chi,X_{L})~{},$ (B.14)
$\displaystyle\frac{\partial}{\partial
X^{n^{\prime}}_{R}}F_{\alpha\beta}^{-}(\phi,\bar{\chi},X_{R})~{}.$ (B.15)

Thus we can use them to construct different affine bundles.

Next we turn to the geometrical setting of the construction of section 5. Let
$\partial_{\pm},\bar{\partial}_{\pm}$ be the Dolbeaut exterior derivatives for
the complex structure $I_{\pm}$. Suppose in each $U_{\alpha}$ there is a (1,0)
form $X_{\alpha}$ with

$X_{\alpha}=X_{\alpha a}dz^{a}+X_{\alpha a^{\prime}}dr^{a^{\prime}}+X_{\alpha
n}d\zeta^{n}$ (B.16)

with glueing relations

$X_{\beta}=X_{\alpha}+\partial_{+}F^{+}_{\alpha\beta}$ (B.17)

These then satisfy (B.8),(B.10),(B.11) and so see the gerbe structure.
Defining $b_{\alpha}=\bar{\partial}_{+}X_{\alpha}$, we have

$b_{\beta}=b_{\alpha}$ (B.18)

so $b$ is a globally defined (1,1) form and

$h\equiv\bar{\partial}_{+}b_{\alpha}$ (B.19)

is zero, so that $b_{\alpha}$ is a holomorphic gerbe connection which is flat
($h=0$), with prepotential $X_{\alpha}$. Now the components $(X_{\alpha
a},X_{\alpha a^{\prime}},X_{\alpha n})$ have exactly the same glueing
relations as $p_{L},r_{L},\pi_{L}$ from (5.13),(5.14), i.e.
$p_{L},r_{L},\pi_{L}$ can be viewed as the components of the prepotential for
a flat holomorphic gerbe connection.

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