# On the Mini-Superambitwistor Space and𝒩=8

Authors: Christian Sämann
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

We construct a new supertwistor space suited for establishing a Penrose-Ward
transform between certain bundles over this space and solutions to the𝒩=8
super-Yang-Mills equations in three dimensions. This mini-superambitwistor
space is obtained by dimensional reduction of the superambitwistor space, the
standard superextension of the ambitwistor space. We discuss in detail the construction
of this space and its geometry before presenting the Penrose-Ward
transform. We also comment on a further such transform for purely bosonic
Yang-Mills-Higgs theory in three dimensions by considering third-order formal
“subneighborhoods" of a miniambitwistor space.


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## Body

## 1. Introduction and Results

A   convenient way of
describing solutions to a wide class of field equations has been developed
using twistor geometry [1–3]. In this picture, solutions to nonlinear field
equations are mapped bijectively via the Penrose-Ward transform to holomorphic
structures on vector bundles over an appropriate twistor space. Such twistor
spaces are well known for many theories including self-dual Yang-Mills (SDYM)
theory and its supersymmetric extensions as well as 𝒩-extended full super-Yang-Mills (SYM)
theories. In three dimensions, there are twistor spaces suited for describing
the Bogomolny equations and their supersymmetric variants. The purpose of this
paper is to fill the gaps for three-dimensional 𝒩=8 super-Yang-Mills theory as well as for
three-dimensional Yang-Mills-Higgs theory; the cases for intermediate 𝒩 follow trivially. The idea we follow in this
paper has partly been presented in [4].Recall that the supertwistor space describing𝒩=3 SDYM theory is the open subset 𝒫3|3:=ℂP3|3∖ℂP1|3;
its anti-self-dual counterpart is 𝒫*3|3≅𝒫3|3,
where the parity assignment of the appearing coordinates is simply inverted.
Furthermore, we denote by 𝒫2|3 the mini-supertwistor space obtained by
dimensional reduction from 𝒫3|3 and used in the description of the
supersymmetric Bogomolny equations in three dimensions.For𝒩=4 SYM theory, the appropriate twistor space L5|6 is now obtained from the product 𝒫3|3×𝒫*3|3 upon imposing a quadric condition reducing the
bosonic dimensions by one (in fact, the field theory described by L5|6 is 𝒩=3 SYM theory in four dimensions, which is
equivalent to 𝒩=4 SYM theory on the level of equations of
motion; in three dimensions, the same relation holds between 𝒩=6 and 𝒩=8 SYM theories). We  perform an analogous construction
for 𝒩=8 SYM theory by starting from the product 𝒫2|3×𝒫*2|3 of two mini-supertwistor spaces. The
dimensional reduction turning the super-self-duality equations in four
dimensions into the super-Bogomolny equations in three dimensions translates
into a reduction of the quadric condition, which yields a constraint only to be
imposed on the diagonal ℂPΔ1=diag(ℂP1×ℂP*1) in the base of the vector bundle 𝒫2|3×𝒫*2|3→ℂP1×ℂP*1.
Thus, the resulting space L4|6 is not a vector bundle but only a fibration,
and the sections of this fibration form a torsion sheaf, as we will see. More
explicitly, the bosonic parts
of the fibers of L4|6 over ℂP1×ℂP*1 are isomorphic to ℂ2 at generic points, but over the diagonal ℂPΔ1,
they are isomorphic to ℂ.As expected, we find a twistor correspondence between
points inℂ3|12 and holomorphic sections of L4|6 as well as between points in L4|6 and certain sub-supermanifolds in ℂ3|12.
After introducing a real structure on L4|6,
one finds a nice interpretation of the spaces involved in the twistor
correspondence in terms of lines with marked points in ℝ3,
which resembles the appearance of flag manifolds in the well-established
twistor correspondences. Recalling that L5|6 is a Calabi-Yau supermanifold
(the essential prerequisite for being the target space of a
topological B-model), we are led to examine an analogous
question for L4|6.
The Calabi-Yau property essentially amounts to a vanishing of the first Chern
class of TL5|6,
which in turn encodes information about the degeneracy locus of a certain set
of sections of the vector bundle L5|6→ℂP1×ℂP*1.
We find that the degeneracy loci of L5|6 and L4|6 are equivalent (identical up to a principal
divisor).A Penrose-Ward transform for𝒩=8 SYM theory can now be conveniently
established. To define the analogue of a holomorphic vector bundle over the
space L4|6,
we have to remember that in the Čech description, a holomorphic vector bundle
is completely characterized by its transition functions, which in turn form a
group-valued Čech 1-cocycle. These objects are still well
defined on L4|6 and we will use such a 1-cocycle to define
what we will call a pseudobundle over L4|6.
In performing the transition between these pseudobundles and solutions to the 𝒩=8 SYM equations, care must be taken when
discussing these bundles over the subset L4|6|ℂPΔ1 of their base. Eventually, however, one
obtains a bijection between gauge equivalence classes of solutions to the 𝒩=8 SYM equations and equivalence classes of
holomorphic pseudobundles over L4|6,
which turn into holomorphically trivial vector bundles upon restriction to any
holomorphic submanifold ℂP1×ℂP*1↪L4|6.Considering the reduction ofL5|6⊂𝒫3|3×𝒫*3|3 to the bodies of the involved spaces
(i.e., putting the fermionic coordinates on all the spaces to
zero), it is possible to find a twistor correspondence for
certain formal neighborhoods of L5|0⊂𝒫3|0×𝒫*3|0 on which a Penrose-Ward transform for purely
bosonic Yang-Mills theory in four dimensions can be built. To improve our
understanding of the mini-superambitwistor space, it is also helpful to discuss
the analogous construction with L4|0.
We find that a third-order subthickening, (i.e., a thickening of the fibers
which are only of dimension one) inside of 𝒫2|0×𝒫*2|0 must be considered to describe solutions to
the Yang-Mills-Higgs equations in three dimensions by using pseudobundles over L4|0.To clarify the role of the
spaceL4|6 in detail, it would be interesting to
establish a dimensionally reduced version of the construction considered by
Movshev in [5]. In
this paper, the author constructs a “Chern-Simons triple” consisting of a
differential graded algebra (A,d) and a d-closed trace functional on a certain space ST related to the superambitwistor space. This
Chern-Simons triple on ST is then conjectured to be equivalent to 𝒩=4 SYM theory in four dimensions. The way the
construction is performed suggests a quite straightforward dimensional
reduction to the case of the mini-superambitwistor space. Besides delivering a
Chern-Simons triple for 𝒩=8 SYM theory in three dimensions, this
construction would possibly shed more light on the unusual properties of the
fibration L4|6.Following Witten's seminal paper [6], there has been growing
interest in different supertwistor spaces suited as target spaces for the
topological B-model (see e.g. [4, 7–14]). Although it is not clear
what the topological B-model on L4|6 looks like exactly (we will present some
speculations in Section 3.7), the mini-superambitwistor space might also prove
to be interesting from the topological string theory point of view. In
particular, the mini-superambitwistor space L4|6 is probably the mirror of the
mini-supertwistor space 𝒫2|4.
Maybe even the extension of infinite dimensional symmetry algebras [12] from the self-dual to the
full case is easier to study in three dimensions due to the greater similarity
of self-dual and full theory and the smaller number of conformal generators.Note that we arenot describing the space of null geodesics
in three dimensions; this space has been constructed in [13].The outline of this paper is as follows. In Section2,
we review the construction of the supertwistor spaces for SDYM theory and SYM
theory. Furthermore, we present the dimensional reduction yielding the
mini-supertwistor space used for capturing solutions to the super-Bogomolny
equations. Section 3, the main part, is then devoted to deriving the
mini-superambitwistor space in several ways and discussing in detail the
associated twistor correspondence and its geometry. Moreover, we comment on a
topological B-model on this space. In Section 4, the Penrose-Ward transform for
three-dimensional 𝒩=8 SYM theory is presented. First, we review both
the transform for 𝒩=4 SYM theory in four dimensions and aspects of 𝒩=8 SYM theory in three dimensions. Then, we
introduce the pseudobundles over L4|6,
which take over the role of vector bundles over the
space L4|6.
Eventually, we present the actual Penrose-Ward transform in detail. In the last
section, we discuss the third-order subthickenings of L4|0 in 𝒫2|0×𝒫*2|0,
which are used in the Penrose-Ward transform for purely bosonic
Yang-Mills-Higgs theory.


## 2. Review of Supertwistor Spaces

We will briefly review some elementary facts on
supertwistor spaces and fix our conventions in this section. For a broader
discussion of supertwistor and superambitwistor spaces in conventions close to
the ones employed here, see [15]. For more details on the mini-supertwistor spaces, we
refer to [4, 14].


### 2.1. Supertwistor Spaces

The supertwistor space ofℂ4|2𝒩 is defined as the rank 2|𝒩 holomorphic supervector bundle (2.1)𝒫3|𝒩:=ℂ2⊗𝒪(1)⊕ℂ𝒩⊗Π𝒪(1) over the Riemann sphere ℂP1.
Here, Π is the parity changing operator which inverts
the parity of the fiber coordinates. The base space of this bundle is covered
by the two patches U± on which we have the standard coordinates λ±∈U±≅ℂ with λ+=(λ−)−1 on U+∩U−.
Over U±,
we introduce furthermore the bosonic fiber coordinates z±α with α=1,2 and the fermionic fiber coordinates ηi± with i=1,…,𝒩.
On the intersection U+∩U−,
we thus have (2.2)z+α=1z−3z−α,ηi+=1z−3ηi−withz±3=λ±. The supermanifold 𝒫3|𝒩 as a whole is covered by the two patches 𝒰±:=𝒫3|𝒩|U± with local coordinates (z±1,z±2,z±3,η1±,…,η𝒩±).Global holomorphic sections of the vector bundle𝒫3|𝒩→ℂP1 are given by polynomials of degree one, which
are parameterized by moduli (xαα˙,ηiα˙)∈ℂ4|2𝒩 via (2.3)z±α=xαα˙λα˙±,ηi±=ηiα˙λα˙±, where we introduced the
simplifying spinorial notation (2.4)(λα˙+):=(1λ+),(λα˙−):=(λ−1).Equations (2.3), the so-called incidence relations, define a twistor
correspondence between the spaces 𝒫3|𝒩 and ℂ4|2𝒩,
which can be depicted in the double  fibration (2.5)
Here, ℱ5|2𝒩≅ℂ4|2𝒩×ℂP1 and the projections are defined
as (2.6)π1(xαα˙,ηiα˙,λα˙±):=(xαα˙,ηiα˙),π2(xαα˙,ηiα˙,λα˙±):=(xαα˙λα˙±,λ±,ηiα˙λα˙±). We can now read off the
following correspondences: (2.7){projectivelinesℂPx,η1in𝒫3|𝒩}⟷{points(x,η)inℂ4|2𝒩},{pointspin𝒫3|𝒩}⟷{null(β−)superplanesℂp2|2𝒩inℂ4|2𝒩}. While the first correspondence
is rather evident, the second one deserves a brief remark. Suppose (x^αα˙,η^iα˙) is a solution to the incidence relations (2.3)
for a fixed point p∈𝒫3|𝒩.
Then, the set of all solutions is given by (2.8){(xαα˙,ηiα˙)}withxαα˙=x^αα˙+μαλ±α˙,ηiα˙=η^iα˙+εiλ±α˙, where μα is an arbitrary commuting 2-spinor and εi is an arbitrary vector with Graßmann-odd
entries. The coordinates λ±α˙ are defined by (2.4) and λ±α˙:=εα˙β˙λβ˙± with ε1˙2˙=−ε2˙1˙=1.
One can choose to work on any patch containing p.
The sets defined in (2.8) are then called null or β-superplanes.The double fibration (2.5) is the foundation of the
Penrose-Ward transform between equivalence classes of certain holomorphic
vector bundles over 𝒫3|𝒩 and gauge equivalence classes of solutions to
the 𝒩-extended supersymmetric self-dual Yang-Mills
equations on ℂ4 (see e.g. [15]).The tangent spaces to the leaves of the projectionπ2 are spanned by the vector fields (2.9)Vα±:=λ±α˙∂∂xαα˙,V±i:=λ±α˙∂∂ηiα˙. Note furthermore that 𝒫3|4 is a Calabi-Yau supermanifold. The bosonic fibers
contribute each +1 to the first Chern class and the fermionic
ones −1 (this is related to the fact that Berezin
integration amounts to differentiating with respect to a Graßmann variable).
Together with the contribution from the tangent bundle of the base space, we
have in total a trivial first Chern class. This space is thus suited as the
target space for a topological B-model [6].


### 2.2. The
Superambitwistor Space

The idea leading naturally to a superambitwistor space
is to “glue together” both the self-dual and anti-self-dual subsectors of𝒩=3 SYM theory to the full theory. For this, we obviously
need a twistor space 𝒫3|3 with coordinates (z±α,z±3,ηi±) together with a “dual” copy
(the word “dual” refers to the spinor indices and not to the line bundles
underlying 𝒫3|3)
𝒫*3|3 with coordinates (u±α˙,u±3,θ±i).
The dual twistor space is considered as a holomorphic supervector bundle over
the Riemann sphere ℂP*1 covered by the patches U±* with the standard local coordinates μ±=u±3.
For convenience, we again introduce the spinorial notation (μα+)=(1,μ+)T and (μα−)=(μ−,1)T.
The two patches covering 𝒫*3|3 will be denoted by 𝒰±*:=𝒫*3|3|U±*,
and the product space 𝒫3|3×𝒫*3|3 of the two supertwistor spaces is thus covered
by the four patches (2.10)𝒰(1):=𝒰+×𝒰+*,𝒰(2):=𝒰−×𝒰+*,𝒰(3):=𝒰+×𝒰−*,𝒰(4):=𝒰−×𝒰−*, on which we have the coordinates (z(a)α,z(a)3,ηi(a);u(a)α˙,u(a)3,θ(a)i).
This space is furthermore a rank 4|6 supervector bundle over the space ℂP1×ℂP*1.
The global sections of this bundle are parameterized by elements of ℂ4|6×ℂ*4|6 in the following way: (2.11)z(a)α=xαα˙λα˙(a),ηi(a)=ηiα˙λα˙(a);u(a)α˙=x*αα˙μα(a),θ(a)i=θαiμα(a).The superambitwistor space is now the subspaceL5|6⊂𝒫3|3×𝒫*3|3 obtained from the quadric condition (the “gluing
condition”) (2.12)κ(a):=z(a)αμα(a)−u(a)α˙λα˙(a)+2θ(a)iηi(a)=0. In what follows, we will denote
the restrictions of 𝒰(a) to L5|6 by 𝒰^(a).Because of the quadric condition (2.12), the bosonic
moduli are not independent of
L5|6,
but one rather has the relation (2.13)xαα˙=x0αα˙−θαiηiα˙,x*αα˙=x0αα˙+θαiηiα˙. The moduli (xαα˙) and (x*αα˙) are, therefore, indeed antichiral and chiral
coordinates on the (complex) superspace ℂ4|12 and with this identification, one can
establish the following double fibration using (2.11): (2.14) where ℱ6|12≅ℂ4|12×ℂP1×ℂP*1 and π1 is the trivial projection. Thus, one has the
correspondences (2.15){subspaces(ℂP1×ℂP*1)x0,η,θinL5|6}⟷{points(x0,η,θ)inℂ4|12},{pointspinL5|6}⟷{nullsuperlinesinℂ4|12}. The above-mentioned null
superlines are intersections of β-superplanes and dual α-superplanes. Given a solution (x^0αα˙,η^iα˙,θ^αi) to the incidence relations (2.11) for a fixed
point p in L5|6,
the set of points on such a null superline takes the form (2.16){(x0αα˙,ηiα˙,θαi)}withx0αα˙=x^0αα˙+tμ(a)αλ(a)α˙,ηiα˙=η^iα˙+εiλ(a)α˙,θαi=θ^αi+ε˜iμ(a)α. Here, t is an arbitrary complex number and εi and ε˜i are both 3-vectors with Graßmann-odd
components. The coordinates λ(a)α˙ and μ(a)α are chosen from arbitrary patches on which
they are both well defined. Note that these null superlines are in fact of
dimension 1|6.The spaceℱ6|12 is covered by four patches 𝒰˜(a):=π2−1(𝒰^(a)) and the tangent spaces to the 1|6-dimensional leaves of the fibration π2:ℱ6|12→L5|6 from (2.14) are spanned by the holomorphic
vector fields (2.17)W(a):=μ(a)αλ(a)α˙∂αα˙,D(a)i=λ(a)α˙Dα˙i,Di(a)=μ(a)αDαi, where Dαi and Dα˙i are the superderivatives defined
by (2.18)Dαi:=∂∂θαi+ηiα˙∂∂x0αα˙,Dα˙i:=∂∂ηiα˙+θαi∂∂x0αα˙.Just as the space𝒫3|4,
the superambitwistor space L5|6 is a Calabi-Yau supermanifold. To prove this,
note that we can count first Chern numbers with respect to the base ℂP1×ℂP*1 of L5|6.
In particular, we define the line bundle 𝒪(m,n) to have first Chern numbers m and n with respect to the two ℂP1s in the base. The (unconstrained) fermionic
part of L5|6 which is given by ℂ3⊗Π𝒪(1,0)⊕ℂ3⊗Π𝒪(0,1) contributes (−3,−3) in this counting, which has to be cancelled by
the body L5 of L5|6.
Consider, therefore, the map (2.19)κ:(z(a)α,λα˙(a),u(a)α˙,μα(a))↦(κ(a)|η=θ=0,λα˙(a),μα(a)), where κ(a) has been defined in (2.12). This map is a
vector bundle morphism and gives rise to the short exact
sequence (2.20)0→L5→ℂ2⊗𝒪(1,0)⊕ℂ2⊗𝒪(0,1)→κ𝒪(1,1)→0. The first Chern classes of the
bundles in this sequence are elements of H2(ℂP1×ℂP1,ℤ)≅ℤ×ℤ,
which we denote by α1h1+α2h2 with α1,α2∈ℤ.
Then, the short exact sequence (2.20) together with the Whitney product formula
yields (2.21)(1+h1)(1+h1)(1+h2)(1+h2)=(1+α1h1+α2h2+⋯)(1+h1+h2), where (α1,α2) label the first Chern class of L5 considered as a holomorphic vector bundle over ℂP1×ℂP*1.
It follows that c1=(1,1),
and taking into account the contribution of the tangent space to the base
(recall that T1,0ℂP1≅𝒪(2))
ℂP1×ℂP*1,
we conclude that the contribution of the tangent space to L5 to the first Chern class of L5|6 is cancelled by the contribution of the
fermionic fibers.SinceL5|6 is a Calabi-Yau supermanifold, this space can
be used as a target space for the topological B-model. However, it is still
unclear what the corresponding gauge theory action will look like. The most
obvious guess would be some holomorphic BF-type theory [16–18] with B a “Lagrange multiplier (0,3)-form."


### 2.3. Reality Conditions on the Superambitwistor
Space

On the supertwistor spaces𝒫3|𝒩,
one can define a real structure which leads to Kleinian signature on the body
of the moduli space ℝ4|2𝒩 of real holomorphic sections of the fibration π2 in (2.5). Furthermore, if 𝒩 is even, one can can impose a symplectic
Majorana condition which amounts to a second real structure which yields
Euclidean signature. We saw above that the superambitwistor space L5|6 originates from two copies of 𝒫3|3 and, therefore, we cannot straightforwardly
impose the Euclidean reality condition. However, besides the real structure
leading to Kleinian signature, one can additionally define a reality condition
by relating spinors of opposite helicities to each other. In this way, we obtain
a Minkowski metric on the body of ℝ4|12.
In the following, we will focus on the latter.Consider the antilinear involutionτM which acts on the coordinates of L5|6 according to (2.22)τM(z±α,λα˙±,ηi±;uα˙,μα±,θ±i):=(u±α˙¯,μα±¯,θ±i¯;z±α¯,λα˙±¯,ηi±¯). Sections of the bundle L5|6→ℂP1×ℂP*1 which are τM-real are thus parameterized by the
moduli (2.23)xαβ˙=xβ˙α¯,ηiα˙=θαi¯. We extract furthermore the
contained real coordinates via the identification (2.24)x11˙=x0+x1,x12˙=x2−ix3,x21˙=x2+ix3,x22˙=x0−x1, and obtain a metric of signature (1,3) on ℝ4 from ds2:=det(dxαα˙).
Note that we can also make the identification (2.24) in the complex case (xμ)∈ℂ4,
and then even on 𝒫3|𝒩.
In the subsequent discussion, we will always employ (2.24) which is consistent,
because we will not be interested in the real version of 𝒫3|𝒩.


### 2.4. The
Mini-Supertwistor Spaces

To capture the situation obtained by a dimensional
reductionℂ4|2𝒩→ℂ3|2𝒩,
one uses the so-called mini-supertwistor spaces. Note that the vector
field (2.25)𝒯3:=∂∂x3=i(∂∂x21˙−∂∂x12˙) considered on ℱ5|2𝒩 from diagram (2.5) can be split into a
holomorphic and an antiholomorphic part when restricted from ℱ5|2𝒩 to 𝒫3|𝒩: (2.26)𝒯3|𝒫3|𝒩=𝒯+𝒯¯,𝒯+=i(∂∂z+2−z+3∂∂z+1),𝒯−=i(z−3∂∂z−2−∂∂z−1). Let 𝒢 be the abelian group generated by 𝒯.
Then, the orbit space 𝒫3|𝒩/𝒢 is given by the holomorphic supervector
bundle (2.27)𝒫2|𝒩:=𝒪(2)⊕ℂ𝒩⊗Π𝒪(1)over ℂP1,
and we call 𝒫2|𝒩 a mini-supertwistor space. We denote the
patches covering 𝒫2|𝒩 by 𝒱±:=𝒰±∩𝒫2|𝒩.
The coordinates of the base and the fermionic fibers of 𝒫2|𝒩 are the same as those of 𝒫3|𝒩.
For the bosonic fibers, we define (2.28)w+1:=z+1+z+3z+2on𝒱+,w−1:=z−2+z−3z−1on𝒱−, and introduce additionally w±2:=z±3=λ± for convenience. On the intersection 𝒱+∩𝒱−,
we thus have the relation w+1=(w−2)−2w−1.
This implies that w±1 describes global sections of the line bundle 𝒪(2).
We parameterize these sections according
to (2.29)w±1=yα˙β˙λα˙±λβ˙±withyα˙β˙=y(α˙β˙)∈ℂ3, and the new moduli yα˙β˙ are identified with the previous ones xαβ˙ by the equation yα˙β˙=x(α˙β˙).
The incidence relation (2.29) allows us to establish a double fibration (2.30) where 𝒦4|2𝒩≅ℂ3|2𝒩×ℂP1.
We again obtain a twistor correspondence (2.31){projectivelinesℂPx,η1in𝒫2|𝒩}⟷{points(y,η)inℂ3|2𝒩},{pointspin𝒫2|𝒩}⟷{2|𝒩−dimensionalsuperplanesinℂ3|2𝒩}. The 2|𝒩-dimensional superplanes in ℂ3|2𝒩 are given by the set (2.32){(yα˙β˙,ηiα˙)}withyα˙β˙=y^α˙β˙+κ(α˙λ±β˙),ηiα˙=η^iα˙+εiλ±α˙, where κα˙ and εi are an arbitrary complex 2-spinor and a vector
with Graßmann-odd components, respectively. The point (y^α˙β˙,η^iα˙)∈ℂ3|2𝒩 is again an initial solution to the incidence
relations (2.29) for a fixed point p∈𝒫2|𝒩.
Note that although these superplanes arise from null superplanes in four
dimensions via dimensional reduction, they themselves are not null.The vector fields along the projectionν2 are now spanned by (2.33)Vα˙±:=λ±β˙∂(α˙β˙),V±i:=λ±α˙∂∂ηiα˙ with (2.34)∂1˙1˙:=∂∂y1˙1˙,∂2˙2˙:=∂∂y2˙2˙,∂(1˙2˙)=∂(2˙1˙):=12∂∂y1˙2˙.The mini-supertwistor space𝒫2|4 is again a Calabi-Yau supermanifold, and the
gauge theory equivalent to the topological B-model on this space is a
holomorphic BF theory [4].


## 2.1. Supertwistor Spaces

The supertwistor space ofℂ4|2𝒩 is defined as the rank 2|𝒩 holomorphic supervector bundle (2.1)𝒫3|𝒩:=ℂ2⊗𝒪(1)⊕ℂ𝒩⊗Π𝒪(1) over the Riemann sphere ℂP1.
Here, Π is the parity changing operator which inverts
the parity of the fiber coordinates. The base space of this bundle is covered
by the two patches U± on which we have the standard coordinates λ±∈U±≅ℂ with λ+=(λ−)−1 on U+∩U−.
Over U±,
we introduce furthermore the bosonic fiber coordinates z±α with α=1,2 and the fermionic fiber coordinates ηi± with i=1,…,𝒩.
On the intersection U+∩U−,
we thus have (2.2)z+α=1z−3z−α,ηi+=1z−3ηi−withz±3=λ±. The supermanifold 𝒫3|𝒩 as a whole is covered by the two patches 𝒰±:=𝒫3|𝒩|U± with local coordinates (z±1,z±2,z±3,η1±,…,η𝒩±).Global holomorphic sections of the vector bundle𝒫3|𝒩→ℂP1 are given by polynomials of degree one, which
are parameterized by moduli (xαα˙,ηiα˙)∈ℂ4|2𝒩 via (2.3)z±α=xαα˙λα˙±,ηi±=ηiα˙λα˙±, where we introduced the
simplifying spinorial notation (2.4)(λα˙+):=(1λ+),(λα˙−):=(λ−1).Equations (2.3), the so-called incidence relations, define a twistor
correspondence between the spaces 𝒫3|𝒩 and ℂ4|2𝒩,
which can be depicted in the double  fibration (2.5)
Here, ℱ5|2𝒩≅ℂ4|2𝒩×ℂP1 and the projections are defined
as (2.6)π1(xαα˙,ηiα˙,λα˙±):=(xαα˙,ηiα˙),π2(xαα˙,ηiα˙,λα˙±):=(xαα˙λα˙±,λ±,ηiα˙λα˙±). We can now read off the
following correspondences: (2.7){projectivelinesℂPx,η1in𝒫3|𝒩}⟷{points(x,η)inℂ4|2𝒩},{pointspin𝒫3|𝒩}⟷{null(β−)superplanesℂp2|2𝒩inℂ4|2𝒩}. While the first correspondence
is rather evident, the second one deserves a brief remark. Suppose (x^αα˙,η^iα˙) is a solution to the incidence relations (2.3)
for a fixed point p∈𝒫3|𝒩.
Then, the set of all solutions is given by (2.8){(xαα˙,ηiα˙)}withxαα˙=x^αα˙+μαλ±α˙,ηiα˙=η^iα˙+εiλ±α˙, where μα is an arbitrary commuting 2-spinor and εi is an arbitrary vector with Graßmann-odd
entries. The coordinates λ±α˙ are defined by (2.4) and λ±α˙:=εα˙β˙λβ˙± with ε1˙2˙=−ε2˙1˙=1.
One can choose to work on any patch containing p.
The sets defined in (2.8) are then called null or β-superplanes.The double fibration (2.5) is the foundation of the
Penrose-Ward transform between equivalence classes of certain holomorphic
vector bundles over 𝒫3|𝒩 and gauge equivalence classes of solutions to
the 𝒩-extended supersymmetric self-dual Yang-Mills
equations on ℂ4 (see e.g. [15]).The tangent spaces to the leaves of the projectionπ2 are spanned by the vector fields (2.9)Vα±:=λ±α˙∂∂xαα˙,V±i:=λ±α˙∂∂ηiα˙. Note furthermore that 𝒫3|4 is a Calabi-Yau supermanifold. The bosonic fibers
contribute each +1 to the first Chern class and the fermionic
ones −1 (this is related to the fact that Berezin
integration amounts to differentiating with respect to a Graßmann variable).
Together with the contribution from the tangent bundle of the base space, we
have in total a trivial first Chern class. This space is thus suited as the
target space for a topological B-model [6].


## 2.2. The
Superambitwistor Space

The idea leading naturally to a superambitwistor space
is to “glue together” both the self-dual and anti-self-dual subsectors of𝒩=3 SYM theory to the full theory. For this, we obviously
need a twistor space 𝒫3|3 with coordinates (z±α,z±3,ηi±) together with a “dual” copy
(the word “dual” refers to the spinor indices and not to the line bundles
underlying 𝒫3|3)
𝒫*3|3 with coordinates (u±α˙,u±3,θ±i).
The dual twistor space is considered as a holomorphic supervector bundle over
the Riemann sphere ℂP*1 covered by the patches U±* with the standard local coordinates μ±=u±3.
For convenience, we again introduce the spinorial notation (μα+)=(1,μ+)T and (μα−)=(μ−,1)T.
The two patches covering 𝒫*3|3 will be denoted by 𝒰±*:=𝒫*3|3|U±*,
and the product space 𝒫3|3×𝒫*3|3 of the two supertwistor spaces is thus covered
by the four patches (2.10)𝒰(1):=𝒰+×𝒰+*,𝒰(2):=𝒰−×𝒰+*,𝒰(3):=𝒰+×𝒰−*,𝒰(4):=𝒰−×𝒰−*, on which we have the coordinates (z(a)α,z(a)3,ηi(a);u(a)α˙,u(a)3,θ(a)i).
This space is furthermore a rank 4|6 supervector bundle over the space ℂP1×ℂP*1.
The global sections of this bundle are parameterized by elements of ℂ4|6×ℂ*4|6 in the following way: (2.11)z(a)α=xαα˙λα˙(a),ηi(a)=ηiα˙λα˙(a);u(a)α˙=x*αα˙μα(a),θ(a)i=θαiμα(a).The superambitwistor space is now the subspaceL5|6⊂𝒫3|3×𝒫*3|3 obtained from the quadric condition (the “gluing
condition”) (2.12)κ(a):=z(a)αμα(a)−u(a)α˙λα˙(a)+2θ(a)iηi(a)=0. In what follows, we will denote
the restrictions of 𝒰(a) to L5|6 by 𝒰^(a).Because of the quadric condition (2.12), the bosonic
moduli are not independent of
L5|6,
but one rather has the relation (2.13)xαα˙=x0αα˙−θαiηiα˙,x*αα˙=x0αα˙+θαiηiα˙. The moduli (xαα˙) and (x*αα˙) are, therefore, indeed antichiral and chiral
coordinates on the (complex) superspace ℂ4|12 and with this identification, one can
establish the following double fibration using (2.11): (2.14) where ℱ6|12≅ℂ4|12×ℂP1×ℂP*1 and π1 is the trivial projection. Thus, one has the
correspondences (2.15){subspaces(ℂP1×ℂP*1)x0,η,θinL5|6}⟷{points(x0,η,θ)inℂ4|12},{pointspinL5|6}⟷{nullsuperlinesinℂ4|12}. The above-mentioned null
superlines are intersections of β-superplanes and dual α-superplanes. Given a solution (x^0αα˙,η^iα˙,θ^αi) to the incidence relations (2.11) for a fixed
point p in L5|6,
the set of points on such a null superline takes the form (2.16){(x0αα˙,ηiα˙,θαi)}withx0αα˙=x^0αα˙+tμ(a)αλ(a)α˙,ηiα˙=η^iα˙+εiλ(a)α˙,θαi=θ^αi+ε˜iμ(a)α. Here, t is an arbitrary complex number and εi and ε˜i are both 3-vectors with Graßmann-odd
components. The coordinates λ(a)α˙ and μ(a)α are chosen from arbitrary patches on which
they are both well defined. Note that these null superlines are in fact of
dimension 1|6.The spaceℱ6|12 is covered by four patches 𝒰˜(a):=π2−1(𝒰^(a)) and the tangent spaces to the 1|6-dimensional leaves of the fibration π2:ℱ6|12→L5|6 from (2.14) are spanned by the holomorphic
vector fields (2.17)W(a):=μ(a)αλ(a)α˙∂αα˙,D(a)i=λ(a)α˙Dα˙i,Di(a)=μ(a)αDαi, where Dαi and Dα˙i are the superderivatives defined
by (2.18)Dαi:=∂∂θαi+ηiα˙∂∂x0αα˙,Dα˙i:=∂∂ηiα˙+θαi∂∂x0αα˙.Just as the space𝒫3|4,
the superambitwistor space L5|6 is a Calabi-Yau supermanifold. To prove this,
note that we can count first Chern numbers with respect to the base ℂP1×ℂP*1 of L5|6.
In particular, we define the line bundle 𝒪(m,n) to have first Chern numbers m and n with respect to the two ℂP1s in the base. The (unconstrained) fermionic
part of L5|6 which is given by ℂ3⊗Π𝒪(1,0)⊕ℂ3⊗Π𝒪(0,1) contributes (−3,−3) in this counting, which has to be cancelled by
the body L5 of L5|6.
Consider, therefore, the map (2.19)κ:(z(a)α,λα˙(a),u(a)α˙,μα(a))↦(κ(a)|η=θ=0,λα˙(a),μα(a)), where κ(a) has been defined in (2.12). This map is a
vector bundle morphism and gives rise to the short exact
sequence (2.20)0→L5→ℂ2⊗𝒪(1,0)⊕ℂ2⊗𝒪(0,1)→κ𝒪(1,1)→0. The first Chern classes of the
bundles in this sequence are elements of H2(ℂP1×ℂP1,ℤ)≅ℤ×ℤ,
which we denote by α1h1+α2h2 with α1,α2∈ℤ.
Then, the short exact sequence (2.20) together with the Whitney product formula
yields (2.21)(1+h1)(1+h1)(1+h2)(1+h2)=(1+α1h1+α2h2+⋯)(1+h1+h2), where (α1,α2) label the first Chern class of L5 considered as a holomorphic vector bundle over ℂP1×ℂP*1.
It follows that c1=(1,1),
and taking into account the contribution of the tangent space to the base
(recall that T1,0ℂP1≅𝒪(2))
ℂP1×ℂP*1,
we conclude that the contribution of the tangent space to L5 to the first Chern class of L5|6 is cancelled by the contribution of the
fermionic fibers.SinceL5|6 is a Calabi-Yau supermanifold, this space can
be used as a target space for the topological B-model. However, it is still
unclear what the corresponding gauge theory action will look like. The most
obvious guess would be some holomorphic BF-type theory [16–18] with B a “Lagrange multiplier (0,3)-form."


## 2.3. Reality Conditions on the Superambitwistor
Space

On the supertwistor spaces𝒫3|𝒩,
one can define a real structure which leads to Kleinian signature on the body
of the moduli space ℝ4|2𝒩 of real holomorphic sections of the fibration π2 in (2.5). Furthermore, if 𝒩 is even, one can can impose a symplectic
Majorana condition which amounts to a second real structure which yields
Euclidean signature. We saw above that the superambitwistor space L5|6 originates from two copies of 𝒫3|3 and, therefore, we cannot straightforwardly
impose the Euclidean reality condition. However, besides the real structure
leading to Kleinian signature, one can additionally define a reality condition
by relating spinors of opposite helicities to each other. In this way, we obtain
a Minkowski metric on the body of ℝ4|12.
In the following, we will focus on the latter.Consider the antilinear involutionτM which acts on the coordinates of L5|6 according to (2.22)τM(z±α,λα˙±,ηi±;uα˙,μα±,θ±i):=(u±α˙¯,μα±¯,θ±i¯;z±α¯,λα˙±¯,ηi±¯). Sections of the bundle L5|6→ℂP1×ℂP*1 which are τM-real are thus parameterized by the
moduli (2.23)xαβ˙=xβ˙α¯,ηiα˙=θαi¯. We extract furthermore the
contained real coordinates via the identification (2.24)x11˙=x0+x1,x12˙=x2−ix3,x21˙=x2+ix3,x22˙=x0−x1, and obtain a metric of signature (1,3) on ℝ4 from ds2:=det(dxαα˙).
Note that we can also make the identification (2.24) in the complex case (xμ)∈ℂ4,
and then even on 𝒫3|𝒩.
In the subsequent discussion, we will always employ (2.24) which is consistent,
because we will not be interested in the real version of 𝒫3|𝒩.


## 2.4. The
Mini-Supertwistor Spaces

To capture the situation obtained by a dimensional
reductionℂ4|2𝒩→ℂ3|2𝒩,
one uses the so-called mini-supertwistor spaces. Note that the vector
field (2.25)𝒯3:=∂∂x3=i(∂∂x21˙−∂∂x12˙) considered on ℱ5|2𝒩 from diagram (2.5) can be split into a
holomorphic and an antiholomorphic part when restricted from ℱ5|2𝒩 to 𝒫3|𝒩: (2.26)𝒯3|𝒫3|𝒩=𝒯+𝒯¯,𝒯+=i(∂∂z+2−z+3∂∂z+1),𝒯−=i(z−3∂∂z−2−∂∂z−1). Let 𝒢 be the abelian group generated by 𝒯.
Then, the orbit space 𝒫3|𝒩/𝒢 is given by the holomorphic supervector
bundle (2.27)𝒫2|𝒩:=𝒪(2)⊕ℂ𝒩⊗Π𝒪(1)over ℂP1,
and we call 𝒫2|𝒩 a mini-supertwistor space. We denote the
patches covering 𝒫2|𝒩 by 𝒱±:=𝒰±∩𝒫2|𝒩.
The coordinates of the base and the fermionic fibers of 𝒫2|𝒩 are the same as those of 𝒫3|𝒩.
For the bosonic fibers, we define (2.28)w+1:=z+1+z+3z+2on𝒱+,w−1:=z−2+z−3z−1on𝒱−, and introduce additionally w±2:=z±3=λ± for convenience. On the intersection 𝒱+∩𝒱−,
we thus have the relation w+1=(w−2)−2w−1.
This implies that w±1 describes global sections of the line bundle 𝒪(2).
We parameterize these sections according
to (2.29)w±1=yα˙β˙λα˙±λβ˙±withyα˙β˙=y(α˙β˙)∈ℂ3, and the new moduli yα˙β˙ are identified with the previous ones xαβ˙ by the equation yα˙β˙=x(α˙β˙).
The incidence relation (2.29) allows us to establish a double fibration (2.30) where 𝒦4|2𝒩≅ℂ3|2𝒩×ℂP1.
We again obtain a twistor correspondence (2.31){projectivelinesℂPx,η1in𝒫2|𝒩}⟷{points(y,η)inℂ3|2𝒩},{pointspin𝒫2|𝒩}⟷{2|𝒩−dimensionalsuperplanesinℂ3|2𝒩}. The 2|𝒩-dimensional superplanes in ℂ3|2𝒩 are given by the set (2.32){(yα˙β˙,ηiα˙)}withyα˙β˙=y^α˙β˙+κ(α˙λ±β˙),ηiα˙=η^iα˙+εiλ±α˙, where κα˙ and εi are an arbitrary complex 2-spinor and a vector
with Graßmann-odd components, respectively. The point (y^α˙β˙,η^iα˙)∈ℂ3|2𝒩 is again an initial solution to the incidence
relations (2.29) for a fixed point p∈𝒫2|𝒩.
Note that although these superplanes arise from null superplanes in four
dimensions via dimensional reduction, they themselves are not null.The vector fields along the projectionν2 are now spanned by (2.33)Vα˙±:=λ±β˙∂(α˙β˙),V±i:=λ±α˙∂∂ηiα˙ with (2.34)∂1˙1˙:=∂∂y1˙1˙,∂2˙2˙:=∂∂y2˙2˙,∂(1˙2˙)=∂(2˙1˙):=12∂∂y1˙2˙.The mini-supertwistor space𝒫2|4 is again a Calabi-Yau supermanifold, and the
gauge theory equivalent to the topological B-model on this space is a
holomorphic BF theory [4].


## 3. The Mini-Superambitwistor Space

In this section, we define and examine the
mini-superambitwistor spaceL4|6,
which we will use to build a Penrose-Ward transform involving solutions to 𝒩=8 SYM theory in three dimensions. We will first
give an abstract definition of L4|6 by a short exact sequence, and present more
heuristic ways of obtaining the mini-superambitwistor space later.


### 3.1. Abstract Definition of the
Mini-Superambitwistor Space

The starting point is the product space𝒫2|3×𝒫*2|3 of two copies of the 𝒩=3 mini-supertwistor space. In analogy to the
space 𝒫3|3×𝒫*3|3,
we have coordinates (3.1)(w(a)1,w(a)2=λ(a),ηi(a);v(a)1,v(a)2=μ(a),θ(a)i) on the patches 𝒱(a) which are Cartesian products of 𝒱± and 𝒱±*: (3.2)𝒱(1):=𝒱+×𝒱+*,𝒱(2):=𝒱−×𝒱+*,𝒱(3):=𝒱+×𝒱−*,𝒱(4):=𝒱−×𝒱−*. For convenience, let us
introduce the subspace ℂPΔ1 of the base of the fibration 𝒫2|3×𝒫*2|3→ℂP1×ℂP*1 as (3.3)ℂPΔ1:=diag(ℂP1×ℂP*1)={(μ±,λ±)∈ℂP1×ℂP*1∣μ±=λ±}. Consider now the map ξ:𝒫2|3×𝒫*2|3→𝒪ℂPΔ1(2) which is defined by (3.4)ξ:(w(a)1,w(a)2,ηi(a);v(a)1,v(a)2,θ(a)i)↦{(w±1−v±1+2θ±iηi±,w±2)forw±2=v±2(0,w(a)2)else, where 𝒪ℂPΔ1(2) is the line bundle 𝒪(2) over ℂPΔ1.
In this definition, we used the fact that a point for which w±2=v±2 is at least on one of the patches 𝒱(1) and 𝒱(4).
Note, in particular, that the map ξ is a morphism of vector bundles. Therefore, we
can define a space L4|6 via the short exact sequence (3.5)0→L4|6→𝒫2|3×𝒫*2|3→ξ𝒪ℂPΔ1(2)→0 (cf. (2.20)). We will call this
space the mini-superambitwistor space.
Analogously to above, we will denote the pull-back of the patches 𝒱(a) to L4|6 by 𝒱^(a).
Obviously, the space L4|6 is a fibration, and we can switch to the
corresponding short exact sequence of sheaves of local
sections: (3.6)0→L4|6→𝒫2|3×𝒫*2|3→ξ𝒪ℂPΔ1(2)→0. Note  the difference in notation:
(3.5) is a sequence of vector bundles, while (3.6) is a sequence of sheaves. To
analyze the geometry of the space L4|6 in more detail, we will restrict ourselves to
the body of this space and put the fermionic coordinates to zero. Similarly to
the case of the superambitwistor space, this is possible as the map ξ does not affect the fermionic dimensions in
the exact sequence (3.5); this will become clearer in the discussion in Section
3.2.Inspired by the sequence defining the skyscraper sheaf
(a sheaf with sections supported only at the pointp)
0→𝒪(−1)→𝒪→𝒪p→0,
we introduce the following short exact sequence: (3.7)0→𝒪(1,−1)⊕𝒪(−1,1)→ζ𝒪(2,0)⊕𝒪(0,2)→𝒪ℂPΔ1(2)⊕𝒪ℂPΔ1(2)→0. Here, we defined ζ:(a,b)↦(aεα˙β˙λα˙μβ˙,bεα˙β˙λα˙μβ˙),
where λα˙ and μα˙ are the usual homogeneous coordinates on the
base space ℂP1×ℂP*1.
The sheaf 𝒪ℂPΔ1(2) is a torsion sheaf (sometimes sloppily
referred to as a skyscraper sheaf) with sections supported only over ℂPΔ1.
Finally, we trivially have the short exact sequence (3.8)0→𝒪ℂPΔ1(2)→α1𝒪ℂPΔ1(2)⊕𝒪ℂPΔ1(2)→α2𝒪ℂPΔ1(2)→0, where α1:(a)↦(a,a) and α2:(a,b)↦(a−b).Using the short exact sequences (3.6), (3.7), and
(3.8) as well as the nine lemma, we can establish the following
diagram: (3.9)000↓↓↓0→𝒪(1,−1)⊕𝒪(−1,1)→ζL4→𝒪ℂPΔ1(2)→0↓id↓↓α10→𝒪(1,−1)⊕𝒪(−1,1)→ζ𝒪(2,0)⊕𝒪(0,2)→𝒪ℂPΔ1(2)⊕𝒪ℂPΔ1(2)→0↓↓↓α20→0→𝒪ℂPΔ1(2)→id𝒪ℂPΔ1(2)→0.↓↓↓000From the horizontal lines of this diagram and the five
lemma, we conclude thatL4⊕𝒪ℂPΔ1(2)=𝒪(2,0)⊕𝒪(0,2).
Thus, L4 is not a locally free sheaf (a
more sophisticated argumentation would use the common properties of the torsion
functor to establish that L4 is a torsion sheaf; furthermore, one can write
down a further diagram using the nine lemma which shows that L4 is a coherent sheaf) but a torsion sheaf,
whose stalks over ℂPΔ1 are isomorphic to the stalks of 𝒪ℂPΔ1(2),
while the stalks over (ℂP1×ℂP*1)∖ℂPΔ1 are isomorphic to the stalks of 𝒪(2,0)⊕𝒪(0,2).
Therefore, L4 is not a vector bundle, but a fibration
(the homotopy lifting property typically included in the definition
of a fibration is readily derived from the definition of L4) with fibers ℂ2 over generic points and fibers ℂ over ℂPΔ1.
In particular, the total space of L4 is not a manifold.The fact that the total space of the bundleL4|6 is neither a supermanifold nor a supervector
bundle over ℂP1×ℂP*1 seems at first slightly disturbing. However,
we will show that once one is aware of this new aspect, it does not cause any
deep difficulties as far as the twistor correspondence and the Penrose-Ward
transform are concerned.


### 3.2. The
Mini-Superambitwistor Space by Dimensional Reduction

In the following, we will motivate the abstract
definition more concretely by considering explicitly the dimensional reduction
of the spaceL5|6,
we will also fix our notation in terms of coordinates and moduli of sections.
For this, we will first reduce the product space 𝒫3|3×𝒫*3|3 and then impose the appropriate reduced
quadric condition. In a first step, we want to eliminate in both 𝒫3|3 and 𝒫*3|3 the dependence on the bosonic modulus x3.
Thus, we should factorize by (3.10)𝒯(a)={∂∂z+2−z+3∂∂z+1on𝒰(1)z−3∂∂z−2−∂∂z−1on𝒰(2)∂∂z+2−z+3∂∂z+1on𝒰(3)z−3∂∂z−2−∂∂z−1on𝒰(4),𝒯(a)*={∂∂u+2˙−u+3∂∂u+1˙on𝒰(1)∂∂u+2˙−u+3∂∂u+1˙on𝒰(2)u−3∂∂u−2˙−∂∂u−1˙on𝒰(3)u−3∂∂u−2˙−∂∂u−1˙on𝒰(4), which leads us to the orbit
space (3.11)𝒫2|3×𝒫*2|3=(𝒫3|3/𝒢)×(𝒫*3|3/𝒢*), where 𝒢 and 𝒢* are  the abelian groups generated by 𝒯 and 𝒯*,
respectively. Recall that the coordinates we use on this space have been
defined in (3.1). The global sections of the bundle 𝒫2|4×𝒫*2|4→ℂP1×ℂP*1 are captured by the
parameterization
(3.12)w(a)1=yα˙β˙λα˙(a)λβ˙(a),v(a)1=y*α˙β˙μα˙(a)μβ˙(a),θ(a)i=θα˙iμα˙(a),ηi(a)=ηiα˙λα˙(a), where we relabel the indices of μα(a)→μα˙(a) and the moduli y*αβ→y*α˙β˙, θiα→θiα˙,
since there is no distinction between left- and right-handed spinors on ℝ3 or its complexification ℂ3.The next step is obviously to impose the quadric
condition, gluing together the self-dual and anti-self-dual parts. Note that
when acting with𝒯 and 𝒯* on κ(a) as given in (2.12), we obtain (3.13)𝒯(1)κ(1)=𝒯(1)*κ(1)=(μ+−λ+),𝒯(2)κ(2)=𝒯(2)*κ(2)=(λ−μ+−1),𝒯(3)κ(3)=𝒯(3)*κ(3)=(1−λ+μ−),𝒯(4)κ(4)=𝒯(4)*κ(4)=(λ−−μ−). This implies that the orbits
generated by 𝒯 and 𝒯* become orthogonal to the orbits of ∂/∂κ only at μ±=λ±.
We can, therefore, safely impose the condition (3.14)(w±1−v±1+2θ±iηi±)|λ±=μ±=0, and the subset of 𝒫2|3×𝒫*2|3 which satisfies this condition is obviously
identical to the mini-superambitwistor space L4|6 defined above.The condition (3.14) naturally fixes the
parametrization of global sections of the fibration L4|6 by giving a relation between the moduli used
in (3.12). This relation is completely analogous to (2.13) and
reads (3.15)yα˙β˙=y0α˙β˙−θ(α˙iηiβ˙),y*α˙β˙=y0α˙β˙+θ(α˙iηiβ˙). We clearly see that this
parameterization arises from (2.13) by dimensional
reduction from ℂ4→ℂ3.
Furthermore, even with this identification, w±1 and v±1 are independent of
λ±≠μ±.
Thus, indeed, imposing the condition (3.14) only at λ±=μ± is the dimensionally reduced analogue of
imposing the condition (2.12) on 𝒫3|3×𝒫*3|3.


### 3.3. Comments on Further Ways of ConstructingL4|6

Although the construction presented above seems most
natural, one can imagine other approaches of defining the spaceL4|6.
Completely evident is a second way, which uses the description of L5|6 in terms of coordinates on ℱ6|12.
Here, one factorizes the correspondence space ℱ6|12 by the groups generated by the vector field 𝒯3=𝒯3* and obtains the correspondence space 𝒦5|12≅ℂ3|12×ℂP1×ℂP*1 together with (3.15). A subsequent projection π2 from the dimensionally reduced correspondence
space 𝒦5|12 then yields the mini-superambitwistor space L4|6 as defined above.Furthermore, one can factorize𝒫3|3×𝒫*3|3 only by 𝒢 to eliminate the dependence on one modulus.
This will lead to 𝒫2|3×𝒫*3|3 and following the above discussion of imposing
the quadric condition on the appropriate subspace, one arrives again at (3.14)
and the space L4|6.
Here, the quadric condition already implies the remaining factorization of 𝒫2|3×𝒫*3|3 by 𝒢*.Eventually, one could anticipate the identification of
moduli in (3.15) and, therefore, want to factorize by the group generated by
the combination 𝒯+𝒯*.
Acting with this sum on κ(a) will produce the sum of the results given in
(3.13), and the subsequent discussion of the quadric condition follows the one
presented above.


### 3.4. Double Fibration

Knowing the
parameterization of global sections of the
mini-superambitwistor space fibered overℂP1×ℂP*1 as defined in (3.15), we can establish a
double fibration, similarly to all the other twistor spaces we encountered so
far. Even more instructive is the following diagram, in which the dimensional
reduction of the involved spaces becomes evident:
(3.16)
The upper half is just the
double fibration for the quadric (2.14), while the lower half corresponds to
the dimensionally reduced case. The reduction of ℂ4|12 to ℂ3|12 is obviously done by factoring out the group
generated by 𝒯3.
The same is true for the reduction of ℱ6|12≅ℂ4|12×ℂP1×ℂP*1 to 𝒦5|12≅ℂ3|12×ℂP1×ℂP*1.
The reduction from L5|6 to L4|6 was given above and the projection ν2 from 𝒦5|12 onto L4|6 is defined by (3.12). The four patches
covering ℱ6|12 will be denoted by 𝒱˜(a):=ν2−1(𝒱^(a)).The double fibration defined by the projectionsν1 and ν2 yields the following twistor
correspondences:(3.17){subspaces(ℂP1×ℂP1)y0,η,θinL4|6}⟷{points(y0,η,θ)inℂ3|12},{genericpointspinL4|6}⟷{superlinesinℂ3|12},{pointspinL4|6withλ±=μ±}⟷{superplanesinℂ3|12}.The superlines and the
superplanes in ℂ3|12 are defined as the sets(3.18){(yα˙β˙,ηiα˙,θα˙i)}withyα˙β˙=y^α˙β˙+tλ(a)(α˙μ(a)β˙),ηiα˙=η^iα˙+εiλ(a)α˙,θα˙i=θ^α˙i+ε˜iμ(a)α˙,{(yα˙β˙,ηiα˙,θα˙i)}withyα˙β˙=y^α˙β˙+κ(α˙λ(a)β˙),ηiα˙=η^iα˙+εiλ(a)α˙,θα˙i=θ^α˙i+ε˜iλ(a)α˙,where t, κα˙, εi,
and ε˜i are an arbitrary complex number, a complex
commuting 2-spinor, and two 3-vectors with Graßmann-odd components,
respectively. Note that in the last line, λ±α˙=μ±α˙,
and we could also have written(3.19){(yα˙β˙,ηiα˙,θα˙i)}withyα˙β˙=y^α˙β˙+κ(α˙μ(a)β˙),ηiα˙=η^iα˙+εiμ(a)α˙,θα˙i=θ^α˙i+ε˜iμ(a)α˙.The vector fields spanning the tangent spaces to the
leaves of the fibrationν2 are for generic values of μ± and λ± given by(3.20)W(a):=μ(a)α˙λ(a)β˙∂(α˙β˙),D˜(a)i:=λ(a)β˙D˜β˙i:=λ(a)β˙(∂∂ηiβ˙+θα˙i∂(α˙β˙)),Di(a):=μ(a)α˙Dα˙i:=μ(a)α˙(∂∂θα˙i+ηiβ˙∂(α˙β˙)),where the derivatives ∂(α˙β˙) have been defined in (2.34). At μ±=λ±,
however, the fibers of the fibration L4|6 over ℂP1×ℂP*1 loose one bosonic dimension. As the space 𝒦5|12 is a manifold, this means that this dimension
has to become tangent to the projection ν2.
In fact, one finds that over ℂPΔ1 besides the vector fields given in (3.20),
also the vector fields(3.21)W˜β˙±=μ±α˙∂(α˙β˙)=λ±α˙∂(α˙β˙)annihilate the coordinates on L4|6.
Therefore, the leaves to the projection ν2:𝒦5|12→L4|6 are of dimension 2|6 for μ±=λ± and of dimension 1|6 everywhere else.


### 3.5. Real Structure onL4|6

Quite evidently, a real structure onL4|6 is inherited from the one on L5|6,
and we obtain directly from (2.22) the action of τM on 𝒫2|4×𝒫*2|4,
which is given by(3.22)τM(w±1,λα˙±,ηi±;v±1,μα˙±,θ±i):=(v±1¯,μα˙±¯,θ±i¯;w±1¯,λα˙±¯,ηi±¯).This action descends in an
obvious manner to L4|6,
which leads to a real structure on the moduli space ℂ3|12 via the double fibration (3.16). Thus, we have
as the resulting reality condition(3.23)y0α˙β˙=y0β˙α˙¯,ηiα˙=θα˙i¯,and the identification of the
bosonic moduli yα˙β˙ with the coordinates on ℝ3 reads as(3.24)y01˙1˙=x00+x01,y01˙2˙=y02˙1˙=x02,y02˙2˙=x00−x01.The reality conditionτM(⋅)=⋅ is indeed fully compatible with the condition
(3.14) which reduces 𝒫2|4×𝒫*2|4 to L4|6.
The base space ℂP1×ℂP*1 of the fibration L4|6 is reduced to a single sphere S2 with real coordinates (1/2)(λ±+μ±)=(1/2)(λ±+λ¯±) and (1/2i)(λ±−μ±)=(1/2i)(λ±−λ¯±),
while the diagonal ℂPΔ1 is reduced to a circle SΔ1 parameterized by the real coordinates (1/2)(λ±+λ¯±).
The τM-real sections of L4|6 have to satisfy w±1=τM(w±1)=v¯±1.
Thus, the fibers of the fibration L4|6→ℂP1×ℂP*1,
which are of complex dimension 2|6 over generic points in the base and complex
dimension 1|6 over ℂPΔ1,
are reduced to fibers of real dimension 2|6 and 1|6,
respectively. In particular, note that θ±iηi±=η¯i±θ¯±i=−θ¯±iη¯i± is purely imaginary and, therefore, the
quadric condition (3.14) together with the real structure τM implies that w±1=v¯±1=w¯±1+2θ¯±iη¯i± for λ±=μ±=λ¯±.
Thus, the body w∘±1 of w±1 is purely real and we have w±1=w∘±1−θ±iηi± and v±1=w∘±1+θ±iηi± on the diagonal SΔ1.


### 3.6. Interpretation of the Involved Real Geometries

For the best-known twistor correspondences (i.e., the
correspondence (2.5)) (more precisely, the compactified version
thereof) its dual, and the correspondence (2.14), there is a nice
description in terms of flag manifolds (see e.g., [3]). For the spaces involved in
the twistor correspondences including minitwistor spaces, one has a similarly
nice interpretation after restricting to the real situation. For simplicity, we
reduce our considerations to the bodies (i.e., drop the fermionic directions)
of the involved geometries, as the extension to corresponding supermanifolds is
quite straightforward.Let us first discuss the double fibration (2.30), and
assume that we have imposed a suitable reality condition on the fiber
coordinates, the details of which are not important. We follow again the usual
discussion of the real case and leave the coordinates on the sphere complex. As
correspondence space on top of the double fibration, we have thus the space ℝ3×S2,
which we can understand as the set of oriented lines (not only
the ones through the origin) in ℝ3 with one marked point. Clearly, the point of
such a line is given by an element of ℝ3,
and the direction of this line in ℝ3 is parameterized by a point on S2.
The minitwistor space 𝒫2≅𝒪(2) now is simply the space of all lines in ℝ3 [19]. Similarly to the case of flag manifolds, the
projections ν1 and ν2 in (2.30) become, therefore, obvious. For ν1,
simply drop the line and keep the marked point. For ν2,
drop the marked point and keep the line. Equivalently, we can understand ν2 as moving the marked point on the line to its
shortest possible distance from the origin. This leads to the space TS2≅𝒪(2),
where the S2
parameterizes
again the direction of the line, which can subsequently be still moved
orthogonally to this direction, and this freedom is parameterized by the
tangent planes to S2 which are isomorphic to ℝ2.Now in the case of the fibration included in (3.16),
we impose the reality condition (3.22) on the fiber coordinates of L4.
In the real case, the correspondence space 𝒦5 becomes the space ℝ3×S2×S2 and this is the space of two oriented lines in ℝ3 intersecting in a point. More precisely, this
is the space of two oriented lines in ℝ3 each with one marked point, for which the two
marked points coincide. The projections ν1 and ν2 in (3.16) are then interpreted as follows. For ν1,
simply drop the two lines and keep the marked point. For ν2,
fix one line and move the marked point (the intersection point) together with
the second line to its shortest distance to the origin. Thus, the space L4 is the space of configurations in ℝ3,
in which a line has a common point with another line at its shortest distance
to the origin.Let us summarize all the above findings in
Table1.Table 1SpaceRelation toℝ3ℝ3marked points inℝ3ℝ3×S2oriented lines with a marked point inℝ3𝒫2≅𝒪(2)oriented lines inℝ3 (with a marked point at shortest distance to the origin.)ℝ3×S2×S2two oriented lines with a common marked
point inℝ3L4two oriented lines with a common marked
point at shortest distance
from one of the lines to the origin inℝ3


### 3.7. Remarks Concerning a Topological B-Model onL4|6

The spaceL4|6 is not well suited as a target space for a
topological B-model since it is not a (Calabi-Yau) manifold. However, one
clearly expects that it is possible to define an analogous model since, if we
assume that the conjecture in [20, 21] is correct, such a model should simply be the mirror
of the minitwistor string theory considered in [14]. This model would
furthermore yield some holomorphic Chern-Simons type equations of motion. The
latter equations would then define holomorphic pseudobundles over L4|6 by an analogue of a holomorphic structure.
These bundles will be introduced in Section 4.3 and in our discussion, they
substitute the holomorphic vector bundles.Interestingly, the spaceL4|6 has a property which comes close to vanishing
of a first Chern class. Recall that for any complex vector bundle, its Chern
classes are Poincaré dual to the degeneracy cycles of certain sets of sections
(this is a Gauß-Bonnet formula). More precisely, to calculate the first Chern
class of a rank r vector bundle, one considers r generic sections and arranges them into an r×r matrix L.
The degeneracy loci on the base space are then given by the zero locus of det(L).
Clearly, this calculation can be translated directly to L4|6.We will now show thatL4|6 and L5|6 have equivalent degeneracy loci (i.e., they
are equal up to a principal divisor) which, speaking about ordinary vector
bundles, would not affect the first Chern class. Our discussion simplifies
considerably if we restrict our attention to the bodies of the two supertwistor
spaces and put all the fermionic coordinates to zero. Note that this will not
affect the result, as the quadric conditions defining L5|6 and L4|6 do not affect the fermionic dimensions: the
fermionic parts of the fibrations L5|6 and L4|6 are identical, which is easily seen by
considering the global sections generating the total spaces of the fibrations.
Instead of the ambitwistor spaces, it is also easier to consider the vector
bundles 𝒫3×𝒫*3 and 𝒫2×𝒫*2 over ℂP1×ℂP*1,
respectively, with the appropriately restricted sets of sections. Furthermore,
we will stick to our inhomogeneous coordinates and perform the calculation only
on the patch 𝒰(1),
but all this directly translates into homogeneous, patch-independent
coordinates. The matrices to be considered are(3.25)LL5=(x11α˙λα˙+x21α˙λα˙+x31α˙λα˙+x41α˙λα˙+x12α˙λα˙+x22α˙λα˙+x32α˙λα˙+x42α˙λα˙+x1α1μα+x2α1μα+x3α1μα+x4α1μα+x1α2μα+x2α2μα+x3α2μα+x4α2μα+),LL4=(y1α˙β˙λα˙+λβ˙+y2α˙β˙λα˙+λβ˙+y1α˙β˙μα˙+μβ˙+y2α˙β˙μα˙+μβ˙+),and one computes the degeneracy
loci for generic moduli to be given by the equations(3.26)(λ+−μ+)2=0,(λ+−μ+)(λ+−ϱ+)=0on the bases of L5 and L4,
respectively. Here, ϱ+ is a rational function of μ+ and, therefore, it is obvious that both
degeneracy cycles are equivalent.When dealing with degenerated twistor spaces, one
usually retreats to the correspondence space endowed with some additional
symmetry conditions [22]. It is conceivable that a similar procedure will help
to define the topological B-model in our case. Also, defining a suitable blowup
of L4|6 over ℂPΔ1 could be the starting point for finding an
appropriate action.


## 3.1. Abstract Definition of the
Mini-Superambitwistor Space

The starting point is the product space𝒫2|3×𝒫*2|3 of two copies of the 𝒩=3 mini-supertwistor space. In analogy to the
space 𝒫3|3×𝒫*3|3,
we have coordinates (3.1)(w(a)1,w(a)2=λ(a),ηi(a);v(a)1,v(a)2=μ(a),θ(a)i) on the patches 𝒱(a) which are Cartesian products of 𝒱± and 𝒱±*: (3.2)𝒱(1):=𝒱+×𝒱+*,𝒱(2):=𝒱−×𝒱+*,𝒱(3):=𝒱+×𝒱−*,𝒱(4):=𝒱−×𝒱−*. For convenience, let us
introduce the subspace ℂPΔ1 of the base of the fibration 𝒫2|3×𝒫*2|3→ℂP1×ℂP*1 as (3.3)ℂPΔ1:=diag(ℂP1×ℂP*1)={(μ±,λ±)∈ℂP1×ℂP*1∣μ±=λ±}. Consider now the map ξ:𝒫2|3×𝒫*2|3→𝒪ℂPΔ1(2) which is defined by (3.4)ξ:(w(a)1,w(a)2,ηi(a);v(a)1,v(a)2,θ(a)i)↦{(w±1−v±1+2θ±iηi±,w±2)forw±2=v±2(0,w(a)2)else, where 𝒪ℂPΔ1(2) is the line bundle 𝒪(2) over ℂPΔ1.
In this definition, we used the fact that a point for which w±2=v±2 is at least on one of the patches 𝒱(1) and 𝒱(4).
Note, in particular, that the map ξ is a morphism of vector bundles. Therefore, we
can define a space L4|6 via the short exact sequence (3.5)0→L4|6→𝒫2|3×𝒫*2|3→ξ𝒪ℂPΔ1(2)→0 (cf. (2.20)). We will call this
space the mini-superambitwistor space.
Analogously to above, we will denote the pull-back of the patches 𝒱(a) to L4|6 by 𝒱^(a).
Obviously, the space L4|6 is a fibration, and we can switch to the
corresponding short exact sequence of sheaves of local
sections: (3.6)0→L4|6→𝒫2|3×𝒫*2|3→ξ𝒪ℂPΔ1(2)→0. Note  the difference in notation:
(3.5) is a sequence of vector bundles, while (3.6) is a sequence of sheaves. To
analyze the geometry of the space L4|6 in more detail, we will restrict ourselves to
the body of this space and put the fermionic coordinates to zero. Similarly to
the case of the superambitwistor space, this is possible as the map ξ does not affect the fermionic dimensions in
the exact sequence (3.5); this will become clearer in the discussion in Section
3.2.Inspired by the sequence defining the skyscraper sheaf
(a sheaf with sections supported only at the pointp)
0→𝒪(−1)→𝒪→𝒪p→0,
we introduce the following short exact sequence: (3.7)0→𝒪(1,−1)⊕𝒪(−1,1)→ζ𝒪(2,0)⊕𝒪(0,2)→𝒪ℂPΔ1(2)⊕𝒪ℂPΔ1(2)→0. Here, we defined ζ:(a,b)↦(aεα˙β˙λα˙μβ˙,bεα˙β˙λα˙μβ˙),
where λα˙ and μα˙ are the usual homogeneous coordinates on the
base space ℂP1×ℂP*1.
The sheaf 𝒪ℂPΔ1(2) is a torsion sheaf (sometimes sloppily
referred to as a skyscraper sheaf) with sections supported only over ℂPΔ1.
Finally, we trivially have the short exact sequence (3.8)0→𝒪ℂPΔ1(2)→α1𝒪ℂPΔ1(2)⊕𝒪ℂPΔ1(2)→α2𝒪ℂPΔ1(2)→0, where α1:(a)↦(a,a) and α2:(a,b)↦(a−b).Using the short exact sequences (3.6), (3.7), and
(3.8) as well as the nine lemma, we can establish the following
diagram: (3.9)000↓↓↓0→𝒪(1,−1)⊕𝒪(−1,1)→ζL4→𝒪ℂPΔ1(2)→0↓id↓↓α10→𝒪(1,−1)⊕𝒪(−1,1)→ζ𝒪(2,0)⊕𝒪(0,2)→𝒪ℂPΔ1(2)⊕𝒪ℂPΔ1(2)→0↓↓↓α20→0→𝒪ℂPΔ1(2)→id𝒪ℂPΔ1(2)→0.↓↓↓000From the horizontal lines of this diagram and the five
lemma, we conclude thatL4⊕𝒪ℂPΔ1(2)=𝒪(2,0)⊕𝒪(0,2).
Thus, L4 is not a locally free sheaf (a
more sophisticated argumentation would use the common properties of the torsion
functor to establish that L4 is a torsion sheaf; furthermore, one can write
down a further diagram using the nine lemma which shows that L4 is a coherent sheaf) but a torsion sheaf,
whose stalks over ℂPΔ1 are isomorphic to the stalks of 𝒪ℂPΔ1(2),
while the stalks over (ℂP1×ℂP*1)∖ℂPΔ1 are isomorphic to the stalks of 𝒪(2,0)⊕𝒪(0,2).
Therefore, L4 is not a vector bundle, but a fibration
(the homotopy lifting property typically included in the definition
of a fibration is readily derived from the definition of L4) with fibers ℂ2 over generic points and fibers ℂ over ℂPΔ1.
In particular, the total space of L4 is not a manifold.The fact that the total space of the bundleL4|6 is neither a supermanifold nor a supervector
bundle over ℂP1×ℂP*1 seems at first slightly disturbing. However,
we will show that once one is aware of this new aspect, it does not cause any
deep difficulties as far as the twistor correspondence and the Penrose-Ward
transform are concerned.


## 3.2. The
Mini-Superambitwistor Space by Dimensional Reduction

In the following, we will motivate the abstract
definition more concretely by considering explicitly the dimensional reduction
of the spaceL5|6,
we will also fix our notation in terms of coordinates and moduli of sections.
For this, we will first reduce the product space 𝒫3|3×𝒫*3|3 and then impose the appropriate reduced
quadric condition. In a first step, we want to eliminate in both 𝒫3|3 and 𝒫*3|3 the dependence on the bosonic modulus x3.
Thus, we should factorize by (3.10)𝒯(a)={∂∂z+2−z+3∂∂z+1on𝒰(1)z−3∂∂z−2−∂∂z−1on𝒰(2)∂∂z+2−z+3∂∂z+1on𝒰(3)z−3∂∂z−2−∂∂z−1on𝒰(4),𝒯(a)*={∂∂u+2˙−u+3∂∂u+1˙on𝒰(1)∂∂u+2˙−u+3∂∂u+1˙on𝒰(2)u−3∂∂u−2˙−∂∂u−1˙on𝒰(3)u−3∂∂u−2˙−∂∂u−1˙on𝒰(4), which leads us to the orbit
space (3.11)𝒫2|3×𝒫*2|3=(𝒫3|3/𝒢)×(𝒫*3|3/𝒢*), where 𝒢 and 𝒢* are  the abelian groups generated by 𝒯 and 𝒯*,
respectively. Recall that the coordinates we use on this space have been
defined in (3.1). The global sections of the bundle 𝒫2|4×𝒫*2|4→ℂP1×ℂP*1 are captured by the
parameterization
(3.12)w(a)1=yα˙β˙λα˙(a)λβ˙(a),v(a)1=y*α˙β˙μα˙(a)μβ˙(a),θ(a)i=θα˙iμα˙(a),ηi(a)=ηiα˙λα˙(a), where we relabel the indices of μα(a)→μα˙(a) and the moduli y*αβ→y*α˙β˙, θiα→θiα˙,
since there is no distinction between left- and right-handed spinors on ℝ3 or its complexification ℂ3.The next step is obviously to impose the quadric
condition, gluing together the self-dual and anti-self-dual parts. Note that
when acting with𝒯 and 𝒯* on κ(a) as given in (2.12), we obtain (3.13)𝒯(1)κ(1)=𝒯(1)*κ(1)=(μ+−λ+),𝒯(2)κ(2)=𝒯(2)*κ(2)=(λ−μ+−1),𝒯(3)κ(3)=𝒯(3)*κ(3)=(1−λ+μ−),𝒯(4)κ(4)=𝒯(4)*κ(4)=(λ−−μ−). This implies that the orbits
generated by 𝒯 and 𝒯* become orthogonal to the orbits of ∂/∂κ only at μ±=λ±.
We can, therefore, safely impose the condition (3.14)(w±1−v±1+2θ±iηi±)|λ±=μ±=0, and the subset of 𝒫2|3×𝒫*2|3 which satisfies this condition is obviously
identical to the mini-superambitwistor space L4|6 defined above.The condition (3.14) naturally fixes the
parametrization of global sections of the fibration L4|6 by giving a relation between the moduli used
in (3.12). This relation is completely analogous to (2.13) and
reads (3.15)yα˙β˙=y0α˙β˙−θ(α˙iηiβ˙),y*α˙β˙=y0α˙β˙+θ(α˙iηiβ˙). We clearly see that this
parameterization arises from (2.13) by dimensional
reduction from ℂ4→ℂ3.
Furthermore, even with this identification, w±1 and v±1 are independent of
λ±≠μ±.
Thus, indeed, imposing the condition (3.14) only at λ±=μ± is the dimensionally reduced analogue of
imposing the condition (2.12) on 𝒫3|3×𝒫*3|3.


## 3.3. Comments on Further Ways of ConstructingL4|6

Although the construction presented above seems most
natural, one can imagine other approaches of defining the spaceL4|6.
Completely evident is a second way, which uses the description of L5|6 in terms of coordinates on ℱ6|12.
Here, one factorizes the correspondence space ℱ6|12 by the groups generated by the vector field 𝒯3=𝒯3* and obtains the correspondence space 𝒦5|12≅ℂ3|12×ℂP1×ℂP*1 together with (3.15). A subsequent projection π2 from the dimensionally reduced correspondence
space 𝒦5|12 then yields the mini-superambitwistor space L4|6 as defined above.Furthermore, one can factorize𝒫3|3×𝒫*3|3 only by 𝒢 to eliminate the dependence on one modulus.
This will lead to 𝒫2|3×𝒫*3|3 and following the above discussion of imposing
the quadric condition on the appropriate subspace, one arrives again at (3.14)
and the space L4|6.
Here, the quadric condition already implies the remaining factorization of 𝒫2|3×𝒫*3|3 by 𝒢*.Eventually, one could anticipate the identification of
moduli in (3.15) and, therefore, want to factorize by the group generated by
the combination 𝒯+𝒯*.
Acting with this sum on κ(a) will produce the sum of the results given in
(3.13), and the subsequent discussion of the quadric condition follows the one
presented above.


## 3.4. Double Fibration

Knowing the
parameterization of global sections of the
mini-superambitwistor space fibered overℂP1×ℂP*1 as defined in (3.15), we can establish a
double fibration, similarly to all the other twistor spaces we encountered so
far. Even more instructive is the following diagram, in which the dimensional
reduction of the involved spaces becomes evident:
(3.16)
The upper half is just the
double fibration for the quadric (2.14), while the lower half corresponds to
the dimensionally reduced case. The reduction of ℂ4|12 to ℂ3|12 is obviously done by factoring out the group
generated by 𝒯3.
The same is true for the reduction of ℱ6|12≅ℂ4|12×ℂP1×ℂP*1 to 𝒦5|12≅ℂ3|12×ℂP1×ℂP*1.
The reduction from L5|6 to L4|6 was given above and the projection ν2 from 𝒦5|12 onto L4|6 is defined by (3.12). The four patches
covering ℱ6|12 will be denoted by 𝒱˜(a):=ν2−1(𝒱^(a)).The double fibration defined by the projectionsν1 and ν2 yields the following twistor
correspondences:(3.17){subspaces(ℂP1×ℂP1)y0,η,θinL4|6}⟷{points(y0,η,θ)inℂ3|12},{genericpointspinL4|6}⟷{superlinesinℂ3|12},{pointspinL4|6withλ±=μ±}⟷{superplanesinℂ3|12}.The superlines and the
superplanes in ℂ3|12 are defined as the sets(3.18){(yα˙β˙,ηiα˙,θα˙i)}withyα˙β˙=y^α˙β˙+tλ(a)(α˙μ(a)β˙),ηiα˙=η^iα˙+εiλ(a)α˙,θα˙i=θ^α˙i+ε˜iμ(a)α˙,{(yα˙β˙,ηiα˙,θα˙i)}withyα˙β˙=y^α˙β˙+κ(α˙λ(a)β˙),ηiα˙=η^iα˙+εiλ(a)α˙,θα˙i=θ^α˙i+ε˜iλ(a)α˙,where t, κα˙, εi,
and ε˜i are an arbitrary complex number, a complex
commuting 2-spinor, and two 3-vectors with Graßmann-odd components,
respectively. Note that in the last line, λ±α˙=μ±α˙,
and we could also have written(3.19){(yα˙β˙,ηiα˙,θα˙i)}withyα˙β˙=y^α˙β˙+κ(α˙μ(a)β˙),ηiα˙=η^iα˙+εiμ(a)α˙,θα˙i=θ^α˙i+ε˜iμ(a)α˙.The vector fields spanning the tangent spaces to the
leaves of the fibrationν2 are for generic values of μ± and λ± given by(3.20)W(a):=μ(a)α˙λ(a)β˙∂(α˙β˙),D˜(a)i:=λ(a)β˙D˜β˙i:=λ(a)β˙(∂∂ηiβ˙+θα˙i∂(α˙β˙)),Di(a):=μ(a)α˙Dα˙i:=μ(a)α˙(∂∂θα˙i+ηiβ˙∂(α˙β˙)),where the derivatives ∂(α˙β˙) have been defined in (2.34). At μ±=λ±,
however, the fibers of the fibration L4|6 over ℂP1×ℂP*1 loose one bosonic dimension. As the space 𝒦5|12 is a manifold, this means that this dimension
has to become tangent to the projection ν2.
In fact, one finds that over ℂPΔ1 besides the vector fields given in (3.20),
also the vector fields(3.21)W˜β˙±=μ±α˙∂(α˙β˙)=λ±α˙∂(α˙β˙)annihilate the coordinates on L4|6.
Therefore, the leaves to the projection ν2:𝒦5|12→L4|6 are of dimension 2|6 for μ±=λ± and of dimension 1|6 everywhere else.


## 3.5. Real Structure onL4|6

Quite evidently, a real structure onL4|6 is inherited from the one on L5|6,
and we obtain directly from (2.22) the action of τM on 𝒫2|4×𝒫*2|4,
which is given by(3.22)τM(w±1,λα˙±,ηi±;v±1,μα˙±,θ±i):=(v±1¯,μα˙±¯,θ±i¯;w±1¯,λα˙±¯,ηi±¯).This action descends in an
obvious manner to L4|6,
which leads to a real structure on the moduli space ℂ3|12 via the double fibration (3.16). Thus, we have
as the resulting reality condition(3.23)y0α˙β˙=y0β˙α˙¯,ηiα˙=θα˙i¯,and the identification of the
bosonic moduli yα˙β˙ with the coordinates on ℝ3 reads as(3.24)y01˙1˙=x00+x01,y01˙2˙=y02˙1˙=x02,y02˙2˙=x00−x01.The reality conditionτM(⋅)=⋅ is indeed fully compatible with the condition
(3.14) which reduces 𝒫2|4×𝒫*2|4 to L4|6.
The base space ℂP1×ℂP*1 of the fibration L4|6 is reduced to a single sphere S2 with real coordinates (1/2)(λ±+μ±)=(1/2)(λ±+λ¯±) and (1/2i)(λ±−μ±)=(1/2i)(λ±−λ¯±),
while the diagonal ℂPΔ1 is reduced to a circle SΔ1 parameterized by the real coordinates (1/2)(λ±+λ¯±).
The τM-real sections of L4|6 have to satisfy w±1=τM(w±1)=v¯±1.
Thus, the fibers of the fibration L4|6→ℂP1×ℂP*1,
which are of complex dimension 2|6 over generic points in the base and complex
dimension 1|6 over ℂPΔ1,
are reduced to fibers of real dimension 2|6 and 1|6,
respectively. In particular, note that θ±iηi±=η¯i±θ¯±i=−θ¯±iη¯i± is purely imaginary and, therefore, the
quadric condition (3.14) together with the real structure τM implies that w±1=v¯±1=w¯±1+2θ¯±iη¯i± for λ±=μ±=λ¯±.
Thus, the body w∘±1 of w±1 is purely real and we have w±1=w∘±1−θ±iηi± and v±1=w∘±1+θ±iηi± on the diagonal SΔ1.


## 3.6. Interpretation of the Involved Real Geometries

For the best-known twistor correspondences (i.e., the
correspondence (2.5)) (more precisely, the compactified version
thereof) its dual, and the correspondence (2.14), there is a nice
description in terms of flag manifolds (see e.g., [3]). For the spaces involved in
the twistor correspondences including minitwistor spaces, one has a similarly
nice interpretation after restricting to the real situation. For simplicity, we
reduce our considerations to the bodies (i.e., drop the fermionic directions)
of the involved geometries, as the extension to corresponding supermanifolds is
quite straightforward.Let us first discuss the double fibration (2.30), and
assume that we have imposed a suitable reality condition on the fiber
coordinates, the details of which are not important. We follow again the usual
discussion of the real case and leave the coordinates on the sphere complex. As
correspondence space on top of the double fibration, we have thus the space ℝ3×S2,
which we can understand as the set of oriented lines (not only
the ones through the origin) in ℝ3 with one marked point. Clearly, the point of
such a line is given by an element of ℝ3,
and the direction of this line in ℝ3 is parameterized by a point on S2.
The minitwistor space 𝒫2≅𝒪(2) now is simply the space of all lines in ℝ3 [19]. Similarly to the case of flag manifolds, the
projections ν1 and ν2 in (2.30) become, therefore, obvious. For ν1,
simply drop the line and keep the marked point. For ν2,
drop the marked point and keep the line. Equivalently, we can understand ν2 as moving the marked point on the line to its
shortest possible distance from the origin. This leads to the space TS2≅𝒪(2),
where the S2
parameterizes
again the direction of the line, which can subsequently be still moved
orthogonally to this direction, and this freedom is parameterized by the
tangent planes to S2 which are isomorphic to ℝ2.Now in the case of the fibration included in (3.16),
we impose the reality condition (3.22) on the fiber coordinates of L4.
In the real case, the correspondence space 𝒦5 becomes the space ℝ3×S2×S2 and this is the space of two oriented lines in ℝ3 intersecting in a point. More precisely, this
is the space of two oriented lines in ℝ3 each with one marked point, for which the two
marked points coincide. The projections ν1 and ν2 in (3.16) are then interpreted as follows. For ν1,
simply drop the two lines and keep the marked point. For ν2,
fix one line and move the marked point (the intersection point) together with
the second line to its shortest distance to the origin. Thus, the space L4 is the space of configurations in ℝ3,
in which a line has a common point with another line at its shortest distance
to the origin.Let us summarize all the above findings in
Table1.Table 1SpaceRelation toℝ3ℝ3marked points inℝ3ℝ3×S2oriented lines with a marked point inℝ3𝒫2≅𝒪(2)oriented lines inℝ3 (with a marked point at shortest distance to the origin.)ℝ3×S2×S2two oriented lines with a common marked
point inℝ3L4two oriented lines with a common marked
point at shortest distance
from one of the lines to the origin inℝ3


## 3.7. Remarks Concerning a Topological B-Model onL4|6

The spaceL4|6 is not well suited as a target space for a
topological B-model since it is not a (Calabi-Yau) manifold. However, one
clearly expects that it is possible to define an analogous model since, if we
assume that the conjecture in [20, 21] is correct, such a model should simply be the mirror
of the minitwistor string theory considered in [14]. This model would
furthermore yield some holomorphic Chern-Simons type equations of motion. The
latter equations would then define holomorphic pseudobundles over L4|6 by an analogue of a holomorphic structure.
These bundles will be introduced in Section 4.3 and in our discussion, they
substitute the holomorphic vector bundles.Interestingly, the spaceL4|6 has a property which comes close to vanishing
of a first Chern class. Recall that for any complex vector bundle, its Chern
classes are Poincaré dual to the degeneracy cycles of certain sets of sections
(this is a Gauß-Bonnet formula). More precisely, to calculate the first Chern
class of a rank r vector bundle, one considers r generic sections and arranges them into an r×r matrix L.
The degeneracy loci on the base space are then given by the zero locus of det(L).
Clearly, this calculation can be translated directly to L4|6.We will now show thatL4|6 and L5|6 have equivalent degeneracy loci (i.e., they
are equal up to a principal divisor) which, speaking about ordinary vector
bundles, would not affect the first Chern class. Our discussion simplifies
considerably if we restrict our attention to the bodies of the two supertwistor
spaces and put all the fermionic coordinates to zero. Note that this will not
affect the result, as the quadric conditions defining L5|6 and L4|6 do not affect the fermionic dimensions: the
fermionic parts of the fibrations L5|6 and L4|6 are identical, which is easily seen by
considering the global sections generating the total spaces of the fibrations.
Instead of the ambitwistor spaces, it is also easier to consider the vector
bundles 𝒫3×𝒫*3 and 𝒫2×𝒫*2 over ℂP1×ℂP*1,
respectively, with the appropriately restricted sets of sections. Furthermore,
we will stick to our inhomogeneous coordinates and perform the calculation only
on the patch 𝒰(1),
but all this directly translates into homogeneous, patch-independent
coordinates. The matrices to be considered are(3.25)LL5=(x11α˙λα˙+x21α˙λα˙+x31α˙λα˙+x41α˙λα˙+x12α˙λα˙+x22α˙λα˙+x32α˙λα˙+x42α˙λα˙+x1α1μα+x2α1μα+x3α1μα+x4α1μα+x1α2μα+x2α2μα+x3α2μα+x4α2μα+),LL4=(y1α˙β˙λα˙+λβ˙+y2α˙β˙λα˙+λβ˙+y1α˙β˙μα˙+μβ˙+y2α˙β˙μα˙+μβ˙+),and one computes the degeneracy
loci for generic moduli to be given by the equations(3.26)(λ+−μ+)2=0,(λ+−μ+)(λ+−ϱ+)=0on the bases of L5 and L4,
respectively. Here, ϱ+ is a rational function of μ+ and, therefore, it is obvious that both
degeneracy cycles are equivalent.When dealing with degenerated twistor spaces, one
usually retreats to the correspondence space endowed with some additional
symmetry conditions [22]. It is conceivable that a similar procedure will help
to define the topological B-model in our case. Also, defining a suitable blowup
of L4|6 over ℂPΔ1 could be the starting point for finding an
appropriate action.


## 4. The Penrose-Ward Transform for the Mini-Superambitwistor Space

### 4.1. Review of the Penrose-Ward Transform on the Superambitwistor Space

Letℰ be a topologically trivial holomorphic vector
bundle of rank n over L5|6 which becomes holomorphically trivial when
restricted to any subspace (ℂP1×ℂP1)x0,η,θ↪L5|6.
Due to the equivalence of the Čech and the Dolbeault descriptions of
holomorphic vector bundles, we can describe ℰ either by holomorphic transition functions {fab} or by a holomorphic structure ∂¯𝒜^=∂¯+𝒜^.
Starting from a transition function fab,
there is a splitting(4.1)fab=ψ^a−1ψ^b,where the ψ^a are smooth GL(n,ℂ)-valued functions (in fact, the collection {ψ^a} forms a Čech 0-cochain) on 𝒰(a) since the bundle ℰ is topologically trivial. This splitting
allows us to switch to the holomorphic structure ∂¯+𝒜^ with 𝒜^=ψ^∂¯ψ^−1,
which describes a trivial vector bundle ℰ^≅ℰ.
Note that the additional condition of holomorphic triviality of ℰ on subspaces (ℂP1×ℂP1)x0,η,θ will restrict the explicit form of 𝒜^.Back at the bundleℰ,
consider its pull-back π2*ℰ with transition functions {π2*fab},
which are constant along the fibers of π2:ℱ6|12→L5|6:(4.2)W(a)π2*fab=D(a)iπ2*fab=Di(a)π2*fab=0.The additional assumption of
holomorphic triviality upon reduction onto a subspace allows for a
splitting(4.3)π2*fab=ψa−1ψbinto GL(n,ℂ)-valued functions {ψa} which are holomorphic on 𝒰˜(a).
Evidently, there is such a splitting holomorphic in the coordinates λ(a) and μ(a) on (ℂP1×ℂP1)x0,η,θ since ℰ becomes holomorphically trivial when
restricted to these spaces. Furthermore, these subspaces are holomorphically
parameterized by the moduli (x0αα˙,ηiα˙,θαi),
and thus the splitting (4.3) is holomorphic in all the coordinates of ℱ6|12.
Due to (4.2), we have on the intersections 𝒰˜(a)∩𝒰˜(b):(4.4)ψaD(a)iψa−1=ψbD(a)iψb−1=:λ(a)α˙𝒜α˙i,ψaDi(a)ψa−1=ψbDi(a)ψb−1=:μ(a)α𝒜αi,ψaW(a)ψa−1=ψbW(a)ψb−1=:μ(a)αλ(a)α˙𝒜αα˙, where 𝒜α˙i, 𝒜αi,
and 𝒜αα˙ are independent of μ(a) and λ(a).
The introduced components of the supergauge potential 𝒜 fit into the linear system(4.5)μ(a)αλ(a)α˙(∂αα˙+𝒜αα˙)ψa=0,λ(a)α˙(Dα˙i+𝒜α˙i)ψa=0,μ(a)α(Dαi+𝒜αi)ψa=0,whose compatibility conditions are(4.6){∇α˙i,∇β˙j}+{∇β˙i,∇α˙j}=0,{∇αi,∇βj}+{∇βi,∇αj}=0,{∇αi,∇α˙j}−2δij∇αα˙=0.Here, we used the obvious
shorthand notations ∇α˙i:=Dα˙i+𝒜α˙i, ∇αi:=Dαi+𝒜αi,
and ∇αα˙=∂αα˙+𝒜αα˙.
However, (4.6) are well known to be equivalent to the equations of motion of 𝒩=3 SYM theory on ℂ4 [23] (note that most of our considerations
concern the complexified case), and, therefore, also to 𝒩=4 SYM theory on ℂ4.We thus showed that there is a correspondence between
certain holomorphic structures onL5|6,
holomorphic vector bundles over L5|6 which become holomorphically trivial when
restricted to certain subspaces and solutions to the 𝒩=4 SYM equations on ℂ4.
The redundancy in each set of objects is modded out by considering gauge
equivalence classes and holomorphic equivalence classes of vector bundles,
which renders the above correspondences one-to-one.


### 4.2.𝒩=8 SYM Theory in Three Dimension

This theory is obtained by dimensionally reducing𝒩=1 SYM theory in ten dimensions to three
dimensions, or, equivalently, by dimensionally reducing four-dimensional 𝒩=4 SYM theory to three dimensions. As a result,
the 16 real supercharges are rearranged in the latter case from four spinors
transforming as a 2ℂ of Spin(3,1)≅SL(2,ℂ) into eight spinors transforming as a 2ℝ of Spin(2,1)≅SL(2,ℝ).The automorphism group of the supersymmetry algebra isSpin(8),
and the little group of the remaining Lorentz group SO(2,1) is trivial. As massless particle content, we,
therefore, expect bosons transforming in the 8v and fermions transforming in the 8c of Spin(8).
One of the bosons will, however, appear as a dual gauge potential on ℝ3 after dimensional reduction, and, therefore,
only a Spin(7) R-symmetry group is
manifest in the action and the equations of
motion. In the mini-superambitwistor formulation, the manifest subgroup of the
R-symmetry group is only SU(3)×U(1)×SU(3)×U(1).
Altogether, we have a gauge potential Aμ with μ=1,…,3,
seven scalars ϕi with i=1,…,7,
and eight spinors χα˙j with j=1,…,8.Moreover, recall that in four dimensions,𝒩=3 and 𝒩=4 super-Yang-Mills theories are equivalent on
the level of field content and corresponding equations of motion. The only
difference is found in the manifest R-symmetry groups which are SU(3)×U(1) and SU(4),
respectively. This equivalence obviously carries over to the three-dimensional
situation. 𝒩=6 and 𝒩=8 super-Yang-Mills theories are equivalent
regarding their field content and the equations of motion. Therefore, it is
sufficient to construct a twistor correspondence for 𝒩=6 SYM theory to describe solutions to the 𝒩=8 SYM equations.


### 4.3. Pseudobundles overL4|6

Because the mini-superambitwistor space is only a
fibration and not a manifold, there is no notion of holomorphic vector bundles
overL4|6.
However, our space is close enough to a manifold to translate all the necessary
terms in a simple manner.Let us fix the covering𝔘 of the total space of the fibration L4|6 to be given by the patches 𝒱(a) introduced above. Furthermore, define 𝔖 to be the sheaf of smooth GL(n,ℂ)-valued functions on L4|6 and ℌ to be its subsheaf consisting of holomorphic GL(n,ℂ)-valued functions on L4|6,
i.e. smooth and holomorphic functions which depend only on the coordinates
given in (3.12) and λ(a),μ(a).We define acomplex pseudo-bundle overL4|6 of rank n by a Čech 1-cocycle {fab}∈Ž1(𝔘,𝔖) on L4|6 in full analogy with transition functions
defining ordinary vector bundles. If the 1-cocycle is an element of Ž1(𝔘,ℌ),
we speak of a holomorphic
pseudo-bundle overL4|6.
Two pseudo-bundles given by Čech 1-cocycles {fab} and {fab′} are called topologically equivalent (holomorphically equivalent), if there is a Čech 0-cochain {ψa}∈Č0(𝔘,𝔖) (a Čech 0-cochain {ψa}∈Č0(𝔘,ℌ)) such that fab=ψa−1fab′ψb.
A pseudo-bundle is called trivial (holomorphically trivial), if it is topologically equivalent
(holomorphically equivalent) to the trivial pseudo-bundle given by {fab}={⊮ab}.In the corresponding discussion of Čech cohomology on ordinary manifolds, one can
achieve independence of the covering if the patches of the covering are all
Stein manifolds. An analogous argument should also be applicable here, but for
our purposes, it is enough to restrict to the covering𝔘.Besides the Čech description, it is also possible to
introduce an equivalent Dolbeault description, which will, however, demand an
extended notion of Dolbeault cohomology classes.


### 4.4. The Penrose-Ward Transform Using the Mini-Superambitwistor Space

With the double fibration contained in (3.16), it is
not hard to establish the corresponding Penrose-Ward transform, which is
essentially a dimensional reduction of the four-dimensional case presented in
Section 4.1.OnL4|6,
we start from a trivial rank n holomorphic pseudo-bundle over L4|6 defined by a 1-cocycle {fab} which becomes a holomorphically trivial vector
bundle upon restriction to any subspace (ℂP1×ℂP1)y0,η,θ↪L4|6.
The pull-back of the pseudo-bundle over L4|6 along ν2 is the vector bundle ℰ˜ with transition functions {ν2*fab} satisfying by definition(4.7)W(a)ν2*fab=D˜(a)iν2*fab=Di(a)ν2*fab=0,at generic points of L4|6 and for λ±=μ±,
we have(4.8)W˜α˙(a)ν2*fab=D˜(a)iν2*fab=Di(a)ν2*fab=0.Restricting the bundle ℰ˜ to a subspace (ℂP1×ℂP1)y0,η,θ↪L4|6⊂ℱ5|12 yields a splitting of the transition function ν2*fab(4.9)ν2*fab=ψa−1ψb,where {ψa} are again GL(n,ℂ)-valued functions on 𝒱˜(a) which are holomorphic. From this splitting
together with (4.7), one obtains that(4.10)ψaD˜(a)iψa−1=ψbD˜(a)iψb−1=:λ(a)α˙𝒜˜α˙i,ψaDi(a)ψa−1=ψbDi(a)ψb−1=:μ(a)α˙𝒜α˙i,ψaW(a)ψa−1=ψbW(a)ψb−1=:μ(a)α˙λ(a)β˙ℬα˙β˙forλ≠μ,ψaWα˙(a)ψa−1=ψbWα˙(a)ψb−1=:λ(a)β˙ℬ˜α˙β˙forλ=μ.These equations are due to a
generalized Liouville theorem, and continuity yields that ℬ˜α˙β˙=ℬα˙β˙.
Furthermore, one immediately notes that a transition function ν2*fab,
which satisfies (4.7) is of the form(4.11)fab=fab(yα˙β˙λα˙(a)λβ˙(a),yα˙β˙μα˙(a)μβ˙(a),λα˙(a),μα˙(a)),and thus condition (4.8) is
obviously fulfilled at points with λ±=μ±.
Altogether, since we neither loose any information on the gauge potential nor
do we loose any constraints on it, we can restrict our discussion to generic
points with λ≠μ,
which simplifies the presentation.The superfieldℬα˙β˙ decomposes into a gauge potential and a Higgs
field Φ:(4.12)ℬα˙β˙:=𝒜(α˙β˙)+i2εα˙β˙Φ.The zeroth-order component in
the superfield expansion of Φ will be the seventh real scalar joining the
six scalars of 𝒩=4 SYM in four dimensions, which are the zeroth
component of the superfield Φij defined in(4.13){Dα˙i+𝒜α˙i,Dβ˙j+𝒜β˙j}=:−2εα˙β˙Φij.Thus, as mentioned above, the Spin(7) R-symmetry group of 𝒩=8 SYM theory in three dimensions will not be
manifest in this description.Equations (4.10) are equivalent to the linear
system(4.14)μ(a)α˙λ(a)β˙(∂(α˙β˙)+ℬα˙β˙)ψa=0,λ(a)α˙(D˜α˙i+𝒜˜α˙i)ψa=0,μ(a)α˙(Dα˙i+𝒜α˙i)ψa=0.To discuss the corresponding
compatibility conditions, we introduce the following differential
operators:(4.15)∇˜α˙i:=D˜α˙i+𝒜˜α˙i,∇α˙i:=Dα˙i+𝒜α˙i,∇α˙β˙:=∂(α˙β˙)+ℬα˙β˙.We thus arrive
at(4.16){∇˜α˙i,∇˜β˙j}+{∇˜β˙i,∇˜α˙j}=0,{∇α˙i,∇β˙j}+{∇β˙i,∇α˙j}=0,{∇α˙i,∇˜β˙j}−2δij∇α˙β˙=0,and one clearly sees that (4.16)
are indeed (4.6) after a dimensional reduction ℂ4→ℂ3 and defining Φ:=A4.
As it is well known, the supersymmetry (and the R-symmetry) of 𝒩=4 SYM theory are enlarged by this dimensional
reduction, and we, therefore, obtained indeed 𝒩=8 SYM theory on ℂ3.To sum up, we obtained a correspondence between
holomorphic pseudobundles overL4|6 which become holomorphically trivial vector
bundles upon reduction to any subspace (ℂP1×ℂP1)y0,η,θ↪L4|6 and solutions to the three-dimensional 𝒩=8 SYM equations. As this correspondence arises
by a dimensional reduction of a correspondence which is one-to-one, it is
rather evident that also in this case, we have a bijection between both the
holomorphic pseudobundles over L4|6 and the solutions after factoring out
holomorphic equivalence and gauge equivalence, respectively.


## 4.1. Review of the Penrose-Ward Transform on the Superambitwistor Space

Letℰ be a topologically trivial holomorphic vector
bundle of rank n over L5|6 which becomes holomorphically trivial when
restricted to any subspace (ℂP1×ℂP1)x0,η,θ↪L5|6.
Due to the equivalence of the Čech and the Dolbeault descriptions of
holomorphic vector bundles, we can describe ℰ either by holomorphic transition functions {fab} or by a holomorphic structure ∂¯𝒜^=∂¯+𝒜^.
Starting from a transition function fab,
there is a splitting(4.1)fab=ψ^a−1ψ^b,where the ψ^a are smooth GL(n,ℂ)-valued functions (in fact, the collection {ψ^a} forms a Čech 0-cochain) on 𝒰(a) since the bundle ℰ is topologically trivial. This splitting
allows us to switch to the holomorphic structure ∂¯+𝒜^ with 𝒜^=ψ^∂¯ψ^−1,
which describes a trivial vector bundle ℰ^≅ℰ.
Note that the additional condition of holomorphic triviality of ℰ on subspaces (ℂP1×ℂP1)x0,η,θ will restrict the explicit form of 𝒜^.Back at the bundleℰ,
consider its pull-back π2*ℰ with transition functions {π2*fab},
which are constant along the fibers of π2:ℱ6|12→L5|6:(4.2)W(a)π2*fab=D(a)iπ2*fab=Di(a)π2*fab=0.The additional assumption of
holomorphic triviality upon reduction onto a subspace allows for a
splitting(4.3)π2*fab=ψa−1ψbinto GL(n,ℂ)-valued functions {ψa} which are holomorphic on 𝒰˜(a).
Evidently, there is such a splitting holomorphic in the coordinates λ(a) and μ(a) on (ℂP1×ℂP1)x0,η,θ since ℰ becomes holomorphically trivial when
restricted to these spaces. Furthermore, these subspaces are holomorphically
parameterized by the moduli (x0αα˙,ηiα˙,θαi),
and thus the splitting (4.3) is holomorphic in all the coordinates of ℱ6|12.
Due to (4.2), we have on the intersections 𝒰˜(a)∩𝒰˜(b):(4.4)ψaD(a)iψa−1=ψbD(a)iψb−1=:λ(a)α˙𝒜α˙i,ψaDi(a)ψa−1=ψbDi(a)ψb−1=:μ(a)α𝒜αi,ψaW(a)ψa−1=ψbW(a)ψb−1=:μ(a)αλ(a)α˙𝒜αα˙, where 𝒜α˙i, 𝒜αi,
and 𝒜αα˙ are independent of μ(a) and λ(a).
The introduced components of the supergauge potential 𝒜 fit into the linear system(4.5)μ(a)αλ(a)α˙(∂αα˙+𝒜αα˙)ψa=0,λ(a)α˙(Dα˙i+𝒜α˙i)ψa=0,μ(a)α(Dαi+𝒜αi)ψa=0,whose compatibility conditions are(4.6){∇α˙i,∇β˙j}+{∇β˙i,∇α˙j}=0,{∇αi,∇βj}+{∇βi,∇αj}=0,{∇αi,∇α˙j}−2δij∇αα˙=0.Here, we used the obvious
shorthand notations ∇α˙i:=Dα˙i+𝒜α˙i, ∇αi:=Dαi+𝒜αi,
and ∇αα˙=∂αα˙+𝒜αα˙.
However, (4.6) are well known to be equivalent to the equations of motion of 𝒩=3 SYM theory on ℂ4 [23] (note that most of our considerations
concern the complexified case), and, therefore, also to 𝒩=4 SYM theory on ℂ4.We thus showed that there is a correspondence between
certain holomorphic structures onL5|6,
holomorphic vector bundles over L5|6 which become holomorphically trivial when
restricted to certain subspaces and solutions to the 𝒩=4 SYM equations on ℂ4.
The redundancy in each set of objects is modded out by considering gauge
equivalence classes and holomorphic equivalence classes of vector bundles,
which renders the above correspondences one-to-one.


## 4.2.𝒩=8 SYM Theory in Three Dimension

This theory is obtained by dimensionally reducing𝒩=1 SYM theory in ten dimensions to three
dimensions, or, equivalently, by dimensionally reducing four-dimensional 𝒩=4 SYM theory to three dimensions. As a result,
the 16 real supercharges are rearranged in the latter case from four spinors
transforming as a 2ℂ of Spin(3,1)≅SL(2,ℂ) into eight spinors transforming as a 2ℝ of Spin(2,1)≅SL(2,ℝ).The automorphism group of the supersymmetry algebra isSpin(8),
and the little group of the remaining Lorentz group SO(2,1) is trivial. As massless particle content, we,
therefore, expect bosons transforming in the 8v and fermions transforming in the 8c of Spin(8).
One of the bosons will, however, appear as a dual gauge potential on ℝ3 after dimensional reduction, and, therefore,
only a Spin(7) R-symmetry group is
manifest in the action and the equations of
motion. In the mini-superambitwistor formulation, the manifest subgroup of the
R-symmetry group is only SU(3)×U(1)×SU(3)×U(1).
Altogether, we have a gauge potential Aμ with μ=1,…,3,
seven scalars ϕi with i=1,…,7,
and eight spinors χα˙j with j=1,…,8.Moreover, recall that in four dimensions,𝒩=3 and 𝒩=4 super-Yang-Mills theories are equivalent on
the level of field content and corresponding equations of motion. The only
difference is found in the manifest R-symmetry groups which are SU(3)×U(1) and SU(4),
respectively. This equivalence obviously carries over to the three-dimensional
situation. 𝒩=6 and 𝒩=8 super-Yang-Mills theories are equivalent
regarding their field content and the equations of motion. Therefore, it is
sufficient to construct a twistor correspondence for 𝒩=6 SYM theory to describe solutions to the 𝒩=8 SYM equations.


## 4.3. Pseudobundles overL4|6

Because the mini-superambitwistor space is only a
fibration and not a manifold, there is no notion of holomorphic vector bundles
overL4|6.
However, our space is close enough to a manifold to translate all the necessary
terms in a simple manner.Let us fix the covering𝔘 of the total space of the fibration L4|6 to be given by the patches 𝒱(a) introduced above. Furthermore, define 𝔖 to be the sheaf of smooth GL(n,ℂ)-valued functions on L4|6 and ℌ to be its subsheaf consisting of holomorphic GL(n,ℂ)-valued functions on L4|6,
i.e. smooth and holomorphic functions which depend only on the coordinates
given in (3.12) and λ(a),μ(a).We define acomplex pseudo-bundle overL4|6 of rank n by a Čech 1-cocycle {fab}∈Ž1(𝔘,𝔖) on L4|6 in full analogy with transition functions
defining ordinary vector bundles. If the 1-cocycle is an element of Ž1(𝔘,ℌ),
we speak of a holomorphic
pseudo-bundle overL4|6.
Two pseudo-bundles given by Čech 1-cocycles {fab} and {fab′} are called topologically equivalent (holomorphically equivalent), if there is a Čech 0-cochain {ψa}∈Č0(𝔘,𝔖) (a Čech 0-cochain {ψa}∈Č0(𝔘,ℌ)) such that fab=ψa−1fab′ψb.
A pseudo-bundle is called trivial (holomorphically trivial), if it is topologically equivalent
(holomorphically equivalent) to the trivial pseudo-bundle given by {fab}={⊮ab}.In the corresponding discussion of Čech cohomology on ordinary manifolds, one can
achieve independence of the covering if the patches of the covering are all
Stein manifolds. An analogous argument should also be applicable here, but for
our purposes, it is enough to restrict to the covering𝔘.Besides the Čech description, it is also possible to
introduce an equivalent Dolbeault description, which will, however, demand an
extended notion of Dolbeault cohomology classes.


## 4.4. The Penrose-Ward Transform Using the Mini-Superambitwistor Space

With the double fibration contained in (3.16), it is
not hard to establish the corresponding Penrose-Ward transform, which is
essentially a dimensional reduction of the four-dimensional case presented in
Section 4.1.OnL4|6,
we start from a trivial rank n holomorphic pseudo-bundle over L4|6 defined by a 1-cocycle {fab} which becomes a holomorphically trivial vector
bundle upon restriction to any subspace (ℂP1×ℂP1)y0,η,θ↪L4|6.
The pull-back of the pseudo-bundle over L4|6 along ν2 is the vector bundle ℰ˜ with transition functions {ν2*fab} satisfying by definition(4.7)W(a)ν2*fab=D˜(a)iν2*fab=Di(a)ν2*fab=0,at generic points of L4|6 and for λ±=μ±,
we have(4.8)W˜α˙(a)ν2*fab=D˜(a)iν2*fab=Di(a)ν2*fab=0.Restricting the bundle ℰ˜ to a subspace (ℂP1×ℂP1)y0,η,θ↪L4|6⊂ℱ5|12 yields a splitting of the transition function ν2*fab(4.9)ν2*fab=ψa−1ψb,where {ψa} are again GL(n,ℂ)-valued functions on 𝒱˜(a) which are holomorphic. From this splitting
together with (4.7), one obtains that(4.10)ψaD˜(a)iψa−1=ψbD˜(a)iψb−1=:λ(a)α˙𝒜˜α˙i,ψaDi(a)ψa−1=ψbDi(a)ψb−1=:μ(a)α˙𝒜α˙i,ψaW(a)ψa−1=ψbW(a)ψb−1=:μ(a)α˙λ(a)β˙ℬα˙β˙forλ≠μ,ψaWα˙(a)ψa−1=ψbWα˙(a)ψb−1=:λ(a)β˙ℬ˜α˙β˙forλ=μ.These equations are due to a
generalized Liouville theorem, and continuity yields that ℬ˜α˙β˙=ℬα˙β˙.
Furthermore, one immediately notes that a transition function ν2*fab,
which satisfies (4.7) is of the form(4.11)fab=fab(yα˙β˙λα˙(a)λβ˙(a),yα˙β˙μα˙(a)μβ˙(a),λα˙(a),μα˙(a)),and thus condition (4.8) is
obviously fulfilled at points with λ±=μ±.
Altogether, since we neither loose any information on the gauge potential nor
do we loose any constraints on it, we can restrict our discussion to generic
points with λ≠μ,
which simplifies the presentation.The superfieldℬα˙β˙ decomposes into a gauge potential and a Higgs
field Φ:(4.12)ℬα˙β˙:=𝒜(α˙β˙)+i2εα˙β˙Φ.The zeroth-order component in
the superfield expansion of Φ will be the seventh real scalar joining the
six scalars of 𝒩=4 SYM in four dimensions, which are the zeroth
component of the superfield Φij defined in(4.13){Dα˙i+𝒜α˙i,Dβ˙j+𝒜β˙j}=:−2εα˙β˙Φij.Thus, as mentioned above, the Spin(7) R-symmetry group of 𝒩=8 SYM theory in three dimensions will not be
manifest in this description.Equations (4.10) are equivalent to the linear
system(4.14)μ(a)α˙λ(a)β˙(∂(α˙β˙)+ℬα˙β˙)ψa=0,λ(a)α˙(D˜α˙i+𝒜˜α˙i)ψa=0,μ(a)α˙(Dα˙i+𝒜α˙i)ψa=0.To discuss the corresponding
compatibility conditions, we introduce the following differential
operators:(4.15)∇˜α˙i:=D˜α˙i+𝒜˜α˙i,∇α˙i:=Dα˙i+𝒜α˙i,∇α˙β˙:=∂(α˙β˙)+ℬα˙β˙.We thus arrive
at(4.16){∇˜α˙i,∇˜β˙j}+{∇˜β˙i,∇˜α˙j}=0,{∇α˙i,∇β˙j}+{∇β˙i,∇α˙j}=0,{∇α˙i,∇˜β˙j}−2δij∇α˙β˙=0,and one clearly sees that (4.16)
are indeed (4.6) after a dimensional reduction ℂ4→ℂ3 and defining Φ:=A4.
As it is well known, the supersymmetry (and the R-symmetry) of 𝒩=4 SYM theory are enlarged by this dimensional
reduction, and we, therefore, obtained indeed 𝒩=8 SYM theory on ℂ3.To sum up, we obtained a correspondence between
holomorphic pseudobundles overL4|6 which become holomorphically trivial vector
bundles upon reduction to any subspace (ℂP1×ℂP1)y0,η,θ↪L4|6 and solutions to the three-dimensional 𝒩=8 SYM equations. As this correspondence arises
by a dimensional reduction of a correspondence which is one-to-one, it is
rather evident that also in this case, we have a bijection between both the
holomorphic pseudobundles over L4|6 and the solutions after factoring out
holomorphic equivalence and gauge equivalence, respectively.


## 5. Purely Bosonic Yang-Mills-Higgs Theory from Third-order Subneighborhoods

In this section, we want to turn to the purely bosonic
situation (in other words, all the superspaces used up to now
loose their Graßmann-odd dimensions) and describe solutions to
the three-dimensional Yang-Mills-Higgs equations using a mini-ambitwistor space
(when speaking about Yang-Mills-Higgs theory, we mean a theory
without quartic interaction term). That is, we will consider the
dimensional reduction of the purely bosonic case discussed in [23, 24] from d=4 to d=3.
In these papers, it has been shown that solutions to the Yang-Mills field
equations are equivalent to holomorphic vector bundles over a third-order
thickening of the ambitwistor space L5 in 𝒫3×𝒫*3.


### 5.1. Thickenings of Complex Manifolds

Given a complex manifoldY of dimension d,
a thickening [25, 26] of a
submanifold X⊂Y with codimension 1 is an infinitesimal
neighborhood of X in Y described by an additional Graßmann-even but
nilpotent coordinate. More precisely, the mth order thickening of X is denoted by X(m) and defined as the manifold X together with the extended structure
sheaf(5.1)𝒪(m)=𝒪YIm+1,where 𝒪Y is the structure sheaf of Y,
and ℐ the ideal of functions on Y which vanish on X.
We can choose local coordinates (x1,…,xd−1,y) on Y such that X is given by y=0.
The mth order thickening X(m),
given by the scheme (X,𝒪(m)) is then described by the coordinates (x1,…,xd−1,y) together with the relation ym+1∼0 (for more details on infinitesimal
neighborhoods, see e.g., [9] and references therein).Note that it is easily possible to mapL5|6 to a third-order thickening of L5⊂𝒫3×𝒫*3 by identifying the nilpotent even coordinate y with 2θiηi (cf. [27]). However, we will not follow this approach for two
reasons. First, the situation is more subtle in the case of L4|6 since L4 only allows for a nilpotent even direction
inside 𝒫2×𝒫*2 for λ±=μ±.
Second, this description has several drawbacks when the discussion of the
Penrose-Ward transform reaches the correspondence space, where the concepts of
thickenings (and the extended fattenings) are not sufficient (see [27]).


### 5.2. Third-Order Thickenings andd=4 Yang-Mills Theory

Consider a vector bundleE over the space ℂ4×ℂ4 with coordinates rαα˙ and sαα˙.
On E,
we assume a gauge potential A=Aαα˙rdrαα˙+Aββ˙sdsββ˙.
Furthermore, we introduce the coordinates(5.2)xαα˙=12(rαα˙+sαα˙),kαα˙=12(rαα˙−sαα˙)on the base of E.
We claim that the Yang-Mills equations ∇αα˙Fαα˙ββ˙=0 are then equivalent to(5.3)[∇αα˙r,∇ββ˙r]=∗[∇αα˙r,∇ββ˙r]+𝒪(k3),[∇αα˙s,∇ββ˙s]=−∗[∇αα˙s,∇ββ˙s]+𝒪(k3),[∇αα˙r,∇ββ˙s]=𝒪(k3),where we define ∗Fαα˙ββ˙r,s:=(1/2)εαα˙ββ˙γγ˙δδ˙r,sFr,sγγ˙δδ˙ separately on each ℂ4
(one could also insert an i into this definition but on ℂ4,
this is not natural).To understand this statement, note that (5.3) are
equivalent to(5.4)[∇αα˙x,∇ββ˙x]=[∇αα˙k,∇ββ˙k]+𝒪(k3),[∇αα˙k,∇ββ˙x]=∗[∇αα˙k,∇ββ˙k]+𝒪(k3),which is easily seen by
performing the coordinate change from (r,s) to (x,k). These equations are solved by the expansion
[23, 24](5.5)Aαα˙k=−12Fαα˙ββ˙x,0kββ˙−13kγγ˙∇γγ˙x,0(∗Fαα˙ββ˙x,0)kββ˙,Aαα˙x=Aαα˙x,0−∗Fαα˙ββ˙x,0kββ˙−12kγγ˙∇γγ˙x,0(Fαα˙ββ˙x,0)kββ˙,if and only if ∇x,0αα˙Fαα˙ββ˙x,0=0 is satisfied. Here, a superscript 0 always denotes an object evaluated at kαα˙=0.
Thus, we saw that a solution to the Yang-Mills equations corresponds to a
solution to (5.3) on ℂ4×ℂ4.As discussed before, the self-dual and anti-self-dual
field strengths solving the first and second
equations of (5.3) can be mapped to certain
holomorphic vector bundles over 𝒫3 and 𝒫*3,
respectively. On the other hand, the potentials given in (5.5) are now defined
on a second-order infinitesimal neighborhood (not a
thickening) of the diagonal in ℂ4×ℂ4 for which 𝒪(k3)=0.
In the twistor description, this potential corresponds to a transition function f+−∼ψ+−1ψ−,
where the Čech 0-cochain {ψ±} is a solution to the equations(5.6)λ±α˙(∂∂rαα˙+Aαα˙r)ψ±=𝒪(k4),μ±α(∂∂sαα˙+Aαα˙s)ψ±=𝒪(k4).Roughly speaking, since the
gauge potentials are defined to order k2 and since ∂/∂rαα˙ and ∂/∂sαα˙ contain derivatives with respect to k,
the above equations can indeed be rendered exact to order k3.
The exact definition of the transition function is given by(5.7)f+−,i:=∑j=0iψ+,j−1ψ−,i−j,where the additional indices
label the order in k.
On the twistor space side, a third-order neighborhood in k corresponds to a third-order thickening
in(5.8)κ∘(a):=z(a)αμα(a)−u(a)α˙λα˙(a).Altogether, we see that a solution to the Yang-Mills
equations corresponds to a topologically trivial holomorphic vector bundle over
a third-order thickening ofL5 in 𝒫3×𝒫*3,
which becomes holomorphically trivial, when restricted to any ℂP1×ℂP*1↪L5.


### 5.3. Third-Order Subthickenings andd=3 Yang-Mills-Higgs Theory

Let us now translate the above discussion to the
three-dimensional situation. First of all, the appropriate Yang-Mills-Higgs
equations obtained by dimensional reduction are(5.9)∇(α˙β˙)F(α˙β˙)(γ˙δ˙)=[ϕ,∇(γ˙δ˙)ϕ],Δϕ:=∇(α˙β˙)∇(α˙β˙)ϕ=0,while the self-dual and
anti-self-dual Yang-Mills equations correspond after the dimensional reduction
to two Bogomolny equations which read as(5.10)F(α˙β˙)(γ˙δ˙)=ε(α˙β˙)(γ˙δ˙)(ε˙ζ˙)∇(ε˙ζ˙)ϕ,F(α˙β˙)(γ˙δ˙)=−ε(α˙β˙)(γ˙δ˙)(ε˙ζ˙)∇(ε˙ζ˙)ϕ,respectively. Using the
decomposition F(α˙β˙)(γ˙δ˙)=εα˙γ˙fβ˙δ˙+εβ˙δ˙fα˙γ˙,
the above two equations can be simplified to(5.11)fα˙β˙=i2∇(α˙β˙)ϕ,fα˙β˙=−i2∇(α˙β˙)ϕ.Analogously to the four
dimensional case, we start from a vector bundle E over the space ℂ3×ℂ3 with coordinates p(α˙β˙) and q(α˙β˙),
additionally we introduce the coordinates(5.12)y(α˙β˙)=12(p(α˙β˙)+q(α˙β˙)),h(α˙β˙)=12(p(α˙β˙)−q(α˙β˙)),and a gauge
potential(5.13)A=A(α˙β˙)pdp(α˙β˙)+A(α˙β˙)qdq(α˙β˙)=A(α˙β˙)ydy(α˙β˙)+A(α˙β˙)hdh(α˙β˙)on E.
The differential operators we will consider in the following are obtained from
covariant derivatives by dimensional reduction and take, for example, the
shape(5.14)∇α˙β˙y=∂∂y(α˙β˙)+[A(α˙β˙)y+i2εα˙β˙ϕy,⋅].We now claim that the
Yang-Mills-Higgs equations (5.9) are equivalent to(5.15)[∇α˙β˙p,∇γ˙δ˙p]=∗[∇α˙β˙p,∇γ˙δ˙p]+𝒪(h3),[∇α˙β˙q,∇γ˙δ˙q]=−∗[∇α˙β˙q,∇γ˙δ˙q]+𝒪(h3),[∇α˙β˙p,∇γ˙δ˙q]=𝒪(h3),where we can use ∗[∇α˙β˙p,q,∇γ˙δ˙p,q]=εα˙β˙γ˙δ˙ε˙ζ˙∇p,qε˙ζ˙ϕp,q.
These equations can be simplified in the coordinates (y,h) to equations similar to (5.3), which are
solved by the field expansion(5.16)A(α˙β˙)h=−12F(α˙β˙)(γ˙δ˙)y,0h(γ˙δ˙)−13h(γ˙δ˙)∇(γ˙δ˙)y,0ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)(∇y,0(σ˙τ˙)ϕ)h(ε˙ζ˙),ϕh=12∇(γ˙δ˙)y,0ϕy,0h(γ˙δ˙)+16h(γ˙δ˙)∇(γ˙δ˙)y,0ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)F(ε˙ζ˙)(σ˙τ˙)y,0h(α˙β˙),A(α˙β˙)y=A(α˙β˙)y,0−ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)(∇y,0(σ˙τ˙)ϕy,0)h(ε˙ζ˙)−12h(γ˙δ˙)∇(γ˙δ˙)y,0(F(α˙β˙)(ε˙ζ˙)y,0)h(ε˙ζ˙),ϕy=ϕy,0+12ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)F(ε˙ζ˙)(σ˙τ˙)y,0h(α˙β˙)+12h(γ˙δ˙)∇(γ˙δ˙)y,0(∇(α˙β˙)ϕy,0)h(α˙β˙),if and only if the
Yang-Mills-Higgs equations (5.9) are satisfied.Thus, solutions to the Yang-Mills-Higgs equations
(5.9) correspond to solutions to (5.15) on ℂ3×ℂ3.
Recall that solutions to the first two equations of (5.15) correspond in the
twistor description to holomorphic vector bundles over 𝒫2×𝒫*2.
Furthermore, the expansion of the gauge potential (5.16) is an expansion in a
second-order infinitesimal neighborhood of diag(ℂ3×ℂ3).
As we saw in the construction of the mini-superambitwistor space L4|6,
the diagonal for which h(α˙β˙)=0 corresponds to L4⊂𝒫2×𝒫*2.
The neighborhoods of this diagonal will then correspond to subthickenings of L4 inside 𝒫2×𝒫*2,
that is, for μ±=λ±,
we have the additional nilpotent coordinate ξ.
In other words, the subthickening of L4 in 𝒫2×𝒫*2 is obtained by turning one of the fiber
coordinates of 𝒫2×𝒫2 over ℂPΔ1 into a nilpotent even coordinate (in a suitable
basis). Then, we can finally state the following:Gauge equivalence classes of solutions to the
three-dimensional Yang-Mills-Higgs equations are in one-to-one correspondence
with gauge equivalence classes of holomorphic pseudobundles over a third-order
subthickening ofL4,
which become holomorphically trivial vector bundles when restricted to a ℂP1×ℂP1 holomorphically embedded into L4.


## 5.1. Thickenings of Complex Manifolds

Given a complex manifoldY of dimension d,
a thickening [25, 26] of a
submanifold X⊂Y with codimension 1 is an infinitesimal
neighborhood of X in Y described by an additional Graßmann-even but
nilpotent coordinate. More precisely, the mth order thickening of X is denoted by X(m) and defined as the manifold X together with the extended structure
sheaf(5.1)𝒪(m)=𝒪YIm+1,where 𝒪Y is the structure sheaf of Y,
and ℐ the ideal of functions on Y which vanish on X.
We can choose local coordinates (x1,…,xd−1,y) on Y such that X is given by y=0.
The mth order thickening X(m),
given by the scheme (X,𝒪(m)) is then described by the coordinates (x1,…,xd−1,y) together with the relation ym+1∼0 (for more details on infinitesimal
neighborhoods, see e.g., [9] and references therein).Note that it is easily possible to mapL5|6 to a third-order thickening of L5⊂𝒫3×𝒫*3 by identifying the nilpotent even coordinate y with 2θiηi (cf. [27]). However, we will not follow this approach for two
reasons. First, the situation is more subtle in the case of L4|6 since L4 only allows for a nilpotent even direction
inside 𝒫2×𝒫*2 for λ±=μ±.
Second, this description has several drawbacks when the discussion of the
Penrose-Ward transform reaches the correspondence space, where the concepts of
thickenings (and the extended fattenings) are not sufficient (see [27]).


## 5.2. Third-Order Thickenings andd=4 Yang-Mills Theory

Consider a vector bundleE over the space ℂ4×ℂ4 with coordinates rαα˙ and sαα˙.
On E,
we assume a gauge potential A=Aαα˙rdrαα˙+Aββ˙sdsββ˙.
Furthermore, we introduce the coordinates(5.2)xαα˙=12(rαα˙+sαα˙),kαα˙=12(rαα˙−sαα˙)on the base of E.
We claim that the Yang-Mills equations ∇αα˙Fαα˙ββ˙=0 are then equivalent to(5.3)[∇αα˙r,∇ββ˙r]=∗[∇αα˙r,∇ββ˙r]+𝒪(k3),[∇αα˙s,∇ββ˙s]=−∗[∇αα˙s,∇ββ˙s]+𝒪(k3),[∇αα˙r,∇ββ˙s]=𝒪(k3),where we define ∗Fαα˙ββ˙r,s:=(1/2)εαα˙ββ˙γγ˙δδ˙r,sFr,sγγ˙δδ˙ separately on each ℂ4
(one could also insert an i into this definition but on ℂ4,
this is not natural).To understand this statement, note that (5.3) are
equivalent to(5.4)[∇αα˙x,∇ββ˙x]=[∇αα˙k,∇ββ˙k]+𝒪(k3),[∇αα˙k,∇ββ˙x]=∗[∇αα˙k,∇ββ˙k]+𝒪(k3),which is easily seen by
performing the coordinate change from (r,s) to (x,k). These equations are solved by the expansion
[23, 24](5.5)Aαα˙k=−12Fαα˙ββ˙x,0kββ˙−13kγγ˙∇γγ˙x,0(∗Fαα˙ββ˙x,0)kββ˙,Aαα˙x=Aαα˙x,0−∗Fαα˙ββ˙x,0kββ˙−12kγγ˙∇γγ˙x,0(Fαα˙ββ˙x,0)kββ˙,if and only if ∇x,0αα˙Fαα˙ββ˙x,0=0 is satisfied. Here, a superscript 0 always denotes an object evaluated at kαα˙=0.
Thus, we saw that a solution to the Yang-Mills equations corresponds to a
solution to (5.3) on ℂ4×ℂ4.As discussed before, the self-dual and anti-self-dual
field strengths solving the first and second
equations of (5.3) can be mapped to certain
holomorphic vector bundles over 𝒫3 and 𝒫*3,
respectively. On the other hand, the potentials given in (5.5) are now defined
on a second-order infinitesimal neighborhood (not a
thickening) of the diagonal in ℂ4×ℂ4 for which 𝒪(k3)=0.
In the twistor description, this potential corresponds to a transition function f+−∼ψ+−1ψ−,
where the Čech 0-cochain {ψ±} is a solution to the equations(5.6)λ±α˙(∂∂rαα˙+Aαα˙r)ψ±=𝒪(k4),μ±α(∂∂sαα˙+Aαα˙s)ψ±=𝒪(k4).Roughly speaking, since the
gauge potentials are defined to order k2 and since ∂/∂rαα˙ and ∂/∂sαα˙ contain derivatives with respect to k,
the above equations can indeed be rendered exact to order k3.
The exact definition of the transition function is given by(5.7)f+−,i:=∑j=0iψ+,j−1ψ−,i−j,where the additional indices
label the order in k.
On the twistor space side, a third-order neighborhood in k corresponds to a third-order thickening
in(5.8)κ∘(a):=z(a)αμα(a)−u(a)α˙λα˙(a).Altogether, we see that a solution to the Yang-Mills
equations corresponds to a topologically trivial holomorphic vector bundle over
a third-order thickening ofL5 in 𝒫3×𝒫*3,
which becomes holomorphically trivial, when restricted to any ℂP1×ℂP*1↪L5.


## 5.3. Third-Order Subthickenings andd=3 Yang-Mills-Higgs Theory

Let us now translate the above discussion to the
three-dimensional situation. First of all, the appropriate Yang-Mills-Higgs
equations obtained by dimensional reduction are(5.9)∇(α˙β˙)F(α˙β˙)(γ˙δ˙)=[ϕ,∇(γ˙δ˙)ϕ],Δϕ:=∇(α˙β˙)∇(α˙β˙)ϕ=0,while the self-dual and
anti-self-dual Yang-Mills equations correspond after the dimensional reduction
to two Bogomolny equations which read as(5.10)F(α˙β˙)(γ˙δ˙)=ε(α˙β˙)(γ˙δ˙)(ε˙ζ˙)∇(ε˙ζ˙)ϕ,F(α˙β˙)(γ˙δ˙)=−ε(α˙β˙)(γ˙δ˙)(ε˙ζ˙)∇(ε˙ζ˙)ϕ,respectively. Using the
decomposition F(α˙β˙)(γ˙δ˙)=εα˙γ˙fβ˙δ˙+εβ˙δ˙fα˙γ˙,
the above two equations can be simplified to(5.11)fα˙β˙=i2∇(α˙β˙)ϕ,fα˙β˙=−i2∇(α˙β˙)ϕ.Analogously to the four
dimensional case, we start from a vector bundle E over the space ℂ3×ℂ3 with coordinates p(α˙β˙) and q(α˙β˙),
additionally we introduce the coordinates(5.12)y(α˙β˙)=12(p(α˙β˙)+q(α˙β˙)),h(α˙β˙)=12(p(α˙β˙)−q(α˙β˙)),and a gauge
potential(5.13)A=A(α˙β˙)pdp(α˙β˙)+A(α˙β˙)qdq(α˙β˙)=A(α˙β˙)ydy(α˙β˙)+A(α˙β˙)hdh(α˙β˙)on E.
The differential operators we will consider in the following are obtained from
covariant derivatives by dimensional reduction and take, for example, the
shape(5.14)∇α˙β˙y=∂∂y(α˙β˙)+[A(α˙β˙)y+i2εα˙β˙ϕy,⋅].We now claim that the
Yang-Mills-Higgs equations (5.9) are equivalent to(5.15)[∇α˙β˙p,∇γ˙δ˙p]=∗[∇α˙β˙p,∇γ˙δ˙p]+𝒪(h3),[∇α˙β˙q,∇γ˙δ˙q]=−∗[∇α˙β˙q,∇γ˙δ˙q]+𝒪(h3),[∇α˙β˙p,∇γ˙δ˙q]=𝒪(h3),where we can use ∗[∇α˙β˙p,q,∇γ˙δ˙p,q]=εα˙β˙γ˙δ˙ε˙ζ˙∇p,qε˙ζ˙ϕp,q.
These equations can be simplified in the coordinates (y,h) to equations similar to (5.3), which are
solved by the field expansion(5.16)A(α˙β˙)h=−12F(α˙β˙)(γ˙δ˙)y,0h(γ˙δ˙)−13h(γ˙δ˙)∇(γ˙δ˙)y,0ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)(∇y,0(σ˙τ˙)ϕ)h(ε˙ζ˙),ϕh=12∇(γ˙δ˙)y,0ϕy,0h(γ˙δ˙)+16h(γ˙δ˙)∇(γ˙δ˙)y,0ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)F(ε˙ζ˙)(σ˙τ˙)y,0h(α˙β˙),A(α˙β˙)y=A(α˙β˙)y,0−ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)(∇y,0(σ˙τ˙)ϕy,0)h(ε˙ζ˙)−12h(γ˙δ˙)∇(γ˙δ˙)y,0(F(α˙β˙)(ε˙ζ˙)y,0)h(ε˙ζ˙),ϕy=ϕy,0+12ε(α˙β˙)(ε˙ζ˙)(σ˙τ˙)F(ε˙ζ˙)(σ˙τ˙)y,0h(α˙β˙)+12h(γ˙δ˙)∇(γ˙δ˙)y,0(∇(α˙β˙)ϕy,0)h(α˙β˙),if and only if the
Yang-Mills-Higgs equations (5.9) are satisfied.Thus, solutions to the Yang-Mills-Higgs equations
(5.9) correspond to solutions to (5.15) on ℂ3×ℂ3.
Recall that solutions to the first two equations of (5.15) correspond in the
twistor description to holomorphic vector bundles over 𝒫2×𝒫*2.
Furthermore, the expansion of the gauge potential (5.16) is an expansion in a
second-order infinitesimal neighborhood of diag(ℂ3×ℂ3).
As we saw in the construction of the mini-superambitwistor space L4|6,
the diagonal for which h(α˙β˙)=0 corresponds to L4⊂𝒫2×𝒫*2.
The neighborhoods of this diagonal will then correspond to subthickenings of L4 inside 𝒫2×𝒫*2,
that is, for μ±=λ±,
we have the additional nilpotent coordinate ξ.
In other words, the subthickening of L4 in 𝒫2×𝒫*2 is obtained by turning one of the fiber
coordinates of 𝒫2×𝒫2 over ℂPΔ1 into a nilpotent even coordinate (in a suitable
basis). Then, we can finally state the following:Gauge equivalence classes of solutions to the
three-dimensional Yang-Mills-Higgs equations are in one-to-one correspondence
with gauge equivalence classes of holomorphic pseudobundles over a third-order
subthickening ofL4,
which become holomorphically trivial vector bundles when restricted to a ℂP1×ℂP1 holomorphically embedded into L4.



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*Source: 784215-2008-11-09.xml*