arXiv:0704.0530v2  [hep-th]  14 Jun 2007
ITP–UH–09/07
Noncommutative Solitons in a
Supersymmetric Chiral Model in 2+1 Dimensions
Olaf Lechtenfeld1 and Alexander D. Popov1,2
1Institut für Theoretische Physik, Leibniz Universität Hannover
Appelstraße 2, 30167 Hannover, Germany
2Bogoliubov Laboratory of Theoretical Physics, JINR
141980 Dubna, Moscow Region, Russia
Email: lechtenf, popov @itp.uni-hannover.de
Abstract
We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from
twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model
with 2N≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model,
we employ the dressing method to explicitly construct multi-soliton configurations on noncom-
mutative R2,1 and analyze some of their properties.
http://arxiv.org/abs/0704.0530v2
1 Introduction
In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on
the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2,
3, 4]. In particular, if the open string has N=2 worldsheet supersymmetry, the tree-level target
space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2
dimensions [5]. Furthermore, open N=2 strings in a B-field background induce on the worldvolume
of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1
dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral
model known as the Ward model [7]. The topological nature of N=2 strings and the integrability
of their tree-level dynamics [8] render this noncommutative sigma model integrable.1
Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering
and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R2.
These are not only a rich testing ground for physical properties such as adiabatic dynamics or
quantization, but also descend to more standard multi-solitons of various integrable systems in
2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There
is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant
or integrable but not both [7, 11]. In fact, Derrick’s theorem prohibits the existence of stable
solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation,
however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted
version. There is another gain: The deformed Ward model possesses not only deformed versions
of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely
noncommutative (multi-)solitons, in particular for the U(1) group [12, 13]! Moreover, this class is
related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15]
by an infinite-stiffness limit of the potential [16].
In [12, 13] and [17]–[20] families of multi-solitons as well as their reduction to solitons of the
noncommutative sine-Gordon equations were described and studied. In the nonabelian case both
scattering and nonscattering configurations were obtained. For static configurations the issue of
their stability was analyzed [21]. The full moduli space metric for the abelian model was computed
and its adiabatic two-soliton dynamics was discussed [16].
Recall that the critical N=2 string theory has a four-dimensional target space, and its open
string effective field theory is self-dual Yang-Mills [8], which gets deformed noncommutatively in
the presence of a B-field [5]. Conversely, the noncommutative SDYM equations are contained [19]
in the equations of motion of N=2 string field theory (SFT) [22] in a B-field background. This
SFT formulation is based on the N=4 topological string description [23]. It is well known that
the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the
twistor space2 [26] CP 3 and the topological N=4 string theory contains twistors from the outset.
The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing
methods were incorporated into the N=2 open SFT in [27, 28]. However, this theory reproduces
only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31]) and integrability properties. It
is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four
dimensions?
1For discussing some other noncommutative integrable models see e.g. [9, 10] and references therein.
2For reviews of twistor theory see, e.g., the books [24, 25].
There are some proposals [33, 34, 35, 36] for extending N=2 open string theory (and its SFT) to
be space-time supersymmetric. Moreover, it was shown by Witten [37] that N=4 supersymmetric
SDYM theory appears in twistor string theory, which is a B-type open topological string with the
supertwistor space CP 3|4 as a target space.3 Note that N<4 SDYM theory forms a BPS subsector
of N -extended super Yang-Mills theory, and N=4 SDYM can be considered as a truncation of the
full N=4 super Yang-Mills theory [37]. It is believed [43, 39] that twistor string theory is related
with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of
N=2 (and topological N=4) string theory which also leads to the N=4 SDYM model.
A dimensional reduction of the above relations between twistor strings and N=4 super Yang-
Mills and SDYM models was considered in [44, 45, 46, 47]. The corresponding twistor string
theory after this reduction is the topological B-model on the mini-supertwistor space P2|4. In [47]
it was shown that the 2N=8 supersymmetric extension of the Bogomolny-type model in 2+1
dimensions is equivalent to an 2N=8 supersymmetric modified U(n) chiral model on R2,1. The
subject of the current paper is an 2N≤8 version of the above supersymmetric Bogomolny-type
Yang-Mills-Higgs model in signature (− + +), its relation with an N -extended supersymmetric
modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type
noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton
configurations on noncommutative R2,1 for the corresponding supersymmetric sigma model field
equations. By studying the scattering properties of the constructed configurations, we prove their
asymptotic factorization without scattering for large times. We also briefly discuss a D-brane
interpretation of these soliton configurations from the viewpoint of twistor string theory.
2 Supersymmetric Bogomolny model in 2+1 dimensions
2.1 N -extended SDYM equations in 2+2 dimensions
Space R2,2. Let us consider the four-dimensional space R2,2 = (R4, g) with the metric
ds2 = gµνdx
µdxν = det(dxαα̇) = dx11̇dx22̇ − dx21̇dx12̇ (2.1)
with (gµν) = diag(−1,+1,+1,−1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2,
α̇ = 1̇, 2̇ are spinor indices. We choose the coordinates4
(xµ) = (xa, t̃) = (t, x, y, t̃) with a, b, . . . = 1, 2, 3 , (2.2)
and the signature (− ++−) allows us to introduce real isotropic coordinates (cf. [19, 6])
x11̇ = 1
(t− y) , x12̇ = 1
(x+ t̃) , x21̇ = 1
(x− t̃) , x22̇ = 1
(t+ y) . (2.3)
SDYM. Recall that the SDYM equations for a field strength tensor Fµν on R
2,2 read
εµνρσF
ρσ = Fµν , (2.4)
3For other variants of twistor string models see [38, 39, 40]. For recent reviews providing a twistor description of
super Yang-Mills theory, see [41, 42] and references therein.
4Our conventions are chosen to match those of [12] after reduction to the space R2,1 with coordinates (t, x, y).
where εµνρσ is a completely antisymmetric tensor on R
2,2 and ε1234 = 1. In the coordinates (2.3)
we have the decomposition
αα̇,ββ̇
= ∂αα̇Aββ̇ − ∂ββ̇Aαα̇ + [Aαα̇, Aββ̇ ] = εαβ Fα̇β̇ + εα̇β̇ Fαβ (2.5)
:= −1
αα̇,ββ̇
and Fαβ := −12ε
α̇β̇F
αα̇,ββ̇
, (2.6)
where εαβ is antisymmetric, εαβε
βγ = δ
α, and similar for ε
α̇β̇, with ε12 = ε1̇2̇ = 1. The gauge
potential (Aαα̇) will appear in the covariant derivative
, · ] . (2.7)
In spinor notation, (2.4) is equivalently written as
= 0 ⇔ F
αα̇,ββ̇
Fαβ . (2.8)
Solutions {Aαα̇} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills
theory on R2,2.
N -extended SDYM in component fields. The field content of N -extended super SDYM is5
N = 0 Aαα̇ (2.9a)
N = 1 Aαα̇, χiα with i = 1 (2.9b)
N = 2 Aαα̇, χiα, φ[ij] with i, j = 1, 2 (2.9c)
N = 3 Aαα̇, χiα, φ[ij], χ̃
[ijk]
with i, j, k = 1, 2, 3 (2.9d)
N = 4 Aαα̇, χiα, φ[ij], χ̃
[ijk]
[ijkl]
with i, j, k, l = 1, 2, 3, 4 . (2.9e)
Here (Aαα̇, χ
[ij], χ̃
[ijk]
[ijkl]
) are fields of helicities (+1,+1
, 0,−1
,−1). These fields obey
the field equations of the N = 4 SDYM model, namely [33, 37]
= 0 , (2.10a)
Dαα̇χ
iα = 0 , (2.10b)
Dαα̇D
αα̇φij + 2{χiα, χjα} = 0 , (2.10c)
Dαα̇χ̃
α̇[ijk] − 6[χ[iα, φjk]] = 0 , (2.10d)
D γ̇α G
[ijkl]
+ 12{χ[iα, χ̃
} − 18[φ[ij ,D
φkl]] = 0 . (2.10e)
Note that the N < 4 SDYM field equations are governed by the first N+1 equations of (2.10),
where F
= 0 is counted as one equation and so on.
5We use symmetrization (·) and antisymmetrization [·] of k indices with weight 1
, e.g. [ij] = 1
(ij − ji).
2.2 Superfield formulation of N -extended SDYM
Superspace R4|4N . Recall that in the space R2,2 = (R4, g) with the metric g given in (2.1)
one may introduce purely real Majorana-Weyl spinors6 θα and ηα̇ of helicities +1
and −1
as anti-
commuting (Grassmann-algebra) objects. Using 2N such spinors with components θiα and ηα̇i for
i = 1, . . . ,N , one can define the N -extended superspace R4|4N and the N -extended supersymmetry
algebra generated by the supertranslation operators
Pαα̇ = ∂αα̇ , Qiα = ∂iα − ηα̇i ∂αα̇ and Qiα̇ = ∂iα̇ − θiα∂αα̇ , (2.11)
where
∂αα̇ :=
∂xαα̇
, ∂iα :=
and ∂iα̇ :=
∂ηα̇i
. (2.12)
The commutation relations for the generators (2.11) read
{Qiα, Qjα̇} = −2δ
iPαα̇ , [Pαα̇, Qiβ ] = 0 and [Pαα̇, Q
] = 0 . (2.13)
To rewrite equations of motion in terms of R4|4N superfields one uses the additional operators
Diα = ∂iα + η
i ∂αα̇ and D
α̇ = ∂
α̇ + θ
iα∂αα̇ , (2.14)
which (anti)commute with the operators (2.11) and satisfy
{Diα,Dj
} = 2δjiPαβ̇ , [Pαα̇,Diβ ] = 0 and [Pαα̇,D
] = 0 . (2.15)
Antichiral superspace R4|2N . On the superspace R4|4N one may introduce tensor fields de-
pending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives LX
etc.. Furthermore, on any such superfield A one can impose the constraint equations LDiαA = 0,
which for a scalar superfield f reduce to the so-called antichirality conditions
Diαf = 0 . (2.16)
These are easily solved by using a coordinate transformation on R4|4N ,
(xαα̇, ηα̇i , θ
iα) → (x̃αα̇ = xαα̇−θiαηα̇i , ηα̇i , θiα) , (2.17)
under which ∂αα̇,Diα and D
α̇ transform to the operators
∂̃αα̇ = ∂αα̇ , D̃iα = ∂iα and D̃
α̇ = ∂
α̇ + 2θ
iα∂αα̇ . (2.18)
Then (2.16) simply means that f is defined on a sub-superspace R4|2N ⊂ R4|4N with coordinates
x̃αα̇ and ηα̇i . (2.19)
This space is called antichiral superspace. In the following we will usually omit the tildes when
working on the antichiral superspace.
6Note that in Minkowski signature the Weyl spinor θα is complex and ηα̇ = εα̇β̇η
β̇ = θα is complex conjugate
to θα. For the Kleinian (split) signature 2 + 2, however, these spinors are real and independent of one another.
N -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in
terms of superfields on the antichiral superspace R4|2N [33, 48]. Namely, for any given 0 ≤ N ≤ 4,
fields of a proper multiplet from (2.9) can be combined into superfields Aαα̇ and Aiα̇ depending on
xαα̇, ηα̇i ∈ R4|2N and giving rise to covariant derivatives
∇αα̇ := ∂αα̇ +Aαα̇ and ∇iα̇ := ∂iα̇ +Aiα̇ . (2.20)
In such terms the N -extended SDYM equations (2.10) read
[∇αα̇,∇ββ̇] + [∇αβ̇ ,∇βα̇] = 0 , [∇
α̇,∇ββ̇ ] + [∇
,∇βα̇] = 0 , {∇iα̇,∇
}+ {∇i
} = 0 , (2.21)
which is equivalent to
[∇αα̇,∇ββ̇] = εα̇β̇ Fαβ , [∇
α̇,∇ββ̇ ] = εα̇β̇ F
β and {∇iα̇,∇
} = ε
F ij , (2.22)
where F ij is antisymmetric and Fαβ is symmetric in their indices.
The above gauge potential superfields (Aαα̇, Aiα̇) as well as the gauge strength superfields
(Fαβ , F iα, F ij) contain all physical component fields of theN -extended SDYMmodel. For instance,
the lowest component of the triple (Fαβ , F iα, F ij) in an η-expansion is (Fαβ , χiα, φij), with zeros
in case N is too small. By employing Bianchi identities for the gauge strength superfields, one
successively obtains [48] the superfield expansions and the field equations (2.10) for all component
fields.
It is instructive to extend the antichiral combination in (2.18) to potentials and covariant
derivatives,
D̃iα̇ = ∂
α̇ + 2 θ
iα ∂αα̇
+ + +
Ãiα̇ := Aiα̇ + 2 θiαAαα̇
‖ ‖ ‖
∇̃iα̇ := ∇iα̇ + 2 θiα∇αα̇
(2.23)
where ∇αα̇, ∇iα̇ and D̃iα̇ are given by (2.20) and (2.18), while Aiα̇ and Aαα̇ depend on xαα̇ and ηα̇i
only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single
{∇̃iα̇, ∇̃
} + {∇̃i
, ∇̃j
} = 0 ⇔ {∇̃iα̇, ∇̃
} = ε
F̃ ij , (2.24)
with F̃ ij = F ij + 4 θ[iαF j]α + 4 θiαθjβFαβ . The concise form (2.24) of the N -extended SDYM
equations is quite convenient, and we will use it interchangeable with (2.21).
Linear system for N -extended SDYM. It is well known that the superfield SDYM equations
(2.21) can be seen as the compatibility conditions for the linear system of differential equations
ζ α̇(∂αα̇ +Aαα̇)ψ = 0 and ζ α̇(∂iα̇ +Aiα̇)ψ = 0 , (2.25)
where (ζ
and ζ α̇ = εα̇β̇ζ
. The extra (spectral) parameter7 ζ lies in the extended complex
plane C∪∞ = CP 1. Here ψ is a matrix-valued function depending not only on xαα̇ and ηα̇i but also
(meromorphically) on ζ ∈ CP 1. We subject the n×n matrix ψ to the following reality condition:
ψ(xαα̇, ηα̇i , ζ)
ψ(xαα̇, ηα̇i , ζ̄)
= 1l , (2.26)
7The parameter ζ is related with λ used in [45] by the formula ζ = i 1−λ
(cf. e.g. [31]).
where “†” denotes hermitian conjugation and ζ̄ is complex conjugate to ζ. This condition guarantees
that all physical fields of the N -extended SDYM model will take values in the adjoint representation
of the algebra u(n). In the concise form the linear system (2.25) is written as
ζ α̇(∇iα̇ + 2θiα∇αα̇)ψ = 0 ⇔ ζ α̇(D̃iα̇ + Ãiα̇)ψ = 0 ⇔ ζ α̇ ∇̃iα̇ ψ = 0 . (2.27)
2.3 Reduction of N -extended SDYM to 2+1 dimensions
The supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained
from the described N -extended super SDYM equations by a dimensional reduction R2,2 → R2,1.
In particular, for the N=0 sector we demand the components Aµ of a gauge potential to be
independent of x4 and put A4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions
(the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can
reduce the N -extended SDYM equations on R2,2 by imposing the ∂4-invariance condition on all
the fields (Aαα̇, χ
[ij], χ̃
[ijk]
[ijkl]
) from the N=4 supermultiplet or its truncation to N<4
and obtain supersymmetric Bogomolny-type equations on R2,1.
Spinors in R2,1. Recall that on R2,2 both N=4 SDYM theory and full N=4 super Yang-
Mills theory have an SL(4, R) ∼= Spin(3,3) R-symmetry group [33]. A dimensional reduction to
2,1 enlarges the supersymmetry and R-symmetry to 2N=8 and Spin(4,4), respectively, for both
theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets
doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)
∼= Spin(2,1)L×Spin(2,1)R of R2,2 to its diagonal subgroup Spin(2,1)D as the local rotation group
of R2,1, the distinction between undotted and dotted indices disappears. We shall use undotted
indices henceforth.
Coordinates and derivatives in R2,1. The above discussion implies that one can relabel the
bosonic coordinates xαβ̇ from (2.3) by xαβ and split them as
xαβ = 1
(xαβ + xβα) + 1
(xαβ − xβα) = x(αβ) + x[αβ] (2.28)
into antisymmetric and symmetric parts,
x[αβ] = 1
εαβx4 = 1
εαβ t̃ and x(αβ) =: yαβ , (2.29)
respectively, with
y11 = x11 = 1
(t− y) , y12 = 1
(x12 + x21) = 1
x , y22 = x22 = 1
(t+ y) . (2.30)
We also have θiα 7→ θiα and ηα̇i 7→ ηαi for the fermionic coordinates on R4|4N reduced to R3|4N .
Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators
∂(αβ) =
(∂αβ + ∂βα) (2.31)
which read explicitly as
∂(11) =
= ∂t−∂y , ∂(12) = ∂(21) = 12
= ∂x , ∂(22) =
= ∂t+∂y . (2.32)
We thus have
= ∂(αβ) − εαβ∂4 = ∂(αβ) − εαβ∂t̃ , (2.33)
where ε12 = −ε21 = −1, ∂4 = ∂/∂x4 and ∂t̃ = ∂/∂t̃.
The operators Diα and D
α̇ acting on t̃-independent superfields reduce to
Diα = ∂iα + η
i ∂(αβ) and D
α = ∂
α + θ
iβ∂(αβ) , (2.34)
where ∂iα = ∂/∂θ
iα and ∂iα = ∂/∂η
i . Similarly, the antichiral operators D̃iα and D̃
α̇ in (2.18)
become
D̂iα = ∂iα and D̂
α = ∂
α + 2θ
iβ∂(αβ) . (2.35)
Supersymmetric Bogomolny-type equations in component fields. According to (2.33),
the components A
of a gauge potential in four dimensions split into the components A(αβ) of a
gauge potential in three dimensions and a Higgs field A[αβ] = −εαβ ϕ, i.e.
Aαβ = A(αβ) +A[αβ] = A(αβ) − εαβ ϕ . (2.36)
Then the covariant derivatives D
reduced to three dimensions become the differential operators
Dαβ − εαβ ϕ = ∂(αβ) + [A(αβ), · ]− εαβ [ϕ, · ] , (2.37)
and the Yang-Mills field strength on R2,1 decomposes as
Fαβ, γδ = [Dαβ , Dγδ] = εαγ fβδ + εβδ fαγ with fαβ = fβα . (2.38)
Substituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent
of x4 = t̃, we obtain the following supersymmetric Bogomolny-type equations on R2,1:
fαβ +Dαβϕ = 0 , (2.39a)
Dαβ χ
iβ + εαβ [ϕ, χ
iβ] = 0 , (2.39b)
Dαβ D
αβφij + 2[ϕ, [ϕ, φij ]] + 2{χiα, χjα} = 0 , (2.39c)
Dαβ χ̃
β[ijk] − εαβ [ϕ, χ̃β[ijk]]− 6[χ[iα, φjk]] = 0 , (2.39d)
[ijkl]
+ [ϕ,G
[ijkl]
] + 12{χ[iα, χ̃jkl]β } − 18[φ[ij ,Dαβφkl]]− 18εαβ [φ[ij , [φkl], ϕ]] = 0 .(2.39e)
Supersymmetric Bogomolny-type equations in terms of superfields. Translations gen-
erated by the vector field ∂4 = ∂t̃ are isometries of superspaces R
4|4N and R4|2N . By taking the
quotient with respect to the action of the abelian group G generated by ∂4, we obtain the reduced
full superspace R3|4N ∼= R4|4N /G and the reduced antichiral superspace R3|2N ∼= R4|2N/G. In the
following, we shall work on R3|2N and R3|2N × CP 1, since the reduced ψ-function from (2.25) and
(2.27) is defined on the latter space.
The linear system stays in the center of the superfield approach to the N -extended SDYM
equations. After imposing t̃-independence on all fields in the linear system (2.27), we arrive at the
linear equations
ζα ∇̂iα ψ ≡ ζα(D̂iα + Âiα)ψ = 0 (2.40)
of the same form but with
D̂iα = ∂
α + 2θ
iβ∂(αβ) and Âiα = Aiα + 2θiβ(A(αβ) − εαβΞ) , (2.41)
where Aiα, A(αβ) and Ξ are superfields depending on yαβ and ηαi only. These linear equations
expand again to the pair (cf. (2.25))
ζβ(∂(αβ) +A(αβ) − εαβΞ)ψ = 0 and ζα(∂iα +Aiα)ψ = 0 . (2.42)
The compatibility conditions for the linear system (2.40) read
{∇̂iα, ∇̂
} + {∇̂iβ, ∇̂jα} = 0 ⇔ {∇̂iα, ∇̂
} = εαβ F̂ ij (2.43)
and present a condensed form of (2.39) rewritten in terms of R3|2N superfields. Similarly, these
equations can also be written in more expanded forms analogously to (2.21) or using the superfield
analog of (2.37). However, we will not do this since all these sets of equations are equivalent.
3 Noncommutative N -extended U(n) chiral model in 2+1 dimensions
As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentz-
invariant or integrable but not both [7, 11]. We will show that the super Bogomolny-type model
discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n)
chiral model (so as to be integrable) first formulated by Ward [7]. Since integrability is compatible
with noncommutative deformation (if introduced properly, see e.g. [9]–[20]) we choose from the
beginning to formulate our super extension of this chiral model on Moyal-deformed R2,1 with
noncommutativity parameter θ ≥ 0. Ordinary space-time R2,1 can always be restored by taking
the commutative limit θ → 0.
Star-product formulation. Classical field theory on noncommutative spaces may be realized
in a star-product formulation or in an operator formalism8. The first approach is closer to the
commutative field theory: it is obtained by simply deforming the ordinary product of classical
fields (or their components) to the noncommutative star product
(f ⋆ g)(x) = f(x) exp{ i
ab −→∂b} g(x) ⇒ xa ⋆ xb − xb ⋆ xa = iθab (3.1)
with a constant antisymmetric tensor θab. Specializing to R2,1, we use real coordinates (xa) =
(t, x, y) in which the Minkowski metric g on R3 reads (gab) = diag(−1,+1,+1) with a, b, . . . = 1, 2, 3
(cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces
introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd
coordinates. Deferring general superspace deformations and their consequences to future work,
we here content ourselves with the simple embedding of the “bosonic” Moyal deformation into
superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann
variables θiα and ηαi .
For later use we consider not only isotropic coordinates and vector fields
u := 1
(t+y) = y22 , v := 1
(t−y) = y11 , ∂u = ∂t + ∂y = ∂(22) , ∂v = ∂t − ∂y = ∂(11) (3.2)
8See [50] for reviews on noncommutative field theories.
introduced in Section 2, but also the complex combinations
z := x+ iy , z̄ := x− iy , ∂z = 12 (∂x − i∂y) , ∂z̄ =
(∂x + i∂y) . (3.3)
Since the time coordinate t remains commutative, the only nonvanishing component of the non-
commutativity tensor θab is
θxy = −θyx =: θ > 0 ⇒ θzz̄ = −θz̄z = −2i θ . (3.4)
Hence, we have
z ⋆ z̄ = zz̄ + θ and z̄ ⋆ z = zz̄ − θ (3.5)
as examples of the general formula (3.1).
Operator formalism. The nonlocality of the star products renders explicit computation cum-
bersome. We therefore pass to the operator formalism, which trades the star product for operator-
valued spatial coordinates (x̂, ŷ) or their complex combinations (ẑ, ˆ̄z), subject to
[t, x̂] = [t, ŷ] = 0 but [x̂, ŷ] = iθ ⇒ [ẑ, ˆ̄z] = 2 θ . (3.6)
The latter equation suggests the introduction of annihilation and creation operators,
ẑ and a† =
ˆ̄z with [a , a†] = 1 , (3.7)
which act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ〉, ℓ = 0, 1, 2, . . .}
such that
a |ℓ〉 =
ℓ |ℓ−1〉 and a† |ℓ〉 =
ℓ+1 |ℓ+1〉 . (3.8)
Any superfield f(t, z, z̄, ηαi ) on R
3|2N can be related to an operator-valued superfield f̂(t, ηαi ) ≡
F (t, a, a†, ηαi ) on R
1|2N acting in H, with the help of the Moyal-Weyl map
f(t, z, z̄, ηαi ) 7→ f̂(t, ηαi ) = Weyl-ordered f
2θa†, ηαi
. (3.9)
The inverse transformation recovers the ordinary superfield,
f̂(t, ηαi ) ≡ F (t, a, a†, ηαi ) 7→ f(t, z, z̄, ηαi ) = F⋆
t, z√
, z̄√
, ηαi
, (3.10)
where F⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl
map, we have
f ⋆ g 7→ f̂ ĝ and
dx dy f = 2π θTrf̂ = 2π θ
〈ℓ|f̂ |ℓ〉 , (3.11)
and the spatial derivatives are mapped into commutators,
∂zf 7→ ∂̂z f̂ = − 1√
[a†, f̂ ] and ∂z̄f 7→ ∂̂z̄ f̂ = 1√
[a , f̂ ] . (3.12)
For notational simplicity we will from now on omit the hats over the operators except when con-
fusion may arise.
Gauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43)
are invariant under a gauge transformation
ψ 7→ ψ′ = g−1ψ , (3.13a)
A 7→ A′ = g−1A g + g−1∂ g (with appropriate indices) , (3.13b)
Ξ 7→ Ξ′ = g−1Ξ g , (3.13c)
where g = g(xa, ηαi ) is a U(n)-valued superfield globally defined on the deformed superspace R
CP 1. Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy
the standard asymptotic conditions (see e.g. [51])
ψ = Φ−1 + O(ζ) for ζ → 0 , (3.14a)
ψ = 1l + ζ−1Υ + O(ζ−2) for ζ →∞ , (3.14b)
where the U(n)-valued function Φ and u(n)-valued function Υ depend on xa and ηαi . This “unitary”
gauge is compatible with the reality condition for ψ,
ψ(xa, ηαi , ζ)
ψ(xa, ηαi , ζ̄)
= 1l , (3.15)
obtained by reduction from (2.26).
Gauge fixing for Âiα. After fixing the unitary gauge (3.14) for ψ and inserting (ζα) =
the linear system (2.40), one can easily reconstruct the superfield given in (2.41) from Φ or Υ via
Âi1 = 0 and Âi2 = Φ−1D̂i2Φ = D̂i1Υ (3.16)
and thus fix a gauge for the superfields Âiα. The operators D̂iα were defined in (2.35). One can
express (3.16) in terms of Aiα and A(αβ) − εαβΞ as
Ai1 = 0 and Ai2 = Φ−1∂i2Φ = ∂i1Υ , (3.17)
A(11) = 0 and A(12) + Ξ = Φ−1∂(12)Φ = ∂(11)Υ , (3.18)
A(21) − Ξ = 0 and A(22) = Φ−1∂(22)Φ = ∂(12)Υ . (3.19)
Using (2.32), we can rewrite the nonzero components as
A := Φ−1∂uΦ = ∂xΥ , B := Φ−1∂xΦ = ∂vΥ , Ci := Φ−1∂i2Φ = ∂i1Υ . (3.20)
Recall that the superfields Φ and Υ depend on xa and ηαi .
Linear system. In the above-introduced unitary gauge the linear system (2.42) reads
(ζ∂x − ∂u −A)ψ = 0 , (ζ∂v − ∂x − B)ψ = 0 , (ζ∂i1 − ∂i2 − Ci)ψ = 0 , (3.21)
which adds the last equation to the linear system of the Ward model [7] and generalizes it to
superfields A(xa, ηαj ), B(xa, ηαj ) and Ci(xa, ηαj ). The concise form of (3.21) reads
ζ D̂i1 − D̂i2 − Âi2
ψ = 0 (3.22)
or, in more explicit form,
∂i1 + 2θ
i1∂v + 2θ
∂i2 + Ci + 2θi1(∂x + B) + 2θi2(∂u +A)
ψ = 0 . (3.23)
N -extended sigma model. The compatibility conditions of this linear system are the N -
extended noncommutative sigma model equations
D̂i1(Φ
−1D̂j2 Φ) + D̂
−1D̂i2 Φ) = 0 (3.24)
which in expanded form reads
(gab + vcε
cab) ∂a(Φ
−1∂bΦ) = 0 ⇔ ∂x(Φ−1∂xΦ) − ∂v(Φ−1∂uΦ) = 0 , (3.25a)
∂i1(Φ
−1∂xΦ) − ∂v(Φ−1∂i2Φ) = 0 , ∂i1(Φ−1∂uΦ) − ∂x(Φ−1∂i2Φ) = 0 , (3.25b)
∂i1(Φ
−1∂j2Φ) + ∂
−1∂i2Φ) = 0 . (3.25c)
Here, the first line contains the Wess-Zumino-Witten term with a constant vector (vc) = (0, 1, 0)
which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimen-
sions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H
and depend on xa and 2N Grassmann variables ηαi . As discussed in Section 2, the compatibility
conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolny-
type equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25)
are equivalent to a gauge fixed form of equations (2.39).
Υ-formulation. Instead of Φ-parametrization of (A,B, Ci) given in (3.17)–(3.20) we may use the
equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the
linear system (3.21) reduce to
(∂2x − ∂u∂v)Υ + [∂vΥ , ∂xΥ] = 0 , (3.26a)
(∂i2∂v − ∂i1∂x)Υ + [∂i1Υ , ∂vΥ] = 0 , (∂i2∂x − ∂i1∂u)Υ + [∂i1Υ , ∂xΥ] = 0 , (3.26b)
(∂i2∂
1 + ∂
1)Υ + {∂i1Υ , ∂
1Υ} = 0 , (3.26c)
which in concise form read
(D̂i2 D̂
1 + D̂
1)Υ + {D̂i1Υ , D̂
1Υ} = 0 . (3.27)
Recall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and
depend on xa and 2N Grassmann variables ηαi .
For N=4, the commutative limit of (3.27) can be considered as Siegel’s equation [33] reduced
to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing
in (2.39) from the prepotential Υ via
∂i1Υ = A
2 , ∂
1Υ = φ
ij , ∂i1∂
1Υ = χ̃
[ijk]
2 , ∂
1Υ = G
[ijkl]
22 , (3.28a)
∂(α1)Υ = A(α2) − εα2ϕ , ∂(α1)∂i1Υ = χiα , ∂(α1)∂(β1)Υ = fαβ , (3.28b)
where one takes Υ and its derivatives at η2i = 0. The other components of the physical fields,
i.e. χ̃
[ijk]
1 , G
[ijkl]
11 , G
[ijkl]
21 , A(11) and A(21)−ϕ, vanish in this light-cone gauge.
Supersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimen-
sions to the form
Qiα = ∂iα − ηβi ∂(αβ) and Q
α = ∂
α − θiβ∂(αβ) . (3.29)
Their antichiral version, matching to D̂iα and D̂
of (2.35), reads
Q̂iα = ∂iα − 2ηβi ∂(αβ) and Q̂
, (3.30)
so that
{Q̂iα , Q̂jβ} = −2 δ
i ∂(αβ) . (3.31)
On a (scalar) R3|2N superfield Σ these supersymmetry transformations act as
δ̂Σ := εiαQ̂iαΣ + ε
αΣ (3.32)
and are induced by the coordinate shifts
δ̂ yαβ = −2εi(αηβ)i and δ̂ η
i = ε
i , (3.33)
where εiα and εαi are 4N real Grassmann parameters. It is easy to see that our equations (3.24)
and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This
is simply because the operators D̂iα and D̂
anticommute with the supersymmetry generators Q̂iα
and Q̂
. Therefore, the equations of motion (3.25) of the modified N -extended chiral model in
2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries
and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this
type of extension is not the standard one since the R-symmetry groups are Spin(N ,N ) in 2+1 and
Spin(N ,N )× Spin(N ,N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry
groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models
the above “noncompact” 2N supersymmetries do not impose any constraints on the geometry of
the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53]. This may be
of interest and deserves further study.
Action functionals. In either formulation of the N -extended supersymmetric SDYM model on
2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for
the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the
equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the
price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space
constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well
as (3.27), where only the operators D̂iα but not D̂iα appear. Note however, that for N = 4 one can
write an action functional in component fields producing the equations (2.39), which are equivalent
to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47]).
One proposal for an action functional stems from Siegel’s idea [33] for the Υ-formulation of the
N -extended SDYM equations. Namely, one sees that ∂i2Υ enters only linearly into the last two
lines in (3.26). Therefore, if we introduce
Υ(1) := Υ|η2
=0 (3.34)
then it must satisfy the first equation from (3.26), and the remaining equations iteratively define
the dependence of Υ on η2i starting from Υ(1). Hence, all information is contained in Υ(1), as can
also be seen from (3.28). In other words, the dependence of Υ on η2i is not ‘dynamical’. For an
action one can then take (cf. [33])
d3x dN η1
Υ(1)∂(αβ)∂
(αβ)Υ(1) +
Υ(1) ε
αβ∂(α1)Υ(1) ∂(β1)Υ(1)
. (3.35)
Extremizing this functional yields the first line of (3.26) at η2i = 0. Except for the Grassmann
integration, this action has the same form as the purely bosonic one [55]. One may apply the same
logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56].
4 N -extended multi-soliton configurations via dressing
The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of theN -extended
U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct
explicit solutions for ψ and hence for Φ† = ψ|ζ=0 and Υ = lim
ζ (ψ−1l). For our purposes the
so-called dressing method [57, 51] proves to be the most practical [12]–[20], and so we shall use it
here for our linear system, i.e. already in the N -extended noncommutative case.
Multi-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new
solution from an old one. More concretely, we rewrite the linear system (3.21) in the form
ψ(∂u − ζ∂x)ψ† = A , ψ(∂x − ζ∂v)ψ† = B , ψ(∂i2 − ζ∂i1)ψ† = Ci . (4.1)
Recall that ψ† := (ψ(xa, ηαi , ζ̄))
† and (A,B, Ci) depend only on xa and ηαi . The central idea is to
demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ.
One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as
well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial
function of ζ globally defined on CP 1. Therefore, it must be a meromorphic function on CP 1
possessing some poles which we choose to lie at finite points with constant coordinates µk ∈ CP 1.
Here we will build a (multi-soliton) solution ψm featuringm simple poles at positions µ1, . . . , µm
with9 Imµk < 0 by left-multiplying an (m−1)-pole solution ψm−1 with a single-pole factor of the
µm − µ̄m
ζ − µm
a, ηαi ) , (4.2)
where the n×n matrix function Pm is yet to be determined. Starting from the trivial (vacuum)
solution ψ0 = 1l, the iteration ψ0 7→ ψ1 7→ . . . 7→ ψm yields a multiplicative ansatz for ψm,
µm−ℓ − µ̄m−ℓ
ζ − µm−ℓ
, (4.3)
which, via partial fraction decomposition, may be rewritten in the additive form
ψm = 1l +
ζ − µk
, (4.4)
9This condition singles out solitons over anti-solitons, which appear for Imµk > 0.
where Λmk and Sk are some n×rk matrices depending on xa and ηαi , with rk ≤ n.
Equations for Sk. Let us first consider the additive parametrization (4.4) of ψm. This ansatz
must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In
particular, the poles at ζ = µ̄k on the left hand sides of these equations have to be removable
since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the
corresponding residues, we learn from (3.15) that
µ̄k − µℓ
Sk = 0 , (4.5)
while from (4.1) we obtain the differential equations
µ̄k − µℓ
A,B,i
Sk = 0 , (4.6)
where L̄
A,B,i
stands for either
L̄Ak = ∂u − µ̄k∂x , L̄Bk = µk(∂x − µ̄k∂v) or L̄ik = ∂i2 − µ̄k∂i1 . (4.7)
Note that we consider a recursive procedure starting from m=1, and operators (4.7) will appear
with k = 1, . . . ,m if we consider poles at ζ = µ̄k.
Because the L̄
A,B,i
for k = 1, . . . ,m are linear differential operators, it is easy to write down
the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η1i , η
i ) to
“co-moving coordinates” (wk, w̄k, sk; η
, η̄i
). The precise relation for k = 1, . . . ,m is [12, 58]
wk := x+ µ̄ku+ µ̄
v = x+ 1
(µ̄k−µ̄−1k )y +
(µ̄k+µ̄
)t and ηik := η
i + µ̄kη
i , (4.8)
with w̄k and η̄
obtained by complex conjugation and the co-moving time sk being inessential
because by definition nothing will depend on it. The kth moving frame travels with a constant
velocity
(vx , vy)k = −
( µk + µ̄k
µkµ̄k + 1
µkµ̄k − 1
µkµ̄k + 1
, (4.9)
so that the static case wk=z is recovered for µk = −i. On functions of (wk, ηik, w̄k, η̄ik) alone the
operators (4.7) act as
L̄Ak = L̄
k = (µk−µ̄k)
=: L̄k and L̄
k = (µk−µ̄k)
. (4.10)
By induction in k = 1, . . . ,m we learn that, due to (4.5), a necessary and sufficient condition for a
solution of (4.6) is
L̄kSk = SkZ̃k and L̄
kSk = SkZ̃
k (4.11)
with some rk×rk matrices Z̃k and Z̃ik depending on (wk, w̄k, η
Passing to the noncommutative bosonic coordinates we obtain
ŵk , ˆ̄wk
= 2θ νkν̄k with νkν̄k =
µk−µ̄k−µ−1k +µ̄
. (4.12)
Thus, we can introduce annihilation and creation operators
and c
so that [ck , c
] = 1 (4.13)
for k = 1, . . . ,m. Naturally, this Heisenberg algebra is realized on a “co-moving” Fock space Hk,
with basis states |ℓ〉k and a “co-moving” vacuum |0〉k subject to ck|0〉k = 0. Each co-moving
vacuum |0〉k (annihilated by ck) is related to the static vacuum |0〉 (annihilated by a) through an
ISU(1,1) squeezing transformation (cf. [12]) which is time-dependent. The fermionic coordinates
ηik and η̄
k remain spectators in the deformation. Coordinate derivatives are represented in the
standard fashion as
7→ −[c†
, · ] and ν̄k
7→ [ck , · ] . (4.14)
After the Moyal deformation, the n×rk matrices Sk have become operator-valued, but are
still functions of the Grassmann coordinates ηi
and η̄i
. The noncommutative version of the BPS
conditions (4.11) naturally reads
ck Sk = Sk Zk and
Sk = Sk Z
k (4.15)
where Zk and Z
k are some operator-valued rk×rk matrix functions of η
and η̄
Nonabelian solutions for Sk. For general data Zk and Z
it is difficult to solve (4.15), but it
is also unnecessary because the final expression ψm turns out not to depend on them. Therefore,
we conveniently choose
Zk = ck ⊗ 1lrk×rk and Zik = 0 ⇒ Sk = Rk(ck, ηik) , (4.16)
where Rk is an arbitrary n×rk matrix function independent of c†k and η̄ik.10 It is known that
nonabelian (multi-) solitons arise for algebraic functions Rk (cf. e.g. [7] for the commutative and [12]
for the noncommutative N=0 case). Their common feature is a smooth commutative limit. The
only novelty of the supersymmetric extension is the ηi
dependence, i.e.
Rk = Rk,0 + η
kRk,i + η
Rk,ij + η
Rk,ijp + η
Rk,ijpq . (4.17)
Abelian solutions for Sk. It is useful to view Sk as a map from C
rk⊗Hk to Cn⊗Hk (momentarily
suppressing the η dependence). The noncommutative setup now allows us to generalize the domain
of this map to any subspace of Cn ⊗Hk. In particular, we may choose it to be finite-dimensional,
say Cqk , and represent the map by an n×qk array |Sk〉 of kets in H. In this situation, Zk and Zik
in (4.15) are just number -valued qk×qk matrix functions of ηjk and η̄
. In case they do not depend
on η̄
, we can write down the most general solution as
|Sk〉 = Rk(ck, ηjk) |Zk〉 exp
) η̄ik
with |Zk〉 := exp
|0〉k . (4.18)
10Changing Zk or Z
k multiplies Rk by an invertible factor from the right, which drops out later, except for the
degenerate case Zk=0 which yields Sk = Rk |0〉k〈0|k.
As before, we may put Zi
= 0 without loss of generality, but now the choice of Zk does matter.
For any given k generically there exists a qk-dimensional basis change which diagonalizes the
ket-valued matrix
|Zk〉 7→ diag
c† , eα
c†, . . . , eα
|0〉k = diag
|α1k〉 , |α2k〉 , . . . , |α
, (4.19)
where we defined coherent states
|αlk〉 := eα
c† |0〉k so that ck |αlk〉 = αlk |αlk〉 for l = 1, . . . , qk and αlk ∈ C . (4.20)
Note that not only the entries of Rk but also the α
k are holomorphic functions of the co-moving
Grassmann parameters η
and thus can be expanded like in (4.17). In the U(1) model, we must
use ket-valued 1×qk matrices |Sk〉 for all k, yielding rows
|Sk〉 =
R1k |α1k〉 , R2k |α2k〉 , . . . , R
for k = 1, . . . ,m , (4.21)
with functions αl
). Here, the Rl
only affect the states’ normalization and can be collected in a
diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally,
we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension
has generalized |Sk〉 → |Sk(ηjk)〉.
Explicit form of Pk. Let us now consider the multiplicative parametrization (4.3) of ψm which
also allows us to solve (4.5). First of all, note that the reality condition (3.15) is satisfied if
Pk = P
= P 2k ⇔ Pk = Tk (T
−1T †
for k = 1, . . . ,m , (4.22)
meaning that Pk is an operator-valued hermitian projector (of group-space rank rk ≤ n) built from
an n×rk matrix function Tk (the abelian case of n=1 is included). The reality condition follows
just because
µk − µ̄k
ζ − µk
µ̄k − µk
ζ − µ̄k
= 1l for any ζ and k = 1, . . . ,m . (4.23)
The rk columns of Tk span the image of Pk and obey
Pk Tk = Tk ⇔ (1l−Pk)Tk = 0 . (4.24)
Furthermore, the equation (4.5) with m = k (induction) rewritten in the form
(1l−Pk)
µk−ℓ − µ̄k−ℓ
µ̄k − µk−ℓ
Sk = 0 (4.25)
reveals that (cf. (4.24))
T1 = S1 and Tk =
1l − µk−ℓ − µ̄k−ℓ
µk−ℓ − µ̄k
Sk for k ≥ 2 , (4.26)
where the explicit form of Sk for k = 1, . . . ,m is given in (4.16) or (4.18). The final result reads
µm−ℓ − µ̄m−ℓ
ζ − µm−ℓ
= 1l +
ζ − µk
(4.27)
with hermitian projectors Pk given by (4.22), Tk given by (4.26) and Sk given by (4.16) or (4.18).
The explicit form of Λmk (which we do not need) can be found in [12]. The corresponding superfields
Φ and Υ are
Φm = ψ
m|ζ=0 =
(1l− ρkPk) with ρk = 1−
, (4.28a)
Υm = lim
ζ (ψm − 1l) =
(µk−µ̄k)Pk . (4.28b)
From (4.22) it is obvious that Pk is invariant under a similarity transformation
Tk 7→ Tk Λk ⇔ Sk 7→ Sk Λk (4.29)
for an invertible operator-valued rk×rk matrix Λk. This justifies putting Zik = 0 from the beginning
and also the restriction to Zk = ck ⊗1lrk×rk in the nonabelian case, both without loss of generality.
Hence, the nonabelian solution space constructed here is parametrized by the set {Rk}m1 of matrix-
valued functions of ck and η
k and the pole positions µk. The abelian moduli space, however, is
larger by the set {Zk}m1 of matrix-values functions of ηik which generically contain the coherent-
state parameter functions {αl
)}. Restricting to ηi
=0 reproduces the soliton configurations of
the bosonic model [12].
Static solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall
that static solutions correspond to the choice m = 1 and µ1 ≡ µ = −i implying w1 = z, so we drop
the index k. Specializing (4.27), we have
ψ = 1l − 2 i
ζ + i
P so that Φ = Φ† = 1l− 2P , (4.30)
where a hermitian projector P of group-space rank r satisfies the BPS equations
(1l−P ) aP = 0 ⇒ (1l−P ) aT = 0 , (4.31a)
(1l−P ) ∂
P = 0 ⇒ (1l−P ) ∂
T = 0 , (4.31b)
with P = T (T †T )−1T † and ηi = η1i + iη
i . In this case T = S, and for a nonabelian r=1 projector
P we get T = T (a, ηi) as an n×1 column. For the simplest case of N=1 we just have (cf. [59])
T = Te(a) + η To(a) with η = η
1 + iη2 , (4.32)
where Te(a) and To(a) are rational functions of a (e.g. polynomials) taking values in the even and
odd parts of the Grassmann algebra. Similarly, an abelian N=1 projector (for n=1) is built from
|T 〉 =
|α1〉 , |α2〉 , . . . , |αq〉
. (4.33)
At θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution
of the standard N=1 supersymmetric CPn−1 sigma model in two dimensions (see e.g. [59]).11 For
11In fact, Φ in (4.30) takes values in the Grassmannian Gr(r, n), and Gr(1, n) = CPn−1.
this reason, one can overcome the previously mentioned difficulty with constructing an action (or
energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions
obeying the BPS conditions (4.31) the topological charge
Q = 2πθ
dη1dη2 Tr tr Φ
D+Φ ,D−Φ
(4.34)
is proportional to the action (BPS bound)
S = 2πθ
dη1dη2 Tr tr
D+Φ ,D−Φ
(4.35)
and is finite for algebraic functions Te and To. Here, the standard superderivatives D± are defined
+ iη ∂z and D− =
+ iη̄ ∂z̄ . (4.36)
One-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain
ψ1 = 1l +
µ− µ̄
ζ − µ P with P = T (T
†T )−1T † (4.37)
Φ = 1l − ρP with ρ = 1− µ
. (4.38)
Now our n×r matrix T must satisfy (putting Zi = 0 and Z = c⊗ 1lr×r)
[c , T ] = 0 and
T = 0 with ηi = η1i + µ̄ η
i , (4.39)
where c is the moving-frame annihilation operator given by (4.13) for k=1.
Recall that the operators c and c† and therefore the matrix T and the projector P can be
expressed in terms of the corresponding static objects by a unitary squeezing transformation (see
e.g. (4.8) and (4.13)). For simplicity we again consider the case N=1 and a nonabelian projector
with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.
T = Te(c) + η To(c) =
T 1e (c) + η T
o (c)...
Tne (c) + η T
o (c)
(4.40)
with polynomials T ae and T
o of order q, say, analogously to the static case (4.32). Note that,
for T ao to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear.
Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing
transformation.
For N=1 the moving frame was defined in (4.8) (dropping the index k) via
w = x + 1
(µ̄−µ̄−1)y + 1
(µ̄+µ̄−1)t and η = η1 + µ̄η2 hence ∂tη = 0 . (4.41)
Consider the moving frame with the coordinates (w, w̄, s; η, η̄) with the choice s = t and the related
change of the derivatives (see [12, 58])
∂x = ∂w + ∂w̄ , (4.42a)
(µ̄−µ̄−1) ∂w + 12(µ−µ
−1) ∂w̄ , (4.42b)
(µ̄+µ̄−1) ∂w +
(µ+µ−1) ∂w̄ + ∂s , (4.42c)
∂η1 = ∂η + ∂η̄ , (4.42d)
∂η2 = µ̄ ∂η + µ∂η̄ . (4.42e)
In the moving frame our solution (4.38) is static, i.e. ∂sΦ = 0, and the projector P has the same
form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore,
by computing the action (4.35) in (w, w̄; η1, η2) coordinates, we obtain for algebraic functions T
in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending
on µ (cf. [12] for the bosonic case).
Large-time asymptotics. Note that in the distinguished (z, z̄, t) coordinate frame (4.41) implies
that at large times w→ κ t with κ = 1
(µ̄+µ̄−1). As a consequence, the tq term in each polynomial
in (4.40) will dominate, i.e.
T → tq
a1 + η b1
an + η bn
=: tq Γ , (4.43)
where Γ is a fixed vector in Cn. It is easy to see that in the distinguished frame the large-time
limit of Φ given by (4.38) is
Φ = 1l − ρΠ with Π = Γ (Γ†Γ)−1Γ† (4.44)
being the projector on the constant vector Γ.
Consider now them-soliton configuration (4.28). By induction of the above argument one easily
arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump
we have
Φm = (1l− ρ1Π1) . . . (1l− ρℓ−1Πℓ−1)(1l− ρℓPℓ)(1l− ρℓ+1Πℓ+1) . . . (1l− ρmΠm) , (4.45)
where the Πm are constant projectors. This large-time factorization of multi-soliton solutions
provides a proof of the no-scattering property because the asymptotic configurations are identical
for large negative and large positive times.
5 Conclusions
In this paper we introduced a generalization of the modified integrable U(n) chiral model with
2N≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model.
It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of
an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory.
The dressing method was applied to generate a wide class of multi-soliton configurations, which
are time-dependent finite-energy solutions to the equations of motion. Compared to the N=0
model, the supersymmetric extension was seen to promote the configurations’ building blocks to
holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic
factorization into a product of single soliton solutions we have shown that no scattering occurs
within the dressing ansatz chosen here.
The considered model does not stand alone but is motivated by twistor string theory [37] with
a target space reduced to the mini-supertwistor space [44, 45, 47]. In this context, the obtained
multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60].
Here 2N appears due to fermionic worldvolume directions of our branes in the superspace de-
scription [60]. Switching on a constant B-field simply deforms the sigma model and D-brane
worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative
abelian solutions.
Restricting to static configurations, the models can be specialized to Grassmannian supersym-
metric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are
invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results
for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler
for admitting two or four supersymmetries, respectively. This difference will be discussed in more
details elsewhere.
We derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction
and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall
that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions
can be obtained from the SDYM equations (or their hierarchy [25]) by suitable dimensional reduc-
tions (see e.g. [61]–[65] and references therein). Moreover, this Ward conjecture [61] was extended
to the noncommutative case (see e.g. [66, 67]). It will be interesting to consider similar reductions
of the N -extended supersymmetric SDYM equations (and their hierarchy [68]) to supersymmetric
integrable equations in three and two dimensions generalizing earlier results [69].
Acknowledgements
We acknowledge fruitful discussions with C. Gutschwager. This work was supported in part by the
Deutsche Forschungsgemeinschaft (DFG).
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  We consider a supersymmetric Bogomolny-type model in 2+1 dimensions
originating from twistor string theory. By a gauge fixing this model is reduced
to a modified U(n) chiral model with N<=8 supersymmetries in 2+1 dimensions.
After a Moyal-type deformation of the model, we employ the dressing method to
explicitly construct multi-soliton configurations on noncommutative R^{2,1} and
analyze some of their properties.

Introduction
In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on
the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2,
3, 4]. In particular, if the open string has N=2 worldsheet supersymmetry, the tree-level target
space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2
dimensions [5]. Furthermore, open N=2 strings in a B-field background induce on the worldvolume
of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1
dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral
model known as the Ward model [7]. The topological nature of N=2 strings and the integrability
of their tree-level dynamics [8] render this noncommutative sigma model integrable.1
Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering
and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R2.
These are not only a rich testing ground for physical properties such as adiabatic dynamics or
quantization, but also descend to more standard multi-solitons of various integrable systems in
2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There
is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant
or integrable but not both [7, 11]. In fact, Derrick’s theorem prohibits the existence of stable
solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation,
however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted
version. There is another gain: The deformed Ward model possesses not only deformed versions
of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely
noncommutative (multi-)solitons, in particular for the U(1) group [12, 13]! Moreover, this class is
related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15]
by an infinite-stiffness limit of the potential [16].
In [12, 13] and [17]–[20] families of multi-solitons as well as their reduction to solitons of the
noncommutative sine-Gordon equations were described and studied. In the nonabelian case both
scattering and nonscattering configurations were obtained. For static configurations the issue of
their stability was analyzed [21]. The full moduli space metric for the abelian model was computed
and its adiabatic two-soliton dynamics was discussed [16].
Recall that the critical N=2 string theory has a four-dimensional target space, and its open
string effective field theory is self-dual Yang-Mills [8], which gets deformed noncommutatively in
the presence of a B-field [5]. Conversely, the noncommutative SDYM equations are contained [19]
in the equations of motion of N=2 string field theory (SFT) [22] in a B-field background. This
SFT formulation is based on the N=4 topological string description [23]. It is well known that
the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the
twistor space2 [26] CP 3 and the topological N=4 string theory contains twistors from the outset.
The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing
methods were incorporated into the N=2 open SFT in [27, 28]. However, this theory reproduces
only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31]) and integrability properties. It
is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four
dimensions?
1For discussing some other noncommutative integrable models see e.g. [9, 10] and references therein.
2For reviews of twistor theory see, e.g., the books [24, 25].
There are some proposals [33, 34, 35, 36] for extending N=2 open string theory (and its SFT) to
be space-time supersymmetric. Moreover, it was shown by Witten [37] that N=4 supersymmetric
SDYM theory appears in twistor string theory, which is a B-type open topological string with the
supertwistor space CP 3|4 as a target space.3 Note that N<4 SDYM theory forms a BPS subsector
of N -extended super Yang-Mills theory, and N=4 SDYM can be considered as a truncation of the
full N=4 super Yang-Mills theory [37]. It is believed [43, 39] that twistor string theory is related
with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of
N=2 (and topological N=4) string theory which also leads to the N=4 SDYM model.
A dimensional reduction of the above relations between twistor strings and N=4 super Yang-
Mills and SDYM models was considered in [44, 45, 46, 47]. The corresponding twistor string
theory after this reduction is the topological B-model on the mini-supertwistor space P2|4. In [47]
it was shown that the 2N=8 supersymmetric extension of the Bogomolny-type model in 2+1
dimensions is equivalent to an 2N=8 supersymmetric modified U(n) chiral model on R2,1. The
subject of the current paper is an 2N≤8 version of the above supersymmetric Bogomolny-type
Yang-Mills-Higgs model in signature (− + +), its relation with an N -extended supersymmetric
modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type
noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton
configurations on noncommutative R2,1 for the corresponding supersymmetric sigma model field
equations. By studying the scattering properties of the constructed configurations, we prove their
asymptotic factorization without scattering for large times. We also briefly discuss a D-brane
interpretation of these soliton configurations from the viewpoint of twistor string theory.
2 Supersymmetric Bogomolny model in 2+1 dimensions
2.1 N -extended SDYM equations in 2+2 dimensions
Space R2,2. Let us consider the four-dimensional space R2,2 = (R4, g) with the metric
ds2 = gµνdx
µdxν = det(dxαα̇) = dx11̇dx22̇ − dx21̇dx12̇ (2.1)
with (gµν) = diag(−1,+1,+1,−1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2,
α̇ = 1̇, 2̇ are spinor indices. We choose the coordinates4
(xµ) = (xa, t̃) = (t, x, y, t̃) with a, b, . . . = 1, 2, 3 , (2.2)
and the signature (− ++−) allows us to introduce real isotropic coordinates (cf. [19, 6])
x11̇ = 1
(t− y) , x12̇ = 1
(x+ t̃) , x21̇ = 1
(x− t̃) , x22̇ = 1
(t+ y) . (2.3)
SDYM. Recall that the SDYM equations for a field strength tensor Fµν on R
2,2 read
εµνρσF
ρσ = Fµν , (2.4)
3For other variants of twistor string models see [38, 39, 40]. For recent reviews providing a twistor description of
super Yang-Mills theory, see [41, 42] and references therein.
4Our conventions are chosen to match those of [12] after reduction to the space R2,1 with coordinates (t, x, y).
where εµνρσ is a completely antisymmetric tensor on R
2,2 and ε1234 = 1. In the coordinates (2.3)
we have the decomposition
αα̇,ββ̇
= ∂αα̇Aββ̇ − ∂ββ̇Aαα̇ + [Aαα̇, Aββ̇ ] = εαβ Fα̇β̇ + εα̇β̇ Fαβ (2.5)
:= −1
αα̇,ββ̇
and Fαβ := −12ε
α̇β̇F
αα̇,ββ̇
, (2.6)
where εαβ is antisymmetric, εαβε
βγ = δ
α, and similar for ε
α̇β̇, with ε12 = ε1̇2̇ = 1. The gauge
potential (Aαα̇) will appear in the covariant derivative
, · ] . (2.7)
In spinor notation, (2.4) is equivalently written as
= 0 ⇔ F
αα̇,ββ̇
Fαβ . (2.8)
Solutions {Aαα̇} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills
theory on R2,2.
N -extended SDYM in component fields. The field content of N -extended super SDYM is5
N = 0 Aαα̇ (2.9a)
N = 1 Aαα̇, χiα with i = 1 (2.9b)
N = 2 Aαα̇, χiα, φ[ij] with i, j = 1, 2 (2.9c)
N = 3 Aαα̇, χiα, φ[ij], χ̃
[ijk]
with i, j, k = 1, 2, 3 (2.9d)
N = 4 Aαα̇, χiα, φ[ij], χ̃
[ijk]
[ijkl]
with i, j, k, l = 1, 2, 3, 4 . (2.9e)
Here (Aαα̇, χ
[ij], χ̃
[ijk]
[ijkl]
) are fields of helicities (+1,+1
, 0,−1
,−1). These fields obey
the field equations of the N = 4 SDYM model, namely [33, 37]
= 0 , (2.10a)
Dαα̇χ
iα = 0 , (2.10b)
Dαα̇D
αα̇φij + 2{χiα, χjα} = 0 , (2.10c)
Dαα̇χ̃
α̇[ijk] − 6[χ[iα, φjk]] = 0 , (2.10d)
D γ̇α G
[ijkl]
+ 12{χ[iα, χ̃
} − 18[φ[ij ,D
φkl]] = 0 . (2.10e)
Note that the N < 4 SDYM field equations are governed by the first N+1 equations of (2.10),
where F
= 0 is counted as one equation and so on.
5We use symmetrization (·) and antisymmetrization [·] of k indices with weight 1
, e.g. [ij] = 1
(ij − ji).
2.2 Superfield formulation of N -extended SDYM
Superspace R4|4N . Recall that in the space R2,2 = (R4, g) with the metric g given in (2.1)
one may introduce purely real Majorana-Weyl spinors6 θα and ηα̇ of helicities +1
and −1
as anti-
commuting (Grassmann-algebra) objects. Using 2N such spinors with components θiα and ηα̇i for
i = 1, . . . ,N , one can define the N -extended superspace R4|4N and the N -extended supersymmetry
algebra generated by the supertranslation operators
Pαα̇ = ∂αα̇ , Qiα = ∂iα − ηα̇i ∂αα̇ and Qiα̇ = ∂iα̇ − θiα∂αα̇ , (2.11)
where
∂αα̇ :=
∂xαα̇
, ∂iα :=
and ∂iα̇ :=
∂ηα̇i
. (2.12)
The commutation relations for the generators (2.11) read
{Qiα, Qjα̇} = −2δ
iPαα̇ , [Pαα̇, Qiβ ] = 0 and [Pαα̇, Q
] = 0 . (2.13)
To rewrite equations of motion in terms of R4|4N superfields one uses the additional operators
Diα = ∂iα + η
i ∂αα̇ and D
α̇ = ∂
α̇ + θ
iα∂αα̇ , (2.14)
which (anti)commute with the operators (2.11) and satisfy
{Diα,Dj
} = 2δjiPαβ̇ , [Pαα̇,Diβ ] = 0 and [Pαα̇,D
] = 0 . (2.15)
Antichiral superspace R4|2N . On the superspace R4|4N one may introduce tensor fields de-
pending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives LX
etc.. Furthermore, on any such superfield A one can impose the constraint equations LDiαA = 0,
which for a scalar superfield f reduce to the so-called antichirality conditions
Diαf = 0 . (2.16)
These are easily solved by using a coordinate transformation on R4|4N ,
(xαα̇, ηα̇i , θ
iα) → (x̃αα̇ = xαα̇−θiαηα̇i , ηα̇i , θiα) , (2.17)
under which ∂αα̇,Diα and D
α̇ transform to the operators
∂̃αα̇ = ∂αα̇ , D̃iα = ∂iα and D̃
α̇ = ∂
α̇ + 2θ
iα∂αα̇ . (2.18)
Then (2.16) simply means that f is defined on a sub-superspace R4|2N ⊂ R4|4N with coordinates
x̃αα̇ and ηα̇i . (2.19)
This space is called antichiral superspace. In the following we will usually omit the tildes when
working on the antichiral superspace.
6Note that in Minkowski signature the Weyl spinor θα is complex and ηα̇ = εα̇β̇η
β̇ = θα is complex conjugate
to θα. For the Kleinian (split) signature 2 + 2, however, these spinors are real and independent of one another.
N -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in
terms of superfields on the antichiral superspace R4|2N [33, 48]. Namely, for any given 0 ≤ N ≤ 4,
fields of a proper multiplet from (2.9) can be combined into superfields Aαα̇ and Aiα̇ depending on
xαα̇, ηα̇i ∈ R4|2N and giving rise to covariant derivatives
∇αα̇ := ∂αα̇ +Aαα̇ and ∇iα̇ := ∂iα̇ +Aiα̇ . (2.20)
In such terms the N -extended SDYM equations (2.10) read
[∇αα̇,∇ββ̇] + [∇αβ̇ ,∇βα̇] = 0 , [∇
α̇,∇ββ̇ ] + [∇
,∇βα̇] = 0 , {∇iα̇,∇
}+ {∇i
} = 0 , (2.21)
which is equivalent to
[∇αα̇,∇ββ̇] = εα̇β̇ Fαβ , [∇
α̇,∇ββ̇ ] = εα̇β̇ F
β and {∇iα̇,∇
} = ε
F ij , (2.22)
where F ij is antisymmetric and Fαβ is symmetric in their indices.
The above gauge potential superfields (Aαα̇, Aiα̇) as well as the gauge strength superfields
(Fαβ , F iα, F ij) contain all physical component fields of theN -extended SDYMmodel. For instance,
the lowest component of the triple (Fαβ , F iα, F ij) in an η-expansion is (Fαβ , χiα, φij), with zeros
in case N is too small. By employing Bianchi identities for the gauge strength superfields, one
successively obtains [48] the superfield expansions and the field equations (2.10) for all component
fields.
It is instructive to extend the antichiral combination in (2.18) to potentials and covariant
derivatives,
D̃iα̇ = ∂
α̇ + 2 θ
iα ∂αα̇
+ + +
Ãiα̇ := Aiα̇ + 2 θiαAαα̇
‖ ‖ ‖
∇̃iα̇ := ∇iα̇ + 2 θiα∇αα̇
(2.23)
where ∇αα̇, ∇iα̇ and D̃iα̇ are given by (2.20) and (2.18), while Aiα̇ and Aαα̇ depend on xαα̇ and ηα̇i
only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single
{∇̃iα̇, ∇̃
} + {∇̃i
, ∇̃j
} = 0 ⇔ {∇̃iα̇, ∇̃
} = ε
F̃ ij , (2.24)
with F̃ ij = F ij + 4 θ[iαF j]α + 4 θiαθjβFαβ . The concise form (2.24) of the N -extended SDYM
equations is quite convenient, and we will use it interchangeable with (2.21).
Linear system for N -extended SDYM. It is well known that the superfield SDYM equations
(2.21) can be seen as the compatibility conditions for the linear system of differential equations
ζ α̇(∂αα̇ +Aαα̇)ψ = 0 and ζ α̇(∂iα̇ +Aiα̇)ψ = 0 , (2.25)
where (ζ
and ζ α̇ = εα̇β̇ζ
. The extra (spectral) parameter7 ζ lies in the extended complex
plane C∪∞ = CP 1. Here ψ is a matrix-valued function depending not only on xαα̇ and ηα̇i but also
(meromorphically) on ζ ∈ CP 1. We subject the n×n matrix ψ to the following reality condition:
ψ(xαα̇, ηα̇i , ζ)
ψ(xαα̇, ηα̇i , ζ̄)
= 1l , (2.26)
7The parameter ζ is related with λ used in [45] by the formula ζ = i 1−λ
(cf. e.g. [31]).
where “†” denotes hermitian conjugation and ζ̄ is complex conjugate to ζ. This condition guarantees
that all physical fields of the N -extended SDYM model will take values in the adjoint representation
of the algebra u(n). In the concise form the linear system (2.25) is written as
ζ α̇(∇iα̇ + 2θiα∇αα̇)ψ = 0 ⇔ ζ α̇(D̃iα̇ + Ãiα̇)ψ = 0 ⇔ ζ α̇ ∇̃iα̇ ψ = 0 . (2.27)
2.3 Reduction of N -extended SDYM to 2+1 dimensions
The supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained
from the described N -extended super SDYM equations by a dimensional reduction R2,2 → R2,1.
In particular, for the N=0 sector we demand the components Aµ of a gauge potential to be
independent of x4 and put A4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions
(the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can
reduce the N -extended SDYM equations on R2,2 by imposing the ∂4-invariance condition on all
the fields (Aαα̇, χ
[ij], χ̃
[ijk]
[ijkl]
) from the N=4 supermultiplet or its truncation to N<4
and obtain supersymmetric Bogomolny-type equations on R2,1.
Spinors in R2,1. Recall that on R2,2 both N=4 SDYM theory and full N=4 super Yang-
Mills theory have an SL(4, R) ∼= Spin(3,3) R-symmetry group [33]. A dimensional reduction to
2,1 enlarges the supersymmetry and R-symmetry to 2N=8 and Spin(4,4), respectively, for both
theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets
doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)
∼= Spin(2,1)L×Spin(2,1)R of R2,2 to its diagonal subgroup Spin(2,1)D as the local rotation group
of R2,1, the distinction between undotted and dotted indices disappears. We shall use undotted
indices henceforth.
Coordinates and derivatives in R2,1. The above discussion implies that one can relabel the
bosonic coordinates xαβ̇ from (2.3) by xαβ and split them as
xαβ = 1
(xαβ + xβα) + 1
(xαβ − xβα) = x(αβ) + x[αβ] (2.28)
into antisymmetric and symmetric parts,
x[αβ] = 1
εαβx4 = 1
εαβ t̃ and x(αβ) =: yαβ , (2.29)
respectively, with
y11 = x11 = 1
(t− y) , y12 = 1
(x12 + x21) = 1
x , y22 = x22 = 1
(t+ y) . (2.30)
We also have θiα 7→ θiα and ηα̇i 7→ ηαi for the fermionic coordinates on R4|4N reduced to R3|4N .
Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators
∂(αβ) =
(∂αβ + ∂βα) (2.31)
which read explicitly as
∂(11) =
= ∂t−∂y , ∂(12) = ∂(21) = 12
= ∂x , ∂(22) =
= ∂t+∂y . (2.32)
We thus have
= ∂(αβ) − εαβ∂4 = ∂(αβ) − εαβ∂t̃ , (2.33)
where ε12 = −ε21 = −1, ∂4 = ∂/∂x4 and ∂t̃ = ∂/∂t̃.
The operators Diα and D
α̇ acting on t̃-independent superfields reduce to
Diα = ∂iα + η
i ∂(αβ) and D
α = ∂
α + θ
iβ∂(αβ) , (2.34)
where ∂iα = ∂/∂θ
iα and ∂iα = ∂/∂η
i . Similarly, the antichiral operators D̃iα and D̃
α̇ in (2.18)
become
D̂iα = ∂iα and D̂
α = ∂
α + 2θ
iβ∂(αβ) . (2.35)
Supersymmetric Bogomolny-type equations in component fields. According to (2.33),
the components A
of a gauge potential in four dimensions split into the components A(αβ) of a
gauge potential in three dimensions and a Higgs field A[αβ] = −εαβ ϕ, i.e.
Aαβ = A(αβ) +A[αβ] = A(αβ) − εαβ ϕ . (2.36)
Then the covariant derivatives D
reduced to three dimensions become the differential operators
Dαβ − εαβ ϕ = ∂(αβ) + [A(αβ), · ]− εαβ [ϕ, · ] , (2.37)
and the Yang-Mills field strength on R2,1 decomposes as
Fαβ, γδ = [Dαβ , Dγδ] = εαγ fβδ + εβδ fαγ with fαβ = fβα . (2.38)
Substituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent
of x4 = t̃, we obtain the following supersymmetric Bogomolny-type equations on R2,1:
fαβ +Dαβϕ = 0 , (2.39a)
Dαβ χ
iβ + εαβ [ϕ, χ
iβ] = 0 , (2.39b)
Dαβ D
αβφij + 2[ϕ, [ϕ, φij ]] + 2{χiα, χjα} = 0 , (2.39c)
Dαβ χ̃
β[ijk] − εαβ [ϕ, χ̃β[ijk]]− 6[χ[iα, φjk]] = 0 , (2.39d)
[ijkl]
+ [ϕ,G
[ijkl]
] + 12{χ[iα, χ̃jkl]β } − 18[φ[ij ,Dαβφkl]]− 18εαβ [φ[ij , [φkl], ϕ]] = 0 .(2.39e)
Supersymmetric Bogomolny-type equations in terms of superfields. Translations gen-
erated by the vector field ∂4 = ∂t̃ are isometries of superspaces R
4|4N and R4|2N . By taking the
quotient with respect to the action of the abelian group G generated by ∂4, we obtain the reduced
full superspace R3|4N ∼= R4|4N /G and the reduced antichiral superspace R3|2N ∼= R4|2N/G. In the
following, we shall work on R3|2N and R3|2N × CP 1, since the reduced ψ-function from (2.25) and
(2.27) is defined on the latter space.
The linear system stays in the center of the superfield approach to the N -extended SDYM
equations. After imposing t̃-independence on all fields in the linear system (2.27), we arrive at the
linear equations
ζα ∇̂iα ψ ≡ ζα(D̂iα + Âiα)ψ = 0 (2.40)
of the same form but with
D̂iα = ∂
α + 2θ
iβ∂(αβ) and Âiα = Aiα + 2θiβ(A(αβ) − εαβΞ) , (2.41)
where Aiα, A(αβ) and Ξ are superfields depending on yαβ and ηαi only. These linear equations
expand again to the pair (cf. (2.25))
ζβ(∂(αβ) +A(αβ) − εαβΞ)ψ = 0 and ζα(∂iα +Aiα)ψ = 0 . (2.42)
The compatibility conditions for the linear system (2.40) read
{∇̂iα, ∇̂
} + {∇̂iβ, ∇̂jα} = 0 ⇔ {∇̂iα, ∇̂
} = εαβ F̂ ij (2.43)
and present a condensed form of (2.39) rewritten in terms of R3|2N superfields. Similarly, these
equations can also be written in more expanded forms analogously to (2.21) or using the superfield
analog of (2.37). However, we will not do this since all these sets of equations are equivalent.
3 Noncommutative N -extended U(n) chiral model in 2+1 dimensions
As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentz-
invariant or integrable but not both [7, 11]. We will show that the super Bogomolny-type model
discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n)
chiral model (so as to be integrable) first formulated by Ward [7]. Since integrability is compatible
with noncommutative deformation (if introduced properly, see e.g. [9]–[20]) we choose from the
beginning to formulate our super extension of this chiral model on Moyal-deformed R2,1 with
noncommutativity parameter θ ≥ 0. Ordinary space-time R2,1 can always be restored by taking
the commutative limit θ → 0.
Star-product formulation. Classical field theory on noncommutative spaces may be realized
in a star-product formulation or in an operator formalism8. The first approach is closer to the
commutative field theory: it is obtained by simply deforming the ordinary product of classical
fields (or their components) to the noncommutative star product
(f ⋆ g)(x) = f(x) exp{ i
ab −→∂b} g(x) ⇒ xa ⋆ xb − xb ⋆ xa = iθab (3.1)
with a constant antisymmetric tensor θab. Specializing to R2,1, we use real coordinates (xa) =
(t, x, y) in which the Minkowski metric g on R3 reads (gab) = diag(−1,+1,+1) with a, b, . . . = 1, 2, 3
(cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces
introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd
coordinates. Deferring general superspace deformations and their consequences to future work,
we here content ourselves with the simple embedding of the “bosonic” Moyal deformation into
superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann
variables θiα and ηαi .
For later use we consider not only isotropic coordinates and vector fields
u := 1
(t+y) = y22 , v := 1
(t−y) = y11 , ∂u = ∂t + ∂y = ∂(22) , ∂v = ∂t − ∂y = ∂(11) (3.2)
8See [50] for reviews on noncommutative field theories.
introduced in Section 2, but also the complex combinations
z := x+ iy , z̄ := x− iy , ∂z = 12 (∂x − i∂y) , ∂z̄ =
(∂x + i∂y) . (3.3)
Since the time coordinate t remains commutative, the only nonvanishing component of the non-
commutativity tensor θab is
θxy = −θyx =: θ > 0 ⇒ θzz̄ = −θz̄z = −2i θ . (3.4)
Hence, we have
z ⋆ z̄ = zz̄ + θ and z̄ ⋆ z = zz̄ − θ (3.5)
as examples of the general formula (3.1).
Operator formalism. The nonlocality of the star products renders explicit computation cum-
bersome. We therefore pass to the operator formalism, which trades the star product for operator-
valued spatial coordinates (x̂, ŷ) or their complex combinations (ẑ, ˆ̄z), subject to
[t, x̂] = [t, ŷ] = 0 but [x̂, ŷ] = iθ ⇒ [ẑ, ˆ̄z] = 2 θ . (3.6)
The latter equation suggests the introduction of annihilation and creation operators,
ẑ and a† =
ˆ̄z with [a , a†] = 1 , (3.7)
which act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ〉, ℓ = 0, 1, 2, . . .}
such that
a |ℓ〉 =
ℓ |ℓ−1〉 and a† |ℓ〉 =
ℓ+1 |ℓ+1〉 . (3.8)
Any superfield f(t, z, z̄, ηαi ) on R
3|2N can be related to an operator-valued superfield f̂(t, ηαi ) ≡
F (t, a, a†, ηαi ) on R
1|2N acting in H, with the help of the Moyal-Weyl map
f(t, z, z̄, ηαi ) 7→ f̂(t, ηαi ) = Weyl-ordered f
2θa†, ηαi
. (3.9)
The inverse transformation recovers the ordinary superfield,
f̂(t, ηαi ) ≡ F (t, a, a†, ηαi ) 7→ f(t, z, z̄, ηαi ) = F⋆
t, z√
, z̄√
, ηαi
, (3.10)
where F⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl
map, we have
f ⋆ g 7→ f̂ ĝ and
dx dy f = 2π θTrf̂ = 2π θ
〈ℓ|f̂ |ℓ〉 , (3.11)
and the spatial derivatives are mapped into commutators,
∂zf 7→ ∂̂z f̂ = − 1√
[a†, f̂ ] and ∂z̄f 7→ ∂̂z̄ f̂ = 1√
[a , f̂ ] . (3.12)
For notational simplicity we will from now on omit the hats over the operators except when con-
fusion may arise.
Gauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43)
are invariant under a gauge transformation
ψ 7→ ψ′ = g−1ψ , (3.13a)
A 7→ A′ = g−1A g + g−1∂ g (with appropriate indices) , (3.13b)
Ξ 7→ Ξ′ = g−1Ξ g , (3.13c)
where g = g(xa, ηαi ) is a U(n)-valued superfield globally defined on the deformed superspace R
CP 1. Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy
the standard asymptotic conditions (see e.g. [51])
ψ = Φ−1 + O(ζ) for ζ → 0 , (3.14a)
ψ = 1l + ζ−1Υ + O(ζ−2) for ζ →∞ , (3.14b)
where the U(n)-valued function Φ and u(n)-valued function Υ depend on xa and ηαi . This “unitary”
gauge is compatible with the reality condition for ψ,
ψ(xa, ηαi , ζ)
ψ(xa, ηαi , ζ̄)
= 1l , (3.15)
obtained by reduction from (2.26).
Gauge fixing for Âiα. After fixing the unitary gauge (3.14) for ψ and inserting (ζα) =
the linear system (2.40), one can easily reconstruct the superfield given in (2.41) from Φ or Υ via
Âi1 = 0 and Âi2 = Φ−1D̂i2Φ = D̂i1Υ (3.16)
and thus fix a gauge for the superfields Âiα. The operators D̂iα were defined in (2.35). One can
express (3.16) in terms of Aiα and A(αβ) − εαβΞ as
Ai1 = 0 and Ai2 = Φ−1∂i2Φ = ∂i1Υ , (3.17)
A(11) = 0 and A(12) + Ξ = Φ−1∂(12)Φ = ∂(11)Υ , (3.18)
A(21) − Ξ = 0 and A(22) = Φ−1∂(22)Φ = ∂(12)Υ . (3.19)
Using (2.32), we can rewrite the nonzero components as
A := Φ−1∂uΦ = ∂xΥ , B := Φ−1∂xΦ = ∂vΥ , Ci := Φ−1∂i2Φ = ∂i1Υ . (3.20)
Recall that the superfields Φ and Υ depend on xa and ηαi .
Linear system. In the above-introduced unitary gauge the linear system (2.42) reads
(ζ∂x − ∂u −A)ψ = 0 , (ζ∂v − ∂x − B)ψ = 0 , (ζ∂i1 − ∂i2 − Ci)ψ = 0 , (3.21)
which adds the last equation to the linear system of the Ward model [7] and generalizes it to
superfields A(xa, ηαj ), B(xa, ηαj ) and Ci(xa, ηαj ). The concise form of (3.21) reads
ζ D̂i1 − D̂i2 − Âi2
ψ = 0 (3.22)
or, in more explicit form,
∂i1 + 2θ
i1∂v + 2θ
∂i2 + Ci + 2θi1(∂x + B) + 2θi2(∂u +A)
ψ = 0 . (3.23)
N -extended sigma model. The compatibility conditions of this linear system are the N -
extended noncommutative sigma model equations
D̂i1(Φ
−1D̂j2 Φ) + D̂
−1D̂i2 Φ) = 0 (3.24)
which in expanded form reads
(gab + vcε
cab) ∂a(Φ
−1∂bΦ) = 0 ⇔ ∂x(Φ−1∂xΦ) − ∂v(Φ−1∂uΦ) = 0 , (3.25a)
∂i1(Φ
−1∂xΦ) − ∂v(Φ−1∂i2Φ) = 0 , ∂i1(Φ−1∂uΦ) − ∂x(Φ−1∂i2Φ) = 0 , (3.25b)
∂i1(Φ
−1∂j2Φ) + ∂
−1∂i2Φ) = 0 . (3.25c)
Here, the first line contains the Wess-Zumino-Witten term with a constant vector (vc) = (0, 1, 0)
which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimen-
sions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H
and depend on xa and 2N Grassmann variables ηαi . As discussed in Section 2, the compatibility
conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolny-
type equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25)
are equivalent to a gauge fixed form of equations (2.39).
Υ-formulation. Instead of Φ-parametrization of (A,B, Ci) given in (3.17)–(3.20) we may use the
equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the
linear system (3.21) reduce to
(∂2x − ∂u∂v)Υ + [∂vΥ , ∂xΥ] = 0 , (3.26a)
(∂i2∂v − ∂i1∂x)Υ + [∂i1Υ , ∂vΥ] = 0 , (∂i2∂x − ∂i1∂u)Υ + [∂i1Υ , ∂xΥ] = 0 , (3.26b)
(∂i2∂
1 + ∂
1)Υ + {∂i1Υ , ∂
1Υ} = 0 , (3.26c)
which in concise form read
(D̂i2 D̂
1 + D̂
1)Υ + {D̂i1Υ , D̂
1Υ} = 0 . (3.27)
Recall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and
depend on xa and 2N Grassmann variables ηαi .
For N=4, the commutative limit of (3.27) can be considered as Siegel’s equation [33] reduced
to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing
in (2.39) from the prepotential Υ via
∂i1Υ = A
2 , ∂
1Υ = φ
ij , ∂i1∂
1Υ = χ̃
[ijk]
2 , ∂
1Υ = G
[ijkl]
22 , (3.28a)
∂(α1)Υ = A(α2) − εα2ϕ , ∂(α1)∂i1Υ = χiα , ∂(α1)∂(β1)Υ = fαβ , (3.28b)
where one takes Υ and its derivatives at η2i = 0. The other components of the physical fields,
i.e. χ̃
[ijk]
1 , G
[ijkl]
11 , G
[ijkl]
21 , A(11) and A(21)−ϕ, vanish in this light-cone gauge.
Supersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimen-
sions to the form
Qiα = ∂iα − ηβi ∂(αβ) and Q
α = ∂
α − θiβ∂(αβ) . (3.29)
Their antichiral version, matching to D̂iα and D̂
of (2.35), reads
Q̂iα = ∂iα − 2ηβi ∂(αβ) and Q̂
, (3.30)
so that
{Q̂iα , Q̂jβ} = −2 δ
i ∂(αβ) . (3.31)
On a (scalar) R3|2N superfield Σ these supersymmetry transformations act as
δ̂Σ := εiαQ̂iαΣ + ε
αΣ (3.32)
and are induced by the coordinate shifts
δ̂ yαβ = −2εi(αηβ)i and δ̂ η
i = ε
i , (3.33)
where εiα and εαi are 4N real Grassmann parameters. It is easy to see that our equations (3.24)
and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This
is simply because the operators D̂iα and D̂
anticommute with the supersymmetry generators Q̂iα
and Q̂
. Therefore, the equations of motion (3.25) of the modified N -extended chiral model in
2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries
and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this
type of extension is not the standard one since the R-symmetry groups are Spin(N ,N ) in 2+1 and
Spin(N ,N )× Spin(N ,N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry
groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models
the above “noncompact” 2N supersymmetries do not impose any constraints on the geometry of
the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53]. This may be
of interest and deserves further study.
Action functionals. In either formulation of the N -extended supersymmetric SDYM model on
2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for
the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the
equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the
price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space
constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well
as (3.27), where only the operators D̂iα but not D̂iα appear. Note however, that for N = 4 one can
write an action functional in component fields producing the equations (2.39), which are equivalent
to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47]).
One proposal for an action functional stems from Siegel’s idea [33] for the Υ-formulation of the
N -extended SDYM equations. Namely, one sees that ∂i2Υ enters only linearly into the last two
lines in (3.26). Therefore, if we introduce
Υ(1) := Υ|η2
=0 (3.34)
then it must satisfy the first equation from (3.26), and the remaining equations iteratively define
the dependence of Υ on η2i starting from Υ(1). Hence, all information is contained in Υ(1), as can
also be seen from (3.28). In other words, the dependence of Υ on η2i is not ‘dynamical’. For an
action one can then take (cf. [33])
d3x dN η1
Υ(1)∂(αβ)∂
(αβ)Υ(1) +
Υ(1) ε
αβ∂(α1)Υ(1) ∂(β1)Υ(1)
. (3.35)
Extremizing this functional yields the first line of (3.26) at η2i = 0. Except for the Grassmann
integration, this action has the same form as the purely bosonic one [55]. One may apply the same
logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56].
4 N -extended multi-soliton configurations via dressing
The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of theN -extended
U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct
explicit solutions for ψ and hence for Φ† = ψ|ζ=0 and Υ = lim
ζ (ψ−1l). For our purposes the
so-called dressing method [57, 51] proves to be the most practical [12]–[20], and so we shall use it
here for our linear system, i.e. already in the N -extended noncommutative case.
Multi-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new
solution from an old one. More concretely, we rewrite the linear system (3.21) in the form
ψ(∂u − ζ∂x)ψ† = A , ψ(∂x − ζ∂v)ψ† = B , ψ(∂i2 − ζ∂i1)ψ† = Ci . (4.1)
Recall that ψ† := (ψ(xa, ηαi , ζ̄))
† and (A,B, Ci) depend only on xa and ηαi . The central idea is to
demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ.
One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as
well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial
function of ζ globally defined on CP 1. Therefore, it must be a meromorphic function on CP 1
possessing some poles which we choose to lie at finite points with constant coordinates µk ∈ CP 1.
Here we will build a (multi-soliton) solution ψm featuringm simple poles at positions µ1, . . . , µm
with9 Imµk < 0 by left-multiplying an (m−1)-pole solution ψm−1 with a single-pole factor of the
µm − µ̄m
ζ − µm
a, ηαi ) , (4.2)
where the n×n matrix function Pm is yet to be determined. Starting from the trivial (vacuum)
solution ψ0 = 1l, the iteration ψ0 7→ ψ1 7→ . . . 7→ ψm yields a multiplicative ansatz for ψm,
µm−ℓ − µ̄m−ℓ
ζ − µm−ℓ
, (4.3)
which, via partial fraction decomposition, may be rewritten in the additive form
ψm = 1l +
ζ − µk
, (4.4)
9This condition singles out solitons over anti-solitons, which appear for Imµk > 0.
where Λmk and Sk are some n×rk matrices depending on xa and ηαi , with rk ≤ n.
Equations for Sk. Let us first consider the additive parametrization (4.4) of ψm. This ansatz
must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In
particular, the poles at ζ = µ̄k on the left hand sides of these equations have to be removable
since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the
corresponding residues, we learn from (3.15) that
µ̄k − µℓ
Sk = 0 , (4.5)
while from (4.1) we obtain the differential equations
µ̄k − µℓ
A,B,i
Sk = 0 , (4.6)
where L̄
A,B,i
stands for either
L̄Ak = ∂u − µ̄k∂x , L̄Bk = µk(∂x − µ̄k∂v) or L̄ik = ∂i2 − µ̄k∂i1 . (4.7)
Note that we consider a recursive procedure starting from m=1, and operators (4.7) will appear
with k = 1, . . . ,m if we consider poles at ζ = µ̄k.
Because the L̄
A,B,i
for k = 1, . . . ,m are linear differential operators, it is easy to write down
the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η1i , η
i ) to
“co-moving coordinates” (wk, w̄k, sk; η
, η̄i
). The precise relation for k = 1, . . . ,m is [12, 58]
wk := x+ µ̄ku+ µ̄
v = x+ 1
(µ̄k−µ̄−1k )y +
(µ̄k+µ̄
)t and ηik := η
i + µ̄kη
i , (4.8)
with w̄k and η̄
obtained by complex conjugation and the co-moving time sk being inessential
because by definition nothing will depend on it. The kth moving frame travels with a constant
velocity
(vx , vy)k = −
( µk + µ̄k
µkµ̄k + 1
µkµ̄k − 1
µkµ̄k + 1
, (4.9)
so that the static case wk=z is recovered for µk = −i. On functions of (wk, ηik, w̄k, η̄ik) alone the
operators (4.7) act as
L̄Ak = L̄
k = (µk−µ̄k)
=: L̄k and L̄
k = (µk−µ̄k)
. (4.10)
By induction in k = 1, . . . ,m we learn that, due to (4.5), a necessary and sufficient condition for a
solution of (4.6) is
L̄kSk = SkZ̃k and L̄
kSk = SkZ̃
k (4.11)
with some rk×rk matrices Z̃k and Z̃ik depending on (wk, w̄k, η
Passing to the noncommutative bosonic coordinates we obtain
ŵk , ˆ̄wk
= 2θ νkν̄k with νkν̄k =
µk−µ̄k−µ−1k +µ̄
. (4.12)
Thus, we can introduce annihilation and creation operators
and c
so that [ck , c
] = 1 (4.13)
for k = 1, . . . ,m. Naturally, this Heisenberg algebra is realized on a “co-moving” Fock space Hk,
with basis states |ℓ〉k and a “co-moving” vacuum |0〉k subject to ck|0〉k = 0. Each co-moving
vacuum |0〉k (annihilated by ck) is related to the static vacuum |0〉 (annihilated by a) through an
ISU(1,1) squeezing transformation (cf. [12]) which is time-dependent. The fermionic coordinates
ηik and η̄
k remain spectators in the deformation. Coordinate derivatives are represented in the
standard fashion as
7→ −[c†
, · ] and ν̄k
7→ [ck , · ] . (4.14)
After the Moyal deformation, the n×rk matrices Sk have become operator-valued, but are
still functions of the Grassmann coordinates ηi
and η̄i
. The noncommutative version of the BPS
conditions (4.11) naturally reads
ck Sk = Sk Zk and
Sk = Sk Z
k (4.15)
where Zk and Z
k are some operator-valued rk×rk matrix functions of η
and η̄
Nonabelian solutions for Sk. For general data Zk and Z
it is difficult to solve (4.15), but it
is also unnecessary because the final expression ψm turns out not to depend on them. Therefore,
we conveniently choose
Zk = ck ⊗ 1lrk×rk and Zik = 0 ⇒ Sk = Rk(ck, ηik) , (4.16)
where Rk is an arbitrary n×rk matrix function independent of c†k and η̄ik.10 It is known that
nonabelian (multi-) solitons arise for algebraic functions Rk (cf. e.g. [7] for the commutative and [12]
for the noncommutative N=0 case). Their common feature is a smooth commutative limit. The
only novelty of the supersymmetric extension is the ηi
dependence, i.e.
Rk = Rk,0 + η
kRk,i + η
Rk,ij + η
Rk,ijp + η
Rk,ijpq . (4.17)
Abelian solutions for Sk. It is useful to view Sk as a map from C
rk⊗Hk to Cn⊗Hk (momentarily
suppressing the η dependence). The noncommutative setup now allows us to generalize the domain
of this map to any subspace of Cn ⊗Hk. In particular, we may choose it to be finite-dimensional,
say Cqk , and represent the map by an n×qk array |Sk〉 of kets in H. In this situation, Zk and Zik
in (4.15) are just number -valued qk×qk matrix functions of ηjk and η̄
. In case they do not depend
on η̄
, we can write down the most general solution as
|Sk〉 = Rk(ck, ηjk) |Zk〉 exp
) η̄ik
with |Zk〉 := exp
|0〉k . (4.18)
10Changing Zk or Z
k multiplies Rk by an invertible factor from the right, which drops out later, except for the
degenerate case Zk=0 which yields Sk = Rk |0〉k〈0|k.
As before, we may put Zi
= 0 without loss of generality, but now the choice of Zk does matter.
For any given k generically there exists a qk-dimensional basis change which diagonalizes the
ket-valued matrix
|Zk〉 7→ diag
c† , eα
c†, . . . , eα
|0〉k = diag
|α1k〉 , |α2k〉 , . . . , |α
, (4.19)
where we defined coherent states
|αlk〉 := eα
c† |0〉k so that ck |αlk〉 = αlk |αlk〉 for l = 1, . . . , qk and αlk ∈ C . (4.20)
Note that not only the entries of Rk but also the α
k are holomorphic functions of the co-moving
Grassmann parameters η
and thus can be expanded like in (4.17). In the U(1) model, we must
use ket-valued 1×qk matrices |Sk〉 for all k, yielding rows
|Sk〉 =
R1k |α1k〉 , R2k |α2k〉 , . . . , R
for k = 1, . . . ,m , (4.21)
with functions αl
). Here, the Rl
only affect the states’ normalization and can be collected in a
diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally,
we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension
has generalized |Sk〉 → |Sk(ηjk)〉.
Explicit form of Pk. Let us now consider the multiplicative parametrization (4.3) of ψm which
also allows us to solve (4.5). First of all, note that the reality condition (3.15) is satisfied if
Pk = P
= P 2k ⇔ Pk = Tk (T
−1T †
for k = 1, . . . ,m , (4.22)
meaning that Pk is an operator-valued hermitian projector (of group-space rank rk ≤ n) built from
an n×rk matrix function Tk (the abelian case of n=1 is included). The reality condition follows
just because
µk − µ̄k
ζ − µk
µ̄k − µk
ζ − µ̄k
= 1l for any ζ and k = 1, . . . ,m . (4.23)
The rk columns of Tk span the image of Pk and obey
Pk Tk = Tk ⇔ (1l−Pk)Tk = 0 . (4.24)
Furthermore, the equation (4.5) with m = k (induction) rewritten in the form
(1l−Pk)
µk−ℓ − µ̄k−ℓ
µ̄k − µk−ℓ
Sk = 0 (4.25)
reveals that (cf. (4.24))
T1 = S1 and Tk =
1l − µk−ℓ − µ̄k−ℓ
µk−ℓ − µ̄k
Sk for k ≥ 2 , (4.26)
where the explicit form of Sk for k = 1, . . . ,m is given in (4.16) or (4.18). The final result reads
µm−ℓ − µ̄m−ℓ
ζ − µm−ℓ
= 1l +
ζ − µk
(4.27)
with hermitian projectors Pk given by (4.22), Tk given by (4.26) and Sk given by (4.16) or (4.18).
The explicit form of Λmk (which we do not need) can be found in [12]. The corresponding superfields
Φ and Υ are
Φm = ψ
m|ζ=0 =
(1l− ρkPk) with ρk = 1−
, (4.28a)
Υm = lim
ζ (ψm − 1l) =
(µk−µ̄k)Pk . (4.28b)
From (4.22) it is obvious that Pk is invariant under a similarity transformation
Tk 7→ Tk Λk ⇔ Sk 7→ Sk Λk (4.29)
for an invertible operator-valued rk×rk matrix Λk. This justifies putting Zik = 0 from the beginning
and also the restriction to Zk = ck ⊗1lrk×rk in the nonabelian case, both without loss of generality.
Hence, the nonabelian solution space constructed here is parametrized by the set {Rk}m1 of matrix-
valued functions of ck and η
k and the pole positions µk. The abelian moduli space, however, is
larger by the set {Zk}m1 of matrix-values functions of ηik which generically contain the coherent-
state parameter functions {αl
)}. Restricting to ηi
=0 reproduces the soliton configurations of
the bosonic model [12].
Static solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall
that static solutions correspond to the choice m = 1 and µ1 ≡ µ = −i implying w1 = z, so we drop
the index k. Specializing (4.27), we have
ψ = 1l − 2 i
ζ + i
P so that Φ = Φ† = 1l− 2P , (4.30)
where a hermitian projector P of group-space rank r satisfies the BPS equations
(1l−P ) aP = 0 ⇒ (1l−P ) aT = 0 , (4.31a)
(1l−P ) ∂
P = 0 ⇒ (1l−P ) ∂
T = 0 , (4.31b)
with P = T (T †T )−1T † and ηi = η1i + iη
i . In this case T = S, and for a nonabelian r=1 projector
P we get T = T (a, ηi) as an n×1 column. For the simplest case of N=1 we just have (cf. [59])
T = Te(a) + η To(a) with η = η
1 + iη2 , (4.32)
where Te(a) and To(a) are rational functions of a (e.g. polynomials) taking values in the even and
odd parts of the Grassmann algebra. Similarly, an abelian N=1 projector (for n=1) is built from
|T 〉 =
|α1〉 , |α2〉 , . . . , |αq〉
. (4.33)
At θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution
of the standard N=1 supersymmetric CPn−1 sigma model in two dimensions (see e.g. [59]).11 For
11In fact, Φ in (4.30) takes values in the Grassmannian Gr(r, n), and Gr(1, n) = CPn−1.
this reason, one can overcome the previously mentioned difficulty with constructing an action (or
energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions
obeying the BPS conditions (4.31) the topological charge
Q = 2πθ
dη1dη2 Tr tr Φ
D+Φ ,D−Φ
(4.34)
is proportional to the action (BPS bound)
S = 2πθ
dη1dη2 Tr tr
D+Φ ,D−Φ
(4.35)
and is finite for algebraic functions Te and To. Here, the standard superderivatives D± are defined
+ iη ∂z and D− =
+ iη̄ ∂z̄ . (4.36)
One-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain
ψ1 = 1l +
µ− µ̄
ζ − µ P with P = T (T
†T )−1T † (4.37)
Φ = 1l − ρP with ρ = 1− µ
. (4.38)
Now our n×r matrix T must satisfy (putting Zi = 0 and Z = c⊗ 1lr×r)
[c , T ] = 0 and
T = 0 with ηi = η1i + µ̄ η
i , (4.39)
where c is the moving-frame annihilation operator given by (4.13) for k=1.
Recall that the operators c and c† and therefore the matrix T and the projector P can be
expressed in terms of the corresponding static objects by a unitary squeezing transformation (see
e.g. (4.8) and (4.13)). For simplicity we again consider the case N=1 and a nonabelian projector
with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.
T = Te(c) + η To(c) =
T 1e (c) + η T
o (c)...
Tne (c) + η T
o (c)
(4.40)
with polynomials T ae and T
o of order q, say, analogously to the static case (4.32). Note that,
for T ao to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear.
Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing
transformation.
For N=1 the moving frame was defined in (4.8) (dropping the index k) via
w = x + 1
(µ̄−µ̄−1)y + 1
(µ̄+µ̄−1)t and η = η1 + µ̄η2 hence ∂tη = 0 . (4.41)
Consider the moving frame with the coordinates (w, w̄, s; η, η̄) with the choice s = t and the related
change of the derivatives (see [12, 58])
∂x = ∂w + ∂w̄ , (4.42a)
(µ̄−µ̄−1) ∂w + 12(µ−µ
−1) ∂w̄ , (4.42b)
(µ̄+µ̄−1) ∂w +
(µ+µ−1) ∂w̄ + ∂s , (4.42c)
∂η1 = ∂η + ∂η̄ , (4.42d)
∂η2 = µ̄ ∂η + µ∂η̄ . (4.42e)
In the moving frame our solution (4.38) is static, i.e. ∂sΦ = 0, and the projector P has the same
form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore,
by computing the action (4.35) in (w, w̄; η1, η2) coordinates, we obtain for algebraic functions T
in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending
on µ (cf. [12] for the bosonic case).
Large-time asymptotics. Note that in the distinguished (z, z̄, t) coordinate frame (4.41) implies
that at large times w→ κ t with κ = 1
(µ̄+µ̄−1). As a consequence, the tq term in each polynomial
in (4.40) will dominate, i.e.
T → tq
a1 + η b1
an + η bn
=: tq Γ , (4.43)
where Γ is a fixed vector in Cn. It is easy to see that in the distinguished frame the large-time
limit of Φ given by (4.38) is
Φ = 1l − ρΠ with Π = Γ (Γ†Γ)−1Γ† (4.44)
being the projector on the constant vector Γ.
Consider now them-soliton configuration (4.28). By induction of the above argument one easily
arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump
we have
Φm = (1l− ρ1Π1) . . . (1l− ρℓ−1Πℓ−1)(1l− ρℓPℓ)(1l− ρℓ+1Πℓ+1) . . . (1l− ρmΠm) , (4.45)
where the Πm are constant projectors. This large-time factorization of multi-soliton solutions
provides a proof of the no-scattering property because the asymptotic configurations are identical
for large negative and large positive times.
5 Conclusions
In this paper we introduced a generalization of the modified integrable U(n) chiral model with
2N≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model.
It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of
an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory.
The dressing method was applied to generate a wide class of multi-soliton configurations, which
are time-dependent finite-energy solutions to the equations of motion. Compared to the N=0
model, the supersymmetric extension was seen to promote the configurations’ building blocks to
holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic
factorization into a product of single soliton solutions we have shown that no scattering occurs
within the dressing ansatz chosen here.
The considered model does not stand alone but is motivated by twistor string theory [37] with
a target space reduced to the mini-supertwistor space [44, 45, 47]. In this context, the obtained
multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60].
Here 2N appears due to fermionic worldvolume directions of our branes in the superspace de-
scription [60]. Switching on a constant B-field simply deforms the sigma model and D-brane
worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative
abelian solutions.
Restricting to static configurations, the models can be specialized to Grassmannian supersym-
metric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are
invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results
for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler
for admitting two or four supersymmetries, respectively. This difference will be discussed in more
details elsewhere.
We derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction
and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall
that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions
can be obtained from the SDYM equations (or their hierarchy [25]) by suitable dimensional reduc-
tions (see e.g. [61]–[65] and references therein). Moreover, this Ward conjecture [61] was extended
to the noncommutative case (see e.g. [66, 67]). It will be interesting to consider similar reductions
of the N -extended supersymmetric SDYM equations (and their hierarchy [68]) to supersymmetric
integrable equations in three and two dimensions generalizing earlier results [69].
Acknowledgements
We acknowledge fruitful discussions with C. Gutschwager. This work was supported in part by the
Deutsche Forschungsgemeinschaft (DFG).
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