# Pullback of the Volume Form, Integrable Models in Higher Dimensions and Exotic Textures

###### Abstract

A procedure allowing for the construction of Lorentz invariant
integrable models living in dimensional space-time and with
an dimensional target space is provided. Here,
integrability is understood as the existence of the generalized
zero-curvature formulation and infinitely many conserved
quantities. A close relation between the Lagrange density of the
integrable models and the pullback of the pertinent volume form on
target space is established. Moreover, we show that the conserved
currents are Noether currents generated by the volume preserving
diffeomorphisms. Further, we show how such models may emerge via abelian
projection of some gauge theories.

Then we apply this framework to the construction of integrable models with
exotic textures. Particularly, we consider integrable models
providing exact suspended Hopf maps i.e., solitons with a nontrivial
topological charge of .

Finally, some families of integrable models with solitons of
type are constructed. Infinitely many exact solutions
with arbitrary value of the topological index are found. In addition, we
demonstrate that they
saturate a Bogomolny bound.

###### keywords:

Zero curvature, classical integrability, higher dimensions, topological solitons, higher rank tensors###### Pacs:

11.27.+d, 11.10.Lm^{†}

^{†}journal: Nuclear Physics B\@xsect

Integrability has proven a valuable concept for the analysis and
solution of nonlinear field theories in 1+1 dimensions, but its generalization
to higher dimensions is a rather difficult endeavour, and a generally accepted
concept of higher-dimensional integrability does not yet exist.
One possible way to generalize integrability to higher dimensions was
proposed in [1], where the zero curvature of Zakharov and Shabat
has been generalized to higher dimensions. Further, it was demonstrated in the
same paper that some known higher-dimensional nonlinear field theories
possess the generalized zero curvature representation and,
at the same time, infinitely many
conservation laws.

It is the main purpose of the present
paper to further develop this investigation.
We shall explicitly construct different families of higher-dimensional
field theories which possess the generalized zero curvature representation.
Further, we will find their infinitely many conservation laws as well as their
exact soliton solutions. A key ingredient will be
that their Lagrangians are related
to the volume forms on their respective target spaces, and the conserved
currents are, in turn, related to the volume preserving diffeomorphisms.
Before introducing our investigation,
it will be useful to review some known results in
order to facilitate some background for the constructions that follow.

Shortly after the general proposal of [1],
some new nonlinear field
theories in 3+1 dimensions with
two dimensional target space and possessing the generalized zero curvature
representation were constructed explicitly.
One first model was
introduced by Aratyn, Ferreira and Zimerman
(AFZ), and they explicitly constructed both infinitely many conservation laws
and infinitely many soliton solutions [2],
[3]. Due to the two-dimensional target space, their solitons are,
in fact, topological and are classified by the Hopf index.
Some integrable generalizations of the model of AFZ were discussed in
[4], where again infinitely many conservation laws and infinitely
many topological (Hopf) solitons were found.
Another model giving rise to Hopf solitons had been originally proposed by
Nicole, who found its simplest Hopf soliton with Hopf index one [5].
This Nicole model again possesses the generalized curvature representation
[6], but it only gives rise to finitely many conservation laws.
Only a submodel of the Nicole model, defined by further first order
equations (“integrability conditions”) in addition to the Euler–Lagrange
equations, possesses infinitely many conservation laws. Further,
for the Nicole model only the simplest soliton may be found in an analytic
form. Higher solitons have to be calculated numerically [7].
Some generalization of the Nicole model have been discussed in
[8], [9],
where again only one analytical soliton solution could be found
in each model. Both the AFZ model and the Nicole model (and their
generalizations) allow for static finite energy solutions because their
kinetic term is chosen non-polynomial in order to have a scale invariant
energy and avoid Derrick’s theorem. This idea of non-polynomial Lagrangians
is originally due to [10], where it was applied to a phenomenological
model of pions.
A more geometric understanding of the conservation laws in the models
mentioned above was developed in [11], [12],
[6], [13],
where it was shown that the conserved currents are
just the Noether currents of the area-preserving diffeomorphisms on
target space. The off-shell divergence of these currents is proportional
to the Euler–Lagrange equations (for AFZ type models) or to a linear
combination of Euler–Lagrange equations and integrability conditions
(for Nicole type models), respectively.
In a parallel development, the generalized zero curvature representation,
integrability and conservation laws of chiral and non-linear sigma models
were investigated, e.g., in [14], [15], [16],
[17].

Soon after this, the investigation was extended to field theories
with a three dimensional target space, the best-known of which is
the Skyrme model [18], [19]. The Skyrme model
again possesses the generalized zero curvature representation, but
only a finite number of conserved currents. But, again, there
exists a submodel of the Skyrme model which has infinitely many
conservation laws [20]. A detailed classification of
the integrability of field theories with three dimensional target
spaces has been performed in [21], [22]. The
abelian projection of SU(2) Yang–Mills dilaton theory, which
effectively has a three dimensional target space, was studied in
[23]. Integrable theories with higher dimensional
target spaces were investigated, e.g., in [15],
[22]. Further, the concept of generalized curvature
representations and integrability was applied to non-linear sigma
models on noncommutative space-time in [24].

After this brief review we will give the outline of the present paper, which
combines and generalizes the ingredients described above: both an algebraic
and a geometrical formulation of the generalized integrability, as well as
the analysis of the geometry and topology of the target space and its
interrelation with the symmetries and conservation laws of the field theories
under investigation.

Our paper is organized as follows. First we briefly recall the idea
of the generalized integrability and discuss it in the case of
target space, concretely for the AFZ model. We find the geometric condition
which makes this model integrable. In section 3 we extend this
geometric approach to models with more complicated target space
manifolds. We show how one can construct integrable models based on the
volume form on target space. We find a
family of infinitely many conserved quantities and explain their
existence by relating them with the pertinent symmetries of the
target space. Section 4 is devoted to the generalization to even higher
dimensional target spaces.
In Section 5 we explain how the models presented in the previous sections may
be related to gauge theories with the help of the abelian projection.
In section 6 we use the methods of Sections 3, 4 to
construct models with exact topological solitons with nontrivial
values of some exotic topological charges like, i.e., or
.
In Section 7 theories with the more conventional textures are
investigated. We construct some families of infinitely many finite
energy solutions and further find that they obey a Bogomolny equation.
Finally, we present our conclusions in Section 8.

The most natural geometrical object in the generalized zero curvature representation is a connection on higher loop space, and the condition of zero curvature for this connection will, in general, not lead to local equations in ordinary space time. There exist, however, sufficient local conditions which ensure the vanishing of the pertinent higher dimensional curvature and, therefore, realize the generalized zero curvature formulation in a local manner. One such sufficient condition is constructed as follows. The starting point is the specification of a Lie algebra and an Abelian ideal together with a connection and a rank antisymmetric tensor . The corresponding curvature vanishes if we assume that the connection is flat and the Hogde dual to is covariantly constant with respect to the connection i.e.,

(2.1) |

We say that a model possesses the generalized zero curvature representation if its equations of motion may be re-expressed in this form. Further, one can notice that for a given field we are able to construct conserved currents which are equal in number to the dimension of the Abelian ideal we used in the construction. Therefore, we say that a model is integrable (within this generalized approach) if the corresponding Abelian ideal has infinite dimensions.

Let us investigate how this general approach works in the case of models with two dimensional target space. Here we identify the target space of the nonlinear model with a two-dimensional manifold . Instead of real coordinates we introduce the complex coordinates . According to the general prescription we fix the Lie algebra and the Abelian ideal. Namely, is the Lie algebra of the Lie group restricted to whereas is the representation space of it with arbitrary angular momentum number and magnetic number restricted to , that is . Then, in the triplet representation

(2.2) |

(2.3) |

where is so far an arbitrary vector depending on the fields as well as their derivatives, is an element of given by

(2.4) |

and . Further, and constitute the basis of the Lie algebra and the Abelian ideal, respectively. are Pauli matrices. The commutators are , , , . The connection is flat by construction. Thus, the only nontrivial condition in the GZC formulation is the covariant constancy of the field. In the triplet representation this results in

(2.5) |

However, in a higher spin representation one gets, in addition to (2.5), the constraint

(2.6) |

So, we can conclude that a dynamical model with two dimensional target space is integrable if one may define a vector quantity such that and the pertinent equations of motion read

(2.7) |

Models with these properties are known as models of the Aratyn-Ferriera-Zimmerman type [3]. They are integrable in the GZC formulation sense: they have the GZC formulation with the infinite-dimensional Abelian ideal. They are given by the following Lagrange density

(2.8) |

where

(2.9) |

is any function of whereas is a positive real parameter. A particular example of such integrable models in four dimensional Minkowski space-time is given by the expression

(2.10) |

where the value of the power is taken to avoid the Derrick arguments for the non-existence of static solitons [2]. The AFZ model describes, in fact, soliton excitations of a three component unit vector field which may be related via the standard stereographic projection with the complex field . As the static solutions are maps from compactified to the target space they carry the corresponding topological charge, i.e., the Hopf index . The lump like structure of the solitons emerges from the fact that the pre-image of a given point on the target sphere is a closed line. For this model such topologically nontrivial solitons (hopfions) have been derived in an exact form [3]. Moreover, one can also construct infinitely many conserved currents

(2.11) |

where

(2.12) |

Further, is an arbitrary function of and , and , etc. In order to understand the geometrical meaning of this model, which may give us a clue how to generalize it to field theories with a more complicated target space, we consider the area two form on the target space manifold

(2.13) |

where is the area density. The pullback of the area two-form in the base Minkowski space-time is

(2.14) |

Now we are able to define two objects. Namely, a rank two anti-symmetric tensor

(2.15) |

and a scalar density which are
exactly the same objects as used in the construction of the AFZ
model. Observe that the density equals the square of the
pullback of the area two-form modulo a multiplicative term depending
on the area density . In other words, the fact which makes the
AFZ model integrable is that it is proportional to a function of
the square of the pullback of the area two-form to the base Minkowski
space-time.

Therefore, one can conjecture that integrable models with higher
dimensional target spaces can be constructed using the square of the
pullback of the pertinent volume form on the target space manifold
into the base Minkowski space-time. In the proceeding sections
we demonstrate this hypothesis by explicit construction.

Following the considerations of the previous section, our starting point for the construction of integrable models with three dimensional target space is to consider the volume three-form on

(3.1) |

where together with a scalar are local coordinates on and is the volume density. Then the pullback of the volume three-form to the base Minkowski space-time is

(3.2) |

Again, we may extract from the last formula a rank three anti-symmetric tensor

(3.3) |

and the corresponding scalar

(3.4) |

The last object is proportional to the square of the pullback of
the volume three-form up to a term which does not contain any
derivatives of the fields. ^{1}^{1}1Such a rank three tensor in a
slightly different parametrization has been previously analyzed in
the context of the so-called generalization of the Goldstone model
in (3+1) dimensions, which solutions are ungauged Higgs analogues
of the Skyrme model solitons [25].

Then, the class of integrable models with tree dimensional target
space is defined as follows

(3.5) |

with a positive parameter (where may, e.g., be chosen to guarantee the
invariance of the model under the scale transformation or to
provide finite energy solutions.)

In order to write the corresponding equations of motion let us
define two vector quantities closely related to the canonical
momenta

(3.6) |

and

(3.7) |

It is easy to verify that they obey the following relations

(3.8) |

(3.9) |

and

(3.10) |

(3.11) |

Therefore, the equations of motion

(3.12) |

(3.13) |

may be rewritten in the following simple form

(3.14) |

(3.15) |

The prime denotes differentiation with respect to .

To express the system of equations (3.14), (3.15) in terms of the generalized zero curvature condition, we specify to be the Lie algebra of the Lie group (restricted to the equator) while is the representation space of it with arbitrary integer angular momentum quantum number , but magnetic quantum number restricted to . As one can see, there is only one change in comparison with the integrable models with two dimensional target space. The Abelian ideal is extended to those representations which carry also zero magnetic number. Then, in spin representation, the flat connection and the Hodge dual field are

(3.16) |

and

(3.17) |

Similar fields are also used in the GZC formulation of the Skyrme model, see [20]. The covariant constancy of the Hodge dual field gives

(3.18) |

Here we used the following important properties obeyed by the objects and . Namely,

(3.19) |

Moreover, if we notice that

(3.20) |

then we arrive at the field equations (3.14),
(3.15). Therefore, we conclude that these models are
integrable. The Abelian ideal we used in the generalized zero
curvature is indeed infinite dimensional.

Observe that the connection belongs to the Lie algebra of
the Lie group restricted to the coset space ,
as is the case for models with two-dimensional target space.
Moreover, the dual field is defined up to an
arbitrary function of and which multiplies
.

Finally, we prove that this family of models possess infinitely many
conserved quantities as is required for the integrable systems.
After some calculations one can verify that there are three families
of infinitely many on-shell conserved currents

(3.21) |

(3.22) |

(3.23) |

Here

(3.24) |

Moreover, there is a good understanding of the geometrical origin of the currents. We show that the conservation laws found for the integrable models are generated by a class of geometric target space transformations. Specifically, they are the Noether currents related to the volume-preserving diffeomorphisms. Let us again consider a three-dimensional target space manifold , parameterized by local coordinates . Then the volume 3-form is given by the expression

(3.25) |

A volume-preserving diffeomorphism is a a coordinate transformation leaving the volume form invariant. For an arbitrary infinitesimal transformation

(3.26) |

invariance of the volume form results in the condition on functions

(3.27) |

As the considered manifold is three dimensional we apply Darboux’s theorem and derive a general (local) solution

(3.28) |

where are arbitrary functions of the local coordinates. Following that we may write a general vector field generated by a volume-preserving diffeomorphisms

(3.29) |

where we assumed the parametrization of the target manifold by a complex field and a real scalar introduced before. These vector fields obey the Lie algebra

(3.30) |

(3.31) |

In a relativistic field theory one can find a general expression for Noether currents corresponding to the vector fields . Namely,

(3.32) |

where are the standard canonical momenta. For the integrable models discussed in the previous section we get (up to an unimportant multiplicative constant)

(3.33) |

The generalization to integrable field theories with a target space of arbitrary dimension is straightforward. As we described in the case of models with two or three dimensional target spaces, one should begin with the pertinent volume form on the target space manifold i.e.,

(4.1) |

for an even number of dimensions or

(4.2) |

for an odd number of dimensions . The local coordinates on are or , respectively for those cases. are complex fields while is a real scalar. Therefore, after performing the pullback into the base Minkowski space-time we can define a rank antisymmetric tensor as

(4.3) |

depending on the dimension of . Here stands for antisymmetrization. In addition we need a scalar quantity built out of this tensor

(4.4) |

The integrable Lagrangian reads

(4.5) |

where is a real function factor depending on the fields i.e., coordinates on the n-dimensional manifold . Then the equations of motion read

(4.6) |

and their complex conjugate together with

(4.7) |

if the dimension of the target space is odd. Here

(4.8) |

are the canonical momenta obeying the following constrains identically

(4.9) |

(4.10) |

(4.11) |

(4.12) |

The families of infinitely many conserved quantities are given by the formulas

(4.13) |

(4.14) |

(4.15) |

where all functions are arbitrary functions of all scalar fields and there is no summation over the indices. Thus, in the case of even dimensional target space we have independent families of infinitely many conserved currents. For odd dimensions this number is .

Remark:

When the dimension of the target space equals the
number of the spatial dimension of the base space , then the
static equations of motion are trivial. In fact, then the quantity
, being the pullback of
the volume form on a target space, is a closed -form also in
the base space. As a consequence, its integral over a
-dimensional manifold only depends on the boundary conditions,
that is, it gives a purely topological action (or energy
functional). Therefore, these integrable models do not seem to lead to
interesting results in the context of solitons. (In
Section 7 we show how one can circumvent this obstacle and
construct actions with solitons of type. The main
idea is to include several pullback tensors in the Lagrangian.) If
, then the models are not longer trivial and may provide
exact soliton solutions with topological charges from the
homotopy group ,
as has been observed in the original AFZ model.

It has been established that two dimensional target space
integrable models of the AFZ type may emerge via the Abelian projection
of the Yang-Mills field. It results in the
observation that the Abelian projection of YM is
an integrable sector of the full theory with magnetic monopoles as
exact solutions [23]. In the case of integrable
models with higher dimensional target space the situation is
analogous. There exists a gauge theory which, after reduction of
degrees of freedom, leads to the corresponding integrable pullback model.

As a first example we consider the following model

(5.1) |

where the gauge fields from the Lie algebra is non-minimally coupled to the scalar field . The field strength tensor is defined in the standard manner . The next step is to use the Cho-Faddeev-Niemi-Shabanov decomposition [31]-[36] and express the gauge fields by means of a new set of degrees of freedom

(5.2) |

where we introduced a three component unit vector field pointing into the color direction, an Abelian gauge potential and a color vector field which is perpendicular to . Now, we restrict the gauge potential to the form

(5.3) |

Then the field strength tensor reads

(5.4) |

Finally, taking into account the stereographic projection

(5.5) |

we arrive at the Lagrange density

(5.6) |

As we claimed, it has exactly the form of the integrable pullback
model with three-dimensional target space (3.5), with
and .

In the case of the four-dimensional integrable models the related
gauge theory reads

(5.7) |

where we have introduced the second, independent gauge potential . Then, and the corresponding abelian projection is performed assuming

(5.8) |

where is another unit, three component vector field. Of course, one may continue this procedure and try to find gauge theories related to higher dimensional integrable pullback models. However, we would like to have a constructive method for generating such models, instead of this guessing-like procedure. Fortunately, such a method exists and is based on the observation that the gauge models (5.1) and (5.7) can be written in a more compact form. Namely, the Lagrange density (5.1) is given by

(5.9) |

where

(5.10) |

In the same way, the model (5.7) may be expressed as

(5.11) |

where

(5.12) |

is the number of the spatial dimensions. Now, we are able to define gauge models which can be reduced via the Abelian projection to the integrable pullback models. Equivalently, one can say that these non-integrable gauge theories possess an integrable sector given by the pertinent integrable pullback model. The specific Lagrange density reads

(5.13) |

where is an even dimension of the target space while is the number of the spatial dimensions. Here

(5.14) |

If the target space has an odd dimension , then we find

(5.15) |

and

(5.16) |

Here each is the field strength
tensor defined by an independent gauge field .

To conclude, we have found theories which give a gauge covering of the integrable
pullback models. The question whether the pullback models may be
immersed in gauge theories of a different type is still an open
problem, which definitely requires further studies.

Remark:

One might conjecture that the rank three tensor
of Section 3 is related by means of some projection
(reduction of degrees of freedom) to a rank three field strength
tensor of a non-Abelian rank two
gauge field in the same way that the rank two tensor
is related to the abelian projection of an SU(2) field
strength tensor.
This would then imply
that there existed an integrable subsector of the corresponding
higher rank nonabelian gauge theory defined by such a projection.
Unfortunately, the issue of nonabelian higher rank gauge theories
is a difficult one, and a satisfactory definition of these theories
has not yet been found. On the other hand, abelian higher rank gauge theories
cf. Kalb–Ramond theories [37] do exist and have already
demonstrated their relevance in numerous applications. Here we just want to
mention that in Minkowski space-time,
our tensor obeys the analog of
the Bianchi identity

as is required for the Abelian Kalb–Ramond tensor [38]

So there may exist embeddings of our pullback model into Kalb–Ramond theories, although perhaps not in such a natural way like the abelian projection in the case of a nonabelian gauge theory.

Experience from integrable models in
dimensions tells us that, in addition to the existence of infinitely
many conservation laws and the zero curvature representation, such
theories allow for exact (soliton) solutions. In this section we
prove that the integrable models described before possess exact
topologically nontrivial solutions. In fact, as our construction is
valid for target and base spaces of any dimension, we are able
to find Lorentz invariant dynamical systems which provide exact
textures carrying rather exotic topological charges.
Higher-dimensional textures with unusual topological charges have already been
investigate, although not from the integrability point of view. Textures for
higher Hopf maps have, e.g., been studied in [39].
A typical field of applications of these higher-dimensional textures is
cosmology, therefore there exist many studies of textures with gravitational
backreaction (i.e., self-gravitating textures). Some examples of the latter are
gravitating 5-dim solitons
[40], gravitating solitons [41], gravitating
monopoles in higher extra dim [42], [43], or a
gravitating 6-dim Abelian Higgs vortex [44].

As we mentioned before, the AFZ model possesses soliton solutions with nontrivial values of the Hopf index . However, one can associated with these Hopf maps a suspended map by mapping the equator of onto the equator of using the Hopf map and then continuing smoothly to the poles. It is known that those new suspended maps may be classified by a topological invariant as the pertinent homotopy group is nonzero, . Examples of the nontrivial representative class are the suspended maps

(6.1) |

where are
coordinates on [45], [46], [47], and gives the extension to . The radius of the base sphere
is . In this subsection we propose a field theoretical model
for which such topologically nontrivial solitons may be found in an
exact form. Let us notice that topological defects of this kind are
relevant for Yang-Mills theory in its euclideanized
dimensional version.

The particular form of the Lagrangian density is

(6.2) |

The value of the parameter has been chosen to render the energies of the solutions finite. For the base space such Lagrangians are admissible. This is due to the fact that the radius of the sphere fixes the scale in the model. Different values of the radius correspond to different theories. On the other hand in base space such Lagrange densities give non-scale invariant static energies and soliton solutions would be unstable according to the Derick argument. From now on we neglect in our calculations as one may always recover it using the dimensional analysis. Moreover, we assume the following Ansatz for static solutions

(6.3) |

Obviously, the complex field is just a Hopf map with