Pullback of the Volume Form, Integrable Models in Higher Dimensions and Exotic Textures
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
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 , 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
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 , 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 , . 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 , 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 . This Nicole model again possesses the generalized curvature representation , 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 . Some generalization of the Nicole model have been discussed in , , 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 , 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 , , , , 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 , , , .
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 , . 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 . A detailed classification of the integrability of field theories with three dimensional target spaces has been performed in , . The abelian projection of SU(2) Yang–Mills dilaton theory, which effectively has a three dimensional target space, was studied in . Integrable theories with higher dimensional target spaces were investigated, e.g., in , . Further, the concept of generalized curvature representations and integrability was applied to non-linear sigma models on noncommutative space-time in .
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.,
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
where is so far an arbitrary vector depending on the fields as well as their derivatives, is an element of given by
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
However, in a higher spin representation one gets, in addition to (2.5), the constraint
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
Models with these properties are known as models of the Aratyn-Ferriera-Zimmerman type . 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
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
where the value of the power is taken to avoid the Derrick arguments for the non-existence of static solitons . 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 . Moreover, one can also construct infinitely many conserved currents
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
where is the area density. The pullback of the area two-form in the base Minkowski space-time is
Now we are able to define two objects. Namely, a rank two anti-symmetric tensor
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
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
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
Again, we may extract from the last formula a rank three anti-symmetric tensor
and the corresponding scalar
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. 111Such 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 .
Then, the class of integrable models with tree dimensional target space is defined as follows
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
It is easy to verify that they obey the following relations
Therefore, the equations of motion
may be rewritten in the following simple form
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
Similar fields are also used in the GZC formulation of the Skyrme model, see . The covariant constancy of the Hodge dual field gives
Here we used the following important properties obeyed by the objects and . Namely,
Moreover, if we notice that
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
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
A volume-preserving diffeomorphism is a a coordinate transformation leaving the volume form invariant. For an arbitrary infinitesimal transformation
invariance of the volume form results in the condition on functions
As the considered manifold is three dimensional we apply Darboux’s theorem and derive a general (local) solution
where are arbitrary functions of the local coordinates. Following that we may write a general vector field generated by a volume-preserving diffeomorphisms
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
In a relativistic field theory one can find a general expression for Noether currents corresponding to the vector fields . Namely,
where are the standard canonical momenta. For the integrable models discussed in the previous section we get (up to an unimportant multiplicative constant)
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.,
for an even number of dimensions or
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
depending on the dimension of . Here stands for antisymmetrization. In addition we need a scalar quantity built out of this tensor
The integrable Lagrangian reads
where is a real function factor depending on the fields i.e., coordinates on the n-dimensional manifold . Then the equations of motion read
and their complex conjugate together with
if the dimension of the target space is odd. Here
are the canonical momenta obeying the following constrains identically
The families of infinitely many conserved quantities are given by the formulas
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 .
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 . 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
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 - and express the gauge fields by means of a new set of degrees of freedom
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
Then the field strength tensor reads
Finally, taking into account the stereographic projection
we arrive at the Lagrange density
As we claimed, it has exactly the form of the integrable pullback
model with three-dimensional target space (3.5), with
In the case of the four-dimensional integrable models the related gauge theory reads
where we have introduced the second, independent gauge potential . Then, and the corresponding abelian projection is performed assuming
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
In the same way, the model (5.7) may be expressed as
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
where is an even dimension of the target space while is the number of the spatial dimensions. Here
If the target space has an odd dimension , then we find
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.
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  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 
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 .
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
, gravitating solitons , gravitating
monopoles in higher extra dim , , or a
gravitating 6-dim Abelian Higgs vortex .
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
coordinates on , , , 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
The particular form of the Lagrangian density is
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
Obviously, the complex field is just a Hopf map with