arXiv:hep-th/9806084v1 11 Jun 1998
Nonlinear Grassmann Sigma Models in Any
An Infinite Number of Conserved Currents
Kazuyuki FUJII,∗Yasushi HOMMA†and Tatsuo SUZUKI‡
We first consider nonlinear Grassmann sigma models in any di-
mension and next construct their submodels. For these models we
construct an infinite number of nontrivial conserved currents.
Our result is independent of time-space dimensions and, therfore, is
a full generalization of that of authors (Alvarez, Ferreira and Guillen).
Our result also suggests that our method may be applied to other
nonlinear sigma models such as chiral models, G/H sigma models in
Nonlinear (Grassmann) sigma models in two dimensions are very interesting
objects to study in the not only classical but also quantum point of view and
we have a great many papers on this topics. See, for example, Zakrzewski
, Mickelsson  and their references.
∗Department of Mathematics, Yokohama City University, Yokohama 236, Japan,
E-mail address: email@example.com
†Department of Mathematics, Waseda University, Tokyo 169, Japan,
E-mail address: firstname.lastname@example.org
‡Department of Mathematics, Waseda University, Tokyo 169, Japan,
E-mail address: email@example.com
But in the dimensions greater than two, we have in general no outstanding
results in spite of much efforts. See, for example, ,,,.
Recently Alvarez, Ferreira and Guillen in  proposed a new approach
to higher dimensional “integrable” theories. Instead of higher dimensional
nonlinear sigma models themselves (these ones are of course non integrable),
they considered their submodels to construct “integrable” theories.
In fact, as a simple example, they considered CP1-model in (1 + 2)-
(1 + |u|2)∂µ∂µu − 2¯ u∂µu∂µu = 0
foru : M1+2→ C
and constructed a submodel
∂µ∂µu = 0and∂µu∂µu = 0
and an infinite number of nontrivial conserved currents for this model.
Soon after their results were reinforced and generalized by Fujii and
Suzuki , and Gianzo, Madsen and Guillen .
But if we consider the submodel more deeply, we find that there is no rea-
son to restrict the submodel to three dimensions. Namely, we may consider
∂µ∂µu = 0and∂µu∂µu = 0
foru : M1+m→ C
in any dimension (m ∈ N). This means a kind of universality of the sub-
After thoroughgoing analysis of the paper , we found that their method
developed there was, more or less, irrelevant to construct submodels and
conserved currents. We of course admit that  is important, suggestive and
instructive to nonexperts in this field.
In this letter, we define submodels of nonlinear Grassmann sigma models
in any dimension and construct an infinite number of nontrivial conserved
Our results is a full generalization of , and . Our method com-
pared to that of  is very simple and easy to understand.
 K. Fujii, T. Kashiwa and S. Sakoda: Coherent States over Grassmann
Manifolds and the WKB Exactness in Path Integral, Jour. Math. Phys.,
 K. Fujii, T. Koikawa and R. Sasaki: Classical Solutions for the Super-
symmetric Grassmannian Sigma Models in Two Dimensions I, Prog.
Theor. Phys., 71(1984), 388-394.
 K. Fujii and R. Sasaki:
Grassmannian Sigma Models in Two Dimensions II, Prog. Theor. Phys.,
Classical Solutions for the Supersymmetric