Nonlinear Grassmann Sigma Models in Any Dimension and An Infinite Number of Conserved Currents

Abstract

We first consider nonlinear Grassmann sigma models in any dimension 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 any dimension. Comment: 11 pages, AMSLaTex

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arXiv:hep-th/9806084v1 11 Jun 1998
Nonlinear Grassmann Sigma Models in Any
Dimension
and
An Infinite Number of Conserved Currents
Kazuyuki FUJII,∗Yasushi HOMMA†and Tatsuo SUZUKI‡
Abstract
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
any dimension.
0Introduction
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
[1], Mickelsson [2] and their references.
∗Department of Mathematics, Yokohama City University, Yokohama 236, Japan,
E-mail address: fujii@yokohama-cu.ac.jp
†Department of Mathematics, Waseda University, Tokyo 169, Japan,
E-mail address: 696m5121@mn.waseda.ac.jp
‡Department of Mathematics, Waseda University, Tokyo 169, Japan,
E-mail address: 695m5050@mn.waseda.ac.jp
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But in the dimensions greater than two, we have in general no outstanding
results in spite of much efforts. See, for example, [3],[4],[5],[6].
Recently Alvarez, Ferreira and Guillen in [7] 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)-
dimensions
(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 [8],[9] and Gianzo, Madsen and Guillen [10].
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
a model
∂µ∂µu = 0and∂µu∂µu = 0
foru : M1+m→ C
in any dimension (m ∈ N). This means a kind of universality of the sub-
model.
After thoroughgoing analysis of the paper [7], we found that their method
developed there was, more or less, irrelevant to construct submodels and
conserved currents. We of course admit that [7] 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
currents.
Our results is a full generalization of [8],[9] and [10]. Our method com-
pared to that of [7] is very simple and easy to understand.
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1Mathematical Preliminaries ··· Grassmann
Manifolds
Let M(m,n;C) be the set of m × n matrices over C and we set M(n;C) ≡
M(n,n;C) for simplicity. For a pair (n,N) with 1 ≤ n < N, we set I, O as
a unit matrix, a zero matrix in M(n;C) and I′, O′as ones in M(N − n;C)
respectively.
We define a Grassmann manifold for the pair (n,N) above as
Gn,N(C) ≡ {P ∈ M(N;C)|P2= P,P†= P,trP = n}.(1.1)
Then it is well-known that
Gn,N(C) =
?
U
?
I
O′
?
U†|U ∈ U(N)
?
(1.2)
∼=
U(N)
U(n) × U(N − n).(1.3)
It is easy to see dimCGn,N(C) = n(N − n) from (1.3). In the case n = 1, we
usually write G1,N(C) = CPN−1and call it the complex projective space. It
is well-known
CPN−1∼=
U(N)
U(1) × U(N − 1)
∼= SN−1
C
/S1.(1.4)
Moreover, in the case N = 2,
G1,2(C) = CP1∼= S2
C/S1∼= S2.(1.5)
Next, let us introduce a local chart for Gn,N(C). For Z ∈ M(N − n;C)
?
O′
a neighborhood of
I
?
in Gn,N(C) is expressed as
P0(Z) =
?
I
Z
−Z†
I′
??
I
O′
? ?
I
Z
−Z†
I′
?−1
.(1.6)
This is also written as
P0(Z) =
?
(I + Z†Z)−1
Z(I + Z†Z)−1
Z†(I′+ ZZ†)−1
(I′+ ZZ†)−1
?
.(1.7)
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We note here relations
Z(I + Z†Z)−1= (I′+ ZZ†)−1Z,(1.8)
(I′+ ZZ†)−1= I′− Z(I + Z†Z)−1Z†.(1.9)
As for these tools, see, for example, [12]. Since any P in Gn,N(C) is writ-
?
O′
neghborhood of P in Gn,N(C) is expressed as
ten as P = U
I
?
U†for some U ∈ U(N) by (1.2), an element of a
P(Z) = UP0(Z)U†. (1.10)
Then,
Lemma 1.1 we have easily
(i) dP0=
?
IZ†
I′
−Z
?−1?
dZ†
dZ
? ?
I
Z
−Z†
I′
?−1
, (1.11)
(ii)[P0,dP0] =
?
IZ†
I′
−Z
?−1?
dZ†
−dZ
??
I
Z
−Z†
I′
?−1
.
(1.12)
2Nonlinear Grassmann Sigma Models and
Submodels
Let M1+mbe a (1+m)-dimensional Minkowski space (m ∈ N). We consider
a nonlinear Grassmann sigma model in any dimension. Let the pair (n,N)
be fixed. The action is
A(P) ≡1
2
?
d1+mx tr∂µP∂µP (2.1)
where
P : M1+m−→ Gn,N(C).
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Its equations of motion read
[P,?P] ≡ [P,∂µ∂µP] = 0.(2.2)
From this
0 = [P,∂µ∂µP] = ∂µ[P,∂µP],(2.3)
so [P,∂µP] are conserved currents (Noether currents).
Next, we look for a local form of our action. By (2.1) and (1.10),(1.11)
we can put
P(Z) = U
?
I
Z
−Z†
I′
??
I
O′
??
I
Z
−Z†
I′
?−1
U†, (2.4)
∂µP(Z) = U
?
IZ†
I′
−Z
?−1?
∂µZ†
∂µZ
??
I
Z
−Z†
I′
?−1
U†,(2.5)
where
Z : M1+m−→ M(N − n,n;C)
and U is a constant unitary matrix. Then,
Lemma 2.1 we have
(i) action
A(Z) =
?
d1+mx tr(I + Z†Z)−1∂µZ†(I′+ ZZ†)−1∂µZ,(2.6)
(ii) the equations of motion
∂µ∂µZ − 2∂µZ(I + Z†Z)−1Z†∂µZ = 0.(2.7)
Let us consider the case n = 1 (CPN−1-model). If we set Z = (u1,··· ,uN−1)t
where uj: M1+m−→ C and remark that
1 + u†u = 1 +
N−1
?
j=1
|uj|2,
(I′+ uu†)−1= I′−
uu†
1 + u†u
from (1.9),
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