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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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