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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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1 Mathematical 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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Corollary 2.2 we have

(i) action

A(u) =

?

d1+mx(1 + u†u)∂µu†∂µu − ∂µu†uu†∂µu

(1 + u†u)2

,(2.8)

(ii) the equations of motion

(1 + u†u)∂µ∂µu − 2u†∂µu∂µu = 0.(2.9)

These formulas are more familiar with us. Moreover in the case N = 2

(CP1-model),

Corollary 2.3 we have

(i) action

A(u) =

?

d1+mx

∂µ¯ u∂µu

(1 + |u|2)2, (2.10)

(ii) the equations of motion

(1 + |u|2)∂µ∂µu − 2¯ u∂µu∂µu = 0.(2.11)

See [7],[8].

Next, we define a submodel ··· a terminology of [7] ··· of our model. Let

us remind equations of motion of the model

[P,?P] = 0.

Since the tensor product P⊗P of P is also projector, we assume the equations

[P ⊗ P,?(P ⊗ P)] = 0.(2.12)

Transforming this, we have

[P,?P] ⊗ P + P ⊗ [P,?P] + [P,∂µP] ⊗ ∂µP + ∂µP ⊗ [P,∂µP] = 0.

(2.13)

Now, let us define our submodel.

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Definition 2.4 We call a nonlinear Grassmann sigma model based on P

satisfying the simultaneous equations

[P,?P] = 0, (2.14)

[P,∂µP] ⊗ ∂µP + ∂µP ⊗ [P,∂µP] = 0 (2.15)

a submodel of this.

Note that (2.14) and (2.15) are equivalent to (2.2) and (2.12).

Next, we want to express our submodel with Z = (zij) in (2.4).

Proposition 2.5 The equations above are equivalent to

∂µ∂µZ = 0 and∂µZ ⊗ ∂µZ = 0 (2.16)

or in each component

∂µ∂µzij= 0 and∂µzij∂µzkl= 0 (2.17)

for any 1 ≤ i,k ≤ N − n, 1 ≤ j,l ≤ n.

In this case n = 1,N = 2 (CP1-model), we have

∂µ∂µu = 0 and∂µu∂µu = 0 (2.18)

with u in (2.10),(2.11). This is a further generalization of that of [7] because

that is restricted to three dimensions.

3An Infinite Number of Conserved Currents

in Submodels

It is usually not easy to construct conserved currents except for Noether

ones in the nonlinear Grassmann sigma models in any dimension, but in our

submodels we can easily construct an infinite number of conserved currents.

This is a feature typical of our submodels.

Our equations of submodel are

[P,?P] = 0,

[P,∂µP] ⊗ ∂µP + ∂µP ⊗ [P,∂µP] = 0

in the global form (2.14),(2.15). Then,

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Theorem 3.1 for j ≥ 1

˜Bj

µ≡

j−1

?

k=0

P ⊗ ··· ⊗ P

????

k

⊗[P,∂µP] ⊗ P ⊗ ··· ⊗ P

? ???

j−1−k

(3.1)

are conserved currents:

∂µ˜Bj

µ= 0. (3.2)

For example,

˜B3

µ= [P,∂µP] ⊗ P ⊗ P + P ⊗ [P,∂µP] ⊗ P + P ⊗ P ⊗ [P,∂µP],

etc. Each component of˜Bj

infinite number of conserved ones.

In particular, in the case n = 1,N = 2 (CP1-model), let us write down

the first column of˜Bj

µis conserved currents, so we constructed an

µwhich is the essential part. In this case

P =

1

1 + |u|2

?

1

u |u|2

¯ u

?

,(3.3)

[P,∂µP] =

1

(1 + |u|2)2

?

∂µu¯ u − u∂µ¯ u

−(∂µu + u2∂µ¯ u) −(∂µu¯ u − u∂µ¯ u)

∂µ¯ u + ¯ u2∂µu

?

.(3.4)

Corollary 3.2 For j ≥ 1,

˜Bj

µ:k=

1

(1 + |u|2)j+1

?

j(∂µu¯ u − u∂µ¯ u)uk− k(1 + |u|2)uk−1∂µu

?

(3.5)

and its complex conjugate are conserved currents, where 0 ≤ k ≤ j and we

put u−1= 0 for k = 0.

This result recovers and, moreover, is simpler than that of [8],[9] and [10].

In the case n = 1 (CPN−1-model), the details of˜Bj

[11].

µwill be published in

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4A Class of Solutions in Submodels

We cannot in general construct general solutions of nonlinear Grassmann

sigma models in any dimension (see, [13],[14] and [1] in two dimensions), but

can construct a class of solutions in our submodels.

Our equations of submodels are

∂µ∂µZ = 0and∂µZ ⊗ ∂µZ = 0

or in each component

∂µ∂µzij= 0 and∂µzij∂µzkl= 0

in the local form (2.16),(2.17).

Proposition 4.1 Let fij (1 ≤ i ≤ N − n,1 ≤ j ≤ n) be any function in

C2-class. Then

zij≡ fij(α0t +

m

?

k=1

αkxk)(4.1)

under

αµαµ≡ α2

0−

m

?

k=1

α2

k= 0(4.2)

is solutions of our submodels.

Since fij is any for 1 ≤ i ≤ N − n,1 ≤ j ≤ n, we constructed an infinite

number of solutions of our models.

This situation is very similar to that of soliton theory.

5 Discussion

We in this paper discussed the constructions of submodels of nonlinear Grass-

mann sigma models and of an infinite number of conserved currents.

Our result is a full generalization of that of [7] and our method is much

simpler than that.

Our discussion is restricted to Grassmann manifolds (these ones are easy

to treat), but it can be generalized to other nonlinear sigma models whose

target spaces are general symmetric spaces G/H instead of Grassmann ones.

This is now under study.

The detail and further developments of our result are published in [11].

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Acknowledgements

Kazuyuki Fujii was partially supported by Grant-in-Aid for Scientific Re-

search (C) No. 10640210. KF is very grateful to Prof. Akira Asada for

useful suggestions. Yasushi Homma and Tatsuo Suzuki are very grateful to

Tosiaki Kori for valuable discussions.

References

[1] W. J. Zakrzewski: Low Dimensional Sigma Models, Adam Hilger, 1989.

[2] J. Mickelsson: Current Algebras and Groups, Plenum Press, 1989.

[3] K. Fujii: Ferretti-Rajeev Term and Homotopy Theory, Commun. Math.

Phys., 162(1994), 273-287.

[4] K. Fujii:

Ferretti-Rajeev model to any dimension: CP-invariance and geometry,

Jour. Math. Phys., 36(1995), 97-114.

Generalizations of the Wess-Zumino-Witten model and

[5] G. Ferretti and S. G. Rajeev: Current Algebra in Three Dimensions,

Phys. Rev. Lett., 69(1992), 2033-2036.

[6] G. Ferretti and S. G. Rajeev: CPN−1Model with a Chern-Simon Term,

Mod. Phys. Lett., A7(1992), 2087-2094.

[7] O. Alvarez, L. A. Ferreira and J. S. Guillen: A New Approach to Inte-

grable Theories in Any Dimension, hep-th/9710147.

[8] K. Fujii and T. Suzuki: Nonlinear Sigma Models in (1+2)-Dimensions

and An Infinite Number of Conserved Currents, hep-th/9802105.

[9] K. Fujii and T. Suzuki:

Models in (1 + 2)-Dimensions, hep-th/9804004.

Some Useful Formulas in Nonlinear Sigma

[10] D. Gianzo, J. O. Madsen and J. S. Guillen: Integrable Chiral Theories

in 2+1 Dimensions, hep-th/9805094.

[11] K. Fujii, Y. Homma and T. Suzuki: in preparation.

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[12] K. Fujii, T. Kashiwa and S. Sakoda: Coherent States over Grassmann

Manifolds and the WKB Exactness in Path Integral, Jour. Math. Phys.,

37(1996), 567-602.

[13] 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.

[14] K. Fujii and R. Sasaki:

Grassmannian Sigma Models in Two Dimensions II, Prog. Theor. Phys.,

71(1984), 831-839.

Classical Solutions for the Supersymmetric

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