Page 1

arXiv:hep-th/0411134v1 12 Nov 2004

T-Duality in Affine NA Toda Models

J.F. Gomes, G.M. Sotkov and A.H. Zimerman

Instituto de F´ ısica Te´ orica - IFT/UNESP

Rua Pamplona 145

01405-900, S˜ ao Paulo - SP, Brazil

Received XXX

The construction of Non Abelian affine Toda models is discussed in terms of its un-

derlying Lie algebraic structure. It is shown that a subclass of such non conformal two

dimensional integrable models naturally leads to the construction of a pair of actions

which share the same spectra and are related by canonical transformations.

PACS: 11.25.Hf, 02.30.Ik,11.10.Lm

Key words: Integrability, Duality, Non Conformal Duality

1Introduction

The affine Toda models consists of a class of relativistic two dimensional inte-

grable models admiting soliton solutions with non trivial topological charge (e.g. the

abelian affine Toda models). Among such models we encounter certain Non Abelian

affine (NA) Toda models admiting electrically charged solitons [1]. In general, the

NA Toda models admit solitons with non trivial internal symmetry structure. The

formulation and classification of such models with its global symmetry structure

is given in terms of the decomposition of an underlying Lie algebraic structure ac-

cording to a grading operator Q and, in terms of a pair of constant generators ǫ±

of grade ±1. In particular, integrable perturbations of the WZW model character-

ized by ǫ±describe the dynamics of fields parametrizing the zero grade subalgebra

G0. The action manifests chiral symmetry associated to the subalgebra G0

due to the fact that Y ∈ G0

0,[Y,ǫ±] = 0. The existence of such subalgebra allows

the implementation of subsidiary constraints within G0

model from the group G0 to the coset G0/G0

viewed according to axial or vector gauging leads to different parametrizations and

different actions, namely axial or vector actions.

0⊂ G0

0and the reduction of the

0. The structure of the coset G0/G0

0

We first discuss the general construction of NA Toda models in terms of the

gauged WZW model. Next, we discuss the structure of the coset G0/G0

U(1)n−1/U(1) and G0/G0

gaugings and explicitly construct the associated lagrangians. Finally, we show that

the axial and vector models are related by canonical transformation (see [2] and

refs. therein) preserving the Hamiltonian which also interchanges the topological

and electric charges.

0= SL(2)⊗

0= SL(3)/SL(2) ⊗ U(1) according to axial and vector

Czechoslovak Journal of Physics, Vol. 54 (2004), No. 0

A1

Page 2

J.F. Gomes et al.

2 General Construction of Toda Models

The basic ingredient in constructing Toda models is the decomposition of a

Lie algebra G of finite or infinite dimension in terms of graded subspaces defined

according to a grading operator Q,

[Q,Gl] = lGl,G = ⊕Gl,[Gl,Gk] ⊂ Gl+k, l,k = 0,±1,···(1)

In particular, the zero grade subspace G0plays an important role since it is parametrized

by the Toda fields. The grading operator Q induces the notion of negative and pos-

itive grade subalgebras and henceforth the decomposition of a group element in the

Gauss form, g = NBM, where N = exp(G<), B = exp(G0) and M = exp(G>).

The action for the Toda fields is constructed from the gauged Wess-Zumino-Witten

(WZW) action,

SG/H(g,A,¯A) = SWZW(g)

?d2xTr(A(¯∂gg−1− ǫ+) +¯A(g−1∂g − ǫ−) + Ag¯Ag−1)

where A = A−∈ G<,¯A =¯A+∈ G>, ǫ±are constant operators of grade ±1. The

action (2) is invariant under

−k

2π

(2)

g′= α−gα+,A′= α−Aα−1

−+ α−∂α−1

−,

¯A′= α−1

+¯Aα++¯∂α−1

+α+,(3)

where α−∈ G<, α+∈ G>. It therefore follows that SG/H(g,A,¯A) = SG/H(B,A′,¯A′).

Integrating over the auxiliary fields A,¯A, we find the effective action,

Seff(B) = SWZW(B) − kø2π

?

Tr(ǫ+Bǫ−B−1)d2x(4)

The equations of motion are given by

¯∂(B−1∂B) + [ǫ−,B−1ǫ+B] = 0,∂(¯∂BB−1) − [ǫ+,Bǫ−B−1] = 0 (5)

It is straightforward to derive from the eqns. of motion (5) that chiral currents are

associated to the subalgebra G0

0⊂ G0defined as G0

0= {X ∈ G0, [X,ǫ±] = 0}, i.e.,

JX= Tr(XB−1∂B),

¯JX= Tr(X¯∂BB−1),

¯∂JX= ∂¯JX= 0 (6)

For the cases where G0

JX =¯JX = 0,X ∈ G0

account the subsidiary constraints (6) reduces the model from the group G0to the

coset G0/G0

0?= 0, we may impose consistently the additional constraints

0. The construction of the gauged WZW action taking into

0and its action is given by

SG0/G0

0(B,A0,¯A0) = SWZW(B) − kø2π

?

Tr(ǫ+Bǫ−B−1)d2x

−kø2π

?

Tr(±A0¯∂BB−1+¯A0B−1∂B ± A0B¯A0B−1+ A0¯A0)d2x

(7)

2A

Czech. J. Phys. 54 (2004)

Page 3

T-Duality in Affine NA Toda Models

where the ± signs correspond to axial or vector gaugings respectively. The action

(7) is invariant under

B′= α0Bα′

0,A′

0= A0− α−1

0∂α0,

¯A′

0=¯A0−¯∂α′

0(α′

0)−1

(8)

where α′

SG0/G0

0= α0(z, ¯ z) ∈ G0

0(B,A0,¯A0) = SG0/G0

0for axial and α′

0(α0Bα′

0= α−1

0,A′0,¯A′0)

0(z, ¯ z) ∈ G0

0for vector cases, i.e.,

0= gf

3The structure of the coset G0/G0

0

We now discuss the structure of the coset G0/G0

and vector gaugings. We shall be considering first the NA Toda models where

G0

0= U(1). The group element of the zero grade subgroup G0is parametrized as

0constructed according to axial

B = e˜ χE−α1eRl1·H+?n

l=1ϕlhle˜ψEα1

(9)

According to the axial gauging we can write B as an element of the the zero grade

subgroup G0is parametrized as

B = e1ø2Rl1·H(gf

0,ax)e1ø2Rl1·H,gf

0,ax= e˜ χe1ø2RE−α1e

?n

l=2ϕlhle˜ ψe1ø2REα1

(10)

The effective action is obtained integrating (7) over A0,¯A0, yielding [1]

Lax

eff= 1ø2

n

?

a,b=2

ηab∂ϕa¯∂ϕb+ 1ø2¯∂ψ∂χø∆e−ϕ2− Vax,∆ = 1 + n + 1ø2nψχe−ϕ2(11)

where ψ =˜ψe1ø2R,χ = ˜ χe1ø2R, and Vax =

ψχe−ϕ2).

The vector gauging can be implemented from the zero grade subgroup G0writ-

ten as

?n

l=2e2ϕl−ϕl−1−ϕl+1+ eϕ2+ϕn(1 +

B = eul1·H(gf

0,vec)e−ul1·H,wheregf

0,vec= e˜ χeuE−α1e

?n

l=1φlhle˜ψe−uEα1(12)

Since u is arbitrary, we may choose u = 1ø2ln(˜ψø˜ χ) so that

gf

0,vec= etE−α1e

?n

l=1φlhletEα1,t2=˜ψ˜ χ(13)

The effective action for the vector model is [2]

Lvec

eff

=1ø2

n

?

a,b=1

ηab∂φa¯∂φb+ ∂φ1¯∂φ1øt2eϕ2−2φ1+ ∂φ1¯∂ln(t) +¯∂φ1∂ln(t) − Vvec.

We now discuss the simplest case in which G0

d, the homogeneous gradation and ǫ±= l2·H(±1). In this case G0

is generated by G0

0= {E±α1,H1,H2} and B is written as

0is nonabelian, i.e. G =ˆSL(3), Q =

0= SL(2)⊗U(1)

B=e˜ χ1E−α1e1ø2(l1·HR1+l2·HR2)(gf

0,ax)e1ø2(l1·HR1+l2·HR2)e˜ ψ1Eα1

gf

0,ax

=eχ1E−α1−α2+χ2E−α2eψ1Eα1+α2+ψ2Eα2

(14)

Czech. J. Phys. 54 (2004)

A3

Page 4

J.F. Gomes et al.

where li,i = 1,2 are the fundamental weights of SL(3). The effective action is then

obtained by integration over the auxiliary matrix fields A0,¯A0yielding

Lax

eff=1ø∆(¯∂ψ2∂χ2(1 + ψ1χ1+ ψ2χ2) +¯∂ψ1∂χ1(1 + ψ2χ2))

−1ø2∆(ψ2χ1¯∂ψ1∂χ2+ χ2ψ1¯∂ψ2∂χ1) − V(15)

where V = 2ø3+ψ1χ1+ψ2χ2and ∆ = (1+ψ2χ2)2+ψ1χ1(1+3ø4ψ2χ2). For the

vector action, the zero grade group element B in (14) is parametrized as

B=e˜ χ1E−α1e1ø2(l1·Hu1+l2·Hu2)(gf

0,vec)e−1ø2(l1·Hu1+l2·Hu2)e˜ψ1Eα1

(16)

where gf

is then [3]

0,vec= e−t2E−α2−t1E−α1−α2eφ1h1+φ2h2et2Eα2+t1Eα1+α2. The effective action

Lvec

=1ø2

2

?

i=1

ηij∂φi¯∂φj+ ∂φ1¯∂φ1øt2

1e−φ1−φ2+¯∂φ1∂ln(t1) + ∂φ1¯∂ln(t1)

−∂φ1¯∂φ1(t2øt1)2e−2φ1+φ2+¯∂(φ2− φ1)∂(φ2− φ1)øt2

¯∂(φ2− φ1)∂ln(t2) + ∂(φ2− φ1)¯∂ln(t2) − V

2eφ1−2φ2

+(17)

where V = 2ø3 − t2

2e−φ1+2φ2− t2

1eφ1+φ2and ηij= 2δij− δi,j−1− δi,j+1.

4Axial-Vector Duality

In this section we shall prove that the axial and vector models are related by a

canonical transformation. Consider the SL(3) vector model

Lvec

=∂φ1¯∂φ1+ ∂φ2¯∂φ2− 1ø2∂φ2¯∂φ1− 1ø2∂φ1¯∂φ2+ ∂φ1¯∂φ1øt2eϕ2−2φ1

∂φ1¯∂ln(t) +¯∂φ1∂ln(t) − (eφ1−2φ2+ e−φ1+2φ2− t2eφ1+φ2).+ (18)

In terms of the new set of more convenient variables a = (1 − t2e2φ1−φ2),

φ1− 2φ2, θ = φ1the lagrangian (18) becomes

f =

4Lvec= (1 + 3aø1 − a)∂θ¯∂θ − ∂θ(¯∂f + 2¯∂aø1 − a) −¯∂θ(∂f + 2∂aø1 − a) + ∂f¯∂f − 4Vvec(19)

where Vvec = ef+ ae−f. The canonical momenta are given by Πρ = δLvecøδ ˙ ρ,

ρ = θ,f,a. The hamiltonian is then given by

Hvec

=−(1 − a)ΠaΠθ+ Π2

1ø4(1 + 3a)ø1 − aθ′2− 1ø2(f′+ 2a′ø1 − a)θ′− 1ø4f′2+ Vvec (20)

f− (1 − a)ΠaΠf− a(1 − a)Π2

a

+

Consider now the following modified lagrangian

Lmod= Lvec−˜θ(∂¯P −¯∂P)(21)

where we identify ∂θ = P,

¯∂θ =¯P [4]. Integrating by parts,

Lmod= (1 + 3aø1 − a)P¯P − 1ø4P(¯∂f + 2¯∂aø1 − a +¯∂˜θ)

−1ø4¯P(∂f + 2∂aø1 − a − ∂˜θ) + 1ø4∂f¯∂f − Vvec

(22)

4A

Czech. J. Phys. 54 (2004)

Page 5

T-Duality in Affine NA Toda Models

Integrating over the auxiliary fields P and¯P we find the effective action

Leff= 1ø4∂f¯∂f − 1ø4(1 − a)ø1 + 3a(¯∂f + 2¯∂aø1 − a +¯∂˜θ)(∂f + 2∂aø1 − a − ∂˜θ) − V (23)

with canonical momenta defined by Πρ= δLefføδ ˙ ρ, ρ =˜θ,f,a. The hamiltonian

becomes

Hmod

=−1ø2(1 − a)Πa˜θ′+ Π2

f− (1 − a)ΠaΠf− a(1 − a)Π2

2− 1ø2(f′+ 2a′ø1 − a)Π˜θ− 1ø4f′2+ Vvec (24)

a

+(1 + 3a)ø1 − aΠ˜θ

The canonical transformation

Πθ= −1ø2˜θ′,θ′= −2Π˜θ

(25)

preserves the Poisson bracket structure and provide the equality of the hamiltanians

Hmod= Hvec. If we now substitute

˜θ = 2ln(ψøχ),a = 1 + ψχe−ϕ2,f = −ϕ2

(26)

in the effective lagrangian Leff (23), we find

Leff= 1ø2∂ϕ2¯∂ϕ2+ ∂χ¯∂ψø∆e−ϕ2− V(27)

which is precisely the axial lagrangian (11) for G0= SL(3). It therefore becomes

clear that the axial and the vector models are related by the canonical transforma-

tion (25) which preserves their hamiltonians.

For the case of G0

0= SL(2) ⊗ U(1) we found the canonical transformation

responsible for the equality of the Hamiltonians to be [3]

Πθα= −2∂x˜θα,Π˜θα= −2∂xθα,α = 1,2 (28)

where θα= ln(ψα/χα),

As a last comment of this section we should like to analyse the topological and

Noether charges of the axial and vector models. Consider the first example where

G0

0= U(1) and with vector and axial lagrangiansgiven by (19) and (23) respectively.

The Noether charges associated to the global transformation θ → θ+c and˜θ →˜θ+˜ c

where c,˜ c = constant, are given by

˜θ1= −1ø2φ1,

˜θ1= −1ø2(φ1+ φ2)

QNoether

vec

=

?

?

(δLvecøδ˙θ)δθdx =

?

?

Πθdx,

QNoether

ax

=(δLaxøδ˙˜θ)δ˜θdx =Π˜θdx (29)

Since the vector and axial models possess respectively the foloowing topological

charges

QTop

vec=

?

(∂xθ)dx,QTop

ax =

?

(∂x˜θ)dx (30)

it is clear that under the canonical transformation (25) their Noether and topo-

logical charges become interchanged. The same can be extended to all isometric

variables within the models described by lagrangians (15) and (17).

Czech. J. Phys. 54 (2004)

A5

Page 6

J.F. Gomes et al. T-Duality in Affine NA Toda Models

5 Concluding Remarks

We have seen that the crucial ingredient which allows the construction of the

axial and vector models is the existence of a non trivial subgroup G0

worked out explicitly examples in which G0

volving one and two isometric variables θα. The same strategy works equally well

for generalized multicharged NA Toda models.

An interesting and intriguing subclass of NA Toda models correspond to the

following three affine Kac-Moody algebras, B(1)

vector actions were constructed in [2] and shown to be identical. In fact, those affine

algebras satisfy the no torsion condition proposed in [2] which is fulfilled by Lie

algebras possesseing Bn-tail like Dynkin diagrams. The very same selfdual models

were shown to possess an exact S-matrix coinciding with certain Thirring models

coupled to affine abelian Toda models in ref. [5].

0. We have

0= U(1) and G0

0= SL(2) ⊗ U(1) in-

n ,A(2)

2nand D(2)

n+1. Their axial and

We are grateful to CNPq, FAPESP and UNESP for financial support.

References

[1] J.F. Gomes, E.P. Gueuvoghlanian, G.M. Sotkov and A.H. Zimerman, Nucl. Phys.

B598 (2001) 615, hepth/0011187; Nucl. Phys. B606 (2001) 441, hepth/0007169

[2] J.F. Gomes, E.P. Gueuvoghlanian, G.M. Sotkov and A.H. Zimerman, Ann. of Phys.

289 (2001) 232, hepth/0007116

[3] J.F. Gomes,G.M. Sotkov and A.H. Zimerman, J. Physics A37 (2004) 4629

[4] T. Busher, Phys. Lett. 159B (1985) 127, Phys. Lett. 194B (1987) 59, Phys. Lett.

201B (1988) 466

[5] V.A. Fateev, Nucl. Phys. B479 (1996) 594

6A

Czech. J. Phys. 54 (2004)