Physics Letters B 277 (1992) 447-452 PHYSIC S LETTERS B North-Holland

Higher spin, W and gauge symmetries in two-dimensional sigma models

George P a p a d o p o u l o s L

Department of Physics, Queen Mary and Westfielcl College, Mile End Road, London El 4NS, UK

Received 12 November 1991

The classical higher spin symmetries of bosonic and supersymmetric gauged sigma models are examined. The necessary and sufficient conditions for the invariance of the action of a gauged sigma model under a higher spin transformation arc derived and the commutator of two such symmetries is presented. In particular, the W symmetries of a gauged sigma model without Wess- Zumino term and a gauged Wess-Zumino-Winen model are examined.

1. I n t r o d u c t i o n

W algebras were introduced by Zamolodchikov in ref. [ 1 ] in the context of conformal field theory. Several realisations o f these algebras have been con- sidered in terms of algebras of symmetries of free two- dimensional field theories [2 -5 ] . The commutator o f any two W transformations closes to W transfor- mations with parameters that depend on the con- served currents of this W algebra of symmetries. Apart from the realizations of W algebras as algebras of symmetries o f two-dimensional free-field theories, some W algebras can be realised as algebras of sym- metries of (non-linear) two-dimensional sigma models [5 -8 ] . In particular, it was observed in refs. [6 -8 ] that there is a W algebra for every N = 1 super- symmetric two-dimensional sigma model with target space an irreducible riemannian manifold.

In this paper, the classical higher-spin (HS) sym- metries of bosonic gauged non-linear sigma models are studied. The necessary and sufficient conditions for the existence of such a kind of symmetries are given and the commutator of two HS transforma- tions is calculated. The HS symmetries o f ( 1, 0) and ( 1, 1 ) supersymmetric sigma models are also consid- ered. The algebra o f HS transformations does not necessarily close. However, there are examples of

Email address: papa@uk.ac.qmw.ph.v 1.

models where their HS symmetries close as W sym- metries. Finally, several examples o f gauged sigma models that admit W symmetries are given. In partic- ular, the W symmetries o f gauged sigma models with- out Wess-Zumino term and the W symmetries of gauged Wess-Zumino-Wit ten models are studied.

Let M be a riemannian manifold equiped with a metric g, I I be a closed three form, and K be a group of isometrics of M. In addition, let us assume that a group a c t i o n f o f the group K leaves the closed three form H invariant. Gauging the group K, we can con- struct a bosonic gauged sigma model with target space M. Indeed a locally defined lagrangian [9,10] of a bosonic gauged sigma model is

L = g i j V , X ~ V=XJ+bij O~X i 0=X j

- A g u j , O=Xi+A"_ui~ i ,~ b . _ O,X +c~bA .A = , ( 1 )

where H = d b , X are sections of a bundle with base space a two-dimensional space-t ime with coordi- nates (y*, y = ) and fiber M, i , j = l .... , d i m M , {Ag, A% } are the components of a connection A (a, b= 1, ..., dim K) , and

i i a i v_x = a~x + A ~ . ,

i i a i v : x = 0 : x +A :~=, (2)

where ~ are the Killing vector fields generated by the group a c t i o n f o f the group K on M,

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~Huk = 2 Ouukl~, (3)

and

C~b = ~ U~b. ( 4 )

In the following, we assume that u is globally defined on M.

The action ( 1 ) is gauge invariant [ 9,10 ] provided that u is equivariant, i.e.

GUo = f ~,,t, uc , (5)

where ~ is the Lie derivative of the vector field G, and

C,b =Ct,,bj . (6)

2. Bosonie model

The equations of motion of the action ( 1 ) are

V ( + ) V ~ X . , .~_ 1 i i a = ~ (~, +u~)F+= = 0 , (7)

.~o= (G +u,DV.X'=O, (8)

and

J=a=(~ia--Uia)V=Xi-~O, (9)

where the connections of the covariant derivatives V ~ ± ) are given by

+--~H ~k. F ~ ± ) % = F ~ k ( g ) 3 i (10)

F ( g ) is the Levi-Civita connection of the metric g. A left-handed HS transformation is

~Xi=a_adti,...ia V:~XiL.. V~ Xia ( 11 )

and

6.4 ~= = a_dDabi,..,ia_ , Fb,. V ,X" . . . V , X i '- ' ,

fiat. =0, (12)

where d is an invariant tensor and D is an equivariant tensor on M, and aa= ad(Y*) is a semilocal parame- ter with Lorentz weight - d. The transformation ( 11 ) is a generalisation of the W transformations of (un- gauged) non-linear sigma models [3,5,4]. The W transformations of (ungauged) sigma models are given by ( 11 ) provided that we use partial deriva- tives instead of covariant ones.

The transformations (11 ) and (12) are symme- tries of the action ( 1 ) provided that

V},+) da...ie+, = 0 , (13)

da...,~÷L = dciL...id÷~ ) , ( 1 4 )

and

(~,~+Uia)diia...ia=Oba~,...id_,(~b+Ub),d). ( 1 5 )

The currents of these transformations are

1 d V , X " V,X ia+' Q ( d ) = ~ - i,...ia+, . . . . (16)

These currents are conserved ( 0 = Q ( d ) = 0 ) subject to the equations of motion (eqs. (7) and (8) ) , pro- vided that eqs. (13) and ( 15 ) are satisfied.

Apart from the left-handed HS symmetries ( 11 ) and (12), we can construct right-handed HS sym- metries of the action (1) by using the tensor d to transform the field X and transform appropriately the component A= of the connection A instead of the component A.. The conserved currents of these sym- metries are given by (16) after replacing the V~ co- variant derivatives with V= ones. It is straightfor- ward to derive the properties of the right-handed symmetries from the properties of the left-handed ones. In the following, we will consider only left- handed symmetries.

The commutator of two HS transformations on X is

[~a, ~e] Xi=rC~ )X' +rc2)X ' +6(3)X~ , (17)

where

~ < l ) X ' = a _ e a _ a P ( d , e)li,...ia+¢ VvXi ' . . .V,X ia+~

=a_ea_a[ dki~...id Vkeiia~-i...id+e

+ deiki,...i,-l V,,dki,.÷ ,...ia+,

-- ( d - , e ) ]V÷:Xa...V,X ~a+" , ( 18 )

~(2)X/= [ e( a_~ O, a_d )eik,...i~_,dkn....,a . . . .

- ( d - , e ) ] V a : X i I . . . V , : X ld+e-I , (19)

and

O ( 3 ) X i = a _ e a _ a d e [ ( eih,...ie_ , dlki,...,d+ ,_ ~ )

- - ( d ~ e ) ] V 3 X J ' V X ~ V X ~÷'-~ (20) r ~- °°" :l: °

If the covariant derivatives V~ in (17) are replaced

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by partial ones, we recover the commutator of two HS (W) transformations generated by the tensors d and e of an (ungauged) sigma model with metric g and Wess-Zumino term H.

The transformation (18) is not always a symmetry of the action ( 1 ). Moreover the tensor P(d, e) may not be expressed as a linear combination of the sym- metric tensors that generate the original HS symme- tries. As a consequence of this, the algebra of HS transformations may not close as a W algebra. How- ever, the tensor P vanishes whenever the tensors e and d are covariantly constant with respect to the Levi- Civita connection of the metric g.

The commutator of two HS transformations on the field A = is

[ad, ~] ,4L

= 6 ( ~ A ~ +8(2)A~ + a ° ) A a +(~t4)AL , (21)

where

8 ( l )A ~= = a_ ~a_ d { dk i l...id Vk E a bid ..... id . . . .

+ eacil . . . i , - l VieOCbie+ l...ta+e- t

+ ( e - 1 )Eabkm. . ie_2 Vie_ tdk i , . . . ia . . . .

- [ (d, D) --,, (e, E) ] }Fb= V , X t l . . . V t X id . . . . ,

8(2)AL = {a_e O.a_d[ Eacit...i~_,DCbi....id+._2

+ (e-- 1 )Eabkit...ie_2dkie_,...ia+._2]

- [(d, D)--, (e, E) ]}F{= V_X"...V.X 'd+"-= ,

(22)

(23)

8(3>A~= =a_ea_d{ ( E%,...i._,DCb,....,~+._=)

-- [(d, D ) ~ (e, E) ] }V.Fb.= V,X"...V.j( '~+'-2 , (24)

and

8 (4)A a_= =. a_ea_d { ( d - 1 )E"c,,...,,_, DCbkie...ia+,_3

+ ( e - 1 )dE%,,...i,_ 2dlki,_ ,...ia+,-,

- [ ( d , D ) - * ( e , E ) ] }

×F~= v 2 x k v ~ X ~' ... V, X u+`-3 • (25)

The commutator of two HS transformations is simplified whenever the curvature F ( A ) of the con- nection A vanishes on shell (this is the case for the gauged Wess-Zumino-Witten models). Then, we

may choose aA = =0. Indeed, the charges (16) are conserved up to the equations of motion ( 7 ) - ( 9 ) and provided that condition (13) is satisfied.

Another special case is whenever d = e and D=E. In this case, ~(I)X=I~(3)X~---0 and ~(I)A=~(3)A=

(4)A = 0 and the transformations ~ (2) X and ~ (2)A are new symmetries of the action ( 1 ).

3. Supersymmetric models

Having studied the HS symmetries of the bosonic gauged sigma models, we now examine HS symme- tries of a gauged supersymmetric sigma model. In particular, we study the HS symmetries for the ( 1, 0) and (1, l ) supersymmetric gauged sigma models. Since the treatment of these symmetries in ( l, 0) and ( l, 1 ) supersymmetric gauged sigma models is simi- lar, we restrict our attention to the HS symmetries of ( l, l ) supersymmetric gauged sigma models.

A locally defined action [9] o fa ( l, l ) supersym- metric gauged sigma model written in ( l, 1 ) super- fields ~1 is

L=gij V+X ~ V_XJ+bo D+X i D _ X j

i a i a b - A + ui~ D_X - A _ ui~ (26) u D+X -t-Cab A _A +,

where X are sections of a bundle with base space a ( 1,1 ) superspace with coordinates (y*, y =, 0_, O+ ) and fiber a manifold M, and {D+, D_, 0., 0=} are the fiat superspace derivatives. The target space M ad- mits a group action, and H, u and c are defined as in the bosonic model. {A +, A_, A~, A = } are the compo- nents of a connection A and {V+, V_, V., V=} are the corresponding covariant derivatives that satisfy the algebra

[V+, V_ ] =F+_,

[V+, V= ] = F + = ,

[V+, V+ ] = iV,,

[V_, V,] = F ~,

[V., V= ] = F, = ,

[V_, V_ ] = i V = . (27)

The rest of the supcrcommutators vanish and F is the curvature of the connection A. The action (26) is gauge invariant provided that u and c obey the con- ditions (5) and (6).

The equations of motion of a ( 1,1 ) supersymmet-

~ The actions ofall (p ,q) supersymmetric gauged sigma models are given in ref. [ 11 ].

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ric gauged sigma model are the following:

V(+) i i i = 0 (28) V+X - ~ ( u ~ + : i ~F" - - % a ] + - - ,

j+~ = (u,~ +¢ ,~)V+X*=0, (29)

and

j - a = ( u , o - ~,~) V _ X ' = O . (30)

Next we consider the HS transformations

OX ~ = a_ 1/2 l'it..., V + Xa... V + X ' , ( 31 )

and

L ~ F b dA ~ - =t / ' - - l /2 I.'il...tl-I +-- V + X i l . . . V + X i l : ,

6A% = 0 . (32)

l is an invariant and L is an equivariant tensor on M under the action of the group K.

The transformations (31 ) and (32) are symme- tries o f the action (26) provided that

V~ + ) l i i t . . .#=O , (33)

lil...il+, = lu,...,+, l , (34)

and

k k (u~ +{o)lk,...i,= - -Lbat i l . . . i t_ , ( l ' lb+~b) i t l • (35)

The currents of these transformations are

1 l Q ( / ) = ~ ,,...o+, v + x i ' . . . v + X'l÷' , (36)

and they are conserved subject to the equations of motion (eqs. (28), (29) ) and provided that the con- ditions (33) and (35) are satisfied.

The commutator o f two HS transformations on the field X is

[am, g l l X i = g ( l ) X i + a ( 2 ) X i + ( ~ ( 3 ) X i , (37)

where

O~ l ) X i = a _ i / z a _ , , / z X ( m, l)ii , . . . , . , ,

X V +Xi ' . . .V + X i'*'~ , (38)

6 ( 2 ) X i = [ a_ 1/2 D +a-,,,/21ml'i,...i,,

×l'k*m÷~...,,,÷~-~-- ( m o l ) I V + X i ' . . . V + X "+"-' (39)

and

J ( 3 ) X i = i l m a _ i / 2a_ml z

X [ ( -- 1 )mmJki,. . . im_, lijim...il+m_2

- ( - 1 )"~(rn- - , l ) ]V.X k V+Xi'. . .V+X *'~÷'-2 ,

(40)

where .iV(m, l) is the Nijenhuis tensor [ 12 ] o f m and l and is given by

./V(m, l)',...,.+t = [ m -/U,...ir,, OlJl lii,,,+ ,---i,-+tl

+ ( - 1 )mlOlt, mki2...i,,,+ll~rkli,,.,+2...i,,,+t ]

-- ( -- 1 y"l(rn--+l) ] . (41)

The commutator (37) of two HS transformations is similar to the commutator o f two H S ( W ) symme- tries o f an (ungauged) N = 1 supersymmetric sigma model [ 6-8 ]. Indeed, the latter can be recovered from (37), if we replace in eqs. ( 3 8 ) - ( 4 0 ) the covariant superspace dcrivatives V+ with partial superspaee ones.

The commutator o f two HS transformations on the connection A_ is

I am, 6AA"_ =g( t )A ~ - +O(2)A~ - + d ( 3 ) A a - + 3 ( 4 ) A a _ _ , (42)

where

6~(t)Aa_ = a _ i / 2 a _ m / 2 { m k i t . . . . , , VkLabi,,,+t_..,,+l+l

+ ( - 1 ) " Vi~MCbi2...,.L~,.i . . . . . . ,.,+l-i

+ ( - 1 ) " ( l - 1 )Vit mk,2...,~+, Labki,,+2...i,,,+t_,

- ( - 1 ):"[ (M, m)o(L, / ) ] } XFb+_ V+XiI. . .V+X im+l-I , (43)

6~(2)A a _ = a _ , /2a_,,,/2 { ( - 1 )mMCoil. . . i , , ,_lL aci,,,...i,+,,,_ 2

- ( - 1 )'m[ (M, m)--,(L, l )]}

xV+F~+_ V + X i ' . . . V + X s'+l-~ , (44)

~(3)Aa__ = {a_ i / 2 D + a _ m / 2 [ MCbit...im_ i Lacn...i,,,+t_2

+ ( 1 - 1 )mkit . . . i , , ,Labki, , , . . . . . i,,+t-2]

- [(M, m ) - , ( L , l) ] } F ~ _ V+X".. .V+X 'm÷'-2 , (45)

and

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J(4)Aa__ = ia_ l/2a_m/2

× { ( - 1 ) r e ( m - 1 )MCbk,...i.,_2 L%,._,_.i,,+,_3

+ ( l - 1 ) m ( -- 1 )"mJki,...i,,L%ji,,+,...i,,+:_3

- ( - l )t"[ ( M , m ) - - , ( L , l ) ]}

xFb+_ V . X k V +X"...V +X "*t- ~ . (46)

As in the bosonic gauged sigma model, if the equa- tions of motion (eqs. ( 2 8 ) - ( 3 0 ) ) imply that F ( A ) =0, we may set 3A =0.

4. Concluding remarks

In both bosonic and supersymmetric gauged sigma models without Wess-Zumino term ( H = 0), we may set u=0 . From eqs. (13) and (33), the symmetric tensor d and the form l are covariantly constant with respect to the Levi-Civita connection. In the bosonic gauged sigma model, this condition imposes strong restrictions on the geometry of the manifold M. In particular it can be shown that M is the product of other riemannian manifolds, i.e. the sigma model manifold is reducible. In the supersymmetric gauged sigma modcls there are irreducible riemannnian manifolds with covariantly constant forms. These manifolds have been classified by Berger [ 13 ] in terms of their holonomy.

To give an example HS symmetries that form a W algebra for gauged sigma models without Wess- Zumino term, let us assume that K = U ( 1 ) (or a product of U( 1 ) 's) acting freely on M. In this case, it can be shown the equations of motion of the gauged sigma model imply that the curvature F of the con- nection A is zero and we may set 3A = = 0 (OA _ = 0). In addition, 6 ~ )X in the commutator (17) ( (37) ) vanishes. The commutator of two HS transforma- lions ( 17 ) o fa bosonic gauged sigma model becomes the same as the commutator of two HS transforma- tions of a linear (ungauged) bosonic sigma model with target manifold a euclidean space. Several ex- amples of HS symmetries of a (ungauged) sigma model that form a W algebra are given in refs. [3,4]. Finally, in the supersymmetric gauged sigma model the commutator (37) of HS transformations reduces to the commutator of the W symmetries of (un- gauged) supersymmetric sigma models with target

spaces manifolds with irreducible holonomy studied in refs. [ 6-8 ].

There are W symmetries in a bosonic non-linear gauged sigma model with a Wess-Zumino term ( H e 0). For example, for every W3 symmetry of an ungauged sigma model with a d tensor in the S O ( n - 1 ) ( d i m M = n ) series of ref. [3], there is a corresponding gauged sigma model with a W3 sym- metry. The commutator of two W3 transformations on both fields A and X closes to a translation (~X~V~X, 6A ~ F) with parameters that 'depend on the energy-momentum tensor of the model.

Another interesting example is a gauged Wess- Zumino-Witten (WZW) model with target mani- fold a semisimple compact Lie group G. Let K be a subgroup of G. The adjoint action of K on G is

g ~ k g k -~ , (47)

where g e G and k~K. This action can be gauged, i.e. the conditions ( 5 ) and (6) are satisfied ~2. The equa- tions of motion ( 1 ) - (3) of this theory imply that the curvature F ( A ) of the connection A is zero. So we can set 314= =0. In addition, if we assume that the symmetric tensors d are both left and right invariant (bi-invariant), they are covariantly constant with re- spect both the Levi-Civita connection of the invar- iant metric on G and the V ~ + ) covariant derivative. Then condition (13) is satisfied, and 3 cI)X= 0 in the commutator ( 17 ). The algebra of all the HS symme- tries generated by the bi-invariant symmetric tensors of G becomes a W (Casimir) algebra (refs. [ 14-17 ] ).

Similarly in a ( 1,1 ) ( ( 1,0) ) supersymmetric WZW model, the equations of motion ( 28 ) - ( 30 ) imply that the curvature F of the connection A is zero and we can set 6.4_ =0. In addition, we may choose the an- tisymmetric tensors I to be the bi-invariant forms of the group G. Then these forms are covariantly con- stant with respect both the Levi-Civita connection of the invariant metric of the group G and the V t + > co- variant derivative. Because of this, condition (33) is satisfied and 8 t l )X=0. The algebra of the symme- tries of the supersymmetric gauged WZW model gen- erated by the bi-invariant forms on G is the same as the algebra of symmetries generated by the same forms in an (ungauged) WZW model [ 18].

,2 For more details see ref. [ 10].

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Acknowledgement

1 wou ld like to thank P.S. Howe, C.M. Hul l and B.

Spence for useful d iscuss ions on W algebras. This

work was suppor ted by SERC.

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