group actions and non-linear sigma models

Volume 238, number 1 PHYSICS LETTERS B 29 March 1990

GROUP ACTIONS AND N O N - L I N E A R S I G M A M O D E L S

George P A P A D O P O U L O S 1 Mathematics Department, King's College London, Strand, London WC2R 2LS, UK

Received 28 October 1989; revised manuscript received 10 January 1990

The Yang-Mills sector of the non-linear sigma models is considered in D dimensions and its symmetries are studied. It is found that a diffeomorphismfof the sigma model manifold M is a symmetry provided that it can be lifted to a mapf T on the principal bundle P associated with the theory and the corresponding connection of P is an invariant Lie-algebra valued one-form with respect to the lifted map. If the group G of transformations of M is compact, connected and the action of G can be lifted to P, it is shown that there is at least one invariant connection. The lifting of the group actions of M to P are studied for U( 1 ) and trivial principal bundles, and for sigma model manifolds which are homogeneous spaces. The number of independent parameters of sigma models on G/H spaces is also examined using Lie-algebra cohomology with applications in the (1, 0) supersymmetric sigma models in two dimensions. Finally, it is shown that there are topological obstructions to gauging the rigid symmetries of a generic bosonic non-linear sigma model.

The symmet r i e s o f the boson ic s igma mode l with ac t ion

Sb = ½ f dDu V/~ 7a~0u¢'O~¢Jgij(0) (1) Z

are the i sometr ies of the s igma mode l man i fo ld M (i, j = l , ..., d im M ) with met r ic g an d the (o r i en t a t i on preserving) d i f feomorphisms of the spacet ime Z with metr ic 7 (P, v= l , ..., d i m Z = D ) . ¢ are the s igma mode l fields, i.e. maps f rom Z to M. In two d i m e n - sions ( D = 2 ) , (1 ) is i n v a r i a n t u n d e r the con fo rma l t r ans fo rma t ions o f the metr ic ~, as well. The act ion of the W e s s - Z u m i n o ( W Z ) t e rm [ 1 ] in D d i m e n s i o n s can be wri t ten as [ 2 ]

Swz = I ( 0 ) -- I (¢o )

1 f dD+l • ~. - (D+ 1 )! e]~l'"'tlD+lO[Al¢t'"'O]AD-l-l~iD+l £

X Ti,...io+, • (2)

where T i s a ( D + 1 ) closed form on M, 4 = (u, t) are the coord ina tes o f ~ = Y X [0, 1 ], 6 (u , 0 ) = 0 o an d

6(u, 1 ) =¢.

e-mail: udahl30 @oak.cc.kcl.ac.uk

The ac t ion of the Yang-Mi l l s sector in D d i m e n - sions is

S~=½ f dDux/@q/°~/bhab. (3)

Z

q/is a sect ion of the vector bund le A ® 0 *e where A is a spin b u n d l e over Z and 0*e is the pull back o f the b u n d l e e f rom M over the space t ime with the m a p 0; a, b = l , ..., r a n k e , h is an inne r p roduc t on e. The covar ian t der iva t ive ~ is

~bra= ~ea"l- J~/a'J¢ - ( ~ *~ )ab~rb , (4)

where F is the spin connec t ion of the metr ic y a n d 12 is the connec t i on o f the b u n d l e e. In two d imens ions , the ac t ion (3) with M a j o r a n a - W e y l fe rmions ~u is the Yang-Mi l l s sector [ 3 ] o f the s igma mode l associated with the ( 1, 0) heterot ic string.

In this paper, the symmet r ies of the act ions (1) , ( 2 ) and (3) i nduced f rom the t r ans fo rma t ions of the target man i fo ld M are s tudied. Par t icu la r a t t en t i on is pa id on the symmet r ies of the Yang-Mi l l s sector.

U n d e r the d i f f e o m o r p h i s m f o f M , S=Sb+Swz+S~, t r ans fo rms as

S(g, T, g2, h , f '¢ , ~,) =S(f*g, f*T, f*12, h, 0, ~). (5)

Eq. (5) impl ies t h a t f i s a symmet ry p rov ided that

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland ) 75


f*g=g, f*T= T

and

(6 ,7 )

f*f2=l-LVl (8)

where l=/(£,¢), h is chosen to be invariant under the rotations I. In particular, the action fg(fg2 (fgl)=fg2g~, £ = IdM ) of the group G (G compact and connected ) on M (geG) is a symmetry of S provided that fg ob- eys (6), (7) and (8) for every gs G. From eq. (6), the diffeomorphisms fg (ge G) are isometries of the metric g. Eq. (7) has been derived under the assump- tion that I(fg~o) - I ( ~ o ) = 0 mod 2re. This is always true if ~o is homotopic to the constant maps from Z to M(M is path connected) [4]. We first study eq. (8) which arises from the symmetries of the action S~,. (6) and (7) are examined later. Non-linear sigma models with target manifolds G / H spaces were examined in ref. [ 5 ].

Eq. (8) has been studied in connection with the theory of invariant connections on G / H spaces [ 6 ], the theory of monopoles [ 7 ] and quantum mechan- ics [ 8 ]. Moreover, invariant connections for the tan- gent bundle of G / H spaces have been studied in special and general relativity (see for a review ref. [ 9 ] ). Next consider P(M, K), a principal bundle over M with group K, and let { U,, h,} a trivialisation of P; h , : ~ - 1 ( g . ) -+ g . X K where re is the projection of P onto M. ~ is an associated vector bundle o fP (M, K). Le t fg(U, ) c U, ~, VgeG, then eq. (8) implies that the action fg of G on M can be lifted to an ac t ionf~ of G on P(M, K) [10]. Indeed

frg(hgl(x,k)):=hg'(fg(x), l- ' (g,x)k), (9)

where x e M and k~K. Observe that zcf~=fgzc and frg(pk) =ftg(p)k, peP and keK. The latter implies that frg commutes with the action Rk of K on P defined by Re(p) =pk. It can be shown t h a t f ~g is glob- ally defined on P(M, K). To show t h a t f tg is a group action, we use the equation

l- '(g2gl,X)=l-'(g2,f(gl,X)) l - ' (g , ,x) , (10)

which is derived from (8) by performing successive

*~ It is not always possible to find a trivialisation {U,, h~} of P(M, K) with the property f,(U,~) c U,~, VgeG. In this case, the group action lifts provided that K is compact and eq. (8) holds.

fg transformations. The lifting of group actions on principal bundles was also considered in ref. [ 11 ] to construct twisted unitary representations of the canonical group.

A connection co of P is an Lie (K)-valued one-form on P (Lie(K) the Lie algebra of the group K) which satisfies the following properties:

R~co=k-~cok k~K, ( l l a )

tO(Xa) = a , ( l l b )

where a~Lie (K) and Xa is the vector field on P generated by the action OfRk(~). k(t) is the one-parameter abelian subgroup of K given by k( t ) = exp (ta). At each point p of P(M, K), Xa, Va~Lie(K), span the vertical subspace V of TP. The horizontal subspace of TP is defined by the directions in TP that are an- nihilated by co, T P = V ~ H . The connection co on P is related to ~2 of eq. (8) by considering the local section a,(x) = hgl(x, e) o fP (e the identity of K) and identifyingO = co~ ~= a* co. (8) can be rewritten [ 10 ] in terms of co and the l if l ingf ~ as

f~,gco=co. (12)

To prove that ( 12 ) is equivalent to ( 8 ), first observe that if a~ is another local section of P such that a'~(x) =%(x)l(x), l ( x ) eK, then co,°' and co,~ are related up to a gauge transformation l(x). Then ap- ply a , on both sides of (12).

The action fg of the group G on M leaves S~, invariant provided that fg lifts to an actionf*g of G on the principal bundle P(M, K) and the connection of co of P, induced from ~2, satisfies eq. (12).

Given a l i f l ingf ~g to P(M, K) of the group action fg and a connection co, it is always possible to construct another (invariant) connection co~ that satisfies ( 12 ) by setting

co,= f f~g*cod~t(g) , (13) G

where d/~ is the Haar measure on G (G compact and connected), co~ satisfies ( 11 ) since the action Rk of K commutes wi th f~ of G.

The group action fg of G does not always lift to the principal bundle P(M, K). First, consider the lifting to P(M, K) with discrete fibers, i.e. P(M, K) is a finite covering of the manifold M ( n , ( M ) ¢ 0 ) . Fi- nite coverings are principal bundles that correspond

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to flat connections [12]. The necessary and suffi- cient condition for such a lifting to exist is p , n ~ ( G ) ~ n l ( P ( M , K ) ) , where p is the map defined b y p ( g ) =fg(X0). Xo is a given point o f M [ 13 ]. I f G is simply connected, there is always a lifting of fg. Next, consider the principal bundle P(M, U( 1 ) ). It can be shown [ 14] that the obstructions for the lifting offg on P(M, U ( 1 ) ) are elements of the cohomology groups H2(BG, H~(M, Z ) ) and H3(BG, Z ). BG is the classifying space of G [ 15 ] ( n, + ~ (BG) = nn ( G ) ) . I f G is semisimple and simply connected there is no obstruction. For G = U ( 1 ) ", BG--- CP(oe )", therefore H 3 ( B G ) = 0 and the obstruction lies in H2(BG, H ~ (M,Z) ) . The latter group vanishes if M is simply connected. Obstructions are expected to appear in the lifting of any group action to a generic principal bundle.

Suppose now a principal bundle P(M, K) with the group a c t i o n f Tg and a connection co that satisfies eq. ( 12 ). I f z is an automorphism of P (M, K) ( z (pk) = Z(p )k and nX=n) then the connection 2*0) solves ( 12 ) with the group action

f gt.z = Z - 'fgX. ( 14 )

U*0) and 0) differ by a gauge transformation, conse- quently the lifts f ~,z and f g~ are equivalent. There is a general theory to compare lifts up to automorphisms [ 14 ].

For example take the trivial bundle P = (S~)"× U ( 1 ), n is a given integer. P is the (n + 1 )-dimensional torus with coordinates (oi, 0), 0 ~< 0, 0 ~< 2n and i ~ i ~ n . The action o f K = U ( 1 ) on P is

R,(O ~, 0) = (¢~, (O+k) mod 2n) , ( 15 )

where k is the parameter of K, 0 < k ~< 2n. The vertical subspace of TP is spanned by the vector fields Xv = 3/ 30. The group G=Yl ~, Yl real numbers, acts on (S ~ )~ as

fr(O i) = (ri +O i) mod 2n , (16)

where the r are the parameters o f G . f can be lifted to P a s

f [ ( 0 , 0 ) = ( ( r ' + ¢ i) m o d 2 n , (O+q,r i) m o d 2 n ) , (17)

where the q are given real numbers (Group actions always lift to trivial principal bundles). Choose the trivializing section o f P as s(~ ~) = (¢i, 0). The auto-

morphisms Z of P correspond to the maps a z : (S l )n-~U( 1 ). Let az(fb ) = (niO ~) rood 2n, ni~Z, the automorphisms of P not connected to identity, from (14) we get

f ~,z(O i, O) = ( (ri +O i) mod 2n,

(0+ (q~+ni)r i) mod 2n) , (18)

which is equivalent to f ~ of eq. (17). This implies that the inequivalent liftings off~ on P are parame- trized by 0~<q~< 1, 1 ~i<~n. Moreover, let ca a connection on P = (S t ) " X U ( 1 ). From ( 11 ), we get

0)=c(¢),dO' +dO . (19)

The connection (19) is invariant unde r f~ provided that cs, i = 1, ... n, are constant functions of P. How- ever, not all the connections co invariant u n d e r f ~r are gauge inequivalent. The gauge inequivalent connections are given by 0 < c~< 1, 1 ~< i~< n. The vector fields

Yi=~/~q)~+qi3/~O, l <~i<~n (20)

generated by the action o f f ~,q on P are not horizontal with respect to co unless c i = - q i . The connection given in (19) by setting c ~ = - q~ is called canonical and it is the unique one with the property 0)(1I,) =0.

Now we turn to study the lifting lTg to a principal bundle P (M, K) of the left action lg of G on the ho- mogenous space M = G / H given by lg,(gH)=g'gH. H is a subgroup of G. Assume that H does not con- tain non-trivial normal subgroups of G. Consider P ( G / H , K) principal bundle that admits a lifting l~g oflg, then P is specified by the group homomorphism o~ : H - , K defined as follows: let g = h s H , then /~eH=eH and this implies l~ (po)~n - I (e l l ) . Thus, l~, (po)=pok where Po is a given point of P on n-~ (e l l ) . Define

a ( h ) = k , (21)

a is a group homomorphism. Suppose that the group homomorphism a : H--,K is given, we can define the principal bundle P = P , = G × K / ~ , where the equiv- alence ~ is defined by setting (g, k) ~ (gh, a ( h - ~ )k ). The lifting of lg on P,~ is

l~g,[g, k] = [g' g, k ] . (22)

The invariant connections co of P , under the action of lg are in one to one correspondence with the linear maps A : L i e (G) -~L ie (K) that satisfy

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A ( h - ~ a h ) = a - ~ ( h ) A ( a ) a ( h ) , (23)

where he H and ae Lie (G) . Indeed, let co be an invariant connection on P, , A is given by

A ( a ) = o g ( Y ~ ) ( p o ) , (24)

r where Ya is the vector field generated by lg~o, g ( t ) = exp ( ta) , tn(Ira) is independent from the point p of P that is evaluated since co is invariant. More- over, if a e L i e ( H ) then A=~x, , where c~. is the induced homomorphism on the Lie algebras L ie (H) and L ie (K) from c~ : H ~ K . Finally, given a linear m a p A : L i e ( G ) ~ L i e ( K ) that satisfies (23), we can recover the invariant connection ~o. Indeed,

og'~=A(o'*(g- 'dg) ) , (25)

where cr is the local section a of the principal bundle G ( G / H , H) and g - l d g is the Maurer -Car tan form of G. (25) can be simplified by decomposing L ie (G) = L i e ( H ) + m , then

o'* ( g - l d g ) = OgLic~n) + O~m. (26)

I f H is compact and connected, it is always possible to arrange such that [L ie (H) , m ] c m and

¢d)°~- a , O)Lic(U) +Am corn, (27)

where An, is the restriction of A on m. The condition (23) in terms of Am is equivalent to

A m ( [ a , c ] ) = [o%(a ) ,Am(c ) ] . (28)

where ae Lie (H) and cz m. (27) and (28) specify all the connections o f P~ invariant under the group action l~ of G. For example consider G = S U ( 2 ) , H = U ( 1 ) and P , a U ( 1 ) principal bundle over M = $2 = SU ( 2 ) / U ( 1 ) with

a ( O ) = e x p ( 2 i ~ n O ) , neT_. (29)

Lie ( S U ( 2 ) ) is generated by a~, a2 and a3 such that

[a3, a l ] = a 2 , [a3, a 2 ] = - a l ,

[ a l , a 2 ] = a 3 . (30)

Let a3 the generator of L ie (H) , then c~. (a3)= ha' where a ' is the generator of Lie (K) = Lie (U ( 1 ) ). In this case Am=0 and the invariant connections are given by

o ~ = n o g ~ n ) a ' . (31)

Next consider eqs. (6) and (7) on the homoge- nous spaces G / H with fg=/g. Eq. (6) implies that g is invariant under the left action of G on G / H , i.e. lg, VgeG, are isometries. Since G acts transitively on G / H and g is invariant under this action, it is enough to know its value evaluated at the origin ell. I f g = he H, then lh(eH) = e l l and eq. (6) gives

f l ( [ a 3 , c l ] , c 2 ) + f l ( c l , [as, c 2 ] ) = 0 , (32)

where a3 e Lie (H) and c l, c2 em. (we have identified m with Tell (G/H) and fl = g at e l l ) . The fl is a bilin- ear symmetric form on m. The invariant metrics on G / H are in one to one correspondence with the solutions fl of (32). For example, using (30) and (32), we can show that there is a unique left invariant metric on S 2 = S U ( 2 ) / U ( 1 ) up to a constant scale. This is a special case of the following more general result: let M be a simply connected symmetric space. If the holonomy of the canonical connection (Am = 0) of M is irreducible, then there is unique left invariant metric on M (up to an overall constant) which is induced from the restriction of the Kill ing-Cartan metric of L ie (G) on m [16]. Note that [Lie(H) , L i e ( H ) ] = L i e ( H ) , [Lie(H) , m ] = r n and Ira, m ] = L ie (H) for symmetric spaces.

Eq. (7) can be solved by using the same argument as in (6). It can be shown that ( D + 1 ) forms T on G / H correspond to skew-symmetric linear maps on m which satisfy the equation

D + I

z(c l , ..., [a, ci], ..., CO+I) = 0 , (33) i = 1

where aeL ie (h ) and ciem, i : 1 .... , D + 1. Apart from ( 33 ), r satisfies the relation

(6~) (Co . . . . , co )

• " = Y~ ( - 1 ) i + J r ( [ c i , c j ] . . . . . , c i . . . . , c j . . . . . c o ) i<j

= 0 , (34)

which is equivalent to d T = 0 for the WZ terms. [ , ] m is the restriction of the Lie brackets [ , ] on m. (34) is a consequence of the Chevalley-Eilenberg theorem that relates the Lie-algebra eohomology [ 17 ] with the cohomology H~' o f invariant forms on a group G. In fact, the operator 6 defined in eq. (34) is the co- boundary operators for the relative Lie-algebra cohomology. Lie-algebra cohomology methods have

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Volume 238, n u m b e r l PHYSICS LETTERS B 29 March 1990

been used in ref. [ 18 ] to study WZ terms that appear in the actions of supersymmetric p-dimensional ex- tended objects.

In order to check if the solutions of eqs. (33) and (34) give topologically non trivial WZ terms, the cohomology H~' ofinvariant forms must be related with the de Rham cohomology H,]R of G / H . If G is compact and connected H~' is isomorphic to H,]g. Indeed, given T~H~R, we can construct a TI EH~' by setting

T~= .f lgTdp(g), (35) G

where d/z is the Haar measure of G. T~ in (35) is closed since lg commutes with the exterior derivative d and T is in the same cohomology class as T~ since G is connected.

The number of couplings of a generic sigma model is infinite. However for invariant models on G / H spaces eqs. (6), (7) and (8) impose severe restrictions on g, T a n d £2. In fact, eqs. (27), (28), (32), (33) and (34) indicate that the number of independent parameters of G / H sigma models is finite. In particular, consider the ( 1, 0) supersymmetric sigma models in two dimensions with action

1 / " i . . 1 S= 2 J goa 'O o .¢ ;y , + ~ f Tij, au~ia~dsx~ke u~x

z £

f gij,~i"[ - V (+=)/~J+ -i f hab• a V_~I b_ +i y x

+½i f RijabA~+AJ+ q/~_ 9/b_ , (36) z

where 2 + are the supersymmetry parameters of the bosons ~ and V ( + ) is a connection on T G / H with tor- sion T. R is the curvature of the connection £2. ~u is the Yang-Mills sector fermion. The fermions 2 and ~u have opposite chiralities. Now if M is an irreducible symmetric space, the number of independent parameters of g is one. Moreover if M is a compact symmetric quaternionic K~ihler manifold (Wolf space) [ 19 ], non-trivial WZ terms do not exist since all the odd Betti numbers of these spaces are zero. Symmet- ric quaternionic K~ihler manifolds play a central role in conformal field theory. For the Yang-Mills sector of the action (36), the number of independent parameters associated with the connection £2, depends on the bundle e.

The left invariant sigma models with (2, 0)-super- symmetries on G / H manifolds can be treated in a similar way. However, the complex structures (re- quired from the additional supersymmetry) of the sigma model manifold G / H and the bundle e must remain invariant under the left action of G. The latter imposes more restrictions on the lifting of the action Ig of G on P ( G / H , K) and the results will be reported elsewhere [20 ].

Another aspect of sigma models with symmetries is the gauging of the symmetry group G [21 ]. How- ever, there are obstructions to gauge rigid symmetries realized non-linearly on the sigma model manifold M even for the standard bosonic sigma model with action ( 1 ) [20]. The obstructions appear when- ever there are no continuous gauged sigma model fields ~. Indeed, after gauging, the fields 6 are sections of the bundle P(Y~, G) XcM where P(Y~, G) is the principal bundle over Z with connection A. The connection A is associated with the gauging of the rigid symmetry G. I f M and Z are not contractible spaces, the bundle P(Z, G)XGM does not always admit global sections. For example, i f M = G / H , then P(Z, G) X aM has a section provided that the structure group of P(Z, G) reduces to a subgroup of H.

In conclusion, we have shown that if the group action is transitive on the target manifold M eqs. ( 6 ) - (8) can be solved and the invariant metrics g, WZ terms T and connections £2 are given by eqs. (32 ) - (34), (27) and (28). If the action of the group G is not transitive on M there does not exist a closed form for the solutions of eqs. (6) and (7), although it is possible to prove that solutions exist if G is compact and connected. Finally, there are solutions for eq. (8) provided that the group action of G on M can be lifted to the principal bundle P(M, K) with connection £2.

I would like to thank P.S. Howe and P. West for useful discussions. This work was supported by the SERC.

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group actions and non-linear sigma models

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