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Gauging of Chern-Simons $p$-branes

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IC/93/236Gauging of Chern-Simons p -Branes Raiko P.Zaikov ??International Centre for Theoretical Physics Strada Costeiera 11,P.O.Box 11,Trieste,Italy Abstract.The Chern-Simons membranes and in general the Chern-Simons p-branes moving in D -dimensional target space admit an in?nite set of secondary constraints.With respect to the Poisson bracket these constraints form a closed algebra which contains classical W 1+∞algebra in p -dimensions as a subalgebra.Corresponding gauged theory in the phase-space is constructed in a Hamilton gauge as an analog of the ordinary W -gravity.7/93

In the previous article[1]it was shown that in the Chern-Simons membrane case there always appears an in?nite set of secondary constraints in contrast to the C-S string theory [2]in which there are two possibilities for the?rst class constraints:there is a?nite or an in?nite number of secondary constraints.There is also another,rather formal,possibility when second class constraints appear also(see Refs.1and2).

When there appears an in?nite set of secondary constrains for the C-S string,they satisfy an in?nite algebra with respect to the Poisson bracket.This algebra contains as a subalgebra the classical(without central term)a?ne SL(2,R)algebra,as well as the classical Virasoro algebra and their higher spin extensions which contain classical W1+∞algebra[4].Through this paper we are dealing only with classical in?nite algebras.

In the case of C-S membrane the in?nite set of constraints gives a linear realization of higher spin extended algebra in two-dimensions which contains a?ne SL(2,R),Virasoro and W1+∞-algebras in two dimensions as subalgebras1[1].We note,that any of these algebras can not be represented as a direct product of two in?nite algebras in one dimension as in the two-dimensional conformal theory[5]–[7]and their W∞extensions2.We have to take into account also that in the C-S membrane theory we are dealing with two spatial dimensions while the time variable appears only as evolution parameter which is not the case on the ordinary2D conformal theory.The generalization of the results for arbitrary C-S p-branes is strigtforward.In that case we have higher spin extension of the a?ne SL(2,C)in p-dimensions.

The polynomial Chern-Simons p-brane action was obtained in[8]from the topological (p+1)-brane action(only for D=p+2)[9]–[12]in the same way as in the ordinary local theory.In the paper[13]a generalization for an arbitrary space-time dimension was found.

In the present article the gauged C-S mebrane theory is constructed in the Hamiltonian approach considering the Lagrange multiplyers as a gauge?elds with arbitrary spin.The W1+∞transformation properties of the gauge?elds are obtained.We note that the”No go theorem”in the case of spin>2(see Ref.[14])does not take place because we are dealing with an in?nite sequence of higher spin gauge?elds.Through this paper we use

basis in which not all the constrains are independent and as a consequence there exists an additional symmetry of Stuckelberg type[15].This symmetry allows us to exclude the corresponding gauge?elds(with odd spin)by means of gauge?xing procedure.Integrating over the momentum variables we?nd W1+∞gauge invariant action on the con?guration space.We remaind that the Lagrangian approach to the W-gravity was considered in a lot of papers among the?rst of which are[16]and[15].

To proceed further we shall remind bri?y some results from the papers[8]and[13], where in order to generalize the C-S p-brane to arbitrary space-time dimension the following notation were introduced:

X,A=?A X=

X if A=?X,a=?a X if A=a,

(a=0,1,2,...,p).The polynomial Lagrangian for the C-S p-brane is given by L=detXμ,A which exists in D=p+2only.To extend this action for target space with arbitrary dimension,a generalized induced metric tensor is introduced

g AB=Xμ,A Xν,Bημν,(1) whereημνis the pseudoeuclidean metric tensor.Formally replaciment of the ordinary induced metric tensor in the Nambu-Goto action with the generalized metric tensor given by Eq.(1)gives the action for the generalized C-S p-branes wich lives on a target space with arbitrary dimension

S=κ dτd pσ

the corresponding ordinary(p+1)-brane Lagrangian and(p+1)-brane constraints substi-tuting?σ

p+1

X by X.We recall,that the ordinary bosonic string has only two(bilinear) constraints while the C-S string has three primary costraints,one of which is of degree four with respect to X.

The appearance of the constraintφ?shows us that some residual symmetry from the p+2-variable di?eomor?sms(under which the action of the p+1-brane is invariant)sur-vives.As a consequence of the appearance of the constraintφ?there arise some secondary constraints too.In the C-S particle case we have only one secondary constraint,while in the C-S string case there are two possibilities:four(three primary constraints and one secondary constraint)?rst class constraints[3]or an in?nite set of?rst class constraints[2]. We note that in the latter case not all of the constraints are independent if we deal with ?nite dimensional target space.Hence we have not dynamical degree of freedom.In that case the dynamical degrees of freedom can take place only if we have in?nite dimensional target space.

For any C-S p-brane the canonical Hamiltonian vanishes identically,i.e.

H0=P˙X?L≡0,(4) which is a property of the ordinary p-brane theory also.

The analyze of the constraint algebra in the case of C-S membrane shows us that there is an in?nite series of secondary constraints[1].An appropriate choice of these constraints is the following:

Ψm,n= P?mσ1?nσ2P ≈0,

Φm,n= X?mσ1?nσ2X ≈0,

Γm,n= P?mσ1?nσ2X ≈0,(m,n=0,1,...).(5)

We note that,as it was mentioned above,if D is?nite we have only a?nite number of independent constraints(5).However,when we are dealing with in?nite dimensional target space it is easy to check that all the constraintsΓare independent as well as those of the constraintsΨandΦfor which m+n=2k(k=0,1,...).To prove the latter statement we use the following identity

(?mσ

1?nσ

2

XY)=

m

p=0n q=0(?)m+n?p?q m p n q ?pσ1?qσ2(X?mσ1?nσ2Y)(6)

which consequend from the Laibniz formula.

Using the Eq.(6)we obtain that the constraintsΨandΦwith arbitrary odd spin can be represented in terms of the constrains with all underlying spins:

1

(XX2k?l+1,l)=

3These W

1+∞-algebras di?er from the ordinary W1+∞-algebras because they are not mutually commuting.

in two dimensional space.Indeed,the Poisson bracket of Γwith the coordinate X μand with the momenta P μgive the transformation laws for the phase space coordinates:

δk,l ΓX μ={Γk,l f ,X μ}P B =?f?k σ1?l σ2

X μ,δk,l ΓP μ={Γk,l f ,P μ}P B =(?)k +l k p =0l q =0

k p l q ?p σ1?q σ2f?k ?p σ1?l ?q σ2P μ.(10)In the same way we obtain also:

δk,l ΦX μ={Φk,l f ,X μ}P B =0,δk,l Φ

P μ={Φk,l f ,P μ}P B =?f?k σ1?l σ2

X μ?(?)k +l k p =0l q =0 k p l q ?p σ1?q σ2f?k ?p σ1?l ?q σ2X μ,δk,l ΨX μ={Ψk,l f ,X μ}P B =f?k σ1?l σ2

P μ(?)k +l

k p =0l q =0 k p l q ?p σ1?q σ2f?k ?p σ1

?l ?q σ2P μ,δk,l ΨP μ={Φk,l f ,P μ}P B =0.(11)

Consequently,δ1,0Γand δ0,1Γare ordinary di?eomorphisms in two-dimensional space.We note,that the assymmetry which appears in the transformation laws of the coordinate X and momentum P is a consequence of the assymetric choise of the constraint basis (5).Taking into account the identity (6)a more symmetric basis can be obtained for the constraints (5)by a simple rede?nition

Λm,n → Λm,n =m

p =0n q =0b mn pq ?p σ1?q σ2Λ

m ?p,n ?q ,where b are constants.By a suitable choise of b the classical algebra (9)can be deformed to an algebra which admits diagonal central extension [4]also,at least for the W 1+∞subalgebra.

Because of the vanishing of the canonical Hamiltonian given by Eq.(4)the ?rst order action can be writen in the form:S = dτd 2σ P ˙X ?αmn XX (m,n ) ?βmn PP (m,n ) ?γmn P X (m,n ) ,(12)

where U (m,n )=?m σ1?n σ2

U .In order to gauge the action givev by Eq.(12)we consider the lagrange multiplyers α,βand γas ?elds depending on the evolition parameter τalso.

Then using the transformation laws for the phase-space coordinates given by Eqs.(10)and (11)with

τdepending

parameters f we obtain the transformation laws for the gauge ?elds:

δΓαmn = k,l ≥0k,l

r,s =0(?)r +s k r l s ?r σ1?s σ2 f kl αm ?k +r,n ?l +s + k,l ≥0k,l r,s =0

k r l s αkl ?r σ1?s σ2f m ?k +r,n ?l +s ,(13)δΓβmn =? k,l ≥0k,l

r,s =0(?)k +l k +r r l +s s f kl ?r σ1?s σ2βm ?k +r,n ?l +s ? k,l ≥0k,l r,s =0(?)k +l k r l s βkl ?r σ1?s σ2f m ?k +r,n ?l +s ,(14)

δΓγmn =?˙f mn ? k,l ≥0k,l r,s =0

k r l s f kl ?r σ1?s σ2γm ?k +r,n ?l +s ? k,l ≥0k,l r,s =0 k r l s γkl ?r σ1?s σ2f m ?k +r,n ?l +s ,(15)

We note,that the action (12)is invariant only with respect to the gauged W 1+∞alge-bra in two dimensions.It is not invariant with respect to the local gauge transformations

(11).

In order to write down the action (12)on the con?guration space we exclude the momentum variables by means of the equation:

δL

2 m,n ≥0 βmn ?m σ1?n σ2+m,n p,q =0?p σ1?q σ2βmn ?m ?p σ1

?n ?q σ2 ,(18)

and the derivatives act on the right.From Eq.(17)we obtain

P μ=Q ?1 ˙X μ? m,n ≥0αmn X m,n μ .

(19)

Incerting the momentum from(19)into(12)we?nd

λ

L=

of Stukelberg type[15]

?δ?α

2M?m+1,m

=u2M?m+1,m,

?δ?α

2M?m,m =? K,l C K,l M,m?2(K?M)?l+m

σ1

?l?m

σ2

u2K+1,l,

?δ?γ

2M?m+1,m

=v2M?m+1,m,

?δ?γ

2M?m,m =? K,l C K,l M,m?2(K?M)?l+m

σ1

?l?m

σ2

v2K+1,l,

(22)

where u m,n and v m,n are arbitrary functions.This invariance allows us to choose the folowing gauge?xing

?α2K?l+1,l=0,

?γ2K?l+1,l=0.

(23) In this gauge the even spin quantities(˙X˙X2K?l+1,l)and(XX2K?l+1,l)are canceled in the action(20).Then the Lagrangian became

L=

λ

References

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preprint INRNE-TH/3/93,hep-th/9304075

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