Volume 240, number 3,4 PHYSICS LETTERS B 26 April 1990

Q U A N T U M C O N S T R A I N T ALGEBRA FOR A C L O S E D B O S O N I C STRING IN A GRAVITATIONAL AND DILATON BACKGROUND

M. D I A K O N O U l, K. FARAKOS 2, G. KOUTSOUMBAS 2 and E. PAPANTONOPOULOS 2 Physics Department, National Technical University of Athens, Zografou Campus, GR-15 7 73 Athens, Greece

Received 15 January 1990

We derive the quantum constraint algebra for a closed bosonic string moving in a gravitational and dilaton background to first order in a ' . The hamiltonian approach is used to directly compute the quantum constraint commutators and calculate the c- and q-number anomalies that arise at the quantum level. The requirement that the algebra preserves the conformal invariance leads to the known background field equations.

1. Introduction

It is well known that all string theories are required to be conformally invariant and this condition im- poses stringent constraints on the background field configuration when we consider motion of strings in the presence of background fields. The way to impose conformal invariance is to demand the vanishing of the associated fl-functions of the theory, which in turn results in the equations of motion that should be sat- isfied by the background fields. In this work we fol- low a hamiltonian approach in quantizing a closed bosonic string in the presence of arbitrary gravita- tional and dilaton fields [ 1,2 ], in contrast to the more conventional lagrangian approach [ 3 ].

At the classical level an arbitrary gravitational field can be introduced without destroying the conformal invariance of the theory. The field G~,~(x) appears in the expression of the constraints in terms of the ca- nonical coordinates and momenta, but does not ap- pear in the constraint algebra [ 1,4 ]. At the quantum level the constraint algebra gets modified. We recall that even in the free case a central charge appears in the algebra [ 1 ]. The presence of the gravitational field modifies the central charge by adding to it an R-term,

Partially supported by the Greek Ministry of Research and Technology.

2 Partially supported by a Bilateral Agreement between Greece and the FRG.

while the requirement of maintaining the conformal invariance of the theory corresponds to Einstein's equations, Ru~ = 0, as expected [ 1 ].

The purpose of our investigation is to extend the previous results including the dilaton field and deriv- ing the full constraint algebra to first order in or' for a closed bosonic string moving in this background. We start our investigation by recalling the action for a closed string in the presence of the gravitational (Gu~) and dilaton (0) fields. We work in the ortho- normal gauge and expand the background fields to first order in c~' using the normal coordinate expan- sion [ 5 ]. At the classical level we write down the en- e rgy-momentum tensor as a function o f the canoni- cal coordinates and momenta, then we check its tracelessness and conservation. At the quantum level considering the commutators of the energy-momen- tum tensor we derive the constraint algebra. The ,8- functions fla and fl, appear as coefficients of the anomalous terms in the algebra [6,3], and setting them equal to zero, the quantum algebra reduces to the classical one. This in turn gives rise to the back- ground field equations.

2. The classical algebra

The action describing the propagation of a string in a background with metric Gu~ and a dilaton field

is given by

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Volume 240, n u m b e r 3,4 PHYSICS LETTERS B 26 April 1990

S=4~.d7~ ~ d2xx/__g(G,,,O~,XaOo, X,,+ot,R~2)¢)

(1)

where R ~ ~ is the scalar curvature of the world sheet and )Co-- r, x~ - a are the world sheet coordinates.

The quantization of the model can only be dis- cussed within the framework of a perturbative method. A well known perturbative method, namely the background field expansion [5], expresses X ~, G, , (X) and ¢(X) as power series in 2 - - ~ . The starting point is the expansion of X ~ around a con- stant field X~ in the target space [ 7 ]:

X~,=X~ +2n , . (2)

Then a change of coordinates is performed, namely n ~' is expressed in terms of the Riemann normal vari- ables, that is the tangent vectors {~ along the geodes- ics of the background field. This is done in order to ensure that the perturbation expansion has manifest reparametrization invariance. The vectors ~" are contravariant, so the coefficients of the expansion of the action are tensors of well defined transformation properties. After integrating out the quantum fluc- tuations, the result will also transform as a tensor un- der arbitrary background field reparametrizations [ 5 ]. In the following we change from the target space indices [denoted by Greek letters in ( 1 ) ] to tangent space ones (denoted by latin letters), using the ap- propriate vielbeins.

We will use the orthonormal gauge in our calcula- tions, that is, we start with the metric

g~a=e~ r/~a (3)

and set <p equal to zero. In this gauge the dilaton term disappears from the action (nevertheless it contrib- utes to the energy-momentum tensor). In anticipa- tion of this fact, we discard from now on the dilaton contribution to the action. The final result for the ex- pansion of the action up to order 2 ~ is [5]

S= [ d2x (½0a~ a O"~ +~2~R,,~,,.a O ' ~ O , ~ a ~ ') •

(4 )

The energy-momentum tensor is defined as

2 8S (5) T.~ = , / ~ a g ~ .

We write the energy-momentum tensor as a sum:

T,~a = --~aTeR~ + T ~ , (6)

where T(R) is the part due to the metric and ~a'r<D) is .t a B

due to the dilaton. T ~ ~ reads, in the basis of x_+ =

(Xo+-X,),

T<R~ rO~)=0 (7) + _ ~ - - - ~ _ +

Since we want to perform the canonical quantiza- tion of the theory, we need the conjugate momentum

of~":

P'~= OL ~ar3,,+,221R,, +R axX.b~c'l 0-~=~" t ~ ~- t ~a ~h~ , , , , , J (8)

for which the fundamental Poisson bracket reads

{~"(r, a), P~(r, o") } =r /~b~(a-a ') . (9)

We then define two quantities related to the momen- tum (0~"/0a= ~'~):

p~ = ~ (e~+~,a) . (10)

Expressing the time derivative of~" in terms of P% T~r_2 becomes

1126 Da Od .~b.~c ( 11 ) T~R_2 =P~ P+_~ -r,^ *,~t,a-+ - - ~ ~ ,

where l~,,b,.a = R,~a + Rd~.,. The energy-momentum tensor defined above has

the properties expected, that is, it is traceless ( r /~aT~ _ T(R~ =0 ) and conserved /a~-r~R~ t,' *aa =0) . The first equality comes from the definition of T~R~ a + _ ,

while the second is implied by the equation of motion. To begin with, we calculate the classical algebra for (.R). T_. ± The result reads

~'~ o")l , = + i ~ ' ( a - o ' ) [ T ~ ( t , a ) + T + 2 ( t ,

{T~R~(r, a), T <R' - . + + _ _ ( . , o - ' ) } = 0 (12)

One expects that the quantum algebra should consist of this quantum algebra plus possible anomalous

terms. On the other hand, the dilaton part of the energy-

momentum tensor is given by

T~D) 1 = _ ~ (0,~ 0aO_r/~a 0~0). (13) ot~ 2n

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Expanding the dilaton field O ( X ) in terms of the quantum field ~ ,

O(X) = ~ ( X o ) +).Dafb(Xo)~a+ ½).2D~Db0(Xo)~ b , (14)

the dilaton part of the energy-momentum tensor, T ~o_+), reads

+ Da~O_+O_+~a+ ~DaDb~O+O+(~a~ b)

"/" ~Dx) = 0 . (15)

The above expression for T~D_2 may be rewritten in terms of the momenta as follows:

± _ D~OP'~ + ~ DaDbO¢~P b

- ~ D~ DbOP~ pb_+. ( 16 )

The part T ~ implies already at its lowest order the inclusion of quantum effects, since it is of order)., so there is no point in calculating its classical algebra, as was done before for the T ~r_2 part.

3. T h e q u a n t u m a lgebra

In order to find the quantum algebra, we have to consider the quantities appearing in the expressions of T_+ ± as quantum operators. Since we have to deal with finite quantities, we consider the normal prod- uct of the operators. However, due to the fact that we encounter operators defined at the same space-time point, the finite part of the energy-momentum ten- sor is not just the normal product. It also contains regularization terms, which arise by applying a pro- cedure of point splitting in order to correctly define the product of operators at the same point. Through this procedure we find that the regularized energy- momentum tensor is given by

)2 02 :~a~b: (17) T ± ± = : T ± ± : - ~ - ~ R,~t,-~O2

in order to calculate the quantum algebra, we need the contractions of the fields appearing in 7~_+ _+. We

use the free field contractions in accordance with the canonical equal time commutators:

[~a(z, e), Ph(r, e ' ) ] = i q a b 6 ( e - - a ') , (18)

which correspond to the Poisson brackets used in the classical case [ cf. eq. (9) ].

The contractions are easily found to be

1 ab 1 (Pa('c'o)~b(T'O'))=+ 4-~ rl J_+ic '

1 r/,~ b 1 ( e ~ ( r , e ) P t ~ _ ( r , e ' ) ) = - ~ n (A + i~ )------------ ~ ,

(e~. (r, e ) e b_ (r, e ' ) ) = 0 . (19)

We have denoted by A the difference e - e ' . The following relation ensures that the previous

contractions are consistent with the canonical com- mutation relations and is useful in manipulations with the commutators:

( A + i ~ ) - ' - ( A - - i ~ ) - I = - 2 ~ i 6 ( A ) . (20)

It turns out that the algebra closes if the regularized energy-momentum tensor /~± ±, eq. (17), is used. However, the enclosure of the quantum algebra that is consistent with the Bianchi identity (see further discussion below) demands the inclusion of one more counterterm in the energy-momentum tensor. This counterterm is thus dictated totally by geometrical reasons. The two components of the energy-momen- tum tensor, with the inclusion of both counterterms, become

).2 a2 ~ : #±._ = :T±± : - 4 ~ n R o ~ :

).2 ( 3 . ~ b. +- -R~h .P±~ . . (21) - 6 ~ ao

The resulting quantum algebra for the final energy- momentum tensor is contained in the next three formulae:

± (r, ¢'± ± (r, a ' ) ]

= +i6' (A) [?,_ ±(t , a) + ? ± ± (t, a ' ) ]

i - 2 6 ' (d) [Z(r , a) + Z ( r, e ' ) ] + - ~ n 6 " ( A ) D

i). 2 +_ 32n2 6 " ( A ) ( R - 4 D a O D a 0 - 4 D ~ D ~ ) , (22)

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[ f'+ ÷ (r, a) , ~__ (r, a ' ) ]

= 6 ( d ) [ Z ' ( r , a ) + Z ' ( z , a ' ) ] . (22 cont 'd )

The q-type Schwinger term Z is the generalization of the function introduced in ref. [ 1 ] when the dilaton field is also present. The expression for Z is

Z = I ~,~ w, (R~b-2D~Dh~) . (23) 24n - + - -

We recall that the term ( i D / 2 4 n ) 6 " ( A ) is cancelled by ghost contributions at D = 2 6 . If we demand the preservation of the classical algebra the vanishing of the coefficients o f the anomaly and the Z-function terms gives two equations for the background fields:

R~b -- 2D~ D~,O = 0 , (24)

R - 4 D ~ D~g~-4D~D~0=0 . (25)

We note that the coefficients of these terms corre- spond to the fl-functions tic and fl~ of the lagrangian approach. These two equations are not independent, since the second may be obtained from the first one through the use o f the Bianchi identity for the Ricci tensor: D~R~a= ~ DbR. We note that the counterterms introduced above are the ones which make the two cquations (24) and (25) consistent, if one takes into account the Bianchi identity.

4. Conclusions

The hamiltonian approach has been used in this work to compute the quantum constraint algebra of a closed bosonic string in the presence o f gravita- tional and dilaton background. This method has the advantage of calculational ease over the equivalent Lagrange formulation. The method can be extended to higher orders in a ' and include string propagation in an antisymmetric tensor background as well. An- other generalization is the inclusion of fermionic de- grees o f freedom in the string, that is, investigation of superstring motion in background fields [ 1 ].

The canonical quantization method for con- strained systems, in spite of its conceptual simplicity, meets with difficulties, when the quantum version of the constraints is considered. More specifically, in our invcstigation we had to modify the energy-momen- tum tensor in two ways. First of all we used the point

splitting method to define a meaningful normal product o f operators at the same point. This proce- dure resulted in the T± ± modification of Tt ±, eq. (17). Another modification comes from purely geo- metrical arguments, namely consistency with the Bianchi identity, eq. (21 ). Similar problems become more severe when one computes the contribution of the antisymmetric tensor in the algebra to order or' or the constraint algebra o f the metric tensor to order a '2. A detailed discussion for these cases will be pre- sented in a future publication.

We conclude by making contract with the previous work on the subject. If we set our ~ equal to zero, we agree with the results o f ref. [ 1 ] (setting their fer- mion contributions equal to zero). There only the re- gularization term was used and this has not caused any consistency problems with the Bianchi identity, because the dilaton field was missing. The relations (24) and ( 25 ) are in agreement with the background field equations of ref. [6] (after setting their anti- symmetric tensor contribution equal to zero).

Acknowledgement

We would like to thank the CERN theory division as well as the DESY theory group, where part of this work has been done, for their hospitality. Enlighten- ing discussions with S. Fubini and G. Vencziano are gratefully acknowledged. We would also like to thank I. Anloniadis, C. Bachas, E. Floratos and C. Kounnas for useful discussions.

References

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[4] J. Maharana and G. Veneziano, Nucl. Phys. B 283 (1987) 126.

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