PHYSICAL REVIEW D 15 NOVEMBER 1996VOLUME 54, NUMBER 10

0556-28

Kramer-Neugebauer transformation for Einstein-Maxwell-dilaton-axion theory

Oleg Kechkin* and Maria YurovaNuclear Physics Institute, Moscow State University, Moscow 119899, Russia

~Received 29 April 1996!

The Kramer-Neugebauer-like transformation is constructed for the stationary axisymmetricD54 Einstein-Maxwell-dilaton-axion system. This transformation directly maps the dualizeds-model equations of the theoryinto the nondualized ones. Also, the new chiral 434 matrix representation of the problem is presented.@S0556-2821~96!02620-3#

PACS number~s!: 04.20.Jb, 04.50.1h

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I. INTRODUCTION

Recently, much attention has been given to the studysymmetries admitted by gravitational models appearingsuperstring theory low energy limit@1–5#. As it has beenestablished, one of such models,D54 Einstein-Maxwell-dilaton-axion~EMDA! theory possesses the Sp(4,R) groupof transformations in the stationary case@6,7# and allows theSp(4,R)/U~2! null-curvature matrix representation@8,9#.

Here, we continue to investigate this theory. It is estalished that its stationary equations can be written on the lguage of two symmetric 232 matrix variables in two differ-ent ways. The first is connected with the employment ofscalar matricesP andQ @9# which are the Gauss decomposition components of the Sp(4,R)/U~2! matrixM . The matrixQ is defined by the set of dualized Pecci-Quinn axion, rotion, and magnetic potential variables. The second formalis defined by the same scalar matrixP and the new vector

matrix VW ; both depend only on the original nondualizestring background fields, i.e., on Kalb-Ramond fields, eltromagnetic vector, and the metric ones.

It is shown that the supplementary imposition of the axsymmetry leads to the vanishing of two components of v

tor matrixVW . The remaining third componentV is connectedwith matrix Q by the system of differential equations~thedualization relations!; their compatibility conditions areequivalent to the matrixQ motion equation.

Subsequently, the algebraical complex transformatwhich directly maps the stationary axisymmetric EMDequations, written using the matricesP andQ, into the onesexpressed in terms ofP andV is constructed. It generalizethe Kramer-Neugebauer transformation@10#, of the pure Ein-stein theory, which was useful to Kinnersley@11# in hisGeroch algebra@12# identification.

Also, it is established that the matricesP andV define theGauss decomposition of the new chiral matrixN which is the434 EMDA analogy of the 232 vacuum one. The chiraequation in terms of this last matrix was the Belinsky aZakharov@13,14# starting point for the 2N-soliton solutionconstruction using inverse scattering transform technique

*Electronic address: kechkin@cdfe.npi.msu.su

5421/96/54~10!/6132~4!/$10.00

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II. MATRIX FORMULATIONS OF THE STATIONARYSTRING GRAVITY EQUATIONS

Let us discuss low energy effective four-dimensional action, which describes the bosonic sector of the heterotistring, taking into account the gravitational, electromagneticdilaton, and Kalb-Ramond fields:

S5E d4xugu1/2~2R12]f22e22fF21 13 e

24fH2!,

~1!

whereR5R•••mnmn is the Ricci scalar (R

•••nlsm 5]lGns

m•••) of

the four-metricgmn , signature1222, m50, . . . ,3, and

Fmn5]mAn2]nAm , Hmnl5]mBnl2AmFnl1cyclic.~2!

Here, the scalar fieldf is the dilaton one, andBmn we un-derstand as the antisymmetric tensor Kalb-Ramond field.

Further on, it will be important to introduce the pseudo-scalar axion fieldk:

]mk51

3e24fEmnlsH

nls. ~3!

Below, we will study the stationary case when both themetric and the matter fields are time independent. As it habeen done@15#, the four-dimensional line element can beparametrized according to

ds25 f ~dt2v idxi !22 f21hi j dx

idxj , ~4!

wherei51,2,3. It has been shown before@6# that in this casepart of the Euler-Lagrange equations can be used for thtransition from both spatial components of the vector potential Ai and functionsv i entered in Eq.~4! to the magneticu and rotationx potentials, respectively. The new and oldvariables are connected by differential relations:

¹u5 f e22f~A2¹3AW 1¹v3vW !1k¹v, ~5!

¹x5u¹v2v¹u2 f 2¹3vW . ~6!

The new notationv5A2A0 is entered and the three-dimensional operator¹ is corresponded to the three-dimensional metrichi j . Also, it has been found that varia-tional equations for the action~1! are at the same time Euler-

6132 © 1996 The American Physical Society

54 6133KRAMER-NEUGEBAUER TRANSFORMATION FOR . . .

Lagrange equations for the three-dimensional action

3S5E d3xh1/2~23R13L !. ~7!

Here, 3R is the curvature scalar constructed accordingthree-metrichi j and

3L is the three-dimensional Lagrangiaexpressed in terms off ,x,u,v,f,k which is invariant underthe ten-parametric continuous transformation group isomphic to Sp(4,R). As it was established@8#, 3L can be writtenwith the aid of the four-dimensional matrixM in the form

3L51

4Tr~JM !2, JM5¹MM21, ~8!

whereM , being the matrix of the coset Sp(4,R)/U~2!, hasthe symplectic and symmetric properties

MTJM5J, MT5M , ~9!

and

J5S 0 2I

I 0 D . ~10!

Here, the matrixM is defined by the Gauss decomposition

M5S P21 P21Q

QP21 P1QP21QD , ~11!

where two symmetric matricesP andQ are @9#

P5S f2v2e22f 2ve22f

2ve22f 2e22f D , ~12!

Q5S 2x1vw w

w 2k D , ~13!

wherew5u2kv.It is easy to see that the chiral matrix equation

¹JM50, ~14!

which follows from Eq.~8!, is equivalent to the system

¹@P21~¹Q!P21#50,

¹@~¹P!P211QP21~¹Q!P21#50. ~15!

After the introduction of two matrix currents

JP5~¹P!P21, JQ5~¹Q!P21 ~16!

in terms of which Eqs.~15! can be rewritten as

¹JQ2JPJQ50, ¹JP1~JQ!250, ~17!

for the Einstein equations one has

3Ri j51

2Tr~Ji

PJjP1Ji

QJjQ!. ~18!

ton

or-

As it can be easily verified, Eqs.~17! and ~18! form theLagrange system for the three-dimensional action

3S5E d3xh1/2$23R1 12 Tr@~J

P!21~JQ!2#%. ~19!

This makes two symmetric matricesP andQ, together withthree-metrichi j , the complete set of Lagrange variables forthe stationary system under consideration.

The first relation from Eqs.~15! can be used for introduc-tion of the new vector matrix variableVW

¹3VW 5P21~¹Q!P21, ~20!

which satisfies the equation

¹3@P~¹3VW !P#50, ~21!

as it immediately follows from this definition. Also, the sec-ond ‘‘material’’ equation~15! can be written only in terms ofthe matricesP andVW :

¹@~¹P!P21#1P~¹3VW !P¹3VW 50. ~22!

Subsequently, using the differential relations~3!, ~5!, and~6!, one can solve the Eq.~20! and express matrixVW in termsof the original ~nondualized! metric, electromagnetic, andKalb-Ramond variables:

VW 5S vW 2A2~AW 1A0vW !

2A2~AW 1A0vW ! 2@BW 1A0~AW 1A0vW !#D , ~23!

whereBi5Bi0 ~and all three-dimensional upper indices areconnected with the metrichi j ). As it can be derived from Eq.~3!, the stationary condition allows us to define the remain-ing componentsBi j of the Kalb-Ramond field. Namely, thetime independence ofk is equivalent to the relation

¹CW 5¹~vW 3BW !2@BW 1A0~AW 1A0vW !#¹3vW 1~AW 1A0vW !¹

3~AW 1A0vW !, ~24!

whereCi5 12E

i jkBjk . Thus, the spatial componentsBi j arenondynamical in the stationary case.

It is convenient to introduce the new matrix current

JVW 5P¹3VW . ~25!

The Einstein equations~18! in view of Eq. ~20! obtain theform

3Rik51

2Tr~Ji

PJkP1Ji

VW JkVW !, ~26!

and for Eqs.~21! and ~22! written above we have

¹JP1~JVW !250, ¹3JVW 2JVW 3JP50. ~27!

The last three equations also form the complete Lagrangesystem for the three-dimensional action,

3S5E d3xh1/2$23R1 12 Tr@~J

P!22~JVW !2#%, ~28!

o

6134 54OLEG KECHKIN AND MARIA YUROVA

that means the existence of the alternative non-s-model ma-trix formulation of the stationary EMDA theory.

III. KRAMER-NEUGEBAUER TRANSFORMATIONIN THE STATIONARY AXISYMMETRIC CASE

Now, let us turn our attention to the stationary and asymmetric case, when the three-dimensional line elemcan be chosen in the Lewis-Papapetrou form@11#

~dl3!25e2g~dr21dz2!1r2dw2, ~29!

where the functiong depends on two space variablesr andz. Also, we assume that all the remaining field componeare independent of the angular coordinatew. So, the equa-tions for matricesP andQ can be rewritten in the form

¹@rP21~¹Q!P21#50, ~30!

¹$r@~¹P!P211QP21~¹Q!P21#%50. ~31!

Here and further on, the operator¹ is connected with thetwo-dimensional metric (dl2)

25dr21dz2, and hence, it isequivalent to a usual partial derivative. These equationsthe variational ones for the two-dimensional action

2S5E drdzrTr@~JP!21~JQ!2#. ~32!

The corresponding Einstein equations are transformedthe system of two relations which define the functiong:

g ,z5r

2Tr@Jr

PJzP1Jr

QJzQ#,

g ,r5r

4Tr@~Jr

P!22~JzP!21~Jr

Q!22~JzQ!2#, ~33!

where the components of the matrix currents are definedbefore.

As in the stationary case, Eq.~30! can be used for theintroduction of the new symmetric matrix variableV,

¹V5rP21~¹̃Q!P21, ~34!

where the dual conjugated operator¹̃, for which ¹̃15¹2

and ¹̃252¹1, is entered in accordance to@17#. The directcalculation ofV leads to its evident form and also, as canseen from Eq.~20!,

V5~VW !3 . ~35!

~Here, it was denoted thatx35w,v3[v, etc.! The definition~34! forms the compatibility conditions for the existencethe matrixQ, so that their fulfillment leads to the equatiofor matrix V:

¹@r21P~¹V!P#50. ~36!

The second ‘‘material’’ equation can be also expressedterms ofP andV:

¹@r~¹P!P211r21P~¹V!PV#50, ~37!

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and it is easy to verify that the system~36! and ~37! is theLagrange one for the two-dimensional action

2S5E drdzTr@r~JP!22r21~JV!2#. ~38!

Subsequenty, the differential relation~34! allows one to ob-tain the equations for the functiong in the form

g ,z51

2Tr@rJr

PJzP2r21Jr

VJzV#,

g ,r51

4Tr$r@~Jr

P!22~JzP!2#1r21@~Jr

V!22~JzV!2#%.

~39!

And, so the EMDA system has two alternative descriptionswhich are connected with the using of the matricesQ andV in the stationary and axisymmetric case.

The sets of two formulations~32! and ~33! and ~38! and~39! have the evident formal analogy with the correspondingsystems of equations~with dualized and nondualized vari-ables! of the pure Einstein theory. Namely, the Einsteintheory can be obtained from Einstein-Maxwell-dilaton-axionone in the stationary axisymmetric case with the aid of re-placements

P→ f , Q→x, ~40!

and

V→v ~41!

@and the generalization for the stationary case is connectedonly with the change of Eq.~41! to VW→vW #.

As it has been pointed out by Neugebauer and Kramer forthe Einstein system@10#, the dualized and nondualized equa-tion systems can bedirectly transformed one into anotherwith the help of the discrete~complex! transformation

f→r f21, x→ iv. ~42!

One can check that the formal analogy expressed by the for-mulas~40! and~41! allows one to suggest the correspondingtransformation for the EMDA system. Namely, the transfor-mation

P→rP21, Q→ iV ~43!

directly maps the pair of equations~30! and~31! into the pair~36! and ~37!. By natural reasons we will name it as theKramer-Neugebauer transformation.

For the complete description it is important to understandthe relation, which reflects the functiong behavior under thistransformation. Let us denote this function asgQ when it isconnected withQ and asgV when it is defined byV. As itcan be established by the straightforward calculation, thefunctiongV is connected to the functiongQ as

e2gV5

r

udetPue2gQ. ~44!

54 6135KRAMER-NEUGEBAUER TRANSFORMATION FOR . . .

The difference of this formula from the Kramer-Neugebauone is defined by the 232 matrix dimension of the EMDAdynamical variables.

As it has been written before, the equations of motiallow the Sp(4,R)/U~2! matrix representation. In two-dimensional case, these equations have the form

¹@rJM#50; ~45!

and

g ,z5r

4Tr@Jr

MJzM#, g ,r5

r

8Tr@~Jr

M !22~JzM !2#. ~46!

The same system for Einstein theory is connectedSL~2,R)/SO~2! coset representation and well known in thliterature.

One can try to unite 232 matricesP and V into thesingle 434 one, as it has been done forP andQ with thehelp of the Gauss decomposition. The same procedurpossible in the case of the pure Einstein system~stationaryand axisymmetric!. The established analogy~40! and ~41!immediately allows one to find the such non-s-model matrixrepresentation. Actually, let us define the real symmetric mtrix N as

N5S P 2PV

2VP VPV2r2P21D . ~47!

It is easy to verify that this matrix satisfies the nongroucondition

NJN52r2J ~48!

and the chiral equation for it

¹@rJN#50 ~49!

is equivalent to the pair of equations~36! and~37!. @Here, thematrix currentJN5(¹N)N21 has been entered.# The rela-tion ~48! is the natural generalization of the equalitdetN52r2 for the vacuum case.

The equation such as Eq.~49! was used by Belinsky andZakharov for the construction of the 2N-soliton solution with

er

on

toe

e is

a-

p

y

the help of the inverse scattering transform technique for thestationary axisymmetric Einstein equations. As the form ofEMDA equations is the same as the Belinsky-Zakharov ones,the EMDA system allows one to make the same procedure asin the Einstein theory. The corresponding results will be pre-sented in forthcoming articles.

For the description completeness, it is necessary to re-write the equations~39! defining the metric functiong intothe form where only the matrixN is used. Such form can beobtained with the aid of Eqs.~40! and ~41! from theBelinsky-Zakharov one and it is defined by the relations

G ,z5r

4Tr@Jr

NJzN#, G ,r5

r

8Tr@~Jr

N!22~JzN!2#, ~50!

where

G5g21

2lnudetPu1 lnr. ~51!

The introduced scalar functionG is evidently connected withBelinsky-Zakharov onef ~which is equal toe2gr f21 in ournotation!.

IV. CONCLUSION

In this article we have presented the discrete complextransformation, which generalizes the well-known Kramer-Neugebauer one of the pure Einstein theory to the case of theEMDA system. This transformation directly maps the~dual-ized! Sp(4,R)/U~2! coset representation of the stationary axi-symmetric EMDA equations into the form based on the useof the original~nondualized! string background field compo-nents. As it is established, the nondualized representationalso admits the chiral 434 matrix form which generalizesthe formulation used by Belinsky and Zakharov for the2N-soliton solution construction in the vacuum case. Thesame procedure for the system under consideration will bepresented in forthcoming publications.

ACKNOWLEDGMENTS

This work was supported in part by the Istituo di ScienzeFisiche Grant No. M79000.

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