quaternionic taub-nut from the harmonic space approach

31 December 1998

Ž .Physics Letters B 445 1998 60–68

Quaternionic Taub-NUT from the harmonic space approach

Evgeny Ivanov a,1, Galliano Valent b,2

a BogoliuboÕ Laboratory of Theoretical Physics, JINR, Dubna, 141 980 Moscow region, Russiab Laboratoire de Physique Theorique et des Hautes Energies, Unite associee au CNRS URA 280, UniÕersite Paris 7, 2 Place Jussieu, 75251´ ´ ´ ´

Paris Cedex 05, France

Received 18 September 1998Editor: L. Alvarez-Gaume

Abstract

We use the harmonic space technique to construct explicitly a quaternionic extension of the Taub-NUT metric. It dependson two parameters, the first being the Taub-NUT ‘mass’ and the second one the cosmological constant. q 1998 ElsevierScience B.V. All rights reserved.

1. An efficient way to explicitly construct hyper-Kahler and quaternionic-Kahler metrics is provided by the¨ ¨Ž . w xharmonic super space method 1–4 .

w xIt was firstly introduced in the context of Ns2 supersymmetry 1 . The basic idea was to extend theŽ . " i qi ystandard Ns2 superspace by a set of internal ‘harmonic’ variables u ,u u s1, parametrizing thei

Ž . w xautomorphism group SU 2 of Ns2 superalgebra. It was shown in 1 that all Ns2 theories admit amanifestly supersymmetric off-shell description in terms of unconstrained superfields given on an analyticsubspace of the Ns2 harmonic superspace, harmonic analytic superfields.

It was soon realized that the harmonics are also relevant to some purely bosonic geometric problems. As isw x Ž .shown in 3 , the constraints defining the hyper-Kahler HK geometry can be given an interpretation of the¨

Ž .integrability conditions for the existence of analytic fields in a SU 2 harmonic extension of the original� im4 Ž . Ž .4n-dimensional HK manifold x , is1,2;ms1, . . . ,2 n . This time, the SU 2 to be ‘harmonized’ is an extra

Ž .SU 2 rotating three complex structures of the HK manifold. The analytic subspace is spanned by the harmonicvariables u" i and half of the initial x-coordinates, xqm . The constraints of HK geometry can be solved via an

q4Ž qm " i. Ž .unconstrained analytic HK potential LL x ,u . It encodes at least, locally all the information about thew xassociated metric. Remarkably, it allows one to explicitly construct the HK metrics by simple rules 3 .

1 E-mail: [email protected] E-mail: [email protected]

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01409-9

( )E. IÕanoÕ, G. ValentrPhysics Letters B 445 1998 60–68 61

w x Ž .In 4 , a generalization of this approach to the quaternionic-Kahler QK manifolds was given. These¨Ž .manifolds generalize the HK ones in that the extra SU 2 which transforms complex structures becomes an

w xessential part of the holonomy group. It was shown in 4 , that the QK geometry constraints can be also solvedq4 Ž .in terms of some unconstrained potential LL living on the analytic subspace parametrized by SU 2

harmonics and half of the original coordinates. The specificity of the QK case is the presence of a non-zeroŽ . q4constant Sp 1 curvature on all steps of the way from LL to the related metric. It is interesting to consider

w xsome examples in order to see in detail how the machinery proposed in 4 works. Only the simplest case of theŽ . Ž . Ž . Ž q4 . w xhomogeneous QK manifold Sp nq1 rSp 1 =Sp n corresponding to LL s0 was considered in Ref. 4 .

The aim of this paper is to demonstrate the power of the harmonic geometric approach on the example of lessŽ . w xtrivial QK metric, a quaternionic generalization of the well-known four-dimensional Taub-NUT TN metric 5 .

w x Ž .Like in the HK case 2 , the computations are greatly simplified due to the U 1 isometry of the quaternionic TNŽ .metric. The metric depends on two parameters, the TN ‘mass’ parameter and the constant SU 2 curvature

parameter which can be interpreted as the inverse ‘radius’ of the corresponding ‘flat’ QK backgroundŽ . Ž . Ž .;Sp 2 rSp 1 =Sp 1 . We end up by performing the identification of the metric with other forms given in the

literature.

w x2. We first recall some salient features of the construction of 4 . One starts with a 4n-dimensional Riemann� mm4manifold parametrized by local coordinates x ,ms1,2, . . . ,2 n; ms1,2, and uses a vielbein formalism. The

QK geometry can be defined as a restriction of the general Riemannian geometry in 4n-dimensions, such thatŽ . Ž . 3the holonomy group of the corresponding manifold is a subgroup of Sp 1 =Sp n . Thus one can choose the

Ž . Ž .tangent space group from the very beginning to be Sp 1 =Sp n and define the QK geometry via appropriateŽrestrictions on the curvature tensor lifted to the tangent space taking into account that the holonomy group is

. w xgenerated by this tensor . As explained in 4 , for the QK manifold of generic dimension the defining constraintscan be written as a restriction on the commutator of two covariant derivatives

DD sy2V RG . 1Ž .a i , DDb k a b Ž i k .Ž .

Here

Emm m mDD se x DD se x q Sp 1 =Sp n yconnections , 2Ž . Ž . Ž . Ž . Ž .a i a i m m a i m mE x

mmŽ .e x being the 4n=4n vielbein with the indices as1,2, . . . ,2 n and is1,2 rotated, respectively, by thea iŽ . Ž . Ž .tangent local Sp n and Sp 1 groups, V is the Sp n -invariant skew-symmetric tensor serving to raise andab

Ž . Ž bg g . Ž .lower the Sp n indices V V sd , G are the Sp 1 generators, and R is a constant, remnant of theab a Ž i k .Ž . ŽSp 1 component of the Riemann tensor its constancy is a consequence of the QK geometry constraint and

.Bianchi identities . The scalar curvature coincides with R up to a positive numerical coefficient, so the casesŽ .R)0 and R-0 correspond to compact and non-compact manifolds, respectively. In the limit Rs0 Eq. 1 is

w xreduced to the constraint defining the HK geometry 3 , in accord with the interpretation of HK manifolds as adegenerate subclass of the QK ones.

w x Ž .Like in the HK case 3 , in order to explicitly figure out which kind of restrictions is imposed by 1 on themmŽ .vielbein e x and, hence, on the metrica i

g mm n s se mm en s a i , g se ea i , 3Ž .a i m m n s m m a i n s

Ž .one should solve the constraints 1 by regarding them as integrability conditions along some complex directionsin a harmonic extension of the original manifold.

3 Ž .For the 4-dimensional case this definition has to be replaced by the requirement that the totally symmetric part of the Sp 1 componentof the curvature tensor lifted to the tangent space is vanishing.

( )E. IÕanoÕ, G. ValentrPhysics Letters B 445 1998 60–6862

Ž .Due to the non-vanishing r.h.s. in 1 , the road to such an interpretation in the QK case is more tricky.� mm4Modulo these peculiarities, the basic step still consists in extending x by a set of some harmonic variables,

� mm4 � mm "4 qi y w x Ž .x ™ x ,w , w w s1. Then, following the general strategy 3,4 , one passes to a new ‘analytic’i i� mm "4basis in x ,wi

x mm ,w" ´ xqm , xym ,w" i 4Ž .� 4 � 4i A A A

x " m sx m i w" qÕ" m x ,w , wqi swqi yRÕqq x ,w wyi , wyi swyi , 5Ž . Ž . Ž .A i A A

" mŽ . qqŽ . qwhere the ‘bridges’ Õ x,w ,Õ x,w are chosen so as to make the w -projection of DD in this basis to bea i

proportional to the partial derivative with respect to xym

Eq qi qi m m m m qDD ;w DD sw e x E q . . . sE x ,w sE x ,w E 6Ž . Ž . Ž . Ž .a a i a i m m a a mymE xA

Ž Ž .simultaneously, one performs an appropriate Sp n rotation of the tangent space index a by a matrixŽ . . qSp n -‘bridge’ . The possibility to reduce DD to this ‘short’ form amounts to the possibility to define analytica

� qm " i4fields living on the analytic subspace x ,w . The original QK geometry constraints prove to be equivalentA Aw xto the existence of such analytic fields and subspace 4 . An essential difference of the QK case from the HK

w x qqcase 3 is the necessity to shift the harmonic variables with the new bridge Õ .q � mn4Besides the opportunity to make DD short, the passing to the harmonic extension of x and further to thea

Ž .analytic basis and frame ‘the l-world’ allows one to exhibit the fundamental unconstrained objects of the QKŽ .geometry, the QK potential. While in the original formulation ‘the t-world’ the basic geometric objects are the

mmŽ . Ž .vielbeins e x properly constrained by Eq. 1 , in the analytic basis such objects are the harmonic vielbeinsa k

covariantizing the derivatives with respect to the harmonic variables. In the original basis the harmonic"" "" " i . i 0 0 qi qi yi yi w qq yyx 0derivatives are D sE sw ErE w , D sE sw ErE w yw ErE w , E ,E sE , i.e. theyw w w w

mn � mm "4contain no partial derivatives with respect to the variables x , because the harmonic space x ,w has thei� mm4 � "4 Ž .structure of the direct product x m w . After passing to the analytic basis by Eq. 5 , the derivativesi

D"" acquire terms proportional to E " 'ErE x . m. Besides, in Dqq there emerges a term proportional to Eyy.m wA

These new terms appear with the appropriate vielbein components Hq3 m, Hyy" m, Hq4 which are related tothe bridges as follows

Eqq qRÕqq xqm sHq3 m , 7Ž .Ž .w A

Eqq qRÕqq ÕqqsyHq4 , 8Ž .Ž .w

1yy " m yy" mE x sH . 9Ž .w Ayy qq1yRE Õw

Note that xym is determined in terms of xqm by the equation

Eqq yRÕqq xym sxqm . 10Ž .Ž .w A A

The original QK geometry constraints require Hq3 m, Hq4 to be analytic

EqHq3 m sEqHq4 s0 ´ Hq3 m sHq3 m xq ,w , Hq4 sHq4 xq ,w , 11Ž .Ž . Ž .m m A A A A

and express Hq3 m in terms of Hq4. Basically, the analytic harmonic vielbein Hq4 is just the unconstrained QKq4 w x q4 Žpotential. To be more precise, the QK potential LL , as it was defined in 4 , is related to H as after

.properly fixing the l-world gauge freedom

Hq4 xq ,w sLLq4 xq ,w qxqHq3 m xq ,w , xq 'V xqn , 12Ž .Ž . Ž . Ž .A A A A m A A m mn


and

1q3n nm y q4 y y q yyˆ ˆH s V E LL , E 'E qRx E . 13Ž .m m m m A2

It can be shown that the only constraint to be satisfied by LLq4 is its analyticity, so this object encodes allthe information about the relevant QK geometry and metrics, whence its name ‘QK potential’. Choosing one or

q4 Ž . Ž . Ž . Ž . " m qqanother explicit LL , and substituting 12 , 13 into Eqs. 7 – 10 , one can solve the latter for x and Õ

as functions of harmonics and the t-world coordinates x mm. Having at hand the explicit form of the variableŽ . q4change 5 , it remains to find the appropriate expression of the l-world vielbeins in terms of LL in order to

be able to restore the t-world vielbein and hence the QK metric itself.Ž w x.Skipping intermediate steps they can be found in 4 , the non-vanishing components of the l-world inverse

QK metric are given by the following expressions

y1 n y1 v y1 Ž mmqny nymq m r myny rs q y3n .ˆ ˆ ˆ ˆg sg sV E H , g sy2 V E H E H E H , 14Ž .Ž . Ž . Ž .Žl. Žl. r Žl. s r v

where

1mq yyqm yy" m yy" m q yyqmˆ ˆ ˆE H 'E H , H ' H , xPH'x H . 15Ž .Ž . n n m1yR xPHŽ .

Ž .Then the t-world metric can be obtained via the change of variables inverse to 4

mm n s vysy q m m q n s vqsy y m m q n s y n m q m sg sg E x E x qg E x E x qE x E x . 16Ž .ž /Žl. v s Žl. v s v s

Ž .In the case of 4-dimensional QK manifolds we will deal with in the sequel m,ns1,2 the t-basis metricŽ .16 , after some algebra, can be put in the form

1 1mm n s m m n sg s G , 17Ž .yy qqˆ 1yR xPH 1yRE ÕŽ . Ž .det E HŽ . w

mm n s l r yy qm m qn sG se E X X q mmln s . 18Ž . Ž .w l r

Here

Xqm m 'Eqx mm 19Ž .r r

are solutions of the system of algebraic equations

Xqm m = xyn sEqxyn sd n , Xqm m = xqn sEqxqn s0 , 20Ž .r m m r r r m m r

Rqq yy= 'E q E Õ E . 21Ž .Ž .mm m m m m wyy qq1yRE Õw

Ž . Ž .As we see, the problem of calculating the QK metric 17 , 18 is reduced to solving the differentialŽ . Ž . Ž . q4 Ž qm " i. " m qqequations 7 , 10 , 8 which define, by the known LL u x ,w , x and Õ as functions of theA

t-basis coordinates x mm and w" i. In general, it is a difficult task. However, it is simplified for the QK metricsw xwith isometries, like in the HK case 2 . We will demonstrate this on the example of the QK analog of the

Taub-NUT metric.

q4 w x3. The QK counterpart of the TN manifold is characterized by the same LL 6

2 2q4 q q qqLL s 2 il x x ' f . 22Ž . Ž . Ž .


Here we introduced the notation 4

q1 q2 q q q qq qx , x s x ,yx , x s x , x syx . 23Ž . Ž . Ž . Ž . Ž .We also assume

qqqqlsl ´ f sf . 24Ž .Ž . Ž . Ž .The basic equations 8 , 7 , 10 for the given case take the form

2 2qq qq qq qqE Õ qR Õ s f , 25Ž . Ž . Ž .EqqqRÕqq xqs2 il xqfqq , 26Ž . Ž .EqqyRÕqq xysxq 27Ž . Ž .

Ž .together with their conjugates . These equations are covariant under two rigid symmetries preserving theq q " i� 4 Ž . Ž .analytic subspace x , x ,w : U 1 Pauli-Gursey PG symmetry¨A

X Xq i a q q yi a qx se x , x se x , 28Ž .Ž . Ž " qqand SU 2 symmetry which uniformly rotates the doublet indices of the harmonic variables x and Õ are

Ž .. Ž .scalars with respect to this SU 2 . They constitute the U 2 isometry group of the QK TN metric.Ž .We will firstly solve Eq. 25 . Defining

qq qq R Õ q q qq q q 2 qqˆÕ sE Õ , v'e , x 'v x , f s2 il x x sv f , 29Ž .ˆ ˆ ˆŽ . Ž .we rewrite 25 , 26 as

2qqfŽ .2qqE vsR , 30Ž . Ž .3v

ˆqqfqq q q q qqE x s2 il x '2 il x k . 31Ž .ˆ 2v

ˆqqŽ .From Eq. 31 and the definition of f one immediately finds

qq ˆqq ˆqq ˆ i k q qE f s0 ´ f sf x w w . 32Ž . Ž .i k

Ž .We observe that Eq. 30 coincides with the pure harmonic part of the equation defining the Eguchi-Hansonw x w xmetric in the harmonic superspace approach 8 . Its general solution was given in 8 , it depends on four

arbitrary integration constants, that is, in our case, on four arbitrary functions of x m i. However, these harmonicŽ . Ž .constants turn out to be unessential due to four hidden gauge symmetries of the set of equations 25 - 27 . One

X Ž . Ž .of them is the scale invariance Õ sÕqb x , while three remaining ones form an extra local SU 2 symmetryw x7 . Using this gauge freedom one can gauge away four integration constants in v and write a solution to Eq.Ž .30 in the following simple form

12 2ˆ ˆ(vs 1qRf ´ Õs ln 1qRf , 33Ž .Ž .

2 R

ˆ ˆqqf fqq qq Ž i k . q yˆ ˆÕ sE Õs , f'f x w w . 34Ž . Ž .i k2ˆ1qRf

w x Ž .One can restore the general form of the solution as it was given in 8 , acting on 33 by a finite form of thew xaforementioned hidden symmetry transformations. In 7 we demonstrate that the whole effect of the full gauge

4 We adopt the convention e sye 12 s1. The complex conjugation is always understood as a generalized one, i.e. the product of the12w xordinary conjugation and Weyl reflection of harmonics 1 .


Ž .SU 2 transformation is reduced to the rotation of the t-world metric corresponding to the fixed-gauge solutionŽ .34 by some harmonic-independent non-singular matrix which becomes identity upon restriction to x-

Ž .independent SU 2 transformations. Thus in what follows we can stick to this solution.Ž . Ž Ž ..Now we are prepared to solve Eq. 26 or 31 . This can be done in a full analogy with the hyper-Kahler¨

w x Ž . Ž .TN case 2 , based essentially upon the PG invariance 28 . Using 34 , we obtain

kqqsEqqk ,

1 1ˆ ˆ' '< <1 R)0 , ks arctan R f ; 2 R-0 , ks arctanh R f . 35Ž . Ž . Ž .' '< <R R

Ž . Ž .For definiteness, in what follows we will choose the solution 1 in 35 . Then, making the redefinition

q q qq� 4 � 4x sexp 2 ik x , x sexp y2 ik x , 36Ž .ˆ ˜ ˆ ˜

Ž .we reduce 31 to

Eqqxqs0 ´˜

q i q q qi i q Ž i k . q yˆx sx w , x sx w syx w , fsy2 il x x w w , 37Ž .˜ ˜i i i i k

ˆ Ž .where, in expressing f, we essentially made use of the PG symmetry 28 .q qŽ . Ž . Ž . Ž .Combining Eqs. 29 , 33 , 36 and 37 we can now write the expressions for x , x in the following form

1 1q i q q i q� 4 � 4x s exp 2 ik x w , x sy exp y2 ik x w , 38Ž .i i

2 2ˆ ˆ( (1qRf 1qRf

i iˆ Ž . Ž . Ž .where k and f are expressed through x , x according to Eqs. 35 and 37 . Comparing 38 with the generali iŽ .definition of the x -bridges 5 , we can identify x , x with the t- world coordinates, i.e. with the coordinates of

the initial 4-dimensional QK manifold.y y i i " Ž .We still need to find x , x as functions of x , x and harmonics w by solving Eq. 27 and its conjugate.i

Ž .Dropping intermediate technical steps they involve a number of redefinitions , its general solution can bepresented in the following form

2ˆ(1 1qRf ˆŽ .y y2 ik i l s 2 ik f yŽ .x s e ye x ,˜ˆ2l ls y ifŽ .

2ˆ(1 1qRf ˆŽ .y y2 ik i l s y2 ik f yŽ .x s e ye x . 39Ž .˜ˆ2l ls q ifŽ .

Here

ily i y y i y i 'x sx w , x syx w , ssx x , k ils 'k s arctanh R ls . 40Ž . Ž . Ž .˜ ˜i i i 0 'R

For what follows it will be convenient to define

A s '1yRl2s2 , B s '1q4l2sqRl2s2 , C s '1qRsqRl2s2 . 41Ž . Ž . Ž . Ž .


Now we are ready to find explicit expressions for the two important quantities entering the general expressionŽ . Ž .for the t-metric 17 , 18 :

ˆ 2 ˆ 21yRf C 1qRfyy qq1yRE Õ sA , 1yR xPH s . 42Ž . Ž .2 2ˆ2 A 1yRfˆ1qRfŽ .

As a next step towards the QK Taub-NUT metric, one needs to find the entries of the matrix Xqm i 'Eqx m in n

Ž .by solving the set of algebraic equations 20 . In the complex notation, this set is divided into the two mutuallyconjugated ones, each consisting of four equations. It is clearly enough to consider one such set, e.g.

qr k y qr k q qr k "X = x s1, X = x s0, X = x s0, 43Ž .rk r k r k

qr k qr k qr k qr kŽ .where X 'X , X 'yX . It is convenient to work with1 2

qr k R Õ qr k 2 qr kˆ ˆ(X se X s 1qRf X . 44Ž ." " Ž .It remains to calculate the transition matrix elements = x ,= x entering Eq. 20 . This can be donerk r k

straightforwardly, the corresponding expressions look rather involved and by this reason we do not quote themˆqr kŽ w x.here explicitly more details are given in 7 . Surprisingly, the expressions for X prove to be much simpler:

1q1 k k l 2 Žk l . q 2 ik 0X s 3 AqB e y4l AqC x x w e ,Ž . Ž . l4

q2 k q 2 k l q 2 ik 0ˆ ˆX s E xsl AqC x x w e 45Ž Ž . Ž . Ž .l

Ž .the remaining components can be obtained by conjugation .ˆqr iŽ . Ž .It will be convenient to rewrite the metric 17 , 18 through Xm

1r i ,lk r i ,lkˆg s G , 46Ž .ˆCdet E HŽ .r i ,lk 2 r i ,lk vb yy qr i qlkˆ ˆ ˆ ˆG s 1qRf G se E X X q r illk . 47Ž . Ž .Ž . v b

ˆŽ .As the last step, one should compute det E H . After some algebra, it can be represented in the followingconcise form

1 A2 a b yy qr k yy ql l qn qmˆ ˆ ˆ ˆdet E H sy 1yRf e E X E X = x = x e . 48Ž .Ž . Ž . a b r k l l nm32 C

As a result of rather cumbersome, though straightforward computation one eventually gets the simple expressionˆŽ .for det E H

B 1q2l2sql2s 2qsRŽ .22 4 ik 2 2 4 ik0 0ˆdet E H sA e s 1yRl s e . 49Ž . Ž .Ž . 3 32 2C 1qRsqRl sŽ .ˆŽ .The harmonic dependence disappeared in det E H , as it should be.

The calculation of this determinant is the most long part of the whole story. Once this has been done, theˆ r i,l lcomputation of the t basis inverse metric amounts to the computation of entries of the matrix G . The final

answer for the metric tensor is as follows

D°2 2g s x x , As1yRl s ,Ž .1k ,1 t k t2C B

D2 2 2~ 50Ž .g s x x , Bs1q4l sqRl s ,Ž .2 k ,2 t k t2C B

12 2 2g s B e qD x x , Cs1qRsqRl s .Ž .¢ 1k ,2 t k t k t2C B


2Ž .Ž . 2Ž .Ž 2 .Here D'l AqC AqB s2l 2qRs 1q2l s . One should observe that this final expression is validŽ Ž . .for any sign of the parameter R even if, in the intermediate steps see relation 35 for instance , the sign choice

plays a significant role.

w x w x4. To compare to the results in the literature 5 one has to use 9

ds s s y is3 2 1 1i i idx sx q i yx , ds s e s n s . 51Ž .i i jk j k2ž / ž /2 s 2 2

i Ž .Using the notation APB'A B relation 51 impliesi

ds s ds2 s3 2 2 2yxPdxs q is , dxPd xs q s qs qs , ssxPx . 52Ž .Ž .1 2 32 2 4 s 4

Ž .The metric given by 50 becomes

21 B sB sA2 2 2 2ds q s qs q s . 53Ž .Ž .1 2 32 2 22 sC C C B

w xThe most general Bianchi IX euclidean Einstein metrics can be deduced from Carter’s results 10 . Aw xconvenient standardization 11 is the following

r 2 y1 D rŽ .22 2 2 2 2 2dt s l dr q4 s q r y1 s qs , 54Ž . Ž . Ž .Ž .3 1 22½ 5D r r y1Ž .

with

yLl 24 2 2 2D r s r q 1q2 Ll r y2 M rq1qLl . 55Ž . Ž . Ž .

3

Ž . 2These metrics are Einstein, with Einstein constant L and isometry group U 2 . If we take Ms4r3Ll q1 themetric simplifies to

2rq1 dr ry1Ž .2 2 2 2 2 2dt Q s l q4 S r s q r y1 s qs , 56Ž . Ž . Ž . Ž .Ž .3 1 2½ 5ry1 S r rq1Ž .

where now

L l 2

S r s1y ry1 rq3 . 57Ž . Ž . Ž . Ž .3

The identifications

ry1 s 4 R2 2s 4l yR , Ll s , 58Ž . Ž .2 2 22 31qRsqRl s 4l yR

give the relation

2 2B sB sA dt QŽ .2 2 2 2 24 4l yR ds q s qs q s s . 59Ž . Ž .Ž .1 2 32 2 2 2sC C C B l


Ž .The quaternionic metric 56 is complete for L-0 and is asymptotically Anti de Sitter. It has been consideredw xrecently in 13 under the name Taub-NUT-AdS metric and reveals itself a useful background for computing

black-holes entropy.

w x5. In this paper we made the first practical use of the harmonic space formulation of the QK geometry 4 tocompute a non-trivial QK metric, the four-dimensional quaternionic Taub-NUT metric. As we were convinced,

w xthe harmonic space techniques, like in the HK case 2,3,8 , allows one to get the explicit form of the QK metricstarting from a given QK potential and following a generic set of rules. It would be interesting to apply thisapproach to find the QK analogs of some other interesting 4- and higher-dimensional HK metrics, in particular,the quaternionic Eguchi-Hanson metric and the quaternionic generalization of the multicenter metrics of

w xGibbons and Hawking 12 .Finally, we note that the HK Taub-NUT metric plays an important role in the modern p-branes realm,

yielding an essential part of one of the fundamental brane-like classical solutions of Ds11 supergravity, theŽ w x.so-called ‘Kaluza-Klein monopole’ see, e.g. 14 . It would be of interest to reveal possible brane implications

of the QK Taub-NUT metric constructed here.

Acknowledgements

E.I. thanks the Directorate of LPTHE, Universite Paris 7, for the hospitality extended to him during the´course of this work. His work was partly supported by the grant of Russian Foundation of Basic Research RFBR96-02-17634 and by INTAS grants INTAS-93-127-ext, INTAS-96-0538.

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quaternionic taub-nut from the harmonic space approach

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