Nuclea~ Physics B222 (1983) 285-308 © North-Holland Publishing Company

SCALAR T E N S O R DUALITY A N D N = 1, 2 NON-LINEAR o-MODELS

u. LINDSTR()M

ITP University of Stockholm, Vanadist~gen 9, 113 46 Stockholm, Sweden

M. ROar_K*

California Institute of Technology, Pasadena, California, 91125, USA

Received 22 November 1982

Using N - 1 superfields, we investigate the duality between chiral multiplets and tensor. multiplets. We find that the relation can be expressed in terms of a Legendre transform. We use this relation to construct new N - 2 non-linear o-models correspoDdln~ to new Einstein-K~aler manifolds. We prove a no-go theorem for the N - 2 hypermultiplet and give off-shell formulations of the existing non-linear o-models in terms of tensor multiplets.

1. Introduction

The s tudy of supersymmetric non-linear a-models has revealed an intimate

relation with complex manifold theory (see e.g. [1], [14]). In this article we exploit

and investigate this relation. In part icular we study the consequences for the duali ty

t ransformations between scalar and ant isymmetric tensor fields. The models we

consider have N = 1 and N = 2 supersymmetry and we use N = 1 superfield formula-

t ion throughout , since this brings out the geometrical significance and seems to be

the most suitable form for future quan tum calculations. Using the scalar tensor

duali ty we discover a general class of N = 2 non-linear o-models. We give the most

general interaction for the N ~= 2 tensor multiplet; this is an off-shell formulation.

For N = 1 we find that the dual forms of a model are related by Legendre t ransformations.

We also prove a no-go theorem for the N = 2 hypermultiplet ; this follows directly

f rom the geometrical interpretation of the model. The article is organized as follows: In sects. 2 and 3 we discuss N = 1 supersym-

metric models. Much of this discussion serves as a preparat ion for the more interesting but analogous results of sects. 4 and 5. In sect. 2 we discuss non-l inear

* Work supported in part by the US National Science Foundation under grant SPI-8018080 and by the Swedish Natural Science Research Council.

285

286 U. Lindstrf~m, M. Ro~ek / Scalar tensor duality

a-models in terms of chiral multiplets and construct non-abelian generalizations of the CP(n) models. In sect. 3 we treat the tensor multiplet. We derive the general result that the action expressed in terms of the tensor multiplet is found by a Legendre transform of the chiral superfield lagrangian. Conversely, we show that any non-linear o-model with a holomorphic Killing vector can be written in terms of a tensor multiplet. Finally we discuss interactions of the tensor multiplet with other multiplets. In sect. 4 we treat N = 2 non-linear o-models, and prove the no-go theorem: There cannot exist a formulation of the hypermultiplet that can be used to describe non-linear o-models such that the transformation laws are independent of the form of the action. Our results on non-linear o-models are analogous to those of sect. 2, but more interesting, since any N -- 2 non-linear o-model corresponds to a Ricci flat K~hler manifold [14]. We give non-abelian generalizations of the known N = 2 models. In sect. 5 we discuss the N = 2 tensor multiplet: we find the most general form of its interactions, including the N = 2 improved tensor multiplet [5]. We use our construction to write an off-shell form of certain N - - 2 non-linear o-models. We use the Legendre transform of sect. 3 to construct the hypermultiplet form of the new general class of non-linear o-models, and give explicitly the (on-shell) N = 2 supersymmetry transformations. Finally, we briefly discuss interac- tions with other multiplets and the nonabelian form of the N = 2 tensor multiplet. Our conventions are listed in an appendix.

2. N ' - 1 ram-linear o-models

We begin this section by a brief review of selected results concerning N 6 1 non-linear o-models. The action for the bosonic (non-supersymmetric) case can be written in the form

f ,v,, Ig,;( ¢ ) (2.1)

where ek(x) are n scalar folds and g~ is the metric of the n-dimensional manifold M on which they lie. The ~,k's are to be considered as coordinates of the internal manifold M (not to be confused with the space-time manifold coordinatized by x). Most often M is taken to be a symmetric space, e.g. the complex projective space CP(n).

A supersymmetric extension of a non-linear o-model exists if and only if M is a Kahler manifold [1]. A ICddaler manifold is a complex manifold; in a suitable coordinate system the line element is ds 2 -- g~jdO~d~ j and, locally, the metric can be written as

a'u( $ ) (2.2)

U. Lind3tr~m, M. Ro~ek / Scalar tensor duality 287

(Holomorphic coordinate transformations preserve this form, whereas non-holomor- phic ones in general do not.) The function U is the ~ e r potential. Supersymmetric o-models are described by the action [ 1]

l = f dSz U( q,i, ~i ) , (2.3)

where ~i is now a chiral supcrfield;

/ ~ = 0. (2.4)

When expanded in components the action (2.3) contains the bosonic action (2.1) where g~j is the KJlhler metric (2.2). An example of a KgLhler manifold is CP(n). Its Kithler potential can be written as

V(~, ~J) --- In(1 + ~ ¢ ~ t ) • (2.5)

Alternatively one can describe this non-linear o-model in terms of n + 1 chiral fields and an auxiliary gauge superfield V: [2]

n+l ) I= f dSz E CY~'eV- V . (2.6)

I

This action has an abelian gauge invarianc¢

, ' - . e'~¢,, ~'-* e-,A~,,

V-~ V+ i ( A - A ) , (2.7)

where A is chiral. Eliminating V using its field equation E¢'~eV = 1, and choosing the gauge ~"+ l = 1, we recover the lagrangian (2.5).

The form (2.6) can be generalized to an arbitrary group

1-- f d'z [¢'(eV);~J- Tr V], (2.8)

where

V~ A = V (TA)j, i = l . . . n + l , A = l . . . 1 , (2.9)

and T a are some matrix representation of the group generators. The gauge transfor-

288 U. Lindstr~m, M, Ro~ek / Scalar tensor duality

mations (2.7) now become

¢--,(e'Al'j,J,

( e V ) ~ - . (e,A-)ik(eV)*,(e-,A~ '. j z j " (2.10)

The field equations for V A are

= (2.11)

For arbitrary groups and representations we cannot give an explicit solution to eqs. (2.11). We now consider some examples.

For the bosonic case, MacFarlane [3] has given a simple non-abelian generaliza- tion of the CP(n) models. We extend his construction to the supersymmetric case. We consider the action (2.8) in the particular case when the group is U(r ) (A -- 1... r 2) and the chiral fields transform as r + q copies of the fundamental representation (i ffi 1.. . r(r + q)). The action becomes

f =m v a b _ V] , 1 . . . r + I - - d S z [ % ( e ) bq#~, Tr mffi q, (2.12)

where Vt~ ffi VA(TA)~b, a = 1... r, and the T A are in the fundamental representation. In addition to the local U( r ) invariance, the action has a global U(r + q) invariance. The field equations (2.11) become

and are solved by

~m V a c _ ck~, ( e ) b*~, - 8~,, (2.13)

(e- v)a b = Obqb~, . = m ~ (2.14)

Substituting the solution (2.14) back into the lagrangian, we find

l = f d s zTr ln (~k~) . (2.15)

The gauge invariance can be used to "gauge away" r 2 fields from ~, . A simple gauge choice is

~ , ffi ( P = 8 a~ = ( 2 . 1 6 ) s, bJ, a, b f f i l . . . r , s r + l . . . r + q .

Then the action (2.15) is

I = f dSz Trln(Sg + / ~ p b ) . (2.17)

U. Lindstrfm, M. Rocek / Scalar tensor duality 289

In an attempt to understand the general case (2.8)-(2.11), we have examined unitary groups and irreducible representations such that n = 1 in (2.9). There are only two such cases: SU(2) and SU(3)*. For SU(2) the chiral fields transform as the isospin-2 representation and for SU(3) as the decuplet. The former case is equivalent to the CP(1) model and it is likely that this is also true in the latter case.

Finally we consider a class of models where eq. (2.11) cannot be solved explicitly, though the metric still can be found. The group is U(1) as for the CP(n) models, but the global SU(n + 1) symmetry is broken. Without loss of generality the generators can be taken to be

T ~ = q , S ] , ( ~ ) , i = l . . . n + l , qn+ ,= 1. (2.18)

In the gauge On+l __ 1 eq. (2.11) becomes

n n + l

eV + Eq,~)ieVq'¢l i= E q,. (2.19) I I

This determines V implicitly in terms of ~' and ~ . If we make the non-holomorphic coordinate transformation (we go to a vector representation)

~r'-- e-~q,e¢, .~ = e½q'Vo~, (2.20)

the relation (2.19) determines V explicitly in terms of the new variables:

in >] v = m •

I (2.21)

This allows us to compute the coordinate transformed metric from (2.2). We have not investigated the global structure of these manifolds.

3. N - I tensor multiplets

Some non-linear o-models can be described in terms of tensor multiplets. We review known results and then present acomplete discussion of the relation.

The tensor multiplet [4] is described by a chirai spinor gauge field, D~4,. -- 0; the field strength is

G z Datka + ~ , D2G = ff2G = O. (3.1)

* We are indebted to Dr. F. Vojdani Ghamsari who checked this matter for us using his algebraic computer program for generating representations of SU(n).

290 U. l.z'ndstr~, M. Ro~ek / Scalar tensor duality

It is invariant under gauge transformations

80po = iD'2DaL, L = L. (3.2)

There are two actions that are (classically) equivalent, or "dual", to the free chiral mulfiplet. The usual action [4] is

I, = - ½ f dSz G 2. (3 .3)

The improved (scale-invariant) action [5] is

12 = - f dSz G In G. (3.4)

The equivalence to the free chiral theory is most easily seen by goIng to the first-order form. We can write I I as [4]

I,= f dgz[½V2- VG]. (3.5)

If we eliminate V by its field equation we recover the action (3.3). On the other hand the G field equation implies that

~O,V = O2~V = O, (3 .6)

which is solved by writing

V - - q ) + ~ , ~q)--- 0. (3.7)

Substituting this into (3.5) gives the usual free action

I= f (3.8)

The equivalence can also be seen from the alternative first-order form

1,=fdSz[-½V ' + V(q, + ~ ) ] . (3.9)

l~.liminating V, we recover (3.8); alternatively, the q~ field equations imply that

D2v= 0, (3.1o)

which is solved by V = G and yields (3.3).

U. Lindstrtm, M. Ro~ek / Scalar tensor duality

The two fn'st-order forms of the improved tensor multiplet are [6]

291

12 ---f dSz (e r - VG), (3.11)

12=-fdSzV[InV-(×+~+1)], ~x=o. (3.12)

In the fwst case (3.11), eliminating V gives (3.4), whereas the G field equation gives Vffi X + X; defining ~ ~ e x we recover (3.8). In the second case (3.12), the X field equations imply Vffi G as in (3.10), whereas eliminating V, with 4,= e x, we once more fmd (3.8).

We can write the action (3.11) in a gauge invariant form by introducing a chiral field

~= f d*z(~,#,er- v6). (3.13)

This resembles the action of the CP(n) models in (2.6). It is known [4] that the tensor multiplet admits self-interactions; in general the

action is

I f p 2 f d S z f ( p - ' G ) , (3.14)

where/t is a constant of dimension mass. This model is classically equivalent (on shell) to a non-linear o-model described by a chiral superfield ~. To fred the lagrangian we use a generalization of the first-order form (3.12)

(3.15)

The form (3.14) follows from the X field equations; the V field equation

f ' ( I~ - 'V )= x + ~, (3.16)

expresses #- ~V in terms of X + X:

/t-IV-- h (x + ~ ) , (3.17)

therefore the action for the chiral superfield corresponding to (3.14) is

1--fdszU(x+ ), - ( × + £ ) h ] (3.18)

Thus U is the Legendre transform of f . The inverse Legendre transform can be

2 9 2 U. Lindstr6m, M. Rocek / Scalar tensor duality

performed whenever U ffi U(X + ~), (or, equivalently, with q~ffi e x, U--U(q,~)) . Geometrically this means that the manifold has a U(1) isometry generated by a holomorphic Killing vector. The first order form is

I= f dSz[U(V)- VG], (3.19)

where G is the tensor multiplet field strength. The metric of the manifold is (from now on we choose units where ~ = 1)

02U --U,,~= O2f v-h<×+~) OXO~ - OV 2 - - fvv . (3.20)

This Legendre transformation generalizes to arbitrary numbers of tensor multi- piers interacting with other matter. For example we consider the CP(n) models and rewrite them in terms of r tensor multiplets and n - r chiral multiplets. We replace the action (2.5) by

= f d'z m 1 + geV" + - voo° I I

(3.21)

From the G field equation we recover (2.5). The V field equations yield

e v°=G" 1+ eV°+ ] ~ i q " = G " 1 + 1 - G" , (3.22) I I

and hence the action

, = f d' , (GalrlGa).~ - I - ~ G a , In 1-~G', - In 1 + ~ ~iOa i .

(3.23)

In particular for n = r we have expressed the CP(n) models in terms of n tensor muhiplets.

There are other interesting examples of the Legendre transformation between the tensor and the chiral multiplets in the presence of interactions. In ref. [6] these transformations are given for one multiplet coupled to N = 1 supergravity; here we give the analogous coupling to the (~, 1) multiplet [7]. There are two forms of this multiplet: the Ogievetsky-Sokatchev form [8] and the de Wit-van Holten form [9]. Both multiplets are described by a gauge spinor superfield ~a, 8q~a = Aa + Dal2; ~ A . = O , a compensating gauge vector superfield H, 8 H = ( ½ i ) ( ~ - 1 2 ) , and a

U. l.andstrJm, M. Ro~ek / Scalar tensor duality 293

further compensating field. In the Ogievetsky-Sokatchev form this last compensating superfield is a tensor multiplet, whereas in the de Wit-van Holten form it is a chiral multiplet. Ref. [7] gives the de Wit-van Holten form in the presence of compensating fields:

I f I(@., H) + ¼ f dSz ( - ~ + ~ D ~ . + ~ / ~ , ) . (3.24)

This is equivalent to the first-order action

I=I(+,,H)-¼fdsz[½V2-V(D'~a+D~,)-VG]. (3.25)

We recover (3.24) if we impose the G field equation (which implies V= ~ + ~). Eliminating V we obtain

lffii(4,a,H)+ ~ f dsz[G+ (D4,+ /~)]2 (3.26)

which is the Ogievetsky-Sokatchev form. Similarly it is possible to reverse the Legendre transform and go from the action (3.26) to (3.24).

We can also perform the Legendre transform if the chiral multiplet has abelian gauge interactions. As an example we consider supersymmetric axial QED with action

I= f dSz(~tkeU-½HDD'2DH), (3.27)

(where H is the gauge field). The first-order action corresponding to this is

I= f dSz(en+V-½nD~Dtt-GV). (3.28)

Eliminating V we find

I--- f dSz(-½HDff2DH - GInG + GH). (3.29)

In general, for an arbitrary number of chiral multiplets with charges q~, we find

(3.30)

where f(G~) is the lagrangian that results in the absence of gauge interactions. It

294 U. 1.2nd~trJm, M. Ro~ek / Scalar tensor duality

would be interesting to study the quantum behaviour of these systems, in particular with respect to anomalies. Do the anomalies remain unchanged, and if so, how are they generated? In the case of one multiplet coupled to supergravity it is known that the anomalies differ in the two versions [10].

The action (3.30) describes the general coupling of the tensor multiplet to an abelian gauge-multiplet. We now consider the non-abelian case. Two forms of the non-abelian tensor [11] multiplets are suggested as generaliTations of (3.5) and (3.11). They are [12]

I I = Tr f dSz (½ V 2 + q,~- VD~eV - ~%VDae- v),

12ffiTr f dSz(e v +¢b~e-VD~eV-~eV~e-v), (3.31)

here ~ is the chiral spinor gauge field of the tensor multiplet. Its transformation law is

8 ¢ = i ~ [e- ~n: v, L], L =/7, (3.32)

and its field equation implies eV= e~Ae-~ which reduces the actions I I and I 2 to (different) non-linear o-models. The V field equation yields

eV/_- (e- VD V, +o) + e- V(eV e- v, ,)eV

-- V ~ + Va~,~, (3.33)

which unfortunately has proven difficult to solve [12].

4. N - 2 non-linear o-models

We now turn to N - - 2 extended supersymmetry; in this section we treat non- linear o-models and in the next, tensor multiplets. Many of our results are generali- zations of the results presented for the N = 1 case in sects. 2 and 3.

The N ffi 2 non-linear o-models are formulated in terms of the hypermultiplet [13]. It can be described by two (N- -1 ) chiral superfields ~ + . Unfortunately this description gives an algebra which only closes on shell (however, see below). Consequently the second supersymmetry transformations depend on the form of the action. For the free case ~--~+~k++ ~ - ~ - ,

**+ = ( + - ) ~ ( ~ + _ ) ,

8~+_= ( + - ) D 2 ( e ¢ + _ ) , D'--* = 0. (4.1)

U. Lindatr~m, M. Ro~ek / Scalar tensor duality 295

Here e is a constant chiral parameter,

= z + OF - 0 2 q , (4.2)

where z generates (off shell) central charge transformations, ~" generates the addi- tional supersymmetry and q generates part of the SU(2) that rotates the two supersymmetries into each other.

In the presence of interactions, the transformations (4.1) are modified:

0q,+_-- (+ - ) f f 2 ( ~ F + _ ) , 8~+_-- (+ - ) D 2 ( e F + _ ) , (4.3)

where F+_ is some function of q, and ~ that depends on the form of the action, and, in general, the chiral parameter e must be further restricted by D2e = 0. (This breaks the SU(2) symmetry between the two supersymmetries).

An N ffi 2 non-linear o-model corresponds to a 2n-complex-dimensional K~thler manifold M. The action is

I = f d S z U ( , + _ , O j + _ ) , i , j = l . . . n . (4.4)

where, as before, U is the K~Lhler potential for the metric on M. An N ffi 2 non-linear o=model can be constructed if and only if M is hyperk~Lhler [14]. A hyperk~Lhler manifold is a Irdthler manifold with three linearly independent coordinate systems, related by non-holomorphic coordinate transformations, in which the metric takes the simple form (2.2). In particular all hyperk~hier manifolds have a vanishing Ricci tensor. (For n ffi 1 (two complex or four real dimensions), the converse is also true: all Ricci fiat KiUaler manifolds are hyperk~thier and self-dual.)

An immediate consequence of these results is a "no-go" theorem for the hyper- multiplet: There cannot exist a formulation of the hypermultiplet that can be used to describe non-linear o-models, such that the transformation laws are independent of the form of the action. If such a formulation could exist, the sum of two invariant actions would also be invariant; however, the sum of the K~aler potentials of two hyperk~der manifolds is not in general the K~hler potential of a hyperk~hler manifold. (This theorem does not exclude formulations such as the relaxed hypermultiplet [16].)

The N ffi 2 non-linear o-models that have been constructed to date [15] are the N -- 2 generalizations of the CP(n) models, (the Calabi metrics). We start with the action [ 17]

l= f dSz(~,,+q,'+eV +e~,_~_e-V-cV)

invariant under the following transformations.

296 U. Lindstr6m, M. Ro~ek / Scalar tensor duality

Abelian gauge transformations

N = 2 supersymmetry

~+_---* e(+-)iA~+_, _ . 7 - _ _

~+ .--, e(+-)la¢,+_ ,

V-~ V+ i ( A - A ) . (4.6)

8V= eg+~S, 8S = -iW*D.e = ff2(D*VD.e), (4.7a)

(4.7b) 8,+_= (+ - ) ~ ( ~ + _ e ~ + - ~ ) , D-~ = D=~ = 0.

(The restriction D2e--0 can be relaxed somewhat to preserve a U(1) subgroup of SU(2) [18].) The constants c (real) and b (complex) parametrize the particular linear combination of the coordinate systems we are using to describe the manifold. If we eliminate the auxiliary fields S and V we find

~i+¢~+ev 7i e - V _ - q , , _ q , _ - c ,

¢ ~ _ q,'+ = b .

Choosing a gauge we solve (4.8b)

,+= ¢~(1 + r+. r_)-'/~( r+, l),

,_=¢~(l + r+.r_)-'/~( X- ).

The solution to (4.8a) is straightforward

M+eV=½[c+ ~c2 +4M+M_ ],

where

M +=-$,+'¢+

= b ~ [ ( 1 + K+-K_)(1 + K+'K_)]-|/2(1 + K+'K,+),

M_=-- cbi_. qi

= ¢~ [ (1 + r+ . r_ )O + to+. x_)]- ' /~(1 + / < . g_ ) .

(4.8a)

(4.8b)

(4.9)

(4.1o)

(4.11)

U. LindstriJm, M. Ro~ek / Scalar tensor duality

The action becomes

297

For c = 0 this action simplifies [17]; for b = 0 we must choose a different gauge than (4.9). For example we choose

q~+= (K+, i K + ~ - ~ _ ), •_= (4.13)

Despite the constraint D2e -- 0, for b = 0, the U(1) subgroup of the SU(2) is realized as a phase rotation of K+ and K_. The supersymmetry transformations in terms of K+_ that leave the action (4.12) invariant are found by substituting (4.9) and (4.10) into (4.7b) and adding a gauge transformation (4.6) to preserve the condition (4.9) or (4.14). (See, e.g., [17].)

Just as in the N = I case, the form (4.5) can be generaliTgd to an arbitrary group

where

l=S dSz[ ~i+(ev)ij+{ + +i-(e-V))~J--cTrV]

+[Sd6z(Oi_Sj¢X+-bTrS)+h.c.],

Vj' -- VA(TA)j,i Sj' = SA(TA)~j , i = 1 . . . . . . n + l, A = 1 l ,

(see (2.8-11)). The transformations that leave (4.14) invariant are

8e v = ege v + e - vS~,

e-V) .

s s = - i w . o . , = [(e-

D-'-e - - D 2 e = 0 .

The equations resulting from varying V and S are

~ , + ( e V ) ' j ( T A ) J k * k - - ¢ , _ ( T A ) i j ( e - V ) J ~ k_ = c(Ta) ~,

*,_(TA ' j )s,+= b(rA),.

(4.14)

(4.15)

(4.16)

(4.17a)

(4.17b)

,=fdSz[~/c2+4M+M-c[ln(c+~/c2+4n+M )-lnM+]]. (4.12)

2 9 8 U. Lindstr~m, M. Ro~ek / Scalar tensor duality

Just as for (2.11) in the N= 1 case, in general these equations cannot be solved expficitly. However, the MacFarlane models can be extended to the N -- 2 case. In the notation of sect. 2 (see eqs. (2.12)-(2.17)), (4.17) become

M+eV-e-VM_-cl =0, (4.18a)

b m _ _ b q~, +q~g_ - bS~, (4.18b)

where

( M + ) b a b 7, . ( M _ ffi . = +:.++."+, (4.18c)

Eq. (4.18a) is solved by

)+.. (4.19)

To solve (4.18b) we choose a gauge

~+= Q(K+, 1)) ° 1

Q~,=vfb[(l + X.r_)- ' /2] a b) ( 4 . 2 0 )

(c.f. (2.16)). The action then becomes

I=Tr f d'z[I/c2+4M+M_ -c[in(c+ vfc2+4M÷M_ )-tnM+]. (4.21)

Just as in the abefian case (4.12) this becomes particularly simple when c -- 0; also the case b -- 0 requires a different gauge choice than (4.20).

Because these models are N-- 2 supersymmetric they correspond to hyperkahler manifolds; to our knowledge, these have not been constructed before. Whereas for N = 1 any scalar function U(#,, ~) gives rise to a non-linear o-model, and so the construction of new examples is not particularly interesting, for N - - 2 there are severe restrictions on the form of the non-linear o-models. This has made construc- tion of new models difficult, and these are the first non-abelian examples.

We could also consider the N - - 2 analogues of the SU(2) and SU(3) models described in sect. 2. It would be interesting to see if they give rise to new manifolds or only to a complicated description of the abelian models described by (4.5).

Finally we consider the class of models analogous to those described by (2.18)-(2.22), where all the definitions needed can be found [19]. In this case, without loss of generality, the charges q~ can be chosen to be positive (cf. (2.18)). We

U. Lindstr6m, M. Ro~ek / Scalar tensor duality 299

consider only the coordinate system corresponding to c - 0 and choose b(E~ '+ Iqi ) -- 1. The relation (4.17b) becomes

n + l

)-". , i _q , , ' += 1, (4.22) I

which is solved by

:/-'' *+_-- ( g + _ , l 1 + iK i_K

1 /

(4.23)

This also determines the gauge; in this gauge (4.17a) becomes

n

e v e V+Eq,(K,~+eq'VK~+-Ki_e-q'vK,~_)=O. I

(4.24)

This determines V implicitly in terms of , + _ and ~+_. As in (2.20) we make non-holomorphic coordinate transformations

~i_= [e-½qtV]gi_ , -- i = [ e - ~ q , V ] j ~ qlr I (4.25)

This determines V explicitly through (4.24) in terms of the new variables

v= ½ sinh-' q,(~; ~'_- ~,+~r . (4.26)

As before, this allows us to compute the coordinate transformed metric. These manifolds are a further class of new hyperKdhler manifolds. In particular they include a one-parameter family of four-(real)-dimensional manifolds. We think it would be worthwhile to study the global aspects of these metrics to determine whether they are complete, have singularities, etc.

5. N - 2 tensor multiplets

In this section we discuss N = 2 tensor multiplets. We begin with a brief review of previous results. Component, N = 1 and N = 2 superfield formulations of the free (linear) tensor multiplet are known [20]. In N = 1 superfields the action is

I = f d ' z ( - ½ C ~ + ~ X ) , (5.0

300 U. Lindstr~m, M. Rocek / Scalar tensor duality

invariant under the N = 2 supcrsymmetry transformations

8x = ~ ( ~ ) , 8¢,,, = - X O , : ,

G - D °,t,,, + ~ , ~ , ~ = (7 , /~. , : = a o : = 0 . (5.2)

These transformations close of f shell; thus the sum of invariant actions is invariant. The improved (non-linear) tensor multiplet has been given in components only [5,211.

We now construct the improved N = 2 using N-- 1 superfields. Since the trans- formations of the N ffi 2 tensor multiplet close off shell, the relations (5.2) remain intact. We extend the action (3 .13) to N ~ 2: the action is

I = f d s z ( ~ + ~ p + e V + o _ ~ _ e - V - V G ) + [ f d t z S ( O + c b + x ) + h . c . ] . (5.3)

This action is invariant under the transformations (5.2) and (4.7). (In this case we can relax the restriction D2e = 0). Varying with respect to X and G we find

S = 0, V = A +/Y, /~.aA = 0. (5.4)

Substituting back, the action (5.3) reduces to the free hyper-multiplet action. Varying with respect to V and S, we find (cf. (4.8))

~ + ¢b + e V - ep_ ~ _ e - V = G , ep_cb + = - X . (5.5)

Solving these equations we find the action (c.f. (4.10)-(4.12))

l=f d'~ [(G' + 4×~ - cin(c + eG'+ 4×~ )]. (5.6)

As emphasized above, this action is invariant under the linear transformations (5.2). (This can be verified directly by an edifying calculation).

As a first application we observe that a linear combination of the actions (5.6) and (5.1) describes an interacting model, even though each action separately corresponds to a free model. As a further application, we construct models corresponding to Calabi metrics in terms of r tensor multiplets and n - r chixal multiplets (see (3.21)). Since we have not been able to extend the Legendre transform to N -- 2, we rederive our N = 1 results using a method that generalizes to the present case. Starting from the action

z=fd'~ (~oeev.-G°Vo)+ ~,~' eV- l - E ~ o v , (5.7)

U. Lindstr~m, M. Rocek / Scalar tensor duality 301

we f'md the usual formulation of the CP(n) model when we vary with respect to G °. If we vary with respect to V and Vo and choose the gauge ¢n-,+l ___ 1, we recover (3.23). The N = 2 action analogous to (5.7) is

I=fdSz +¢~ eV° + ¢o_ ~"_e- v° - G"V.)

+ .r-

E [(~,+*'.)e V+(*,-~'-)e-V]- c - G° I

[J [ [( ) ( , t l} ] + d6z E S a ( q l . } a + I O a + X a ) + s n--~+lcpa_l.~a_ -- X a - b + h . c . . 1 1

(5.8)

(This is invariant under the obvious generalizations of (5.2) and (4.7).) Eliminating V,, V, So and S we Find

I.v°=O=*~'o÷O"+eV°-~"-%-e - v ' = G ' , ( ~ )

n - r + l r 7i - .~ c _ ~ G a ' i .v=O= . y" [(~,++,_)eV_(C_O,_)e v]

1 I

° " - ( Z ) I . s o = O ~ x + 0 + % _ - 0 ,

• n - r + I

l , s = O ~ Z x ° + b - E 0'÷*, " - = 0 . (5.9) I 1

Defining

n - r + ! n - r + I

M+= E ~,+¢+, M_= E *,-~,'- (5.10) I I

and choosing the gauge

,[(, )l 0'+ = (K.~, 1 xa+b 1+ EKe+K,_ , I

,[(" )J( . . )1,/2 *,_ = ( K , _ , 1 Y~. X ° + b 1 + ~ K~_ K i_ , I I

(5.11)

302 U. l..indstr~m, M. Ro~ek / Scalar tensor duality

we find

+ c - G" +4M+M_ c - G"

× 1( r /:) ]] In c-~-' .Ga+ c - G a + 4 M + M _ - l n M . . I

(5.1z)

In particular, if n = r we find

I = f dSz[~[~/G2*+4~"x°-Galn(G"+~/G:'*+4~"X")]

(( ) (~ )(~ )),,,~ ( r ) c - t 2 + 4 X a + b x a + b c-~'l G" + ~('G a

X In c - Ga+ c - G a + 4 x a + b I

(5.13)

(5.14)

and require invariance under the transformations (5.2) assuming D2e = 0. (This is the most general form of the N = 2 transformations). We find that this action is

l---f dSzf(G,x,£),

The action (5.13) is the first off shell formulation of the N = 2 non-linear o-models. It would be interesting to find an expression in terms of N -- 2 superfields.

We now find the most general self-interaction of an N = 2 tensor multiplet. Since we have an off shell algebra, where the transformations are known, this is a straightfor- ward computation. We consider a general ansatz

U. Lindatri~m, M. Ro~ek / Scalar tensor duality 303

invariant if and only if

o2f + oy (5.15) OG 2 OX O~

This is just the three-dimensional Laplace equation. The solutions are thus well known. They can be written in terms of Green functions or expanded over a complete set of eigenfunctions, etc. The reader can verify that (5.1) and (5.6) satisfy (5.15).

It is straightforward to generalize this result to higher (internal) dimensions. An action

(5.16)

is invariant under transformations

8q/. = - x ' D o e =~ 8 G ' = - D*(x~D,,e) + h.c.

8x'= (5.17)

if and only if

fa'a, + f%,'x j = O. (5.18)

This computation is somewhat tedious; in particular, certain terms cancel only when the 0 integration is completely performed. This is most easily done by converting the integration into spinor derivatives D, a n d / ~ . The system of equations (5.18) is a system of elementary linear partial differential equations, whose solutions are no more difficult to find than those of (5.15).

To investigate the geometry of these models we need to replace the tensor multiplets by chiral superfields. Ideally we would want an N = 2 duality transforma- tion; we have not found one. Instead we perform an N = 1 dualization on each of the N = 1 tensor mnltiplets, and then find the N = 2 supersymmetry transformations by hand.

We begin with a single N = 2 tensor multiplet. The first-order action is

l=f dSz[.f(V,x,2)-V(q,+~,)]. (5.19)

Varying with respect to ~ and ~ gives (5.14). Variation with respect to V gives the Legendre transform

U( ep + ~, X, X) = f ( h , X, X) - (tb + ~)h, (5.20)

3O4

where

U. Lindstr~m, M. Ro~ek / Scalar tensor duality

f e ( h , x , ~ ) - g p + ~ , x , h ( f v , X ,~) = V,

To find the consequences of (5.15) in terms of U, we use

fvvh , •fvvh; ffi 1,

/vx + fvvh× =fv~ +fvvh~ = O.

Thus we find

= u z = - h ,

( f v = ~ V ) . (5.21)

Ux = fx, Ux~= ( f vv ) - ' ( f x~ fv} ' - f vx fv~) ,

u~ ; = - ( / ~ ) - ' ,

and hence

det[ U"] - -fx~ ( f v v ) - ' ,

or, using (5.15),

(5.22)

(5.23)

(5.24)

~ e ffi Oa~effiD2e ffi O. (5.26)

Invafiance can be proven either directly using (5.25), or using (5.23). Thus we have found the most general hyperKdhler manifoM (corresponding to the most general

This is familiar to mathematicians as the Monge-Amp~re equation. It is the form of the Einstein equation, without a cosmological constant, on a two-complex-dimen- sional K~h|er manifold. (Recall that every two-dimensional Ricci flat Kahler mani- fold is hyperk~aler, and vice versa.) This result is in accord with the theorem of ref. [14].

It is straightforward to find the N = 2 supersymmetry transformations leaving the action corresponding to (5.20) invariant. They are

det[ U"I = 1. (5.25)

U. Lindstrtm, M. Rocek / Scalar tensor duality 305

N = 2 non-linear o-model) with (at least) one holomorphic Killing vector. Because the transformations to the tensor multiplet involves a N = 1 Legendre transform (5.19), we can go freely between the two forms. Whereas the Monge-Amptre equation (5.23) is non-linear and difficult to solve, the Legendre transformed form (5.15) can be solved trivially.

In particular this gives the explicit solution to the four-real-dimensional self-dual- ity condition for all metrics with at least one holomorphic Killing vector. (U-- U(qb + ~, X, X))- What remains to be investigated are global aspects and what conditions on f determines the signature of these metrics.

The generalization to an arbitrary number of tensor multiplets is straightforward: From the first-order action

I f f d S z [ f ( V i , × ' , £ O - V ' ( ¢ , + 7 ~ , ) ] , i = l . . . n , (5.27)

corresponding to (5.16), we fred the Legendre transform

v( , , x', x ' )=/(h' , (,, + h', (5.28)

where

V'. (5.29)

As before we find

det[fx,~, ] det[ U"] = det[ - f v , vJ ] " (5.30)

The Monge-Amp~re equation corresponds to det[U"] -- 1; in our case, N -- 2 super- symmetry of the original tensor form of the action implies the stronger condition (5.18). Expressed in terms of U, these linear conditions become non-linear partial differential equations

= + V×,~,,.(V,,.~,.) V,.~,. (5.31)

We believe that (at least when there are n abelian holomorphic Killing vectors) (5.31) i s satisfied if and only if the manifold is hyperkiihler. In any case this construction yields a general class of 2n-complex.dimensional hyperKilhler manifolds for all n. Of course the global properties of these manifolds remain to be investigated.

T o find the N-- 2 supersymraetry transformations leaving the action correspond- ing to (5.28) invariant, we consider the first-order action (5.27). The transformations that leave it invariant can be found by modifying the tensor multiplet transforma-

306 U. Lmdstr6m, M. Ro~ek / Scalar tensor duality

tions (5.2); in particular the variation of ~'i is chosen to cancel terms cx ff2V~. The new transformations are:

8v ,= -z>O(x,O:,)- ~(~%0,

8~'=D2(,w), 8x'=~(~'),

8~' = z,~ [.[s~, + v,(s~,, . , - s~,,.,)]],

8+, = ~ [~[s., + v , (s , , , . , - s.,,.,)]] (5.32a)

(the proof is analogous to the original proof of the invariance of the action (5.16), and requires use of the identities fv~vtJxkl = fvt,vJl~k = 0, which follow from (5.18)). Eliminating V ~ by its field equation, and using (5.28) to reexpress the transforma- tions in terms of U, we find:

8~'= -D~(,u;,), 8x'= -~(~uo,),

(5.32b)

Actually, because the Killing vector implies U,-- Ug, we cannot tell which form of the derivative is needed, but we believe (though we have not verified) that (5.31) and (5.32b) are general.

We now return to N = 2 duality transformations. We can find the duality transformation in the presence of interactions with other multiplets. In ref. [21] it has been claimed that a form of N = 2 supergravity using the improved N = 2 tensor multiplet has been constructed. Here we consider N -- 2 supersymmetric QED (c.f. (3.28)). We start with the first order form

l=S dSz [~+ e##+ vltll+ +~I-e-H-VlII--½HDD2DH + S~-GV]

+ +,.o.]. (5.33)

Varying with respect to X and G yields the usual N - - 2 supersymmetric QED in terms of a gauge multiplet (H, ~) and the hypermultiplet (~+, 9-) ; varying with

U. Lindstr6m, M. Ro~ek / Scalar tensor duality

respect to V and S yields (c.f. (3.29))

307

' - - f dSz [~/G 2 + 4X~ - GIn[G + v/G 2 + 4X~ ] + ~ -½HDffZDH + GH]

(5.34)

This gives an off shell formulation of N = 2 supersymmetric QED. Finally, we consider the N = 2 non-abelian tensor multiplet. The first order action

takes the form (c.f. (3.31))

1=Tr f dSz[q,+eV¢+ +4,- e-V¢-+¢"e-VD~eV-~%VD~e -v]

+ (5.35)

where ¢+_ are in (conjugate) fundamental representations and all other fields in the adjoint representation. As in the N = 1 case, we cannot solve the equations that result from eliminating the vector multiplet (V, S).

M.R. would like to thank the Institute of Theoretical Physics at the University of Stockholm for its generous hospitality.

AEpendix

NOTATION AND CONVENTIONS

We use the conventions of ref. [18]. Relevant formulae are

(D., ~ ) = ia. , ,

0 2 =- ½O°O., ~ - ½5' /~ = (O2) ?,

D~D# = 8 ~ D 2 = - D#D °,

[] = ½ a .~ a ~" ,

fd'~-fd'~d'O-fd',~-fd'~d'O~. means no implicit summation over repeated indict .

(A.I)

(A.2)

(A.3)

(A.4)

(A.5)

308 U. findstrcim, M. Roiek / Scalar tensor duahty

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