Volume 190, number 1,2 PHYSICS LETTERS B 21 May 1987

H Y P E R I ( ~ L E R METRICS FROM (4,0) SUPERSPACE ~

Mark EVANS and Burt A. OVRUT Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-6396, USA

Received 16 January 1987

We construct (4,0) supersymmetric non-linear sigma models using the superspace formalism. An explicit expression for hyper- k~ihler metrics in terms of quaternionic structures results. These metrics are vacuum solutions to Einstein's equation, satisfy the tree level equations of motion of the superstring and are analogs of the Calabi-Yau manifolds favored for superstring compactification.

Compact K~ihler [ 1 ] Ricci-flat manifolds have long been of interest to mathematicians. Necessary and sufficient conditions for their existence were conjec- tured by Calabi [2] and proven by Yau [3], but the problem of constructing such metrics remained unsolved. These manifolds are of interest to physi- cists for a variety of reasons: they satsify the vacuum Einstein equations and have been much discussed in relation to the compactification of superstring the- ories from ten to four dimensions [4].

An interesting subclass of these manifolds is termed hyperk~ihler. These are Ricci-flat K~ihler manifolds for which the complex structure has been generalized to a quaternionic structure. Hyperk~ihler manifolds correspond to finite supersymmetric non-linear sigma models [5,6], and hence are exact solutions to the tree level superstring equations of motion [ 7 ].

In this letter we present a technique for construct- ing explicit hyperk~ihler metrics from a given qua- ternionic structure. The construction is algebraic; (1 +4)-dimensional non-linear sigma models with hyperk~ihler target spaces are constructed in super- space, and the metric read off from the action expanded in component fields. Related ideas have been pursued by others [8].

We begin by discussing pertinent notions from dif- ferential geometry. We then discuss (p,0) supersym- metric non-linear sigma models, and explain the

'~ Work supported in part by the Department of Energy Contract number DOE-AC02-76-ER0-307 I.

relationship between (4,0) models and hyperk~ihler manifolds. We proceed with the construction of these models by exhibiting the appropriate superfield con- straints, and show how the covariant constancy of the quaternionic structure follows from R invari- ance. The sigma model action is then constructed in superspace, and the hyperkahler metric read off from its component expansion.

A manifold is complex if it admits an integrable complex structure [ 1 ]. That is, it must admit a glob- ally defined tensor J"~ such that

JU J ~ = _ ~ a (1)

NU,a = ( O~JUx -OaJUo) J ~

- (O,~JU. -O.JU,~) J ~ =0. (2)

N",a is called the Nijenhuis tensor, and its vanishing implies that it is possible to choose real coordinates x" such that everywhere

From this coordinate system one can construct com- plex coordinates

z'~ = x~' + i J '~x" , z'~ = x'~ - i J ' ~ x ~. (4)

We shall refer to j u as the complex structure of the manifold.

A complex manifold is said to be hermitean if its metric satisfies

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Volume 190, number 1,2 PHYSICS LETTERS B 21 May 1987

g~jUjy,~ =gzo, ( 5 )

and a hermitean manifold is K~ihler if the complex structure is covariantly constant with respect to the riemannian (i.e. torsion free) connection

VuJV~ =0. (6)

A hyperk~ihler manifold is a K~hler manifold on which the complex structure is generalized to a qua- ternionic structure. That is, there exist three integr- able complex structures, j(~)uy, such that the matric is K~ihler with respect to each J(a) (eqs. (1), (2), (5) and (6) hold) and

{J(a) , J(b) } = -- 2¢~ab'~. ( 7 )

This algebra constrains hyperk~ihler manifolds to be of dimension 4k (k an integer). Eq. (6) tells us that the holonomy group leaves the quaternionic struc- ture invariant, and so is contained in Sp(k). Since Sp(k) =_ SU(2k), these manifolds are Ricci flat.

We construct hyperk~_hler manifolds as target spaces for non-linear sigma models with (4,0) super- symmetry. To work only with irreducible represen- tations or the (1 + 1 )-dimensional Lorentz group, we employ light-cone coordinates x +, and take all spi- nors to be Majorana-Weyl (hence real and one-com- ponent). The (p,0) supersymmetry algebra [9,10] is generated by p Majorana-Weyl supercharges Q+~ and the generator of right translations P+,

{Q+~, Q+j}-~zsoP+, I <~i,j<~p. (8)

Under Lorentz transformations,

P+ ---)e-~P+, Q+,~e-'~/2Q+~, (9)

where we follow the conventions of refs. [10,11 ], in which spinorial indices are distinguished from vec- tor indices by an additional integer label.

The component field action for the (1,0) super- symmetric non-linear sigma model is [9]

s= f d2x {[G~(Xo)+B~(Xo)] O+XUoO_X%

-iG~gtu(O_ ~f +FY~,,O_ X~o~°) } (10)

where XUo and q/u are, respectively, scalar and spinor fields. Guy is symmetric, interpreted as the metric for the target space, and Bu~ is an antisyrnmetric tensor on the target space. FYao is a connection that includes a torsion piece constructed out of Bu~.

FY~ ={L}+½ CWc(O,~Bzo+O~Bo,~+O~,B,~).

(11)

The (1,0) supersymmetry transformations are

8X%=ie qf, 8V u = - { 0+XUo, (12)

where ~ is a grassmannian parameter. The action (10) will have additional supersym-

metrics [ 5,12 ] if it possesses certain chiral symme- tries (X(a) are parameters)

8g/u=x(a)J(,o(Xo)uyg/v, a = l ..... p - l , (13)

the additional supersymmetry transformations being

8 (a)XUo = i(- J(a) uy~ tu,

8 (a)(J(a)Uv~//y) = - - ~ 0+ X'Uo . (14)

Transformation (14) will satisfy the supersymmetry algebra (8) if

{J(a) , g(b) } ~--" - - 2aab{, (15 )

N(,~),,ua=O, a = l ..... p - 1 (16)

where (16) is precisely the integrability condition (2) for J(~) to be a complex structure. In particular (4,0) supersymmetry is associated with target manifolds that possess an integrable quaternionic structure (eqs. (2) and (7)).

This discussion of extended supersymmetry was predicated upon our complex structures J(a) gener- ating chiral symmetries. Substituting (13) into the action (10), we discover that we do indeed have a symmetry iff

Guy j(,) uj(a) u,~ = G~,,, (17)

~J<~> y~ =0. (18)

Eq. (I 7) says that the target space is hermitean with respect to each complex structure and eq. (I 8) that each complex structure is covariantly constant with respect to the connection (I I ). Since this connection contains a torsion piece constructed out of Bw,, (I 8) reduces to (6) when

Buy =0. (19)

Stated differently, eqs. (15)- (18) and condition (19) imply that the target space is K~hler for (2,0) and

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Volume 190, number 1,2 PHYSICS LETTERS B 21 May 1987

hyperk~ihler for (4,0) supersymmetric non-linear sigma models. Thus constructing (2,0) or (4,0) models in superspace and imposing condition (19) will enable us to read off metrics that are, respec- tively, K~ihler and hyperk~ihler. The K~ihler case is described in ref. [11 ], and yields the well known result

G,~ =O,~O~K. (20)

We therefore proceed with the (4,0) construction. Superspace [ 13] is the group manifold corre-

sponding to the supersymmetry algebra (8), and has coordinates (x +, x - , 0 + z,..., 0+0 . Functions defined on superspace are superfields and transform under the algebra (8) through translations on superspace induced by left or right group multiplication. The generators of these motions are denoted Q+t and D+~, respectively:

Q+i=O+i-iO+iO+, (21)

O+i =O+i +iO+iO+. (22)

Q+~ and D+~ each satisfy the algebra (8) and anticommute

{Q+,, D+j) =0. (23)

We take the Q+~ to represent supersymmetry trans- formations, and use the D+~ to constrain superfields in a supersymmetric way.

Real scalar (4,0) superfields have the component expansion

X~_.~ Xa 0 + iO+i X~t+i + iO+i O+JX,U+i+j

--(1/3!) O+iO+JO+kx'u+i+j+k

- (1/4!) O+iO+JO+kO+lXu+i+j+k+l. (24)

Each superfield contains sixteen independent real component fields, only two of which (the lowest component and its fermionic superpartner) appear in the action (10). It is therefore necessary to impose constraints on the superfields X u such that the lowest component and one superpartner are unconstrained and all other components are determined in terms of these physical fields.

An appropriate set of constraints is as follows. Choose three real 4 × 4 matrices Lt~) that satisfy the Clifford algebra (7),

{Lo~),L(b)}=--26ab'~, a = l , ..., 3.

A convenient choice is

', - l \ '1 _) . . . . . . . . . . i . . . . . . . . ,

L I - - - - 1 II

\ l i /

. . . . . . . . . ~1 L 2 - / 1 i ] '

\ '! /

(25)

i !

L3= ~- - - - ~ - - . (26) i i i i

Then impose the constraint

D + iX u =L(~)ij J( X)%D +jX". (27)

Note that here, and throughout this paper, we do not employ the summation convention for the index (a) in parentheses - constraint (27) is imposed sepa- rately for each a= 1 .... , 3. It is apparent from (25) and (27) that D+iXu=O unless the J~a) satisfy (7): indeed, they must satisfy

3

J(a)J(b) =--~ab'~-- ~ (.abcJ(c). ( 2 8 ) c = l

With this choice of algebra for the J(a), (27) reduces to three independent constraints:

D+ 1X u =J(3) UvD+2XV, (29a)

D+2 Xu = J ( l ) UvD+3XV, (29b)

D +4X u =J(l) %D + IX ~. (29c)

Wepresent the solution of these constraints for the bosonic components of the superfields X u.

The O(0) terms of constraints imply

3 XU+i+j= ~ L(a)OJ(a)UvO+XVo. (30)

a = l

To 0 ( 0 3 ) we learn that

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Volume 190, number 1,2 PHYSICS LETTERS B 21 May 1987

X'U + 1 +2+ 3+4 = 02 X'Uo --Jfa) u,¢ O~J(a) '~

X J(b)'~oJ(b)Vz+t~'~ot~u~ O+X~oO+X':o, 1

(31)

as well as various relations involving Xu+i+j, which are consistent with (30) iff

N(~)%~ =0. (32)

The right-hand side of (31 ) is also the same for each value of (a) if (32) holds. We thus have a consistent solution to the constraints (27) if the J(~) form an integrable quaternionic structure for the target space.

For the target space to be hyperk~ihler, it was cru- cial that the quaternionic structure generate chiral symmetries (13). With the construction of the supersymmetry multiplet through constraints (27), we may now understand these chiral symmetries as an R invariance. The supersymmetry algebra ((8) with p = 4 ) has an SO(4) automorphism group

6 Q +i = ~.(,-) T ! f ) Q +j . (33)

where T ! f ) ( r = l , ..., 6) are real antisymmetric matrices. Eq. (33) induces a coordinate transfor- mation on superspace,

5 0 +i = ~(,) T ! j ) 0 +J, (34)

which in turn induces identical transformations on the differential operators Q+~ and D+~. Demanding that the superfields X u be invariant under (34) leads to a transformation on the fermionic component fields

~ X~+ ~ = E(~) T ! j ) XU + j , (35)

while the lowest components, X%, remain invariant. We shall construct an action by integrating an R invariant lagrangian over superspace using an R invariant measure. Thus the chiral transformation (35) will be a symmetry of the theory if the super- field constraints are R invariant. However, con- straints (27) are not invariant under the full SO(4) R transformations because the L t ~ ) are not invariant tensors of SO(4). SO(4) is locally isomorphic to SU(2) ×SU(2) , and the L(~) are generators of one SU (2). The generators of the other SU (2), which we denote Kta), commute with the L~):

[K~), L~b)] =0. (36)

Therefore, the R transformations generated by the K(a) do leave constraints (27) invariant. Thus the original SO(4) R invariance is explicitly broken to SU(2) by the constraints. The remaining R trans- formations are

~X/ ' +i = ¢(a)K(a)ij X'u +j. (37)

Eq. (37) is the desired chiral invarianee, but it is not yet in the form (13). To clarify the meaning of (37) we examine the lowest component of con- straint (27),

X u +i = L ~ a ) i J ( , , ) ( X o ) % X " +j, (38)

which tells us that only one quarter of the compo- nent fields XU+~ are independent. We take the phys- ical fields to be

q t " = X U + l . (39)

Then in terms of physical fields (using an explicit representation for the K(a) and (38)) (37) becomes (13 ), as desired.

We now proceed to the construction of the action in (4,0) superspace. That is, we seek to represent the action (10) as

J" d 2 x d 4 0 . ~ _ _ (40) S--

where ~ _ _ is a superfield with the Lorentz charge needed to cancel the charge of the superspace inte- gration measure.

Unfortunately, it is not possible to construct the desired lagrangian out of the superfields X u because the only way to produce the correct Lorentz charge is to introduce two derivatives O , which would lead to a higher derivative theory. We therefore introduce a charged superfield V + and write the most general acceptable action constructed out of V + and X u,

d 2 x d 4 O f u ( X ) ( O _ X u) V + , (41 ) S =

wherefu(X) is an arbitrary target space covector field. We must now constrain the superfield V + in such

a way that we preserve the SU(2) R invariance and do not introduce into the component action any fields beyond the physical sigma model multiplet. Appro- priate constraints are

h(a) ( X ) = - iL (a ) i jD + i D + j V + , (42)

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Volume 190, number 1,2 PHYSICS LETTERS B 21 May 1987

g(X)=O+ V + . (43)

These constraints may be solved for the compo- nent fields of V +. The bosonic components satisfy

O+ V+o=g(Xo), (44)

L(~)oV + + , + j = -h(a)(Xo), (45)

V + + i+2+3+4 = (O~g(Xo)) O+X~o. (46)

As usual, solving these constraints leads to a series of consistency conditions, and it can be shown that they are all satisfied if

J(a) uO~h(a), a = l ..... 3. (47)

Eqs. (47) are solved most easily in the coordinate systems specified by eq. (3). It is not hard to see that g must be the real part of a holomorphic function of the complex coordinates (eq. (4)), and that this must be true for each complex structure.

The explicit solution of the constraints ((30), (31) and (44) - (46) ) enables us to evaluate the bosonic piece of the action (41 ) in terms of component fields. We obtain

+~=l ~ J(a)~ (h(a) O[,oCul-½f~Oj, h(~)))

× O_ X%0+ X%

+A,~O +X%O + O_X'% V+o

+Bp,,~O_ XZ oO+ X"oO+ X ~ o V + o ], (48)

where

X [J(a) ~ (4 O~) (J(a) 'uy L ) - - Otuf2 ] , (49 )

Baor = O a A ( . ~ )

XJ(a) exO;~(J(a)v~,OIpfu])

3 - 2 ~ J(b) ~ (o 10P Jib) u j~) a [~f~l- (50)

b=l

Clearly action (48) is not a non-linear sigma model unless we choose f~ such that

AT~ =0, Ba~=O. (51)

One solution is

L = 0 j , (52)

wherefis the real part ofa holomorphic function with respect to any one complex structure.

Finally, to get a hyperkiihler target space, we must set the torsion to zero; that is we impose condition (19). Assuming (52), this implies

f=g. (53)

Then the metric, read off from (48), is

(54)

Eq. (54) is our final result, giving a hyperkiihler metric in terms of the quaternionic structure and a function g that is, to repeat, the real part of a hol- omorphic function with respect to each complex structure.

References

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Volume 190, number 1,2 PHYSICS LETTERS B 21 May t987

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