〉 PostScript processed by the SLAC/DESY Libraries on 21 May 1995. HEP-TH-9505119 , 1995 y ector eld R/95/15 Ma o-dimensional c w (1) WZW mo del. ector eld on the sigma U h is the space of orbits of w that the metric couplings (2) os SU e sho et CT y a tri-holomorphic Killing v t and nd that the only non-singular e t. W arian adopoul arian v v ap D.A.M.T.P Silver Str ABSTRA (3)-in G. P Cambridge CB3 9EW SO University of Cambridge y the tri-holomorphic Killing v e also examine the global structure of a sub class of these sigma mo dels with torsion c monop oles and (4,0)-sup ersymmetri (1) symmetry generated b U Ellipti es in addition the torsion in t under a v e explicitly construct the metric and torsion couplings of t W arian v arise from magnetic monop oles on thethe three-spheregroup action whic generated b mo del target manifold. W metrics that are in addition one, for mo dels with non-zero torsion, is that of (4,0)-sup ersymmetric sigma mo dels within target space athat four-manifold lea that are
1. Intro duction
It has b een known for sometime that there is an interplaybetween the num-
b er of sup ersymmetries which leave the action of a sigma mo del invariant and the
geometry of its target space. More recently, sigma mo dels with symmetries gener-
ated by Killing vector elds are a fertile area for investigation of the prop erties of
T-duality. The couplings of two-dimensional sigma mo dels are the metric g and a
lo cally de ned two-form b on the sigma mo del manifold M. The closed three-form
?
3
H = db is called Wess-Zumino term or torsion . A sp ecial class of sup ersym-
2
metric sigma mo dels are those with (4,0) sup ersymmetry [1, 2]. These mo dels are
ultra-violet nite [1, 3] and arise naturally in heterotic string compacti cations
[4]. It is known that the target space of (4,0)-sup ersymmetric sigma mo dels ad-
mits three complex structures that ob ey the algebra of imaginary unit quaternions.
When the torsion H vanishes the target space is hyp er-Kahler with resp ect to the
metric g . In the presence of torsion, the geometry of the target space of (4,0)-
sup ersymmetric sigma mo dels may not b e hyp er-Kahler and new geometry arises.
It can b e shown that given a (4,0)-sup ersymmetric sigma mo del with target space
a four-dimensional hyp er-Kahler manifold with metric g (H = 0), it is p ossible to
F F
construct another one with metric g = e g and torsion H = dF provided that
F is a harmonic function with resp ect to the metric g [4]. The converse though
it is not necessarily true. For example the b osonic Wess-Zumino-Witten (WZW)
mo del with target space SU (2) U (1) admits a (4,0)-sup ersymmetric extension
[5] but its target manifold is not hyp er-Kahler as it can b e easily seen by observing
that the second deRham cohomology group of SU (2) U (1) vanishes. To con-
struct (4,0)-sup ersymmetric sigma mo dels with torsion that are not conformally
related to hyp er-Kahler ones, the authors of ref. [6] employed harmonic sup erspace
metho ds and found a generalisation of the Eguchi-Hanson and Taub-NUT geome-
tries that have non-zero torsion. Subsequently,itwas shown in [7] that the ab ove
? In two-dimensions, the sigma mo del action can b e extended to include terms with couplings
other than g and b.For example, the fermionic sector in (p,0)-sup ersymmetric sigma mo dels
has as a coupling a Yang-Mills connection on M. 2
Eguchi-Hanson geometry with torsion is conformally equivalent to the standard
Eguchi-Hanson one.
One result of this letter is to give the most general metric and torsion couplings
of a (4,0)-sup ersymmetric sigma mo del with target space a four-dimensional mani-
fold M that admits a tri-holomorphic Killing vector eld X that in addition leaves
the torsion H invariant. Let N b e the space of orbits of the U (1) group action
generated by the vector eld X on the sigma mo del target manifold M (away from
the xed p oints of X ). We shall show that the non-vanishing comp onents of the
metric g and torsion H are given as follows:
2 1 0 i 2 i j
ds = V (dx + ! dx ) + V dx dx
i ij
(1:1)
H = V; i; j; k =1;2;3 ;
ij k ij k
where ! is a one-form, V is a scalar function and is a three-metric on N ,soall
i
dep end on the co-ordinates fx ; i =1;2;3g of N , and is a real constant. The
tensors , V and ! satisfy the following conditions:
k
2@ ! = @ V
ij
k
[i j ]
(1:2)
(3) 2
R =2 ;
ij k l
k[i j]l
where is adopted to the metric . The scalar function V is therefore a harmonic
ij k
function of N with resp ect to the metric and the metric has p ositive constant
curvature. The latter implies that the three-manifold N is elliptic and therefore
lo cally isometric to the (round) three-sphere. In the case that the constant is
zero one recovers the Gibb ons-Hawking metrics [8, 9] that are asso ciated with
(3) 3
monop oles on the at ( R = 0) three-space N = R . In the case that 6= 0 and
V constant, we recover the (4,0)-sup ersymmetric WZW mo del with target space
the manifold SU (2) U (1). We shall also study the global prop erties of a sub class
of these metrics that are invariant under an isometry group with Lie algebra u(2)
acting on M with three-dimensional orbits (u(2)-invariant metrics), for 6=0,
and we shall nd that apart from the metric of SU (2) U (1) WZW mo del all the
remaining ones are singular. 3
The material of this letter is organised as follows: In section two, the geometry
of (4,0)-sup ersymmetric sigma mo dels is brie y reviewed. In section three, the
derivation of equations (1.1) and (1.2) is presented. In section four, the singularity
structure of a sub class of the metrics (1.1) that are u(2)-invariantisinvestigated.
Finally in section vewe give our conclusions.
2. Geometry and (4,0) sup ersymmetry
Let M b e a Riemannian manifold with metric g and a lo cally de ned two-form
b. The action of the (1,0)-sup ersymmetric sigma mo del with target space M is
Z
2 +
I = i d xd (g + b )D @ ; (2:1)
+ =
= = + (1;0)
where (x ;x ; ) are the co-ordinates of (1,0) sup erspace and D is the
+
2 = =
sup ersymmetry derivative(D = i@ ); (x ;x )=(x+t; t x) are light-cone
=
+
co-ordinates. The elds of the sigma mo del are maps from the (1,0) sup erspace,
(1;0)
,into the target manifold M.
To construct sigma mo dels with (4,0) sup ersymmetry,weintro duce the trans-
formations
= a I D (2:2)
r r +
I
written in terms of (1,0) sup er elds, where I , r=1,2,3, are (1,1) tensors on M
r
and a are the parameters of the transformations. The commutator of these
r
?
transformations closes to translations provided that
I I = + I
r s rs rst t
(2:3)
N (I ) =0 ;
r
? Due to the classical sup erconformal invariance of the mo del the parameters a of the sup er-
r
symmetry transformations can b e chosen to b e semi-lo cal in which case the commutator of
sup ersymmetry transformations closes to translations and sup ersymmetry transformations. 4
where
N (I ) = I @ I I @ I ( $ ) (2:4)
r r r r r
is the Nijenhuis tensor of I . The conditions (2.3) imply that I , r =1;2;3, are
r r
complex structures that satisfy the algebra of imaginary unit quaternions.
The action (2.1) of (1,0)-sup ersymmetric sigma mo del is invariant under the
(4,0) sup ersymmetry transformations (2.2) provided that, in addition to the con-
ditions obtained ab ove for the closure of the algebra of these transformations, the
following conditions are satis ed:
(+)
g I =0 ; r I =0 ; (2:5)
r r
( )
()
where r is the covariant derivative of the connection
()
= f gH ; (2:6)
and
3
@ b : (2:7) H =
[ ]
2
Note that if I are integrable, N (I ) = 0, and covariantly constant with resp ect to
r r
(+) (+)
the r covariant derivative, r I = 0, then the torsion H is (2,1)-and (1,2)-
r
form with resp ect to all complex structures I . Note also that, if the torsion H is
r
zero, the ab ove conditions simply imply that the sigma mo del target manifold is
hyp er-Kahler with resp ect to the metric g
Next consider the transformations
a
= X ( ) ; (2:8)
a
a
of the sigma mo del eld , where f ; a =1;2;:::g are the parameters of these
transformations and fX ;;a =1;2;:::g are vector elds on the sigma mo del man-
a
ifold M. The action (2.1) is invariant under these transformations up to surface 5
terms provided that
r X + r X =0; X H = @ u : (2:9)
a a
[ a ]
a
These conditions imply that fX ;;a =1;2;:::g are Killing vector elds on the
a
sigma mo del manifold M and leave the closed three-form H invariant . The com-
mutator of the (2.8) transformations with the (4,0) sup ersymmetry transformations
is
b
[ ; ] = a L I D (2:10)
r r +
I X
b
This commutator closes on the existing symmetries of the theory,ifwe take
L I =0 ; (2:11)
r
X
a
where L is the Lie derivative with resp ect to the vector eld X . All three com-
a
X
a
plex structures I are invariant under the vector eld X , i.e. X is tri-holomorphic.
r a a
Note that it is p ossible to relax the ab ove condition. For example, we can take
that the complex structures I rotate under the isometries but this p ossibility will
r
not b e considered here. Finally the commutator of the transformations (2.8) with
themselves closes provided that
c
X (2:12) [X ;X ]=f
c a
b ab
c
are the structure constants of a Lie algebra. where f
ab 6
3. (4,0) sup ersymmetry and four-dimensional geometry
To nd the geometry of the target space of sigma mo dels with (4,0) sup er-
symmetry, one has to solve the conditions (2.3) and (2.5) of the previous section.
For this we will restrict ourselves to four-dimensional target spaces M and we
will assume that there is a Killing vector eld X on M that leaves the torsion H
and the complex structures I invariant. As wehave seen in the previous section,
r
these conditions on the vector eld X are those required for the invariance of the
action (2.1) under the transformations (2.8) of the sigma mo del elds and the
closure of the commutator these transformations with the (4,0) sup ersymmetry
transformations.
0 i
Next we adopt co ordinates fx ;x ;i =1;2;3g on the sigma mo del manifold
along the Killing vector eld X , i.e.
@
X = : (3:1)
0
@x
Then the metric g and the torsion H are written as follows:
2 1 0 i 2 i j
ds = V (dx + ! dx ) + V dx dx ; (3:2)
i ij
and
H = @ u ; H = U; (3:3)
0ij
ij k ij k
[i j ]
where , u, V and U are tensors of the space of orbits, N , of the group action
generated by the Killing vector eld X on M (away from the xed p oints of X )
i
and dep end only up on the co-ordinates fx ; i =1;2;3g. The tensor is a metric
on N and the tensor is adopted to the three-metric .
ij k 7
Weintro duce the frame
1
0 i 0
2
(dx + ! dx ) ; e = V
i
(3:4)
1
r
r i
2
e = V E dx ; r =1;2;3
i
asso ciated with the metric (3.2), and its dual
1
2
e = V @
0 0
(3:5)
1
i
2
e @ ! @ ; = V E
r i i 0
r
@ @
r
where @ = , @ = , and E and E is a frame of the metric and its dual,
0 0 i i r
@x @x
resp ectively. Next weintro duce three invariant (1,1) tensors on M as follows:
r 0 t
I = e e e e e e (3:6)
r 0 r rst s
Contracting with the metric g ,we can show that
0 r s t
I I dx dx =2e ^e e e : (3:7)
r r rst
are two-forms on M and so the rst condition of eqn. (2.5) is satis ed. It is
straightforward to verify that the tensors I are almost complex structures that
r
satisfy the algebra of imaginary unit quaternions and so the rst condition of
eqn. (2.3) is also satis ed. Furthermore, it can b e shown that fI ; r =1;2;3g
r
(eqn. (3.6)) is the most general set of almost complex structures, up to an SO(3)
r
(gauge) rotation of the frame E of the three-metric , that ob ey the algebra of
imaginary unit quaternions and are invariant under the group action generated
by the Killing vector eld X . Therefore X is a tri-holomorphic vector eld and
satis es eqn. (2.11). So it remains to nd the conditions on V ; U; u and in order
the metric (3.2), antisymmetric tensor (3.3) and the almost complex structures I
r
satisfy the second condition of (2.3) and the second condition of (2.5). Combining 8
the second condition of (2.5) and (2.11), we can show that
(+)
r X (3:8)
j
i
is a (1,1)-form with resp ect to all almost complex structures I which in turn
r
implies that
k
@ V: (3:9) 2@ ! 2V@ u =
ij
k
[i j ] [i j]
Next using the fact that the torsion H is (2,1)-and (1,2)-form with resp ect to all
almost complex structures I , the condition
r
(+)
r
=0 (3:10) r I
]
[
implies that
1
du =0; UV = ;
(3:11)
(3) 2
R =2 ;
ij k l
k[i j]l
(3)
where R is the curvature of the metric and is a real constant. After some
computation, it can b e shown that the almost complex structures are integrable
without further conditions, i.e. their Nijenhuis tensor vanishes. Finally, the second
condition of (2.5) follows from the integrabilityof I , the condition (3.10) and the
r
fact that H is (2,1)- and (1,2)-form with resp ect to all complex structures I .
r
Substituting the equations (3.9) and (3.11) backinto (3.2) and (3.3) we get the
metric g and torsion H of eqn. (1.1).
Apart from the sp ecial cases for which either =0 or V constant mentioned
in the intro duction, new metrics can b e found by taking
N
X
i
V (x )=c + c G(x; x ) (3:12)
0 n n
n=1
where fG(x; x ); n =1;:::;Ng are the Green's functions of the Laplace-Beltrami
n
op erator asso ciated with the metric , fc ;c ;n =1;:::;Ng are real constants
0 n
and fx ; n =1;:::;Ng are N p oints in N .
n 9
The metric g of (1.1) is not conformally equivalenttoa hyp er-Kahler one if
0
6= 0 and x is an angular co-ordinate. To show this, let
F F
g = e g (3:13)
F
where F is a function of M. Since I are complex structures, to prove that g is
r
F F
hyp er-Kahler it is enough to show that the three two-forms I g I are
r
r
F F
closed, i.e. dI =0. After some computation, it can b e shown that dI =0
r r
implies that
1
@ F = 2V
0
(3:14)
1
@ F = 2V ! :
i i
It is clear that, if = 0 (the torsion H is zero), the most general solution of (3.14)
F
is that F is equal to a real constant. In this case g is hyp er-Kahler and so is g .
In the case that 6=0,we di erentiate the second equation in (3.14) with resp ect
to @ and we get
0
@ @ F =0 (3:15)
0 i
but
1
@ @ F @ @ F = 2@ V =0 ; (3:16)
0 i i 0 i
which implies that V is constant. So for the metric g to b e conformally equivalent
to a hyp er-Kahler metric, V must b e a real constant. However, the case that V
0
equal to a constant can also b e excluded if the co-ordinate x is an angle. To see this
0
observe that all solutions F of (3.14) are linear in x and so they cannot b e scalar
functions of M. For example, the metric g of the (4,0)-sup ersymmetric WZW
mo del with target space the group SU (2) U (1) is not conformally equivalentto
ahyp er-Kahler one. Using a similar argument and under the assumptions that
0
is an angle, we can also show that the metric g of eqn (1.1) is not 6= 0 and x
conformally equivalenttoa Kahler one. 10
4. u(2)-invari ant metrics
The metrics given in eqn. (1.1) mayhave singularities. To examine such global
prop erties of these metrics, we consider the following example. Let us write the
three-metric in spherical p olar co-ordinates
2 2 2 2 2
d = R d + sin d (4:1)
3 2
where R is the radius of the three-sphere and
2 2 2 2 2
d = R d + sin d' (4:2)
2
is the metric on a two-sphere. Next consider the case that V is equal to the Green's
function G that dep ends only on . Such V is
V = c cot + c (4:3)
1 0
where c and c are real constants. Then it can b e arranged that
1 0
! = c cos d' : (4:4)
1
The metric g with V given in (4.3) admits an isometry group with Lie algebra
u(2) acting on M with three-dimensional orbits. We will consider three cases the
following. Case(i) the constant c = 0 and the constant c 6= 0, in this case the
0 1
metric g exhibits singular b ehaviour at = 0 and = . The singularityat
2
=0isanut singularity and can b e removed by the standard metho ds of ref. [10]
0
x
provided that 0 < 4 (c > 0). However the singularityat = cannot b e
1
c R 2
1
removed as it can b e easily seen bychanging the co-ordinate to u = cot and
by studying the b ehaviour of the metric at u = 0. Case (ii) the constants c 6=0
0
and c 6= 0 , in this case it can b e arranged by rescaling the metric to set c =1.
1 0
1
The metric g has singular b ehaviour at = 0 and cot = . The singularityat
c
1 11
0
x
=0 isanut singularity and again it can b e removed provided that 0 < 4
c R
1
1
(c > 0) as in case (i). But the singularity at cot = cannot b e removed
1
c
1
1
as it can b e easily seen bychanging the co-ordinate to u = cot + and by
c
1
studying the b ehaviour of the metric at u = 0. Another way to see that the metric
1
is singular at cot = is to observe that the geo desics of M that dep end only
c
1
up on reach the singularity at nite prop er time and they cannot b e extended
beyond it. The manifold M is therefore geo desically incomplete.
Next de ne the new co-ordinate s = tan , then, the metric g is rewritten as
2 2
R s
2
0 1 2 1 1 2 2
dx + c cos d' +(c +c s ) ds =(c +c s ) : ds + d
1 0 1 0 1
2
2 2
1+s 1+s
(4:5)
This metric as s ! +1 (or ! ) b ehaves as
2
0
x 1
2 2 2 2 2 2 2 2 2
ds R dw + c R (d( ) + cos d ) + (d + sin d ) (4:6)
1
2
Rc
c
1
1
1
where w = s . Note that, if c = 1, the manifold M near = b ecomes
1
2
3 3
RS and S has radius 2R but eventually the metric g will b ecome singular at
cot + 1 = 0 as wehave mentioned ab ove. The metric g in the parameterisation
0
(4.5) involving the co-ordinates fs; x ;; g is the same as the metric given by the
authors of ref. [6] (up to a gauge choice for the one-form ! , and a relab elling
of some of parameters of the metric and some of the co-ordinates of M) for the
Taub-NUT geometry with torsion. The same authors had also observed that the
space of orbits N for this metric is the three-sphere. But our conclusion that this
metric is singular is opp osite from that of ref. [6]. The torsion is
3 2
H = c R sin sin (c cot + c )d d d ; (4:7)
1 1 0
and it is non-singular. Finally, case (iii) the constant c 6= 0 and the constant
0
c = 0, in this case the metric g and the torsion H b ecome those of the SU (2) U (1)
1
WZW mo del. 12
5. Concluding Remarks
Wehave determined the metric and torsion couplings of (4,0)-sup ersymmetric
sigma mo dels with a No ether symmetry generated by a tri-holomorphic Killing
vector eld and target space a four-dimensional manifold. These metrics are nat-
urally asso ciated to monop oles on the three-sphere which is the space of orbits
of the group action generated by the tri-holomorphic Killing vector eld on the
sigma mo del target manifold. In relation to this result, it is worth p ointing out
that the hyp er-Kahler Gibb ons-Hawking metrics are asso ciated with monop oles on
the at three-space, and the scalar at Kahler LeBrun metrics [11] are asso ciated
with monop oles on the hyp erb olic three-space. Wehave also studied the global
prop erties of a sub class of these metrics that admit an isometry group with Lie
algebra u(2) acting on the sigma mo del manifold with three-dimensional orbits, for
mo dels with non-zero torsion, and wehave found that, apart from the metric of
the WZW mo del with target space the group SU (2) U (1), are singular. It also
seems likely that all the remaining metrics, for mo dels with non-zero torsion, are
singular as well.
The class of the ab ove metrics that are singular cannot b e thoughtasgravi-
tational instantons asso ciated with some string inspired extension of the Einstein
gravity. However they may serve as couplings of sup ersymmetric sigma mo dels
way from the singularities. Such sigma mo dels will b e ultra-violet nite by the
arguments of refs. [1, 3]. It may also b e that the (4,0) sup ersymmetry of the
ab ove sigma mo dels can b e extended to a (4,4) one by an appropriate addition
of a fermionic sector. This suggestion is supp orted by the fact that the (4,0)-
sup ersymmetric WZW mo del with target space the group SU (2) U (1) admits
such an extension.
The T-duality transformation can b e easily applied to the couplings g and b of
the (4,0)-sup ersymmetric sigma mo del with target space a four-dimensional mani-
fold with resp ect to the tri-holomorphic Killing vector eld X to give the couplings
0 0 0
, b and dilaton of the dual mo del [12] (see also [13]). In fact the couplings g and g 13
0
b can b e simpli ed in this case b ecause, as wehave shown in section 3, X H =0.
In addition, the T-dual theories of the ab ove (4,0)-sup ersymmetric sigma mo dels
are exp ected to b e ultra-violet nite since the latter are ultra-violet nite. Fi-
nally, the application of the new metrics that wehave derived from consideration
of (4,0)-sup ersymmetric mo dels to string theory and conformal eld theory needs
further study. Suchaninvestigation will involve the construction of the asso ciated
sup erconformal eld theory.
Acknowledgments: Iwould like to thank Gary Gibb ons and Paul Townsend for
many useful discussions on the global prop erties of four-metrics with singularities,
Chris Hull for his comments on T-duality and Emery Sokatchev for corresp ondence.
I am supp orted by a University ResearchFellowship from the Royal So ciety.
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