R/95/15 May, 1995 Elliptic monop oles and (4,0)-sup ersymmetric sigma mo dels with torsion G. Papadopoulos D.A.M.T.P University of Cambridge Silver Street Cambridge CB3 9EW ABSTRACT We explicitly construct the metric and torsion couplings of two-dimensional processed by the SLAC/DESY Libraries on 21 May 1995. 〉 (4,0)-sup ersymmetric sigma mo dels with target space a four-manifold that are invariant under a U (1) symmetry generated by a tri-holomorphic Killing vector eld PostScript that leaves in addition the torsion invariant. We show that the metric couplings arise from magnetic 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. We also examine the global structure of a sub class of these metrics that are in addition SO(3)-invariant and nd that the only non-singular one, for mo dels with non-zero torsion, is that of SU (2) U (1) WZW mo del. HEP-TH-9505119 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 .