A twistor sphere of generalized Kahler potentials on hyperkahler manifolds
aa r X i v : . [ h e p - t h ] N ov A twistor sphere of generalized K¨ahlerpotentials on hyperk¨ahler manifolds
Malte Dyckmanns C.N.Yang Institute for Theoretical Physics, Stony Brook University,Stony Brook, NY 11794-3840,USA
Abstract
We consider the generalized K¨ahler structures ( g, J + , J − ) that arise on a hyperk¨ahler mani-fold ( M, g, I, J, K ) when we choose J + and J − from the twistor space of M . We find arelation between semichiral and arctic superfields which can be used to determine thegeneralized K¨ahler potential for hyperk¨ahler manifolds whose description in projective su-perspace is fully understood. We use this relation to determine an S -family of generalizedK¨ahler potentials for Euclidean space and for the Eguchi-Hanson geometry. Cotangentbundles of Hermitian symmetric spaces constitute a class of hyperk¨ahler manifolds whereour method can be applied immediately since the necessary results from projective super-space are already available. As a non-trivial higher-dimensional example, we determinethe generalized potential for T ∗ C P n , which generalizes the Eguchi-Hanson result. E-mail: [email protected]
Introduction
Hyperk¨ahler manifolds admit various generalized K¨ahler structures. The correspondinggeneralized K¨ahler potentials can be used to reconstruct the hyperk¨ahler geometry. Thesegeneralized potentials are in general quite different from the ordinary K¨ahler potential andthus provide a new way of studying hyperk¨ahler geometry and finding hyperk¨ahler metrics.To gain insight into this new way of looking at hyperk¨ahler geometry, one first needs tostudy some examples. In this paper, we make use of the twistor space of hyperk¨ahlermanifolds to develop a general method for determining their generalized K¨ahler potentialsand we also explicitly work out some examples.In the next section, we review the relevant features of generalized K¨ahler geometryin its bihermitian formulation. This geometry involves two complex structures J + , J − on a Riemannian manifold ( M, g ) and can locally be described by a generalized K¨ahlerpotential. In this paper, we consider the case where the kernel of [ J + , J − ] is trivial.Then the potential is defined as the generating function for a symplectomorphism betweencoordinates ( x L , y L ) and ( x R , y R ) that are holomorphic w.r.t. J + and J − respectively.Generalized K¨ahler geometry was initially found as the target space geometry of 2 D N =(2 ,
2) supersymmetric sigma models, where the potential is the superspace Lagrangian andthe coordinates x L , x R describe semichiral superfields.In section 3, we review aspects of hyperk¨ahler geometry and its twistor space. Weparametrize the twistor sphere of complex structures by a complex coordinate ζ andintroduce holomorphic Darboux coordinates Υ( ζ ) , ˜Υ( ζ ) for a certain holomorphic sym-plectic form. This construction is relevant for the projective superspace description of 2 D N = (4 ,
4) sigma models, where Υ , ˜Υ are arctic superfields.Using its twistor space, a hyperk¨ahler manifold can be seen as a generalized K¨ahlermanifold in various ways while keeping the metric fixed. In section 4, we consider atwo-sphere of generalized K¨ahler structures on a hyperk¨ahler manifold and express thecoordinates x L , x R , y L , y R in terms of Υ , ˜Υ. This enables us to determine the generalizedK¨ahler potential on a hyperk¨ahler manifold if we can find the decomposition of the arcticsuperfields Υ , ˜Υ in terms of their N = (2 ,
2) components, i.e. in terms of coordinates on M . In section 5, we consider four-dimensional hyperk¨ahler manifolds and explicitly deter-mine the partial differential equations that the coordinates describing those arctic super-fields have to fulfill. In section 6, we determine the potential for Euclidean space, wherethe differential equations for Υ , ˜Υ are easy to solve. In section 7, we look at the Eguchi-Hanson metric, where the relevant coordinates Υ , ˜Υ have been found previously in [3].We give an explicit expression for the S -family of generalized K¨ahler potentials for this1eometry, which belongs to the family of gravitational instantons and is thus of interestto physicists.The Eguchi-Hanson geometry lives on the cotangent bundle of C P . For hyperk¨ahlerstructures on cotangent bundles over arbitrary K¨ahler manifolds, projective superspacecan be used to determine the coordinates Υ , ˜Υ. This has been done in particular for allHermitian symmetric spaces. In section 8, we review this procedure and as a non-trivialhigher-dimensional example, we use the results for T ∗ C P n to determine its generalizedK¨ahler potential, which generalizes the Eguchi-Hanson result.In an appendix, we extend our results from section 4 and consider the full S × S -family of generalized K¨ahler structures on a hyperk¨ahler manifold, i.e. we let both J + and J − be an arbitrary point on the twistor sphere of complex structures. We also give theexplicit potential depending on two complex parameters ζ + , ζ − for the simplest hyperk¨ahlermanifold, namely for Euclidean space. Generalized K¨ahler geometry first appeared in the study of 2D N = (2 ,
2) nonlinear σ -models [5] and was later rediscovered by mathematicians as a special case of generalizedcomplex geometry [4]. In its bihermitian formulation, generalized K¨ahler geometry consistsof two (integrable) complex structures J + , J − on a Riemannian manifold ( M, g ), wherethe metric is hermitian with respect to J + and J − . Furthermore, the forms ω ± := gJ ± have to fulfill [7] d c + ω + + d c − ω − = 0 , dd c + ω + = 0 , (2.1)where d c ± = i ( ¯ ∂ ± − ∂ ± ). This allows us to define the closed three-form H := d c + ω + = − d c − ω − , whose local two-form potential we denote by B ( H = dB ). In general, ω ± is notclosed and thus ( M, g, J ± ) is not K¨ahler. In this paper, we will however consider the casewhere H = 0. Then ∂ ± ω ± , ¯ ∂ ± ω ± have to vanish separately, so dω ± = 0, i.e. ( M, g, J ± ) isK¨ahler.In [6] it was shown that like ordinary K¨ahler geometry, generalized K¨ahler geometry islocally described by a single function, the generalized K¨ahler potential. On a generalizedK¨ahler manifold, one can define the Poisson structure σ := [ J + , J − ] g − [8]. Here, we con-sider the case where [ J + , J − ] is invertible and recall how the generalized K¨ahler potentialis defined in this case [13]:Inverting σ gives Ω G := σ − = g [ J + , J − ] − , (2.2)2hich is a real, closed and non-degenerate two-form that fulfills J T ± Ω G J ± = − Ω G [8], i.e.it is a real holomorphic symplectic form both w.r.t. J + and w.r.t. J − . This means thatΩ G can be split into the sum of a (2 , , J + and w.r.t. J − :Ω G = Ω (2 , + Ω (0 , = Ω (2 , − + Ω (0 , − , (2.3)where ¯ ∂ ± Ω (2 , ± = 0 and Ω (0 , ± = Ω (2 , ± (here the complex conjugate is taken w.r.t. J + and J − respectively).One then introduces Darboux coordinates x pL and y Lp , holomorphic w.r.t. J + , for Ω (2 , ;and x pR and y Rp , holomorphic w.r.t. J − , for Ω (2 , − ( p = 1 , ..., n , where dim R M = 4 n ) [13].Then Ω G = Ω (2 , + Ω (0 , = dx pL ∧ dy Lp + d ¯ x pL ∧ d ¯ y Lp , Ω G = Ω (2 , − + Ω (0 , − = dx pR ∧ dy Rp + d ¯ x pR ∧ d ¯ y Rp , (2.4)i.e. the coordinate transformation from { x L , ¯ x L , y L , ¯ y L } to { x R , ¯ x R , y R , ¯ y R } is a symplecto-morphism (canonical transformation) preserving Ω G . It is thus described by a generatingfunction P ( x L , x R , ¯ x L , ¯ x R ) such that (omitting indices from now on) ∂P∂x L = y L , ∂P∂x R = − y R , ∂P∂ ¯ x L = ¯ y L , ∂P∂ ¯ x R = − ¯ y R . (2.5)This generating function is the generalized K¨ahler potential and can be used to locallyreconstruct all the geometric data of generalized K¨ahler geometry [13], i.e. the two complexstructures J + , J − , the metric g and the B -field. It also turns out to be the superspaceLagrangian for the N = (2 , σ -models that led to the discovery of generalized K¨ahlergeometry [6]. In this paper, we consider generalized K¨ahler structures ( g, J + , J − ) on a hyperk¨ahler mani-fold M and investigate their generalized K¨ahler potentials. For the choice of the twocomplex structures J + and J − , we will make use of the twistor space Z = M × S of M .Hyperk¨ahler manifolds appear for instance as the target spaces for hypermultipletscalars in four-dimensional nonlinear σ -Models with rigid N = 2 supersymmetry on the We can choose to let the potential depend on other combinations of old and new coordinates aswell. The potentials corresponding to the four different choices of variables are then related via Legendretransforms. In previous papers, the roles of x R and y R were interchanged. However, formulas in previouspapers for reconstructing g , J + , J − and B from the potential remain unchanged when using our convention. M, g, I, J, K ), where g is a Riemannian metric on M that is K¨ahler with respect to the three complex structures I, J, K , which fulfill the quaternion algebra (i.e. IJ = K = − J I ). In fact, there exists awhole two-sphere of complex structures on M with respect to which g is a K¨ahler metric,namely ( M, g, J = v I + v J + v K ) is K¨ahler for each ( v , v , v ) ∈ S . Using (the inverseof) the stereographic projection, we parametrize this family of complex structures on M in a chart of S including the north-pole by a complex coordinate ζ : J ( ζ ) := v ( ζ ) I + v ( ζ ) J + v ( ζ ) K := 11 + ζ ¯ ζ (cid:2) (1 − ζ ¯ ζ ) I + ( ζ + ¯ ζ ) J + i ( ¯ ζ − ζ ) K (cid:3) . (3.6)We define the complex two-forms ω (2 , := ω + iω , ω (0 , := ω − iω ; (3.7)where ω = gI , ω = gJ , ω = gK are the three K¨ahler forms. Then for each ζ ∈ C Ω H ( ζ ) := ω (2 , − ζ ω − ζ ω (0 , (3.8)turns out to be a holomorphic symplectic form with respect to the complex structure J ( ζ )[2]. In particular, ω (2 , = Ω H ( ζ = 0) is a (2 , I = J ( ζ = 0).Starting from ζ = 0, we can locally find holomorphic Darboux coordinates Υ p ( ζ ) and˜Υ p ( ζ ) ( p = 1 , ..., n , where dim R M = 4 n ) for Ω H ( ζ ) that are analytic in ζ such that [3]Ω H ( ζ ) = i d Υ p ( ζ ) ∧ d ˜Υ p ( ζ ) . (3.9)These canonical coordinates Υ , ˜Υ( ζ ) for Ω H are crucial for the projective superspace for-mulation of σ -models with eight real supercharges , where they describe ”arctic” super-fields. They have been determined for instance in [3] for the Eguchi-Hanson metric and wewill use them in this paper to determine the generalized K¨ahler potential for hyperk¨ahlermanifolds. We want to transport the idea of a twistor space from hyperk¨ahler to generalized K¨ahlergeometry, namely we interpret a hyperk¨ahler manifold (
M, g, I, J, K ) as a generalizedK¨ahler manifold (
M, g, J + , J − ), where we fix the left complex structure J + = I and let If we define ˘Υ( ζ ) := ¯Υ( − ζ ), ˘˜Υ( ζ ) := ¯˜Υ( − ζ ), then (Υ , ˜Υ) and ( ˘Υ, ˘˜Υ) are related by a ζ -twistedsymplectomorphism whose generating function f (Υ , ˘Υ; ζ ) can be interpreted as the projective superspaceLagrangian [3]. ζ : J − = J ( ζ ) (see eq. (3.6)). So for a givenhyperk¨ahler manifold, we consider an S -family of generalized K¨ahler structures whosegeneralized K¨ahler potentials we now try to determine.First, we need an explicit expression for the symplectic form Ω G (eq. (2.2)), whichnow depends on ζ . The anticommutator of two complex structures on a locally irreduciblehyperk¨ahler manifold is equal to a constant times the identity, { J + , J − } = c (see, e.g.,[9]). If J + = ± J − , then | c | < √ − c [ J + , J − ] is another complex structure, so inparticular it squares to − . Using this, we haveΩ G = g [ J + , J − ] − = − − c g [ J + , J − ] , (4.10)which in our case, where we have c = − v = − − ζ ¯ ζ ζ ¯ ζ and [ J + , J − ] = 2 v K − v J , givesΩ G ( ζ ) = − − v ( v ω − v ω ) = − ζ ¯ ζ ζ ¯ ζ (cid:2) ( ζ + ¯ ζ ) ω − i ( ¯ ζ − ζ ) ω (cid:3) . (4.11)This can be split into the sum of the holomorphic form Ω (2 , = i ¯ ζ ζ ¯ ζ ζ ¯ ζ ω (2 , and theantiholomorphic form Ω (0 , = − iζ ζ ¯ ζ ζ ¯ ζ ω (0 , with respect to J + (see equation (3.7)). Com-bining equations (3.8) and (3.9), we can choose the following Darboux coordinates forΩ G ( ζ ): x pL = Υ p ( ζ = 0) , y Lp = − ¯ ζ ζ ¯ ζ ζ ¯ ζ ˜Υ p ( ζ = 0);¯ x pL = ¯Υ p ( ζ = 0) , ¯ y L p = − ζ ζ ¯ ζ ζ ¯ ζ ¯˜Υ p ( ζ = 0) . (4.12)With respect to J − = J ( ζ ), Ω G splits into the sum of Ω (2 , − = i ¯ ζ ζ ¯ ζ Ω H ( ζ ) and Ω (0 , − = − iζ ζ ¯ ζ Ω H ( ζ ). Consequently, we can choose x pR = Υ p ( ζ ) , y Rp = − ¯ ζ ζ ¯ ζ ˜Υ p ( ζ );¯ x pR = Υ p ( ζ ) , ¯ y R p = − ζ ζ ¯ ζ ˜Υ p ( ζ ) . (4.13)We are thus able to express the coordinates x L,R and y L,R that describe semichiral su-perfields in N = (2 ,
2) models in terms of the coordinates Υ( ζ ), ˜Υ( ζ ) describing arcticsuperfields in the projective superspace formulation of N = (4 ,
4) supersymmetric sigmamodels. This will enable us to determine the ζ -dependent generalized K¨ahler potential forhyperk¨ahler manifolds whose projective superspace description is known. We denote the complex conjugate of Υ( ζ ) by ¯Υ ≡ ¯Υ(¯ ζ ) ≡ Υ( ζ ) which is not to be confused with thenotation in [3], where ¯Υ is shorthand for ˘Υ( ζ ) = ¯Υ( − ζ − ). The four-dimensional case
In this section, we consider the four-dimensional case and explicitly determine the partialdifferential equations for Υ( ζ ) and ˜Υ( ζ ) in order to be holomorphic w.r.t. J ( ζ ) and tofulfill equation (3.9). A four-dimensional K¨ahler manifold ( M, g, I ) is hyperk¨ahler if andonly if around each point there are holomorphic coordinates ( z, u ) on M such that theK¨ahler potential K ( z, u ) fulfills the following Monge-Amp`ere equation [10]: K u ¯ u K z ¯ z − K u ¯ z K z ¯ u = 1 . (5.14)From a K¨ahler potential fulfilling this equation, we can construct the three K¨ahler forms: ω = − i ∂ ¯ ∂K,ω = i dz ∧ du − d ¯ z ∧ d ¯ u ) ,ω = 12 ( dz ∧ du + d ¯ z ∧ d ¯ u ) . (5.15)Together with the metric g whose line element is ds = K u ¯ u dud ¯ u + K u ¯ z dud ¯ z + K z ¯ u dzd ¯ u + K z ¯ z dzd ¯ z, (5.16)we get the three complex structures, where equation (5.14) ensures that J = g − ω and K = g − ω indeed square to − .We find the following basis for the (1 ,
0) forms w.r.t. J ( ζ ): θ = dz − ζ K u ¯ u d ¯ u − ζ K u ¯ z d ¯ z, θ = du + ζ K z ¯ u d ¯ u + ζ K z ¯ z d ¯ z. (5.17)For Υ , ˜Υ to be holomorphic w.r.t. J ( ζ ), d Υ( ζ ) and d ˜Υ( ζ ) must be linear combinations of θ and θ (here the differential does not act on ζ ): d Υ = ∂ Υ ∂z θ + ∂ Υ ∂u θ , d ˜Υ = ∂ ˜Υ ∂z θ + ∂ ˜Υ ∂u θ . (5.18)Here the coefficients have been determined by comparing the dz - and du -terms on bothsides. From equations (5.17) and (5.18), we get the requirement that both Υ and ˜Υ haveto fulfill the following two PDEs: ∂ Ψ ∂ ¯ z = ζ (cid:18) K z ¯ z ∂ Ψ ∂u − K u ¯ z ∂ Ψ ∂z (cid:19) , ∂ Ψ ∂ ¯ u = ζ (cid:18) K z ¯ u ∂ Ψ ∂u − K u ¯ u ∂ Ψ ∂z (cid:19) (Ψ = Υ , ˜Υ) . (5.19)Furthermore, we find thatΩ H ( ζ ) = idz ∧ du + iζ ∂ ¯ ∂K + iζ d ¯ z ∧ d ¯ u = iθ ∧ θ , (5.20)so using (5.18), we obtain that equation (3.9) corresponds to the requirement ∂ Υ ∂z ∂ ˜Υ ∂u − ∂ Υ ∂u ∂ ˜Υ ∂z = 1 . (5.21)6 Euclidean space
We now use the relation between x, y and Υ , ˜Υ derived in section 4 to determine thegeneralized K¨ahler potential for Euclidean space. Here the K¨ahler potential is given by K = u ¯ u + z ¯ z, (6.22)which clearly fulfills equation (5.14). Assuming that ( z, u ) are holomorphic coordinatesw.r.t. I and setting ω (2 , = idz ∧ du , we get the complex structures as described insection 5. They are the differentials (pushforwards) of the left action of the imaginarybasis quaternions i, j, k on H ≈ C , where we make the identification ( z, u ) = ( x + ix , x + ix ) x + ix + jx + kx .Υ( ζ ) = z − ζ ¯ u, ˜Υ( ζ ) = u + ζ ¯ z (6.23)fulfill equations (5.19) and (5.21), i.e. they are holomorphic w.r.t. J ( ζ ) and satisfyequation (3.9). Using equations (4.12) and (4.13), we make the identifications x L = Υ( ζ = 0) = z, x R = Υ( ζ ) ≡ Υ and y L = − ζ ¯ ζ ζ u, y R = − ζ ˜Υ . (6.24)Solving for y L , y R in terms of x L , x R , we get y L = − ζ ¯ ζ ζ ¯ ζ (¯ x L − ¯ x R ) , y R = − ζ ¯ ζ (cid:0) (1 + ζ ¯ ζ )¯ x L − ¯ x R (cid:1) , (6.25)which leads (up to an additive constant) to the generating function (see equation (2.5)) P = − ζ ¯ ζ (cid:2) x R ¯ x R + (1 + ζ ¯ ζ ) · ( x L ¯ x L − x L ¯ x R − ¯ x L x R ) (cid:3) . (6.26)This is the generalized K¨ahler potential for Euclidean space, where J + = I and J − is anarbitrary point on the twistor-sphere of complex structures, J − = ± I . However, we noticethat P only involves the combination ζ ¯ ζ , i.e. it only depends on the angle between J + and J − in the space spanned by the three complex structures ( I, J, K ). Also P turns out to beasymmetric between left- and right-coordinates. This can be resolved however, as there arevarious ambiguities in the generalized K¨ahler potential. For instance, we could distributefactors differently in (6.24) or even perform a more complicated symplectomorphism, goingto new coordinates x ′ L/R , y ′ L/R . If we make the identifications x ′ L = i s ζ ¯ ζ ζ z, x ′ R = i r ζ Υ and y ′ L = i s ζ ¯ ζ ζ u, y ′ R = i r ζ ˜Υ , (6.27) We stick to the convention from previous papers and include the i -factor in the choice of ω (2 , .This interchanges the complex structures J and K , s.t. J corresponds to left multiplication by k and K corresponds to left multiplication by − j . x ′ R and y ′ R , we arrive at the potential P ′ = q ζ ¯ ζ · ( x ′ L y ′ R + ¯ x ′ L ¯ y ′ R ) + q ζ ¯ ζ · ( x ′ L ¯ x ′ L + y ′ R ¯ y ′ R ) . (6.28) The real function K = p u ¯ u (1 + z ¯ z ) + 12 log u ¯ u (1 + z ¯ z ) (cid:16) p u ¯ u (1 + z ¯ z ) (cid:17) (7.29)in the two complex variables z, u fulfills the Monge-Amp`ere equation (5.14). It thusdefines a hyperk¨ahler metric, where the K¨ahler forms are given by equation (5.15). Thefirst K¨ahler form takes the form ω = − i z ¯ z p u ¯ u (1 + z ¯ z ) " (1 + z ¯ z ) du ∧ d ¯ u + 2 u ¯ z dz ∧ d ¯ u +2 z ¯ u du ∧ d ¯ z + (cid:18) z ¯ z ) + 4 u ¯ u (cid:19) dz ∧ d ¯ z , (7.30)from which the metric can be read off. This is the well-known Eguchi-Hanson geometry .The holomorphic Darboux coordinates for Ω H ( ζ ) (fulfilling equations (5.19) and (5.21))can be chosen as [3]˜Υ = u + ζ ¯ z ¯ u + ¯ zζ z ¯ z p u ¯ u (1 + z ¯ z ) , Υ = z − uζ (1 + z ¯ z ) p u ¯ u (1 + z ¯ z ) + 2¯ u ¯ zζ (1 + z ¯ z ) . (7.31)We solve Υ , ¯Υ( z, ¯ z, u, ¯ u ) for u and ¯ u to get u ( z, ¯ z, Υ , ¯Υ) = ζ z ¯ z · (¯ z − ¯Υ)(1 + Υ¯ z ) ζ ¯ ζ (1 + ¯Υ z )(1 + Υ¯ z ) − ( z − Υ)(¯ z − ¯Υ) (7.32)and its complex conjugate. Using this and the identifications derived in section 4 (equation(6.24)), we get y L ( x L , x R ). We then integrate y L ( x L , x R ) w.r.t. x L to get the generalized Setting u = u ′ , z = z ′ u ′ and r := √ u ′ ¯ u ′ + z ′ ¯ z ′ gives the familiar K¨ahler potential K = √ r +log r √ r for the Eguchi-Hanson metric [3]. x L : P = Z y L ( x L , x R ) dx L = − ¯ ζ ζ ¯ ζ ζ ¯ ζ Z u ( z, Υ) dz = − · log 1 + x L ¯ x L ζ ¯ ζ (1 + x L ¯ x R )(1 + ¯ x L x R ) − ( x L − x R )(¯ x L − ¯ x R ) (7.33)Plugging u ( z = x L , Υ = x R ) into ˜Υ( z = x L , u ) (equation (7.31)) gives y R ( x L , x R ) = − ¯ ζ ζ ¯ ζ ˜Υ( x L , x R ) = − · (1 + ζ ¯ ζ ) · ¯ x L − (cid:0) − ζ ¯ ζ · x L ¯ x L (cid:1) ¯ x R ζ ¯ ζ (1 + x L ¯ x R )(1 + ¯ x L x R ) − ( x L − x R )(¯ x L − ¯ x R )(7.34)which is indeed equal to − ∂P∂x R . P is real, so ∂P∂ ¯ x L = ¯ y L and ∂P∂ ¯ x R = − ¯ y R are also fulfilledand thus equation (7.33) gives indeed the ζ -dependent generalized K¨ahler potential forthe Eguchi-Hanson geometry: P ( x L , ¯ x L , x R , ¯ x R ) = − · log 1 + | x L | ζ ¯ ζ · | x L ¯ x R | − | x L − x R | . (7.35)Again, the generalized K¨ahler potential turns out to depend only on the combination ζ ¯ ζ ,i.e. on the angle between J + and J − . Of course, there are again many ambiguities in thepotential, but (7.35) seems to be already in its simplest form. The target space of 4D N = 2 sigma models is constrained to be a hyperk¨ahler manifold [1].This corresponds to 2 D N = (4 ,
4) sigma models without B -field . Projective superspaceprovides methods to construct such models and for a large class of examples it can beused to extract the arctic superfields Υ and ˜Υ [18],[20],[21] (see [17] for a review) thatwe need in order to determine the generalized K¨ahler potential of the hyperk¨ahler targetspace using the method derived in chapter 4. This has been done in particular for thehyperk¨ahler structure on cotangent bundles of Hermitian symmetric spaces ([18]-[22]). Asan example, we use the results from [18],[20],[21] to determine the generalized K¨ahlerpotential for T ∗ C P n = T ∗ ( SU ( n + 1) /U ( n )). The special case n = 1 then corresponds tothe Eguchi-Hanson geometry that we considered in the last section. For vanishing B -field, all the results from 4 D N = 2 projective superspace can be immediatelytransferred to 2 D N = (4 ,
4) projective superspace. Actually the results that we are using only dependon the target space geometry, not on the number of space-time dimensions.
9t is a well-known fact that a hyperk¨ahler metric exists on (some open subset of) thecotangent bundle of every K¨ahler manifold M [15],[16],[19]. In projective superspace, thiscorresponds to models where the projective superspace Lagrangian f (Υ , ˘Υ; ζ ) (see footnote3) does not explicitly depend on ζ [18]. To obtain the coordinates Υ and ˜Υ for T ∗ M , onetakes the N = (4 ,
4) projective superspace Lagrangian f (Υ , ˘Υ) to be the K¨ahler potential K ( φ, ¯ φ ) of the base space M , where the arctic and antarctic superfields Υ , ˘Υ replace thechiral and antichiral superfields φ, ¯ φ of the N = (2 ,
2) description, i.e. the holomorphiccoordinates on M . In N = (2 ,
2) components, Υ decomposes asΥ = φ + ζ Σ + ∞ X j =2 ζ j X j , (8.36)where φ is a chiral superfield describing the coordinates on M , Σ is a complex linearsuperfield describing coordinates in the fiber of the tangent bundle T M and all higher orderterms are unconstrained auxiliary superfields [19],[22]. Solving the algebraic equations ofmotion for the auxiliary superfields yields Υ, and thus the Lagrangian in terms of the N = (2 ,
2) superfields ( φ, Σ) and the auxiliary complex variable ζ . Integrating out ζ anddualizing the action, i.e. performing a Legendre transform replacing the complex linearsuperfields Σ by chiral superfields ψ , gives the transformation Σ( ψ ) from which we obtainΥ( φ, ψ ). Here, ψ describes coordinates in the fiber of the cotangent bundle T ∗ M . ˜Υ( φ, ψ )can then be obtained from f and Υ( φ, ψ ) via ˜Υ = ζ ∂f∂ Υ [3]. One can also read off theordinary K¨ahler potential of T ∗ M from the dualized action [19],[22]. T ∗ C P n The crucial step in the above procedure is to eleminate the infinite tower of unconstrainedauxiliary N = (2 ,
2) superfields using their algebraic equations of motion. This has beendone for instance in [20] for T ∗ C P n .For the projective sigma model with target space T ∗ C P n , we take the projective su-perspace Lagrangian f to be the K¨ahler potential of the Fubini-Study metric on C P n : f (Υ i ( ζ ) , ˘Υ i ( ζ )) = a log j ˘Υ j a ! . (8.37)Here, a is a real parameter. The equations of motion for the auxiliary superfields have been The duality between chiral ψ and complex linear superfields Σ is just an ordinary coordinate trans-formation that does not change the target space geometry. N = (2 ,
2) superfields :Υ i = z i + ζ Σ i − ζ ¯ z ¯ k Σ k a + z l ¯ z ¯ l . (8.38)Here, we change notation and let z ≡ φ parametrize the base space and u ≡ ψ the fibersof the cotangent bundle.Dualizing the action of the sigma model to go from complex linear coordinates Σ tochiral coordinates u gives the equations [20] u i = − g i ¯ j ¯Σ ¯ j − g k ¯ l Σ k ¯Σ ¯ l a , ¯ u ¯ i = − g ¯ ij Σ j − g k ¯ l Σ k ¯Σ ¯ l a ; (8.39)which have to be solved for the old coordinates Σ, ¯Σ in terms of u , ¯ u . Here g i ¯ j is g i ¯ j = a δ ij a + z k ¯ z ¯ k − a ¯ z ¯ i z j ( a + z l ¯ z ¯ l ) , (8.40)the Fubini-Study metric on C P n , and g i ¯ j is its inverse. We find the following solution:Σ i = − g i ¯ j ¯ u ¯ j q g k ¯ l u k ¯ u ¯ l a , ¯Σ ¯ i = − g ¯ ij u j q g k ¯ l u k ¯ u ¯ l a . (8.41)Plugging this into (8.38) gives the arctic superfields Υ in terms of chiral N = (2 , z and u : Υ i = z i − ζ u ¯ j g i ¯ j q g k ¯ l u k ¯ u ¯ l a + 2 ζ g p ¯ q ¯ z p ¯ u ¯ q a + z m ¯ z ¯ m . (8.42)Together with ˜Υ i = ζ ∂f∂ Υ i = ζ ˘Υ i Υ j ˘Υ j a , (8.43)this is all the information we need to determine the generalized K¨ahler potential for T ∗ C P n using the identifications found in section 4.Solving (8.42) and its complex conjugate for u and ¯ u gives u ( z, Υ): u i = ζ · a ( a + z k ¯ z ¯ k )( a + ¯ z ¯ k Υ k ) · g i ¯ j (¯ z ¯ j − ¯Υ ¯ j ) a ζ ¯ ζ ( a + ¯ z ¯ k Υ k )( a + z k ¯Υ ¯ k ) − ( z l − Υ l ) g l ¯ m (¯ z ¯ m − ¯Υ ¯ m )( a + z k ¯ z ¯ k ) . (8.44) This result was already obtained in the preparation of [19] and later independently derived and firstpublished in [20]. Repeated indices are always summed over 1 , ..., n and the metric is always written out explicitly, i.e.we never use it to raise or lower indices. z i gives: Z u i dz i = ζ a ζ ¯ ζ log a + z T ¯z a ζ ¯ ζ ( a + ¯z T Υ )( a + z T ¯Υ ) − ( z − Υ ) T g ( ¯z − ¯Υ )( a + z T ¯z ) . (8.45)Here, no sum is implied on the left-hand side and on the right-hand side we use vectornotation ( z := ( z , ..., z n ) T , etc.) and g := ( g i ¯ j ) ≤ i,j ≤ n . So, up to an additive term c ( x R , ¯ x R ), the generalized K¨ahler potential for T ∗ C P n is P = − a a + x L T ¯x L a ζ ¯ ζ ( a + ¯x T L x R )( a + x L T ¯x R ) − ( x L − x R ) T g ( ¯x L − ¯x R )( a + x L T ¯x L ) . (8.46)For n = 1 and a = 1, we have g z ¯ z = z ¯ z ) and all the results from section 7 arereproduced. Therefore, we assume that c ( x R , ¯ x R ) can be set to zero. There is an increasing number of examples, most notably among Hermitian symmetricspaces, where the decomposition of the N = (4 ,
4) arctic superfields Υ, ˜Υ in terms oftheir N = (2 ,
2) components ( z, u ) has been determined. In these cases, one can applythe methods developed in this paper to determine more examples of generalized K¨ahlerpotentials on hyperk¨ahler manifolds. Having whole classes of manifolds available for ouranalysis, one could try to find more general statements about the generalized K¨ahlerpotential in the case of hyperk¨ahler manifolds.The Eguchi-Hanson geometry is one of the hyperk¨ahler manifolds that can be obtainedfrom the generalized Legendre transform construction in [2] (generalized T-duality). Themanifolds stemming from that construction are 4 n -dimensional hyperk¨ahler manifoldsadmitting n commuting tri-holomorphic killing vectors. They are called toric hyperk¨ahlermanifolds and have been classified in [14]. It should be possible to determine the relevantcoordinates Υ( z, u ; ζ ) and ˜Υ( z, u ; ζ ) for toric hyperk¨ahler manifolds. For four-dimensionaltoric hyperk¨ahler manifolds, [11] gives a formula for the generalized K¨ahler potential as acertain threefold Legendre transform in the special case ζ ¯ ζ = 1. One could compare thisconstruction with our results at least for the examples given in this paper or try to relatethe two methods in general for four-dimensional toric hyperk¨ahler manifolds. As a furtherexplicit example, one could for instance consider the Taub-NUT geometry and determineits generalized K¨ahler potential.The generalized K¨ahler potential for the Eguchi-Hanson geometry can also be obtainedfrom a generalized quotient of Euclidean 8-dimensional space by a U (1)-isometry and inthis setting turns out to be exactly (7.35) as well [12].12 D N = (2 ,
2) sigma models have a target space that is a generalized K¨ahler manifoldand in general, they are described by chiral, twisted chiral and semichiral superfields.The models with hyperk¨ahler target space and J + = ± J − are described purely in termsof semichiral superfields. These models do not in general admit off-shell N = (4 , N = (4 ,
4) supersymmetry if thetarget space is 4 n -dimensional with n >
1. However, they are always dual to models with N = (4 ,
4) supersymmetry that are parametrized by chiral and twisted chiral superfields[11]. The exact relation between the N = (4 ,
4) sigma models described by chiral andtwisted chiral superfields, and their dual semichiral models will be described in [25].The relation between the coordinates x L/R , y L/R and Υ, ˜Υ has been obtained inthis paper from a purely differential geometric approach. x L/R , y L/R describe left- andright-semichiral superfields in 2 D N = (2 ,
2) sigma models. For a target space that ishyperk¨ahler, these models are dual to models with N = (4 ,
4) supersymmetry that areparametrized by chiral and twisted chiral superfields. The coordinates Υ( ζ ), ˜Υ( ζ ) howeverdescribe arctic superfields in N = (4 ,
4) sigma models in projective superspace. The fieldtheoretical interpretation and understanding of this relation between arctic N = (4 , N = (4 ,
4) models remains an openproblem. The complex coordinate ζ is an auxiliary variable that gets integrated out inprojective superspace, but for ordinary superspace it is just a constant parameter. Thus inour relation, an arctic model corresponds to a two-sphere (or more precisely to a cylinder)of presumably equivalent semichiral models.In this paper, we mainly focused on the special case, where the bihermitian structure( J + , J − ) only depends on one complex parameter ζ and established a relation to projectivesuperspace. In the appendix, we show that the results from section 4 can be generalizedto the full S × S -family of generalized complex structures on a hyperk¨ahler manifoldparametrized by two complex parameters ζ + and ζ − . Many hints point towards an intimaterelation of this formulation to doubly-projective superspace [26],[27]. Acknowledgements :The author owes a lot to Martin Roˇcek, who initiated the project and kept it alive, provid-ing motivation and knowledge. The author would like to thank P. Marcos Crichigno forvaluable discussions during the creation of this paper. Ulf Lindstr¨om and Malin G¨otemanhave provided comments helping to put the paper into its final form.13
Appendix: S × S -family of generalized complexstructures In this appendix, we generalize our results from section 4. Instead of fixing one complexstructure, we can also let both J + and J − depend on an individual complex coordinateand thus consider an S × S -family of generalized complex structures ( M, g, J + , J − ) ona given hyperk¨ahler manifold ( M, g, I, J, K ). We parametrize vectors ~u , ~v ∈ S \ ( − , , ζ + , ζ − like in equation (3.6) and define J + := J ( ζ + ) = u I + u J + u K, J − := J ( ζ − ) = v I + v J + v K. (A.1)The anticommutator depends only on the angle θ between ~u and ~v : { J + , J − } = − ~u · ~v ) = − θ . (A.2)The commutator turns out to be perpendicular to J + and J − in the space spanned by( I, J, K ):[ J + , J − ] = 2( u v − u v ) I − u v − u v ) J +2( u v − u v ) K = 2( ~u × ~v ) · ( I, J, K ) T . (A.3)In order to determine the coordinates x R , y R , we need to split Ω G into a (2 , , J − . Indeed, we find that g [ J + , J − ] = i (1 + ζ − ¯ ζ − ) (cid:16) ( ~a ( ζ − ) · ~u ) Ω H ( ζ − ) − ( ~a ( ζ − ) · ~u ) Ω H ( ζ − ) (cid:17) , (A.4)where ~a ( ζ ) = ( − ζ , − ζ , i (1 + ζ )) T , i.e. Ω H ( ζ ) = ~a ( ζ ) · ~ω (see eq. (3.8)). So we findΩ G = − − ~u · ~v ) g [ J + , J − ] = Ω (2 , − + Ω (0 , − , (A.5)where Ω (2 , − = − i ( ~a ( ζ − ) · ~u )4 sin θ (1 + ζ − ¯ ζ − ) Ω H ( ζ − ) ≡ − ic − Ω H ( ζ − ) , Ω (0 , − = Ω (2 , − . (A.6)Thus knowing that Ω H ( ζ − ) = id Υ p ( ζ − ) ∧ d ˜Υ p ( ζ − ), we can choose (omitting indices) x R = Υ( ζ − ) , y R = c − ˜Υ( ζ − ) . (A.7)to get Ω G = dx R ∧ dy R + d ¯ x R ∧ d ¯ y R .Exchanging the roles of ~u , ~v and ζ + , ζ − respectively and considering the antisymmetryof [ J + , J − ], we get the following splitting w.r.t. J + :Ω (2 , = i ( ~a ( ζ + ) · ~v )4 sin θ (1 + ζ + ¯ ζ + ) Ω H ( ζ + ) ≡ − ic + Ω H ( ζ + ) , Ω (0 , = Ω (2 , , (A.8)14hich allows us to choose x L = Υ( ζ + ) , y L = c + ˜Υ( ζ + ) . (A.9)The constants c + , c − ( ζ + , ζ − ) can be written as c + = 1 + ζ − ¯ ζ − ζ + ¯ ζ − )( ζ + − ζ − ) , c − = 1 + ζ + ¯ ζ + ζ + ζ − )( ζ + − ζ − ) . (A.10)We see that by exchanging ζ + with ζ − , we exchange x L with x R and y L with − y R . Inthe special case ζ + = 0 (i.e. ~u = (1 , , ζ − = ζ , we have sin θ = ζ ¯ ζ (1+ ζ ¯ ζ ) and (A.7),(A.9) reduce to the results (4.12), (4.13) from section 4.Using (A.7) and (A.9), we can now also determine an S × S -family of generalizedK¨ahler potentials P ζ + ,ζ − for hyperk¨ahler manifolds. For Euclidean space, we find the ζ + -and ζ − -dependent generalized potential to be P = − ζ + − ζ − )( ¯ ζ + − ¯ ζ − ) " (1 + ζ − ¯ ζ − ) x L ¯ x L + (1 + ζ + ¯ ζ + ) x R ¯ x R − (1 + ζ + ¯ ζ + )(1 + ζ − ¯ ζ − ) (cid:18) x L ¯ x R ζ + ¯ ζ − + ¯ x L x R ζ + ζ − (cid:19) . (A.11) References [1] L. Alvarez-Gaume and D. Z. Freedman, “Geometrical Structure And UltravioletFiniteness In The Supersymmetric Sigma Model,” Commun. Math. Phys. , 443(1981).[2] N. J. 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