c--Map,very Special Quaternionic Geometry and Dual Ka\"hler Spaces

We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kaehler Geometry which is ``dual'' to the original very special Kaehler geometry with metric G_{a\bar{b}}= - \partial_a \partial_b \ln V (V={1/6}d_{abc}\lambda^a \lambda^b \lambda^c). The dual metric g^{ab}=V^{-2} (G^{-1})^{ab} is Kaehler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in Type IIB compactifications on Calabi--Yau orientifolds.


Isometries of dual quaternionic manifolds
One of the basic constructions in dealing with the low energy effective Lagrangians of Type IIA and Type IIB superstrings is the so called c-map [1], which associates to any Special Kähler manifold of complex dimension n a "dual" quaternionic manifold of quaternionic dimension n H = n + 1.
In particular it was shown [2] that "dual" quaternionic manifolds always have at least 2n + 4 isometries: one scale isometry ǫ 0 and 2n + 3 shift isometries β I , α I , ǫ + (I = 0, · · · , n), whose generators close a Heisenberg algebra [3]: (1.1) The corresponding generators can be written according to their ǫ 0 weight as [4,5,6,7]: However it was shown in [6,7] that when the Special Kähler manifold has some isometries, then some "hidden symmetries" are generated in the c-map spaces which are classified by In particular, for a generic very special geometry, with a cubic polynomial prepotential with generic d abc , with no additional isometries, it was shown that: Since the isometries of a generic very Special Geometry of dimension n are n + 1, the dual manifold has then 3n + 6 isometries, where the n + 2 additional isometries lie, n + 1 in V 0 , denoted by ω I , (I = 0, · · · , n), and oneβ 0 in V − 1 2 . For symmetric spaces the upper bound in equation (1.3) is saturated so that dimG Q = dimG SK + 4n + 7 where G SK and G Q are the isometry groups of the Special Kähler and Quaternionic spaces respectively.
2 The very Special σ-model Lagrangian and its N = 1 reduction The quaternionic "dual" σ-model for a generic Special Geometry was derived in [2] by dimensional reduction of a N = 2 Special Geometry to three dimensions. By adapting the conventions of [2] to those of [6] and [8] we call the special coordinates z a as z a = x a + iy a and define: The 2n + 4 additional coordinates are denoted by ζ I ≡ (ζ 0 , ζ a ),ζ I ≡ (ζ 0 ,ζ a ), D,Φ.
The α I , β I isometries act as shifts on the 2n + 2 coordinates ζ I ,ζ I : while the ω a shift isometries of the special geometry, δx a = ω a , act as duality rotations on the ζ I ,ζ I symplectic vector: On the other hand theβ 0 isometry rotates ζ a into x a so that the x a ,ζ a variables are related by quaternionic isometries. It is immediate to see that the full σ-model Lagrangian [2,6,8] is invariant under the following parity operation Ω: so that, restricting to the plus-parity sector is a consistent truncation, giving rise to the following Lagrangian for 2n + 2 (real) variables: By a change of variables we can decouple the (D, ζ 0 ) fields from the rest as follows: define two new variables (Φ, λ a ): Thus it follows that V (λ)e 2D = e Φ 2 and the Lagrangian becomes: Therefore in the (t a ,ζ a ) variables we finally get Therefore by defining the complex variables we get for the 2n-dimensional σ-model: The previous Lagrangian is Kähler provided This condition is achieved by settingK = −2 log V (λ). Indeed

Isometries of the N = 1 reduction
The σ-model isometries of the c-map, using the notations of [7] are parametrized by ǫ + , ǫ 0 , α I , β I , ω a , ω 0 ,β 0 . (3.1) The N = 1 reduction projects out ǫ + , α a , β 0 , ω a , so the remaining isometries are n + 4, namely: β a , ω 0 , ǫ 0 , α 0 ,β 0 . Three of the latter generate a SL(2, R) symmetry (otherwise absent in generic dual quaternionic manifolds), the others generate a shift symmetry in ℑη a and a scale symmetry in the η a variables. The dual manifold has the same isometries of the original Special Kähler. Since theζ a variables are related to the x a variables by quaternionic isometries, the two manifolds need in fact not be distinct. Even though theζ a variables are related to the x a variables by quaternionic isometries, the two manifolds are in general distinct. However, in the particular case of homogeneous-symmetric spaces [9], it turns out that the dual manifold coincide with the original one. The proof of this statement will be given elsewhere.

Connection with Calabi Yau orientifolds
The c-map was originally studied in relation to the Type II A → Type II B mirror map in Calabi-Yau compactifications. In Calabi Yau orientifolds of Type II B strings with D-branes present, the bulk Lagrangian is obtained combining a world-sheet parity with a manifold parity which, for generic spaces [10], is precisely doing the truncation we have encountered in this note. For certain Calabi Yau manifolds more generic orientifoldings are possible where the set of special coordinates z A is separated in two parts with opposite parity, z A ± (n + + n − = n) such that [11] y ± → ± y ± and then consequently However in this case one must demand in order for the N = 1 reduction to be consistent [12]. In this case the σ-model Lagrangian acquires more terms and can be symbolically written as: where for the sake of simplicity space-time indices have been suppressed from partial derivatives and contraction over them is understood. In (4.4) G ++ is as before since d +++ = 0, G +− = 0 and G −− = −6 (d −−+ y + )/(d +++ y + y + y + ). The total set of coordinates are: y + , x − , ζ − ,ζ + and (Φ, ζ 0 ). Since in this case some of the y coordinates, namely y − , have been replaced by x − , the new variables define a Kähler manifold of complex dimension n + 1 certainly distinct from the original one.
Also cases in which a further splitting appears are realized if the orientifold projection acts [18] differently on T p−3 ×T 9−p (p = 3, 5, 7, 9). This is the analogue of the y ± , x ± splitting [11]. In all these cases the dual manifolds coincide, as predicted by N = 4 supergravity.
5 Properties of the dual Special Kähler spaces and noscale structure.
The dual Kähler space, obtained by a N = 1 truncation of the (c-map) very special quaternionic space has a metric that satisfies a "duality" relation with the original very special Kähler space: Moreover it can be shown that its affine connection is simply related to the affine connection of original Kähler space: Actually in the one-dimensional case the two connections coincide. These dual spaces are also no-scale [20,21,22]. Indeed it is sufficient to prove that But this is indeed the case since λ a G ab λ b = 3. From a Type II B perspective, this was anticipated in [23] 6 Concluding remarks.
In this note we have shown that for an arbitrary very special geometry, through the c-map, it is possible to construct a "dual" Kähler geometry which has a dual metric, it is Kähler and it provides a dual no-scale potential. Recently such constructions have found applications in Calabi Yau orientifolds [24,11] but the procedure considered here is intrinsic to the four dimensional context. We have not shown that the final Lagrangian is supersymmetric but, using the reduction techniques of [12], it can be shown that this is indeed the case. It is reassuring that the SL(2, R) symmetry, related to the Type II B interpretation, comes out in a pure four dimensional context, thanks to the results of [6,7]