ar X iv : h ep - t h / 04 01 16 1 v 1 2 2 Ja n 20 04 CERN - PH - TH / 2004 - xxx 010 c – Map , very Special Quaternionic

We show that for all very special quaternionic manifolds a different N = 1 reduction exists, defining a Kähler Geometry which is “dual” to the original very special Kähler geometry with metric Gab̄ = −∂a∂b lnV (V = 1 6dabcλaλbλc). The dual metric g = V (G) is Kähler 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. 1 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 nH = n+ 1. In particular it was shown [2] that “dual” quaternionic manifolds always have at least 2n+4 isometries: one scale isometry ǫ0 and 2n+ 3 shift isometries βI , α I , ǫ+ (I = 0, · · · , n), whose generators close a Heisenberg algebra [3]: [β , ǫ] = [αI , ǫ ] = 0; [β , αJ ] = δ I Jǫ ; [ǫ, αI ] = 1 2 αI ; [ǫ , β ] = 1 2 β ; [ǫ, ǫ] = ǫ (1.1) The corresponding generators can be written according to their ǫ weight as [4, 5, 6, 7]: V = V0 + V 1 2 + V1. (1.2) 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 V−1,V− 1 2 , with dim(V−1) ≤ 1; dim(V− 1 2 ) ≤ 2n + 2. (1.3) In particular, for a generic very special geometry, with a cubic polynomial prepotential F (z) = 1 48 dabcz zz (1.4) with generic dabc, with no additional isometries, it was shown that: dim(V−1) = 0; dim(V− 1 2 ) = 1; dim(V0) = n + 2. (1.5) 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 V0, denoted by ωI , (I = 0, · · · , n), and one β̂0 in V− 1 2 . For symmetric spaces the upper bound in equation (1.3) is saturated so that dimGQ = dimGSK +4n+7 where GSK and GQ 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