S-Duality and the Universal Isometries of q-Map Spaces

Abstract

The tree-level q-map assigns to a projective special real (PSR) manifold of dimension n-1ge 0, a quaternionic Kähler (QK) manifold of dimension 4n+4. It is known that the resulting QK manifold admits a (3n+5)-dimensional universal group of isometries (i.e. independently of the choice of PSR manifold). On the other hand, in the context of Calabi–Yau compactifications of type IIB string theory, the classical hypermultiplet moduli space metric is an instance of a tree-level q-map space, and it is known from the physics literature that such a metric has an mathrm {SL}(2,{mathbb {R}}) group of isometries related to the mathrm {SL}(2,{mathbb {Z}}) S-duality symmetry of the full 10d theory. We present a purely mathematical proof that any tree-level q-map space admits such an mathrm {SL}(2,{mathbb {R}}) action by isometries, enlarging the previous universal group of isometries to a (3n+6)-dimensional group G. As part of this analysis, we describe how the (3n+5)-dimensional subgroup interacts with the mathrm {SL}(2,{mathbb {R}})-action, and find a codimension one normal subgroup of G that is unimodular. By taking a quotient with respect to a lattice in the unimodular group, we obtain a quaternionic Kähler manifold fibering over a projective special real manifold with fibers of finite volume, and compute the volume as a function of the base. We furthermore provide a mathematical treatment of results from the physics literature concerning the twistor space of the tree-level q-map space and the holomorphic lift of the (3n+6)-dimensional group of universal isometries to the twistor space.