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

Abstract

AbstractThe tree-level q-map assigns to a projective special real (PSR) manifold of dimension $$n-1\ge 0$$ n - 1 ≥ 0 , a quaternionic Kähler (QK) manifold of dimension $$4n+4$$ 4 n + 4 . It is known that the resulting QK manifold admits a $$(3n+5)$$ ( 3 n + 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 $$\mathrm {SL}(2,{\mathbb {R}})$$ SL ( 2 , R ) group of isometries related to the $$\mathrm {SL}(2,{\mathbb {Z}})$$ SL ( 2 , 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 $$\mathrm {SL}(2,{\mathbb {R}})$$ SL ( 2 , R ) action by isometries, enlarging the previous universal group of isometries to a $$(3n+6)$$ ( 3 n + 6 ) -dimensional group G. As part of this analysis, we describe how the $$(3n+5)$$ ( 3 n + 5 ) -dimensional subgroup interacts with the $$\mathrm {SL}(2,{\mathbb {R}})$$ SL ( 2 , 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 $$(3n+6)$$ ( 3 n + 6 ) -dimensional group of universal isometries to the twistor space.