S-duality and the universal isometries of q-map spaces

TL;DR

This paper proves that all tree-level q-map spaces possess an $ ext{SL}(2, ext{R})$ isometry group, extending the known universal isometry group and connecting mathematical structures with string theory dualities.

Contribution

It provides a rigorous mathematical proof of the $ ext{SL}(2, ext{R})$ symmetry in q-map spaces, enlarging the universal isometry group and analyzing its interaction with existing symmetries.

Findings

Established $ ext{SL}(2, ext{R})$ action as isometries of q-map spaces.

Described the interaction between the new $ ext{SL}(2, ext{R})$ symmetry and the existing universal group.

Constructed quotients of q-map spaces leading to finite-volume quaternionic Kähler manifolds.

Abstract

The tree-level q-map assigns to a projective special real (PSR) manifold of dimension $n-1\geq 0$, a quaternionic K\"{a}hler (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 K\"ahler 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.