We construct classes of smooth metrics which interpolate from Bianchi attractor geometries of Types II, III, VI and IX in the IR to Lifshitz or AdS 2 × S 3 geometries in the UV. While we do not obtain these metrics as solutions of Einstein gravity coupled to a simple matter field theory, we show that the matter sector stress-energy required to support these geometries (via the Einstein equations) does satisfy the weak, and therefore also the null, energy condition. Since Lifshitz or AdS 2 × S 3 geometries can in turn be connected to AdS 5 spacetime, our results show that there is no barrier, at least at the level of the energy conditions, for solutions to arise connecting these Bianchi attractor geometries to AdS 5 spacetime. The asymptotic AdS 5 spacetime has no non-normalizable metric deformation turned on, which suggests that furthermore, the Bianchi attractor geometries can be the IR geometries dual to field theories living in flat space, with the breaking of symmetries being either spontaneous or due to sources for other fields. Finally, we show that for a large class of flows which connect two Bianchi attractors, a C-function can be defined which is monotonically decreasing from the UV to the IR as long as the null energy condition is satisfied. However, except for special examples of Bianchi attractors (including AdS space), this function does not attain a finite and non-vanishing constant value at the end points.

We study the Borel summation of the Gromov–Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson–Thomas (DT) invariants of the resolved conifold, having a direct relation to the Riemann–Hilbert problem formulated by Bridgeland (Invent Math 216(1), 69–124, 2019). There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann–Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by Jafferis and Moore (Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909, 2008).

We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra g of infinitesimal automorphisms of the initial hyper-Kähler data, we associate a central extension of g, acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the c-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed c-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous, then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic Kähler metric on SU (n, 2)/S(U (n) × U (2)), which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet (n = 1), for which we determine the full isometry group. Contents

We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete 1/4-pinched negatively curved quaternionic Kähler (i.e. half conformally flat Einstein) metrics g c , c ≥ 0, on R 4 . The metric g 0 is the complex hyperbolic metric whereas the family (g c ) c>0 is equivalent to a family of metrics (h b ) b>0 depending on b = 1/c and smoothly extending to b = 0 for which h 0 is the real hyperbolic metric. In this sense the one-loop deformation interpolates between the real and the complex hyperbolic metrics. We also determine the (singular) conformal structure at infinity for the above families.

We classify Einstein metrics on $$\mathbb {R}^4$$ R 4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.