We study the relations between two-dimensional Yang–Mills theory on the torus, topological string theory on a Calabi–Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern–Simons theory on torus bundles. The chiral partition function of the Yang–Mills gauge theory in the large $N$ limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang–Mills theory as an efficient tool for the computation of Gromov–Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar–Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured non-perturbative relation between two-dimensional Yang–Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson–Thomas theory of the Calabi–Yau background. We match the instanton representation of Yang–Mills theory on the torus with the non-abelian localization of Chern–Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of $BPS$ black holes in the local Calabi–Yau background.

Several Einstein–Sasaki seven-metrics appearing in the physical literature are fibred over four-dimensional Kähler–Einstein metrics. Instead we consider here the natural Kähler–Einstein metrics defined over the twistor space $Z$ of any quaternion Kähler four-space, together with the corresponding Einstein–Sasaki metrics. We work out an explicit expression for these metrics and we prove that they are indeed tri-Sasaki. Moreover, we present a squashed version of them which is of weak $G2$ holonomy. We focus in examples with three commuting Killing vectors and we extend them to supergravity backgrounds with T³ isometry, some of them with $AdS_4 × X_7$ near horizon limit and some others without this property. We would like to emphasize that there is an underlying linear structure describing these spaces. We also consider the effect of the $SL(2,R)$ solution-generating technique presented by Maldacena and Lunin to these backgrounds and we find some rotating membrane configurations reproducing the $E–S$ logarithmic behaviour.

We match collapsing inhomogeneous as well as spatially homogeneous but anisotropic spacetimes to vacuum static exteriors with a negative cosmological constant and planar or hyperbolic symmetry. The collapsing interiors include the inhomogeneous solutions of Szekeres and of Barnes, which in turn include the Lemaître–Tolman and the McVittie solutions. The collapse can result in toroidal or higher genus asymptotically $AdS$ black holes.