This paper discusses the relation between topological M-theory, self-dual Yang–Mills and general relativity. We construct a topological membrane field action from Witten’s cubic string field theory, which reduces to topological Yang–Mills on a one-parameter family of conifolds. It turns out that this can be interpreted as the twistor space of the four-dimensional Lagrangian submanifold M for large momenta. From the viewpoint of the target, we find that A-model and B-model on M unify in the topological membrane theory through the Penrose–Ward transform. The partition function is constructed and it is shown that, in the weak-coupling regime, it is equal to the partition function of Donaldson-Witten theory. Additionally, homological mirror symmetry, background independence as well as role of knot cobordisms as topological two-branes is discussed. It is outlined that all types of Floer homology are part of the topological membrane theory. Additionally, we find evidence that in the non-perturbative regime, the partition function of the membrane field action and that of the partially twisted (2,0) SU(N) superconformal field theory on the worldvolume of N topological fivebranes must coincide.

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We show how a foliated 4-dimensional FLRW-metric becomes a gravitational instanton, if the spatial metric minimizes a three-dimensional Einstein–Hilbert action with positive cosmological constant, which is equal to the demand, that the scale factor satisfies the Bogomolny-equation, where the curvature parameter varies over the one-parameter family of hyperslices and takes the role of a potential depending on the scale factor. Additionally, we draw the connection to SO(4)-Chern–Simons theory and show how the established interpolating solutions describe the gradient flow between the minima of the vacuums of the Einstein–Hilbert action, as well as how they can be used to calculate tunnelling-amplitudes of gravitons and trivialize the calculations of path integrals in quantum gravity. All the calculations are carried out particularly for k admitting a $${\mathbb {Z}}_{2}$$ Z 2 -symmetry.

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