Losely speaking, a topological twist has the effect that it exchanges some of the worldsheet supersymmetry with topological symmetry, which can be much easier investigated. Building a BRST charge from these symmetries, we find in the two-dimensional case that in its cohomology are either Kähler deformations, or deformations of the conplex structure of the Kähler target. So there are two possible ways to twist the N=2 superconformal algebra, resulting in TQFTs with critical dimension 6. Another cool effect of the twisting is that the theories are semi-classical exact and that they are related by topological T-Duality (mirror symmetry). As in the phyisical case there is also a theory connecting both models.
We need to think of such a geometric transtition as a continuous physical process. This can be described with the help of a Narain lattice, which is the composition of a lattice and its dual. The authomorphisms of this lattice are elements of O(d,d,Z) and they lift to the symmetries of the CFT.
Additionally, A-model and B-model are also S-dual. This means that while making a geometric transition we vary the coupling continiously from the A-model geometry to the B- model geometry. For example we go from the deformed conifold over the singular to the resolved conifold. This is, the theory has four sectors. Let's start in the limit of zero coupling. Here the geometry is the deformed conifold which is isomomorphic to the cotangent bundle of some three-manifold T*Y. With Wittens string field theory on calculates that the A-model reduces to Chern -Simons theory because the higher modes of the string decouple and only zero modes contribute. Therefore the string field is just a U(N) Lie algebra-valued gauge potential. The group comes from the N branes we have to wrap around the zero locus were the open A-strings end. Now we increase the coupling to 1 and move towards the singular geometry. The three-manifold decribes a path and shrinks, the CFT is getting deformed and the geometry looks like T*M for M a four-manifold. Hence we need something else to describe the theory on the worldsheet times R in this regime, as well as the worldvolume the three-branes describe while transitioning. Here, with the help of the membrane field theory one shows that topological M-theory becomes twisted N=2 super Yang-Mills on Y×R in the finite perturbative regime and the membrane field is an instanton. This theory is a four-dimensional twisted cohomological theory, which computes the Donaldson polynomials. Moving further into the non-perturbative regime, a two-sphere grows instead and the geometry is M×T*S^2. We have 5-branes wrapped around M×S^2, which is the twistorspace of M. The worldvolume theory is partially twisted U(N) (0,2) superconformal field theory in 6d and one can show that this is topological M-theory on M×T*S^2 and the membrane field is a torsion-free coherent sheaf over the twistor space. This fits also well because it is known that the 4d N=2 theory is the infrared limit of the 6d theory. Topological M theory is a way to understand why. If the coupling approaches infinity the boundary looks like the resolved conifold were the membrane field takes values in the string field, which is a section of holomorphic bundle. Thus the B-model reduces to holomorphic Chern-Simons as it should be.
The whole transition is a membrane field and the dynamics are described by the membrane field action. Very cool. The claim is now that this can be generalized to arbitrary 4 and 6 manifolds. One can also show that the 6d theory is the holomorphic dual to M theory on AdS_7×S^4.
In this way we have a theory whose UV regime is a quantum theory of non-abelian gerbes. Furthermore, since the topological string is part of the physical one, both are described by the string field. We should expect the same from the topological membrane.
With regard to the second part of your post, it is known that the gradient flow lines of the CS functional are instantons. I have shown, that in the case of FLRW metrics with semi-definite Ricci curvature, the EH action has the same structure as topological Yang-Mills on Y×R. Its vacuum looks like the background of N infinite D3-branes. This made me wonder if one could apply the open/closed string duality in this case. Indeed, in the category of A branes the morphisms are strings stretched between them, mathematically described with symplectic Floer homology. Using the Atiyah-Floer conjecture I can instead consider instantons. So the situation in general relativety is mirrored by the equivalence of the wrapped Fukaya category and the (1,infinity)-category with morphisms as Lagrangian cobordisms. Understanding membrane fields as instantons over M or semi-stable coherent sheaves over the twistorspace opens a deep connection between GR and topologcial M-theory. In this szenario our expanding spacetime is an instanton as a tunneling process of closed strings between two 3-branes . It also follows that GR beyond the Planck length is somehow a 6d superconformal theory.
