Gromov-Witten/Hilbert versus AdS3/CFT2 Correspondence

We consider the boundary dual of AdS3xS3xK3 for NS5-flux Q5=1, which is described by a sigma model with target space given by the d-fold symmetric product of K3. Building on results in algebraic geometry, we address the problem of deforming it away from the orbifold point from the viewpoint of topological strings. We propose how the 't Hooft expansion can be geometrized in terms of Gromov-Witten invariants and, in favorable settings, how it can be summed up to all orders in closed form. We consider an explicit example in detail for which we discuss the genus expansion around the orbifold point, as well as the divergence in the strong coupling regime. We find that within the domain of convergence, scale separation does not occur. However, in order for the mathematical framework to be applicable in the first place, we need to consider "reduced" Gromov-Witten invariants that fit, as we argue, naturally to topologically twisted N=4 strings. There are some caveats and thus to what extent this toy model captures the physics of strings on AdS3xS3xK3 remains to be seen.