The topological string is a simplified version of physical string theory. It is of interest because it computes the BPS spectrum of relevant string theory compactifications, but also because it shares structural properties of physical string theory, Dualities and symmetries which often must be argued for arduously in the physical string can often be verified by computation in the topological setting. The central observable of the theory is the topological string partition function Z_top. This quantity has an eerie habit of making surprise appearances in many areas of mathematical physics. Numerous techniques exist for its computation in various expansions in parameters of the theory, yet to date, no satisfactory closed form for this quantity is known. In this talk, after reviewing notions of topological string theory with an emphasis on the interplay between worldsheet and target space physics (one of the structural similarities between the physical and the topological string alluded to above), I will report on progress in computing Z_top in settings where it is related to enigmatic 6d theories.

In this talk, I will discuss work in progress with Bertrand Eynard, in which we derive the BKMP remodelling the B-model conjecture, in the large radius limit. This is the claim that Gromov-Witten invariants of any toric Calabi-Yau 3-fold coincide with the spectral invariants of the mirror curve. Our method consists in explicitly constructing a matrix model which reproduces the topological string partition function obtained via the vertex formalism, and then demonstrating that the spectral curve of this matrix model coincides with the mirror geometry.

Suppose we could calculate the string partition function to all genera using worldsheet methods. What could we learn from this expression about a potentially underlying (non-perturbative) target space description of the theory? We address this question in the context of the open topological A-model.