I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I am specifically looking at toric Calabi-Yau threefolds for which the relevant data can be described by a quiver (possibly with a superpotential). It seems that there are several different methods that have been employed, but I am having a difficult time sorting out which methods have been found to work in what scenarios. For a given method, I'd appreciate an answer to the following related questions:
- In what chambers of the Kähler moduli space can this method be employed?
- What manifolds or type(s) of manifolds has this method been used on?
- What types of invariants can this method compute? By that I mean, is it only good for numerical Donaldson-Thomas invariants, or can it be refined to get e.g. motivic Donaldson-Thomas invariants.
- Is there a known relationship between wall-crossing formulas and this model?
I am still a little foggy on the exact relationship between BPS invariants and Donaldson-Thomas invariants, and so unfortunately I am phrasing my question to be about both of them instead of a particular type.
An example answer I am looking for would be something like this. Given a quiver diagram corresponding to a toric Calabi-Yau threefold, you can use the path algebra of the quiver to build a "crystal", and we consider the process of melting this crystal from the point of view of classical statistical mechanics. The partition function of this melting crystal model is a generating function for BPS invariants. This model can be constructed in chambers X, Y and Z of the Kähler moduli space for manifolds A, B, and C, and how this model relates to wall crossing phenomena has been interpreted in cases D, E, and F, and it has been refined to compute motivic Donaldson-Thomas invariants in the case of G, H, and I.
There are other methods that have been found to compute these invariants, such as the topological vertex. I'd like answers about these other methods if possible (I probably do not even know about all of them), but if that makes this question too broad then I will be happy with an explanation of what is known about the melting crystal model. If there is a recent reference that gives a broad review of these types of questions, I would be very pleased to have that as an answer as well. I am relatively proficient in the mathematical language used in this area, but I only understand the basics of string theory, so I'd appreciate a mathematically-focused answer. Thanks!