For arithmetic connections to string theory, look at recent papers of Candelas, de la Ossa and collaborators.

Another useful review is "Update on Modular Non-Rigid Calabi-Yau Threefolds" by Edward Lee arxiv.org/pdf/0803.0006.

The review "Modularity of Calabi-Yau Varieties" arxiv.org/abs/math/0601238 by Klaus Hulek, Remke Kloosterman and Matthias Schütt, while somewhat outdated, will be a useful starting point.

This is a late comment, and you may know this already, but (for $\mu=0$ at least) such expressions, including the substitution in 2, show up in our work on Hilbert schemes of points of ADE singularities, see e.g. arxiv.org/pdf/1512.06844 which has some explanation of the connection to affine Lie algebra reps also.

One (big) step up from this example, this Stoppa-Thomas paper works out one concrete but pertinent example in great detail. arxiv.org/abs/0903.1444

If we are really collecting mayors, one must mention en.m.wikipedia.org/wiki/Sergio_Cecotti of BCOV fame.

There is "Moduli spaces M_{g,n}(W) for surfaces" by Valery Alexeev arxiv.org/pdf/alg-geom/9410003, which constructs a relevant moduli space. Perhaps forward searching for papers referring to this one will turn up something useful.

...with isolated fixed point set, right?

Duplicate of mathoverflow.net/questions/457256/…

Have you looked at the recent paper arxiv.org/abs/2311.07539 by Kaplan and Schedler? Seems highly relevant.

In this case (and indeed $z$ central, I forgot to say this earlier), you can still only have $z=0$ for finite dimensional quotient for the affine case (or point-like Hilbert function in the projective case), for the usual reason of taking the trace (in char zero). The commutative part only exists (for point-like objects) along the line $z=0$.

...In the context of my remark, you would have to homogenise first, as you would do in standard commutative algebraic geometry, to get the equation $[x,y]=z^2$ in three variables. Now you have a graded non-commutative algebra, with all the right conditions so that the Artin et al theory applies. And in this case, all point modules must have $z=0$, so ''lie at infinity'', and they are simply points on the ''classical'' ${\mathbb P}^1$ inside the noncommutative ${\mathbb P}^2$.

@AlexanderChervov no, they are not. Note that from the point of view of my comment, the Weyl algebra is affine, not projective; it's a noncommutative deformation of ${\mathbb A}^2$, not of ${\mathbb P}^2$. It has no finite-dimensional quotients, so no points...