Question

Do all $\mathcal{N}=2$ Gauge Theories “Descend” from String Theory?

Answer 1

I'm thinking about the beautiful story of "geometrical engineering" by Vafa, Hollowood, where various types of N=2N=2 SYM gauge theories on R4=C2R4=C2 arise from considering string theory on certain local, toric Calabi-Yau threefolds.

More specifically, the topological string partition function (from Gromov-Witten or Donaldson-Thomas theory via the topological vertex) equals the Yang-Mills instanton partition function which is essentially the generating function of the elliptic genera of the instanton moduli space. (In various settings you replace elliptic genus with χyχy genus, χ0χ0 genus, Euler characteristic, or something more fancy.)

From my rough understanding of the Yang-Mills side, we can generalize this in a few ways. Firstly, we can consider more general N=2N=2 quiver gauge theories where I think the field content of the physics is encoded into the vertices and morphisms of a quiver. And there are various chambers where one can define instanton partition functions in slightly different ways, though they are expected to agree in a non-obvious way. For a very mathy account see Of course, the second way to generalize is to consider not R4R4, but more general four dimensional manifolds like a K3 surface, a four-torus, or the ALE spaces arising from blowing up the singularities of R4/ΓR4/Γ.

My questions are the following:

Is it expected that this general class of N=2N=2quiver gauge theories comes from string theory? In the sense that their partition functions may equal Gromov-Witten or Donaldson-Thomas theory on some Calabi-Yau threefold. If not, are there known examples or counter-examples? Specifically, I'm interested in instantons on the ALE spaces of the resolved R4/ΓR4/Γ.

In a related, but slightly different setting, Vafa and Witten showed that the partition function of topologically twisted N=4N=4 SYM theory on these ALE manifolds give rise to a modular form. Now I know Gromov-Witten and Donaldson-Thomas partition functions often have modularity properties related .