I should preface this my saying that I'm an algebraic geometer, and am not terribly knowledgeable about physics, nor the physics literature.

I want to consider the singular surfaces $\mathbb{C}^{2}/\Gamma$ where $\Gamma$ is a finite ADE type subgroup of $SU(2)$. We can blow up these non-compact surfaces to get *smooth* non-compact surfaces which are remarkably, Calabi-Yau. It's well known that inside the smooth surface, we get a bunch of $\mathbb{P}^{1}$'s linked together nodally, precisely like the corresponding Dynkin diagram. For the easiest example, note that for $\mathbb{C}^{2}/\mathbb{Z}_{2}$, the resulting smooth surface is the total space of the bundle $\mathcal{O}(-2) \to \mathbb{P}^{1}$. This is both non-compact and Calabi-Yau.

I was watching an old talk of Vafa's and he vaguely mentioned that in the six "extra dimensions" of string theory, you can take four of the six to come from one of these ADE spaces. It's unclear to me whether he meant the singular ones, or the smooth ones. This intrigued me. I know the physicists want these six dimensional spaces to be Calabi-Yau, so you can take for example, $X \times E$, where $X$ is one of these ADE surfaces and $E$ is an elliptic curve. Here $X$ can be either the smooth or the singular surface; both are Calabi-Yau as the resolution is 'Crepant'.

**So my question is,** have there been results on string theory compactified on ADE surfaces times an elliptic curve, or something similar? If so, I'd be extremely interested. I know there's been recent papers on the elliptic genera of the surfaces, but I'm curious if people have thought about topological string theory on a threefold intimately related to these ADE surfaces.

This post imported from StackExchange Physics at 2017-02-05 11:29 (UTC), posted by SE-user spietro