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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.

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  • $\begingroup$ see "ALE spaces" $\endgroup$ – Mitchell Porter Feb 4 '17 at 7:46
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M-theory compactified on a ADE singularity space is rather nice to describe qualitatively: Let $\Gamma\subset\mathrm{SU}(2)$ be an ADE group and consider the compactification of M-theory on $\mathbb{C}^2/\Gamma \times \mathbb{R}^{6,1}$. (Well, the actual compactification happens on $K3\times\mathbb{R}^{1,6}$ since the ALE space itself isn't compact, and when the $K3$ surface is singular, we locally model the singularity as $\mathbb{C}^2/\Gamma$.)

While the singularity is resolved by the crepant resolution, we have $\mathrm{rk}(\Gamma)$ homologically non-trivial 2-spheres which give a gauge group $\mathrm{U}(1)^{\mathrm{rk}(\Gamma)}$ by dimensional reduction of the M-theory 3-form since we get one harmonic 2-form for each non-trivial 2-sphere since homology and cohomology with real coefficients have the same dimension. In the singular limit the particles associated to M2-branes wrapped around the sphere become massless since the volume of the spheres goes to zero, and their transformation properties are precisely such that they behave analogously to W-bosons and we therefore get an enhanced gauge group in the singular limit, which is precisely the Lie group associated to $\Gamma$ by the ADE classification.

Of course, this seven-dimensional theory may be further reduced to a four-dimensional theory by compactifying three additional dimensions. For more details, see e.g. "M theory, $G_2$-manifolds and four-dimensional physics" by Acharya.

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  • $\begingroup$ Thanks a lot for this. I'm not sure this is a good question, but is there a topological sector to the theory you describe? When you compactify some string theory on a Calabi-Yau, in a topological sector you can consider mapping the string worldsheets into the Calabi-Yau itself. I'm curious, because you can't map a compact curve into a non-compact affine space (unless the whole curve maps to a point), so the worldsheets must land in the K3 surface you describe? Or maybe on the homologically non-trivial spheres? $\endgroup$ – Benighted Feb 6 '17 at 1:44
  • $\begingroup$ @spietro There is a notion of topological M-theory but I don't know much about it. The analogue to worldsheets here are 3d and 6d worldvolumes of the M2 and M5 branes. $\endgroup$ – ACuriousMind Feb 6 '17 at 2:22
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    $\begingroup$ There is another answer at physicsoverflow.org/38399 $\endgroup$ – Arnold Neumaier Feb 10 '17 at 13:34
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In the literature, we refer to an ALE space as being a solution to the field equations which is a blow up of $\mathbb C^2 / \Gamma$ for a finite subgroup $\Gamma \hookrightarrow \mathrm{SU}(2)$.

A paper by C.V. Johnson and others, Aspects of Type IIB Theory on ALE Spaces, considers the case of compactifying type $\mathrm{IIB}$ string theory on $\mathbb R^6 \times \mathcal M$ where $\mathcal M$ is an ALE space, which can be interpreted as a gravitational instanton.

The construction of the spaces $\mathcal M$ in the context of the theory is shown to be equivalent to how four-dimensional ALE spaces were constructed by Kronheimer in his classification, using hyper-Kahler quotients, viewing the spaces as Riemannian endowed with a hyper-Kahler structure.

More specifically, the vacua of two-dimensional $\mathcal N = 4$ supersymmetric field theories on D1-brane world volumes possess a spectrum related to the classification in terms of hyper-Kahler quotients, and these two-dimensional theories in turn are related to how ALE spaces are probed in the full theory considered.

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  • $\begingroup$ Thanks a ton. So it sounds like from this answer, and the excellent one below, that in physics you compactify string theory on the two (complex) dimensional space $\mathcal{M}$. How then do you get to the famous 10 dimensions, or is this not needed for some reason? I was really hoping something like $\mathcal{M} \times E$ had been studied in physics where $E$ is an elliptic curve. $\endgroup$ – Benighted Feb 4 '17 at 18:23

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