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I am studying Ben Craps' lecture notes Big Bang Models In String Theory for my undergraduate thesis. I am not well versed in topology, and would like intuitive explanations for certain ideas regarding Asymptotically Locally Euclidean (ALE) singularities.

Firstly, the author states.

The orbifold has a singularity at the fixed point $(z_{1}, z_{2}) = (0, 0)$; mathematically, it is known as an $A_{n−1}$ singularity, a special case of an ALE singularity. Once again, perturbative string theory turns out to be completely smooth due to the twisted closed strings. It is well-known that geometrically, the $A_{n−1}$ singularity can be resolved into $n−1$ intersecting two-spheres. Before saying more about this singularity, I have to briefly introduce the concept of D-branes.

What does it mean for a singularity to be resolved into $n-1$ intersecting two spheres?

Then the author introduces two types of D-branes: bulk and fractional.

For the orbifolds we are considering, one finds that there exist two types of D-branes. The first type are “bulk” D-branes. On the covering space, they correspond to a $\mathbb{Z}_n$ symmetric configuration of $n$ D-branes. Bulk branes have the property that they can move anywhere in the orbifold: the images move in such a way that the configuration remains symmetric under the $\mathbb{Z}_n$ orbifold group.

What is the covering space of an orbifold, intuitively? Are these images under the orbifold identification of the brane?

The second type are “fractional” D-branes. On the covering space, they correspond to just a single D-brane placed at the fixed point, which is a symmetric configuration by itself. Fractional branes are stuck at the orbifold sin- gularity: to move away from it, they would need $n−1$ companions to keep the configuration symmetric, but those companions aren’t there. The fact that fractional branes are stuck at the singularity makes them ideal probes of the structure of the singularity.We have seen before that an $A_{n−1}$ singularity can be viewed as a limit of the resolved $A_{n−1}$ singularity, where the $n − 1$ two-spheres collapse into a single point.

I don't understand what he means by needing $n-1$ companions to keep a configuration symmetric.

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What does it mean for a singularity to be resolved into n−1 intersecting two spheres?

Just to be sure, the mathematically precise of idea of "resolving a singularity" is given in the realm of algebraic geometry under the name blow up. Now, what is a blow up intuitively speaking? Algebraic geometers have sets of rules to replace singular spaces transmuting them into smooth manifolds in (a more or less) canonical way.

Let's analize the $TN_{k}$ ($k$-centered Taub-NUT space) case. Its singularities are arranged in the pattern of the roots of the $A_{k-1}$ Lie algebra. You must think on those singularities as points where the curvature becomes infinite. Resolving them just means replacing points by spheres of finite size (smaller that the separation between points) making the background smooth.

enter image description hereResolving the base $\Rightarrow$enter image description here

Is this replacement cannonical in some way? It is. The general rule is to replace simple classes of singularities by the projectivization of its normal bundle. In our case the projectivization of a point in $\mathbb{R}^{3}$ is a 2-sphere.

I urge you to read the classic Strings on orbifolds to understand how string theory is consistent over backgrounds with orbifold singularities. The classic McKay correspondence to learn about blow ups, Orbifold Resolution by D-Branes to understand in full detail the physics of the stringy resolution of orbifold singularities and my answer for a roadmap to interesting applications of orbifolds in string theory.

What is the covering space of an orbifold, intuitively? Are these images under the orbifold identification of the brane? You can think on the covering space of an orbifold is any of its smooth resolutions.

I don't understand what he means by needing n−1 companions to keep a configuration symmetric. Imagine that you resolve just one point-singularity among a large number of them, leaving the rest of them untouched. That's not a valid operation in the theory of orbifolds. The reason is that any allowed birrational transformation must preserve the action of $\mathbb{Z}_{n}$ intact. Something that the sort I mentioned does not treat democratically all the symmetry generators (a symmetry breaking is taking place) and that's not what we want, the paper you are reading is interested in exploring the stringy dynamics of a particular spacetime without spoiling its symmetries.

In plain english: Any transformation made over a point singularity must be performed in exactly the same way for over any other one. Otherwise the global symmetry group $\mathbb{Z}_{n}$ is broken.

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  • $\begingroup$ Ah thank you. This makes things somewhat clearer. I was going to study the Vafa paper - even though this isn’t the focus of my thesis and I probably won’t have to explain this.However, I suspect I will have to study Topology by Nakahara in graduate school to make rigorous sense of this. Do you have any resource recommendations - as in math textbooks - for someone at my level (studied GR and group theory in addition to undergrad math methods)? $\endgroup$ – saad Aug 10 '20 at 9:39
  • $\begingroup$ It was a pleasure, I hope it was helpful. Concerning the resource recommendations: Nakahara's book is the safe recommendation. Concise, understandable and exactly covers what is required to go deeper on those topics (essentially characteristic classes and self-dual connections).Taubes book "Differential Geometry: Bundles, Connections, Metrics and Curvature" is a little bit more precise on the topic of instantons if you want more details. $\endgroup$ – Ramiro Hum-Sah Aug 12 '20 at 4:13
  • $\begingroup$ See Uranga's book "String Theory and Particle Physics" for a pedagogical dicussion of the Taub-NUT solution. Witten's Branes, Instantons, And Taub-NUT Spaces is very readable and explicity constructs Taub-NUT solutions from brane configurations. Black Holes in String Theory is a wonderful resource to understand specifics of the Taub-NUT soultions and their applications to black hole physics and the counting of D-brane bound states. $\endgroup$ – Ramiro Hum-Sah Aug 12 '20 at 4:25
  • $\begingroup$ Now, If you are interested in begin to understand some algebraic geometry. The canonical resource is the book "Mirror Symmetry" (Hori,Katz,Klemm,Vafa etc.). It's not an easy book, but serves as a wonderful roadmap to the literature always on the string theory way, a great option to begin the adventure. $\endgroup$ – Ramiro Hum-Sah Aug 12 '20 at 4:29

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