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To my understanding, given a rank $r$ and degree $d$, we can fix a $C^{\infty}$ vector bundle $\mathcal{V}$ over some curve $\Sigma$ of genus $g$. We then want to study the moduli space $M_{g}(r,d)$ of holomorphic structures on this fixed bundle $\mathcal{V}$. Let $\mathcal{C}$ denote the space of holomorphic structures on $\mathcal{V}$. Naively, we would simply use $\mathcal{C} / \rm{Aut}(\mathcal{V})$ as the moduli space, but apparently this space has terrible properties, i.e. isn't even Hausdorff, etc. So I've heard Mumford's Geometric Invariant Theory is one way around this problem. And this is where you see the familiar definition of stable vector bundles: $E$ is stable if $\mu(F) < \mu(E)$ for all proper, holomorphic sub-bundles $F \subset E$, where $\mu$ is of course the slope of the bundle.

Personally, I don't have any intuition as to why this strange definition of stability leads to well-defined moduli spaces of bundles. Is there any intuition anyone can help me with, or perhaps is it just the sort of thing where you say, with hindsight, it's simply the correct thing to do to force the Geometric Invariant Theory to work?

EDIT: So restricting to simple bundles (where $\rm{Aut}\mathcal{V} = \mathbb{C}^{*}$) makes sense to me. It seems to be analogous to asking that a group action not have fixed points; thus, avoiding singularities. Maybe it would be helpful for me to consider how semi-stable bundles relate to simple bundles? (I know stable implies simple)

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    $\begingroup$ You may want to look first at simpler examples of GIT quotients, like configuration spaces of tuples of points on the projective line, starting with, say, the case of triples when everything is almost trivial and proceeding from there to the case of n-tuples. This example will give you a good idea of what can go wrong and how GIT takes care of the problem. $\endgroup$ – Moishe Kohan Mar 20 '16 at 19:33
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    $\begingroup$ I second Moishe's claim. Given any action of $G$ on $X$ GIT cooks up a way of removing the points of $X$ that stop the quotient from being reasonable (or, reasonable as a scheme--it'll be fine as a stack) and these points it removes are the stable/semi-stable ones. So, from that perspective semi-stable vector bundles are just the stable/semi-stable points of the natural atlas for $\text{Vect}$. $\endgroup$ – Alex Youcis Nov 20 '16 at 14:52
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In pure mathematics the slope stability condition is motivated only from the fact that it works. But if you are after a good intuition as for what is conceptually going on here, it helps to look at the physics analog of this, which is Douglas's "Pi-stability". This was, in turn, the inspiration for Birdgeland's general concept of stability conditions, which includes slope stability of coherent sheaves as a special case.

So, the physical interpretation of slope stability of vector bundles is revealed once one thinks of the vector bundles as being the "Chan-Paton gauge fields" on D-branes. Then the rank of the vector bundle is proportional to the mass density of a bunch of coincident D-branes, while the degree, being the Chern-class, is a measure for the RR-charge carried by the D-branes.

This reveals that the "slope of a vector bundle" is nothing but the charge density of the corresponding D-brane configuration.

Now a D-brane state is supposed to be stable if it is a "BPS-state", which is the higher dimensional generalization of the classical concept of a charged black hole being an extremal black hole in that it carries maximum charge for given mass.

Hence the stable D-branes are those which maximize their charge density, hence the "slope" of their Chan-Paton vector bundles.

The condition that every sub-bundle have smaller slope hence means that smaller branes can increase their charge density, hence their slope, by forming "bound states" into the larger, stable object.

Hence slope-stability of vector bundles/coherent sheaves is the BPS stability condition on charged D-branes.

This idea is really what underlies Michael Douglas's discussion of "Pi-stability" of D-branes, which then inspired Tom Bridgeland to his general mathematical definition, now known as Bridgeland stability, which subsumes slope-stability/mu-stability of vector bundles as a special case. But, unfortunately, this simple idea is never quite stated that explicitly in Douglas's many articles on the topic.

For more along these lines and more pointers see the discussion at

**nLab: Bridgeland stability -- As stability of BPS D-branes **

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  • $\begingroup$ Thank you! Perhaps I can clarify one thing. To my understanding, to get a B-type BPS D-brane, you want a vector bundle supported on a holomorphic submanifold. You want holomorphic because BPS requires mass minimizing. To get a SUSY state, you further need the vector bundle to carry a HYM connection, which means it is (poly)stable by DUY. You mention BPS D-branes maximizing their charge density...should I interpret this as the physical intuition behind requiring the bundle carry a HYM connection? $\endgroup$ – Benighted Oct 5 '18 at 18:42
  • $\begingroup$ @Benighted yes, I think so. BPS means to maximize charge over mass, hence, dually, to minimize mass for given charge. And the connection is a way to reflect the Chern character, hence the charge. $\endgroup$ – Urs Schreiber Oct 6 '18 at 9:39

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