The motivation of stability via Mumford slopes

Question

A vector bundle $F$ is called (semi)stable if Mumford slopes $\mu=c_1 / rk$ of all the subbundles are less (or equal) then Mumford slope of $F$. Can you explain the motivation of this definition or give me a good reference?

Answer 1

Assuming your vector bundles are over a projective scheme $X$, the answer has to do with creating a nice finite type, projective, integral moduli scheme $\mathcal{M}$ parametrizing vector bundles of given Hilbert polynomial $P$ on $X$. One thing that becomes clear is that you'll not have much luck if you try to force $\mathcal{M}$ to parametrize all such vector bundles. For example, if $\mathcal{M}$ parametrizes all bundles on $\mathbb{P}^1$ with Hilbert polynomial $P(t)=2t+2$, then $\mathcal{M}$ has an infinite filtration $\mathcal{M}\supset M_1 \supset M_2 \supset...$ by proper closed subsets $M_a$ given by bundles with $h^1\geq a$. The inclusions are proper because $H^1(O(-a-1)\oplus O(a+1))=a$. No finite dimensional integral scheme could have such a filtration. Therefore, one must invariably throw some stuff out.

Once we accept that we must throw things out, we have a good candidate for $\mathcal{M}$ in Grothendieck's quot scheme of coherent quotients with Hilbert polynomial $P$ of a trivial bundle twisted by $O(m)$ (call this guy $\mathcal{W}$). By demanding our bundles be quotients of $\mathcal{W}$, we prevent things like the unbounded filtration above. But still there is a problem in that the quot scheme will have infinitely many points corresponding to a single isomorphism class of vector bundles. You want to identify these points, so you impose an equivalence relation.

The right equivalence relation turns out to be the one given by identifying orbits of the action of the determinant $1$ subgroup $G$ of $\text{Aut}(\mathcal{W})$ on the quot scheme. In algebraic geometry, you don't just pass to the quotient space like in topology, because that won't be a scheme in general. So what you do is "linearize" the action in some natural way, throw out the closed set of nonstable points with respect to the linearization (there is a good thread here: Geometric intuition for linearizing algebraic group action), and then pass to the GIT quotient. This GIT quotient is an $\mathcal{M}$, which parametrizes all SEMISTABLE vector bundles, and $\mathcal{M}$ has exactly one point for each strictly stable vector bundle. This GIT semistability coincides with Mumford's notion above.

A good reference might be the Geometry of Moduli of Sheaves by Huybrechts.

Answer 2

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 then 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 stae 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 **