Stability

We have seen in Section 5.2 that as a consequence of the requirement that a D-brane configuration preserve supersymmetry, the sheaf that describes this configuration must be semi-stable. This was encoded in (5.7b). Roughly speaking, if this requirement is not satisfied, the configuration is unstable and will decay into stable, supersymmetric constituents. From this point of view it is very interesting to observe that these two totally different concepts of stability – mathematical and physical – agree. We are therefore led to investigate semi-stable sheaves which will be the content of this section.

We begin with the definition of semi-stable sheaves for which we first need to introduce some technicalities. We assume that all our sheaves are over a toric Calabi-Yau threefoldX. For a coherent sheafFwe define its Chern character ch(F) by means of a projective resolution (5.13) as follows

ch(F) =

n X i=0

ch(Ei) (5.15)

This definition is independent of the choice of the resolution. Furthermore we define the degree ofFto be

degω(F) = Z

X

c1(F)∧ω2 (5.16)

whereω is the (uncomplexified) K¨ahler form and we define the normalized degree or the slope ofFto be

µω(F) = degω(F)

5.3. Vector bundles versus sheaves

We will suppress the dependence onω from now on. A coherent sheafEis said to be µ-semi-stable, if for every coherent subsheafFwith rk(F)>0 we have

µ(F)≤µ(E) (5.18)

If strict inequality holds for every subsheaf Fwith 0 <rk(F)<rk(E) then we say that E is µ-stable. If equality holds then we say that E is strictly µ-semi-stable. Since µ-stability is the only notion of stability we will use, we will drop the µ from now on. A holomorphic vector bundle E is said to be semi-stable (stable) if the sheaf of holomorphic sectionsO(E) is semi-stable (stable). Note that even if we are only interested in vector bundles we need to consider not only subbundles but also subsheaves. This is a further motivation why we need to introduce coherent sheaves.

Note that there are different notions of stability in mathematics, e.g. there is also Gieseker sta- bility [235]. It is not yet clear which one is physically relevant, e.g. Gieseker stable objects appeared in [219] and [236]. It is conceivable that string theory needs both of them as limits of Π-stability [237]. Let us give a few examples and simple criteria for stability. A torsion-free coherent sheaf of rank 1 is always stable. IfF is a torsion free coherent sheaf and Lis a line bundle thenF⊗Lis semi-stable (stable) if and only if Fis semi-stable (stable). F is semi-stable (stable) if and only if its dualF∨ _is

semi-stable (stable). Furthermore, if

0−→L0−→F−→L1−→0 (5.19)

is a non-trivial extension with line bundlesL0 andL1 of degree 0 and 1, respectively, thenFis stable.

IfF1 andF2 are torsion-free coherent sheaves thenF1⊕F2is semi-stable if and only if F1and F2 are

both semi-stable with µ(F1) =µ(F2). However, if F1 and F2 are nonzero, thenF1⊕F2 can never be

stable.

One more notion that we will need is S-equivalence. Suppose that E is a semi-stable torsion-free sheaf with µ(E) = µ. Then there is a filtration {0} = F0 _{⊂ F}1 _{⊂ · · · ⊂F}k _{= E such that F}i_/Fi−1

is torsion-free and stable for every i and µ(Fi_/Fi−1_{) = µ for alli. Such a (generally non-canonical)}

filtration is called a Jordan-H¨older filtration of E. The associated graded sheaf grE=L_iFi_/Fi−1 _is

independent of the choice of the filtration. Two sheaves E1 and E2 are S-equivalent if grE1 = grE2.

This has the following meaning. Points on a moduli space of sheaves that are strictly semi-stable do not necessarily correspond to unique semi-stable sheaves but to S-equivalence classes of strictly semi-stable sheaves. What will be important for us is that each S-equivalence class contains a unique representative that is split, i.e. is a direct sum of stable sheaves [235]. Such a sheaf is also called polystable. The physical relevance of S-equivalence classes has been pointed out in different contexts in [238] and [239]. We have seen that the vector bundles we are interested in satisfy the Hermitian Yang-Mills equa- tion (5.7b). The Donaldson-Uhlenbeck-Yau theorem [240], [241], [115] (see also [209]) now states that if the vector bundle Eadmits an irreducible Hermitian Yang-Mills connection thenE is µ-stable. More- over, if the connection is reducible, thenE is strictly semi-stable and is split, i.e. E=L_iEi where Ei

admit irreducible Hermitian Yang-Mills connections and are therefore stable. Hence, the representative of the S-equivalence class that is relevant for the physics of D-branes is the split representative. This fact will be often used in Section 6.3.

The most important necessary criterion for stability is the Bogomolov inequality [235]. If F is a semi-stable torsion free coherent sheaf, then

Z X

∆(F)∧J ≥0 (5.20)

where ∆(F) ≡ 2 rk(F) c2(F)−(rk(F)−1) c1(F)2 = c2(EndF). On manifolds with h1,1(X) > 1 this

describes an explicit dependence on the K¨ahler classω as described in [239]. Equality in (5.20) defines a boundary within the K¨ahler cone on which stability degenerates to semi-stability [242]. Physically, this means that the connection on the D-brane becomes reducible, and an enhanced gauge symmetry

appears. Furthermore, as sheaves generally do not admit connections, this allows us to define the analog of reducible connections on vector bundles for sheaves and hence to consider those objects which represent both kinds of singularities on MD discussed in Section 5.3.1. There is a beautiful relation

between these sheaves leading to enhanced gauge symmetry and certain boundary states in the Gepner model [184]. This will be explained in Section 6.3.