Objective B-Fields and a Hitchin-Kobayashi Correspondence

A simple trick invoking objective B-fields is employed to refine the concept of characteristic classes for twisted bundles. Then the objective stability and objective Einstein metrics are introduced and a new Hitchin-Kobayashi correspondence is established between them. As an application the SO(3)-instanton moduli space is proved to be always orientable.


Introduction
Recently there has been intensive literature on gerbes starting from Brylinski's book [4], in which the Chern-Weil theory of gerbes was carried out. The geometry of abelian gerbes was further refined and clarified by Murray [22] and Chatterjee [6], Hitchin [13] from two different view-points. These objects have found applications in Physics for the descriptions of the so-called B-fields and the twisted K-theory according to [1,3,18,24] among many others. A real version of gerbes is presented in [28].
In one aspect gerbes are utilized to define twisted bundles where the usual cocycle condition fails to hold. The concept of twisted principal bundles was explicitly proposed in [19,20] where their holonomy is studied, while the authors in [2,23] introduced bundle gerbe modules in a different setting for the generalization of the index theory. In the context of algebraic geometry these are Azumaya bundles via the description of gerbes in terms of Brauer groups, see [21,31,5] for example.
The first main purpose in our paper is to propose a new concept of Chern classes for twisted vector bundles as a refinement of existing definitions from [2,5]. In fact the earlier definitions are not completely satisfactory in the sense that the one give in [2] is not quite topological because of the dependence on different curvings, while the definition in [5] involves an un-specified choice of a second twisted bundle. Our approach can be viewed as a balance of the two. To explain the key idea behind, recognizing that the underlying gerbe L of a rank m-twisted vector bundle E is an m-torsion, we select a trivialization L of L m . To remove the dependence on the B-fields B (namely curvings), we restrict to a special B so that mB is objective with respect to L. Given a twisted connection D on E, twisted over a gerbe connection A, we use their curvatures and B together to define closed global Chern forms φ k (E, L; A, B, D). (The usage of the word "objective" throughout the paper is inspired by [6].) It seems likely that Theorem 0.1 will provide an alternative starting point to the current research [3,2,23,31] concerning a host of index theorems and twisted K-theory. With objective Chern classes and properly defined twisted subbundles, we further introduce objective stability. Moreover by fixing a metric on L, we are able to characterize objective Einstein metrics so that we can formulate and generalize the Hitchin-Kobayashi correspondence as follows. (A possible statement as such was alluded in [7].) The original correspondence for untwisted bundles was a corner-stone result that had been established in [17,8,26,9].
Theorem 0.2. Suppose E is a rank-m indecomposable holomorphic bundle twisted over a gerbe L and L is a holomorphic trivialization of L m . Then E is L-objective stable iff E admits an H-objective Einstein metric for some Hermitian metric H on L.
The theorem is new only if the base manifold has complex dimension at least 2. For the case of a complex curve, any gerbe L must be trivial hence no twisting occurs. But one might still ask about stable bundles and Yang-Mills connections on non-orientable Riemann surfaces, compare [27] for example.
Our consideration of trivializations L of L m is rather natural and motivated in part by the lifting of SO(3)-bundles. More precisely when an SO(3)-bundle S is spin c , lifting its structure group from SO(3) to U(2) one has the spinor bundle E with determinant line bundle det E. In general if S is not spin c , one can only have a twisted U(2)-bundle E over a non-trivial gerbe L. Then a trivialization L of L 2 is a replacement of the determinant. In this regard, we get the following result as an application of our theory: Proposition 0.3. The moduli space of anti-self dual connections on any SO(3)-bundle S over a 4-manifold is orientable, whenever it is smooth.
This is a small generalization of Donaldson's theorem [11] where S is assumed to be spin c . Donaldson's theorem has proved to be quite crucial for various gauge theoretic applications.
Here is a brief description of the paper. After reviewing and setting up notations for gerbes and so on in Section 1, we develop the objective Chern-Weil theory and prove Theorem 0.1 in Section 2. The fundamental concepts of objective subbundles, Einstein metrics and stability are laid out in Sections 3 and 4. Then we devote Sections 5, 6 to the proofs of the two directions in Theorem 0.2. Finally Section 7 contains the proof of Proposition 0.3.

Review of gerbes, B-fields and twisted bundles
All vector bundles, differential forms will be defined on the complex field C unless otherwise indicated.
To set up notations, first recall the definition of gerbes from [6,13]. Let X be a smooth manifold X bearing an open cover {U i }. A gerbe L = {L ij } consists of line bundles L ij → U i ∩ U j with given isomorphisms (1) L ij = L −1 ji , L ij ⊗ L jk = L ik on their common domains of definition. Using a good or refinement cover if necessary, we can assume L ij are trivial bundles. Choose trivializations ξ ij for L ij . They should be compatible with (1) in that where z = {z ijk } forms aČech 2-cycle of the sheaf C * of nowhere vanishing complex functions on X. One imposes that z be co-closed: δz = 1.
Thus the gerbe class [L] := [z] ∈Ȟ 2 (X, C * ) is defined. Two gerbes are isomorphic if they become the same on a common refinement. This is equivalent to their gerbe classes being the same. As in the case of bundles, it is often convenient not to distinguish a gerbe L from its class [L]. One can view [L] ∈ H 3 (X, Z) according to the natural iso-morphismȞ 2 (X, C * ) → H 3 (X, Z), which comes from the short exact sequence Z → C → C * . Gerbes form an Abelian group under the tensor product.
From (1), one has a trivialization s ijk of L ijk := L ij ⊗ L jk ⊗ L ki as part of the gerbe definition. In [6,13], a gerbe connection {A ij , B i } on L consists of two parts: connections A ij on L ij and complex 2-forms B i on U i , such that 1) s ijk is a covariant constant trivialization of L ijk under the induced product connection using A ij , and 2) B j − B i = F A ij on U i ∩ U j for the curvature of A ij . However in this paper we will separate them and call A = {A ij } alone a gerbe connection and give a more prominent role to B = {B i }, which we call a (A-compatible) B-field from its physics interpretation. The closed local 3-form {dB i } can be patched together to yield a global form G, called the gerbe curvature of the pair A, B. The class [G] ∈ H 3 (X, R) is the real image of [L] ∈ H 3 (X, Z), hence independent of the choice of A, B. Moreover every representative of the real image can be realized by some B-field and connection. Here is a collection of useful facts to recall: 3) The real gerbe class of L is trivial, namely the image there exists a gerbe connection A with a compatible B-field B such that the curvature 3-form G = 0.
Next suppose L is trivial, i.e. its class [L] = 0 ∈Ȟ 2 (X, C * ). Then z is a coboundary of aČech 1-cocycle, which leads to a trivialization L = {L i } of L, namely line bundles L i on U i such that L j = L i ⊗ L ij when restricted to U i ∩ U j . Given a second trivialization L ′ = {L ′ i }, we have a global line bundle ℓ on X, called the difference bundle of L and L ′ . We will denote ℓ = L ⊖ L ′ . A trivialization of a gerbe connection A is a family of connections A = {A i } on {L i } subject to an analogous condition. Since the 1-form sheaf is a fine sheaf, trivializations always exist for any gerbe connection but not unique. Given a second trivializing connection A ′ on L ′ , we have a difference connection A ⊖ A ′ on the bundle ℓ.
Given a trivializing connection A of A, the collection of local curvature forms {F A i } is a B-field compatible with A; but not every compatible B-field has this form (i.e. from some trivializing connection of A). Given a compatible B-field B and a trivializing connection A, the difference ǫ := B i − F A i is a global 2-form on X. This error form ǫ is essential in defining the gerbe holonomy of A. Following [6] we call B objective if ǫ = 0 for some A. A necessary condition is that B i be closed. Suppose so, then the error form ǫ with any trivializing connection is also closed hence gives rise to a class [ǫ] ∈ H 2 (X, R). In this language, objective B-fields are exactly those closed B-fields such that the error class [ǫ] lives in the lattice H 2 (X, Z) ⊂ H 2 (X, R). Here of course a gerbe connection A has been fixed throughout. Now is a good time to set out the following guide for our notations in the paper: A trivialization L = {L i } of L can also be viewed as a twisted line bundle over L. In general, given an arbitrary gerbe L, an L-twisted vector bundle E = {E i } on X consists of a collection of local bundles of the same rank such that E j = E i ⊗ L ij on U i ∩ U j . We will also say that E is over the twisting gerbe L. (Twisted principal bundles are first defined in [19].) In a similar spirit, one defines a twisted connection D = {D i } on E over a gerbe connection A. We will adopt the following convenient notations to indicate the twisting gerbe L and the twisting connection A: Twisted bundles/connections are not necessarily mysterious: simply put, these are so defined that their projections are just regular fiber bundles and connections respectively. For any 1-dimensional vector space V , V ⊗ V * carries a canonical basis. It follows that the local endomorphism bundles {End(E i )} naturally fit together to produce a global vector bundle on X, in view of End(E i ) = E i ⊗ E * i . We denote it by End(E). Note that the wedge product twisted bundle is over L r : In particular det E gives rise to a trivialization of L m , where m = rankE. In other words, the underlying gerbe L of E must be an mtorsion.
Other operations can be introduced as well: one has the dual E * ≺

Objective Chern classes for twisted bundles
We now develop a new version of Chern-Weil theory for rank mtwisted vector bundles. Fix a gerbe L with connection A and B-field B. Assume L is an m-torsion. By Proposition 1.1, we can choose B to be flat so that all B i are closed 2-forms. For a twisted rank mvector bundle E ≺ L with connection D ≺ A, it is easy to check that B-twisted curvature of D: is a global section of Ω 2 (End(E)). In order to get the expected topological results we will need to place a crucial restriction on B. 2) The class [φ k ] ∈ H 2k (X, R) is independent of the choices of A, B, D. 3 Then the following holds: Proof. 1) Following the standard Chern-Weil theory (for example from Chapter III of [30]), it suffices to show a Bianchi type identity still holds: for some θ ∈ Ω 1 (End(E)) on any small open set V ⊂ U i . To prove the identity, if V is small enough, there is a line bundle K → V such that for the curvature of the tensor product connection D = D ⊗ G −1 on E i ⊗ K −1 . Now the standard Bianchi identity for D yields (3), where θ is the connection matrix of D under a local frame.
2) Note that A and B are both determined uniquely by A (although L is not so by L). It is enough to show more generally that [φ k ] is independent of the choices of D, A. For a second pair of choices Then A(t) determines families of gerbe connections A(t) and objective B-fields B(t) on L. Moreover D(t) is a twisted connection over A(t) for each t.
for the connection matrix θ(t) of D(t) under a frame. Consequentlẏ As in the standard Chern-Weil theory [30], the above formula together with the Bianchi formula (3) shows that 3) Here it is important to be able to keep the same gerbe connection A on L and hence the same twisted connection D on E. In other words a different trivialization L ′ will only impact on the B-field via A ′ . Now where α is the difference connection of A, A ′ on ℓ. Under a frame, write F D − BI as a matrix (F i j ) of 2-forms. Then F D − B ′ I is given by the matrix (F i j + 1 m δ i j F α ). Applying the classical formula to det(I + F D − BI), one has Doing the same for det(I + F D − B ′ I) will give These formulas together will give us the desired result for their classes after noting c 1 (ℓ) = [F α ]. q.e.d.
In particular if L ′ is flat equivalent to L, namely if ℓ is flat, then c 1 (ℓ) = 0 ∈ H 2 (X, R) and the class [φ k ] remains the same. This is the case if L ′ is isomorphic to L (i.e. ℓ is trivial). Thus it makes sense to introduce: Definition 2.2. Suppose E ≺ L is a rank-m twisted bundle and L ≺ L m is a trivialization. The L-objective Chern class of E is defined as follows: , where A is any gerbe connection on L, D is any twisted connection on E and B is a B-field compatible with A such that mB is objective. Remark 2.3. Alternatively one could define a twisted bundle as a pair (E, L) where E ≺ L and L ≺ L m in our notations. Then one could define the Chern classes for (E, L) without the cumbersome prefix "Lobjective". For us the advantage of separating L from E is that we can illustrate better the dependence on L as shown in the next corollary.
The first formula is a translation of Part 3) of Theorem 2.1. To show the third formula, note that rank ∧ r E = C(m, r) and Hence L w is a trivialization of L rC(m,r) , which carries the B-field C(m, r)rB. Let ∧D denote the induced twisted connection on ∧ r E. One has Remark 2.5. One could in particular use the special trivialization For any other trivialization L ≺ L m , the previous corollary then yields Note c 1 (E) = 0 and the formula above implies that c L 1 (E) = c 1 (ℓ) is always an integer class. The following example from gauge theory has been a useful guide for us and will also show why it is important to expand c k (E) to c L k (E) with the incorporation of trivializations other than detE.
Example 2.6. Let Q be an SO(3)-vector bundle on X. Suppose Q is spin c so that w 2 (Q) has integral lifts in H 2 (X, Z). Fix such a lift L viewed as a line bundle as well as a connection A on it. Then Q together with connection ∇ will lift to a unique U(2)-vector bundle Q with connection ∇ such that det( Q) = L, det ∇ = A. We can interpret Q, ∇ and especially the Chern classes c k ( Q) in terms of our twisted vector bundle theory. For this purpose choose any open cover (The gerbe L, first observed in [29], can be formally viewed as being given by √ L, namely L is the obstruction to the existence of a global squared root of L. Similarly there is a gerbe given by the n-th root n √ L. More on real gerbes can be found in [27].) Moreover A leads to a unique twisted connection i is global since L 2 is trivial as aČech cycle not just as aČech class.) When Q is not spin c , the line bundle L will not exist any more. Nonetheless, Q still lifts locally to a twisted SU (2) .) The essence of our theory above is to replace L with a trivializing twisted bundle of L 2 , and such a trivialization always exits! More generally, the lesson for our entire paper is this: Given twisted data such as E, D that are defined on local open sets only, one should couple them with objective gerbe data in order to get global data on X.
Remark 2.7. We now compare our Definition 2.2 with several existing definitions of Chern classes in the literature.
1) To get the topological invariance, it was important and indeed necessary for us to impose the condition that mB be objective in Theorem 2.1. For an arbitrary flat B-field B, without invoking L or assuming mB to be objective, one can still prove that φ(E; A, B, D) = det(I + 1 2πi F D ) is closed. (Indeed one has a Bianchi identity directly from the following computation where θ is the connection matrix of D i under a local frame and the last equation follows from [B i I, θ] = 0 since B i is a 2-form.) One could then try to define the Chern class of E as c k (E) = [φ k (E; A, B, D)]. In terms of bundle gerbe modules, this seems to correspond to the Chern character using an arbitrary curving f in Section 6.3 of Bouwknegt et al [2]. However such an expanded definition in either place has the issue that c k (E) is not well-defined but depends on the choices made for A, B, D. Without invoking L, it is not possible to characterize such dependence. The indeterminacy is essentially due to the fact that the compatibility between B and A only mildly constrains B by A.
2) In [5], Cȃldȃraru proposed to define the Chern classes c W (E) of E as the regular Chern classes c(E ⊗ W ), where W is a fixed twisted vector bundle over L −1 . The main problem in this approach is that no natural choices were given for W in applications. In comparison, our choice L as a trivialization of L m is pertinent and natural, and does indeed reflect the fact that L is an m-torsion as the underlying gerbe of E. Compare also with [14], where B is taken to be a global closed 2-form on X. However this works only because the gerbe L is trivial for their applications in K3-complex surfaces X.
3) When the gerbe connection A is flat, namely all F A ij = 0, one can take B i = 0 for all i. (This is a constraint on L.) Thus {F D i } is already a global section of End(E) and our Chern classes c L k (E) are given by det(I + 1 2πi F D i ). Passing over to bundle gerbe modules, this special case should correspond to the Chern classes in [21,23].
It is an interesting problem to apply the objective Chern-Weil theory to various index theorems and twisted K-theory, which we hope to return in a future work.

Objective sections and subbundles of twisted bundles
We begin by the following definition; the second part seems to be brand new.
2) Note L −1 ≺ L −m and the latter carries local sections ij . We will rephrase s as a t-objective section of E in this context, in order to emphasize the relevance of {t i }.

The main point of introducing objective sections is that
whenever m √ t i is properly defined. Of course here t is just a twisted section of L −1 over L −m . We use L −1 instead of L, since t i may vanish so t −1 i may not exist. Since generally ξ −m is not objective, the existence of t does constrain s via ξ. In fact one checks readily the existence of a nowhere vanishing t is equivalent to that theČech 2-cycle z = {z ijk } of L has order m: z m = 1. By Proposition 1.1, such a 2-cycle exists only on special covers.
From sections we can define twisted bundle homomorphisms in a rather formal way.
2) From a twisted homomorphism g : L n → K m , one has then a g-objective homomorphism f . Here L n ≺ L mn , K m ≺ K nm are the associated twisted bundles. (Note that g maps in the opposite direction as f .) More clearly, the gerbes K, L come with local trivializations η ij , ξ ij as in Equation (2). They give rise to bundle isomorphisms χ ij : In particular we can define a twisted or g-objective subbundle F ≺ K of E ≺ L. However in this approach, it is unclear how K should be tied to L. Obviously one can not always choose the same gerbe K = L for all subbundles. For example for a twisted line bundle F , the gerbe K must be trivial. Thus if L is non-trivial, then E does not allow any twisted line subbundles if one insists on using the same gerbe.
To handle the issue more decisively, we shall adopt a different approach for twisted subbundles and make a comparison with the above general approach at the end. We start by the following arithmetic result. All integers are assumed to be positive. The key issue in defining twisted subbundles is to spell out what should be the underlying twisting gerbes.
The definition makes sense since each L d is indeed an n-torsion gerbe by the definition of T . If n = 1, then m * = m * = m and any twisted line subbundle in this case is over the gerbe L m , hence a trivialization of L m . For other d ∈ T , an L d -twisted line subbundle of E may be subject to a bigger gerbe L ms for some s. The proposition below exhibits certain compatibilities with wedge product and subbundle operations.
Proposition 3.5. Suppose E ≺ L is a rank m twisted bundle and F, W are twisted bundles of ranks n, r.
. This in turn follows from the identity: C(n, k)C(m, n) = C(m − k, n − k)C(m, k).
(However m * = (m ′ ) * , the latter being C(m ′ , n ′ ) by definition.) 2) This amounts to the natural map: which can be confirmed directly. q.e.d. Note that statement 1) is false if L m * is replaced by L m * . Namely for an L m * -twisted subbundle F ⊂ E, the wedge product ∧ k F ≺ S m * may not be a twisted subbundle of ∧ k E, as m * / ∈ T n ′ ,m ′ in general. (For instance take m = 6, n = 4, k = 3.) Remark 3.6. For applications of subbundles, one often fixes a choice of d ∈ T . Proposition 3.5 indicates that there is an advantage in selecting d = m * . For example, given an L m * -twisted subbundle F ⊂ E, its determinant det F is then a twisted line bundle over the gerbe S m * . This is consistent with det F ⊂ ∧ n E being an S (m ′ ) * -twisted line subbundle, where (m ′ ) * = m ′ = rank(∧ n E) = m * .
One can easily interpret L d -twisted subbundles using twisted homomorphisms of 3.2.
1) Then F is the same as an injective twisted homomorphism f : Choose a compatible B-field B ∈ Ω 1,1 ∩ iΩ 2 . Let E ≺ L be a twisted vector bundle of rank m with a Hermitian metric (which restricts to H on L). Definition 4.1. Suppose D ≺ A is a (1, 1)-Hermitian connection on E, namely all its curvature F D i ∈ Ω 1,1 (End(E i )). Then D is called B-twisted Hermitian-Yang-Mills, if its curvature satisfies (4) iΛ for some real constant c on X. Here Λ F D = F D · Φ is the usual projection Ω 1,1 → Ω 0 along the direction of Φ ∈ Ω 1,1 .
Let us do some preliminary analysis of the nature of the equation (4). Since BI is diagonal with equal entries, (4) splits into a pair of equations: By taking a refinement cover we may further assume D to be an SU(m)connection. Then (5) in turn reduces to equations (6) ΛF D = 0, ΛB = ic.
In fact only the first equation is essential, since B is already fixed at the beginning and now it also needs to satisfy the second equation (which just means B i = icΦ| U i ). More plainly, the first equation in (6) is Here F H is the curvature of the H-compatible connection on L; the Einstein constant factor c of E (with respect to L) is given by (7).

Remark 4.3.
Since we don't assume the associated connection of h to be SU(m), we will not attempt to solve the two equations in (6) separately with B = 1 m F H . Instead, we fix H as above and consider the consequences when the single equation (8) does have a solution for h. As a matter of fact it will prove quite useful to combine h, H together and view them as a single mteric locally in small neighborhood of any point.
Naturally E is called objective Hermitian-Einstein if it admits an objective Einstein metric with respect to some H and L. A few basic facts are recorded here for later use.  Next we consider the algebro-geometric counterpart. Definition 4.6. Suppose L, L, E are as in 4.2. Then E is called Lobjective stable (or just L-stable) if for every holomorphic L mw -twisted line subbundle K ⊂ ∧ r E , 1 ≤ r < m, we have where w = C(m − 1, r − 1) as in Corollary 2.4.
The word "objective" is employed in the definition because of the involvement of L. To be sure we have the degrees Note that the same trivialization L w is used for both Chern classes here. By Corollary 2.4, it is possible to express (9) as One can also define the notion of objective semi-stability by replacing < with ≤ in the formulas (9) or (10).
(But it can not be L-stable unless their degrees are zero.) Proof. All statements are generalizations of the standard stability and can be proved in a similar fashion.
1) This is evident since ∧ 1 Q = Q does not have any nontrivial proper sub-twisted bundle.
2) As in the usual case, one can use quotient twisted bundles (sheaves) and a short exact sequence to prove the statement.
3) Note ∧ r (E ⊗ Q) = ∧ r E ⊗ Q r and any sub-twisted bundle K ⊂ ∧ r E ⊗ Q r can be written uniquely as K = K ′ ⊗ Q r for some sub-bundle The rest of the proof is clear by working componentwise. q.e.d.
The L-stability of a bundle does depend on the choice of the trivialization L up to a point. More precisely we have the following: 2) Let S be a gerbe given by 3) When the gerbe (class) L is trivial, the objective stability corresponds to the standard stability.
Proof. 1) Let K ⊂ ∧ r E be any L mw -twisted line subbundle. By The results follows easily.
2) One just needs to note that Q m is identified naturally with ℓ −1 . Then by Proposition 4.7 3) By assumption, there is a trivialization J ≺ L −1 . Thus W := E i ⊗ J i is a global holomorphic bundle. Then one shows, via Proposition 4.7, that E is objective stable iff W is stable in the usual sense. (The usual stability condition actually involves all proper coherent subsheaves V ⊂ W of arbitrary ranks r, from which det V ⊂ ∧ r E. Then one needs to apply the familiar fact that any reflexive rank-1 sheaf is locally free, i.e. a line bundle. Compare also with the Appendix of [7].) q.e.d.

Stability of objective Hermitian-Einstein bundles
We establish here one direction of the generalized Hitchin-Kobayashi correspondence that every objective Hermitian-Einstein bundle is objective stable. More precisely we have the following result. We follow the proof of Lübke [17] closely as described by Chapter V of Kobayashi [15]. Proof. Take any L mw -twisted holomorphic line subbudle K ⊂ ∧ r E for each r such that 1 ≤ r < m. In view of (7), the semistable version of (10) says that to show E is L-twisted semistable, we need to prove (11) rc − c ′ ≥ 0 always holds. Here c, c ′ are respectively the constant factors of the H-objective Einstein metrics on E and K.
The inclusion K ⊂ ∧ r E yields a nowhere vanishing twisted section s = {s i } ofÊ = ∧ r E ⊗ K * ≺L, whereL = L r ⊗ L −mw . SetL ≺Lm to be the trivialization induced by L, wherem = C(m, r) is the rank ofÊ. By taking refinement cover if necessary, we can assume s to be objective (see Definition 3.1), so there are nowhere vanishing local holomorphic sections ξ = {ξ ij } ofL and t = {t i } ofL such that (12) s ≺ ξ, t ≺ ξm.
(Here we adjust the notation t slightly based on the fact that the holomorphic gerbeL is of orderm.) LetĤ be the metric onL, that is induced by H. By Proposition 4.4, E carries anĤ-objective Einstein metricĥ with the constant factor rc − c ′ . Define a family of local functions f = {f i } on our manifold X, where each . Note thatĥ ≺Ĥ,Ĥ ≺Ĥm, whereĤ is the gerbe metric onL. This and relations (12) together yield the key fact that f is a globally welldefined (smooth) function on X. Thus f must have a maximum q on X.
Take any point x 0 ∈ f −1 (q) and a coordinate neighborhood V . Introduce the functional on V , where g αβ are the components of the metric Φ. Now choose V so small that V ⊂ U i for some i andm √ t i exists as a holomorphic section of m L i on V . Thus we have a nowhere vanishing holomorphic section on and f = š 2 h , whereȟ =ĥ ⊗Ĥ −1/m . Apply the standard Weitzenböck formula to the bundleĚ i =Ê i ⊗L −1/m i over V : Here Dȟ is the unique compatible connection associated toȟ (recallš is holomorphic), andŘ is its mean curvature. More preciselyŘ is the skew symmetric form corresponding to iΛFȟ ∈ End(Ě i ), where Fȟ is the curvature of Dȟ as before.
It is not hard to check thatȟ isȞ-objective Einstein with factor rc −c ′ by Proposition 4.4. (Any metric H on L is certainly H-objective Einstein with factor 0 as deg L (L) = 0, andĤ −1/m is some power of H hence also objective Einstein with factor 0.) However by definition, H =Ĥ ⊗ (Ĥ −1/m )m = 1 is the trivial metric on the trivial twisting Thusȟ is Einstein in the usual sense with the same factor rc − c ′ . This means iΛFȟ = (rc − c ′ )I and (13) becomes here We now return to the proof of (11). Suppose this were not the case but rc − c ′ < 0. Then the formula (14) above would yield that J(f ) ≥ 0 on V , in fact J(f ) > 0, asš never vanishes. Since J(f ) attains its maximum at the interior point x 0 ∈ V , the Hopf maximum principle says that f must be a constant on V . Hence J(f ) = 0 on V , which is a contradiction.
Hence E is L-twisted semistable. Suppose E is however not quite L-objective stable, namely rc − c ′ = 0. Then we will seek the desirable decomposition as stated in the theorem. First (14) still says J(f ) = Dȟš 2 h ≥ 0 on V and the maximum principle again implies f is a constant on V , which means V ⊂ f −1 (q). This is done for each x 0 ∈ f −1 (q), so f −1 (q) is open in X hence must be X as it is certainly closed.
In other words, f is actually a constant function on X.
Now that f is a constant function, J(f ) = 0 on X. To each x ∈ X, from (14), one has Dȟš = 0 on a sufficiently small neighborhood V x ⊂ U i for some i. Namelyš is Dȟ-flat on V x . Since we can arrange easily t i to beĤ i -flat (as L i is a line bundle), we see that s i , namely the inclusion x for some subbundles of ranks r, m − r on V x , with E ′ x , E ′′ x both preserving the connection D h . Namely the metric h splits with respect to the decomposition: E ′ x ⊥ E ′′ x . As x ∈ X varies, we obtain a refinement cover {V x } of {U i } as well as two bundles x }, both of which are twisted over the gerbe L restricted to the refinement cover. To this end we have the decomposition E = E ′′ ⊕ E ′′ on X, in which E ′ , E ′′ ≺ L both carry the induced Einstein metric by h. Repeat the same process for E ′ , E ′′ . After finite many steps, we will have to stop. The final decomposition of E is what is required in the statement of the theorem. q.e.d.
Corollary 5.2. If E is also indecomposable, namely E = E ′ ⊕ E ′′ for any twisted bundles E ′ , E ′′ ≺ L of positive ranks, then E is L-objective stable.
In fact, more is true. One can weaken the assumption by requiring additionally that E ′ , E ′′ are objective stable in the direct sum.

Existence of objective Einstein metrics on stable bundles
Now we work on the other direction of the correspondence, following the original papers [8,9,26] and the expositions [25,15,16,12]. As in [25], we will adapt the approach that combines [8] and [26] together to our situation. Each h leads to a unique gerbe metric H on L and the compatibility means that H ≺ H m . In a small neighborhood of any point, it will be essential to view h ∈ M as an ordinary (untwisted) Hermitian where-ever existing). In this way the local picture matches that of the standard case in [8,9,26,25], and similar local computations, including pointwise algebraic operations, can be carried over to our case without essential changes. The proof can then be repeated almost verbatim. We outline below the main steps and leave the detailed checking to the interested reader.
is a globally well defined function on X by the internal compatibility of k, h and End( E i ) = End(E i ) canonically. Let h(t) = (h(t), H(t)) be a curve in M joining h to k. Likewise we see that h(t) −1 ∂ t h(t) ∈ Ω 0 (End(E)) is a global section, where ∂ t h(t) = d dt h(t). The associated connection of h i on E i has the curvature F e h i equal to the twisted curvature As it has used several times before, the collection { F h i } yield a global section in Ω 2 (End(E)). Hence we have a global section F e h = {F e h i } as well and we can introduce a 2-form with c the (potential) Einstein constant given in (7). To verify Q 2 ( h) is independent of the paths h(t), it is sufficient to check c Tr[( h(t) −1 ∂ t h(t)) · F e h(t) ]dt ∈ Im∂ + Im∂ for any closed path c = h(t) ⊂ M. The latter depends on the same kind of local computations as in the standard case [8] and they can be transplanted over directly.
Step 2. Establish the main properties of the grading flows for D.
Given two tangent vectors v, w ∈ B = T e h M, h −1 v and h −1 w are global sections of End(E). The Riemann metric on M is defined by the inner product (v, w) = X Tr( h −1 v · h −1 w)Φ n . For a smooth family of h(t) ⊂ M, one has d dt Q 1 ( h(t)) = Tr( h(t) −1 ∂ t h(t)) and d dt Q 2 ( h(t)) − iTr( h(t) −1 ∂ t h(t) · F e h(t) ) ∈ Im∂ + Im∂ by the same local computations as in [8,15]. Hence one has the variation of the functional where the ∆ ′ 0 , ∂ 0 , ∂ 0 are the operators associated with the initial metric pair h(0). This is a non-linear parabolic equation of the same type as in [8] and can be solved essentially in the same way. Briefly, by linearizing the equation and the Fredholm theory one obtains the short time existence, and by a continuity argument, one has the long time existence. In the end, together with the maximum principle one shows that starting at any h(0) ∈ M, the evolution equation (15) has a unique solution h(t) defined for all time t together with a uniform bound (16) max For the remaining proof we will set h(0) = k so det( k −1 h(t)) = 1 at t = 0. By the maximum principle again, we have det( k −1 h(t)) = 1 identically along the entire flow.
Step 3. Under the previous assumption prove the following key estimate for all t ≥ 0, where C 1 , C 2 are positive constants independent of t.
The L-stability is used in this step. The estimate (17) substitutes for a weaker version in [9] for the case of projective varieties, whose proof requires a Mehta-Ramanathan's restriction theorem on stable bundles. Here we follow [25] closely, based on analytic results from [26]. It goes very briefly as follows.
Write u(t) = log( k −1 h(t)) ∈ Ω 0 (End( E)); this is a self-adjoint endomorphism so with all real eigenvalues. (We point out that although E = E i ⊗ ( m √ L i ) * involves the choice of the root m √ L i , the eigenvalues and the various norms of u are well-defined. This note should be kept in mind for the rest of the argument.) Because of (16), there are constants A 1 , A 2 independent of t such that max X |u| ≤ A 1 + A 2 u L 1 .
We will show (17) using proof by contradiction. Suppose it were not true. Then there are sequences of t i → ∞ and constants B i → ∞ such that (18) u Re-normalize v i = u i / u i L 1 so that v i L 1 = 1. Moreover by (18), ∂v i is bounded in L 2 . Thus there is a subsequence v i weakly convergent to v ∞ ∈ L 2 1 . The limit v ∞ is self-adjoint almost everywhere and it can be shown that its eigenvalues λ 1 , · · · , λ r are constants. Let {γ} be the set of intervals between the eigenvalues and for each γ choose a function p γ : R → R such that p γ (λ i ) = 1 for λ i < γ and p γ (λ i ) = 0 otherwise.
Define π γ = p γ (v ∞ ) ∈ L 2 1 (End( E)). This can be viewed as an L 2 1 subbundle S γ of E through projection in the sense that π 2 γ = π γ . By the main regularity theorem of [26], S γ is actually a smooth subsheaf of E. Since the proof is purely local, there is a smooth subsheaf S γ of which is an L mw -twisted line subbundle of ∧ b E, where b = rank S γ = Tr(π γ ) and w = C(m − 1, b − 1). We will show for one of the K γ , That is to say, . This however violates the L-stability of E, yielding our expected contradiction.
To show (19) let k γ be the restriction of the base metric k to S γ . By direct computations, the curvature satisfies ΛF e kγ = π γ ΛF e k π γ + Λ(∂π γ ∂π γ ) for every γ. It follows that deg L w (K γ ) = i X Tr(π γ ΛF e k )Φ n − X |∂π γ | 2 Φ n . Let a denote the biggest eigenvalue of v ∞ and a γ the width of the interval γ. Then v ∞ = aId − a γ π γ and a combination of degrees can be computed: where the last inequality comes from Lemma 5.4 of [25]. On the other hand, a · m − Tr(π γ ) = Tr(v ∞ ) = 0.
Together with the inequality above, this shows that (19) must hold for at least one γ.
Step 4. Existence of the objective Einstein metric as a limit metric.
Owing to the strong estimate (17), unlike [8], here we can avoid Uhlenbeck's theorems on removable singularities and Columbo gauges (which do not hold anyway in higher dimensions).
(ii) s(t) L p is bounded above for each p. To see (ii), just note D is decreasing along the downward flow h(t), hence s(t) is bounded in C 0 -as well as L p -norms.
From a minimizing sequence of D by (i) and the mean value theorem, we have a sequence t i → ∞ such that Then from the expression of D and noting Q 1 ( h(t)) = 0 identically here, one sees ∂(s(t i )) is bounded in L p . Together with (ii), this implies that s(t i ) is bounded in L p 1 . Choose p > n and by the Kondrakov compactness of L p 1 ֒→ C 0 there is a subsequence s(t i ) convergent to s ∞ in C 0 . Then by (16) together with the gradient equation (15) and the maximum principle, it is possible to show that |∆s(t i )| is uniformly bounded on X in the sequence. (Compare the proof of Lemma 19 in [8].) Thus for any p, ∆s(t i ) is bounded in L p . By the ellipticity of the Laplace operator, this and (ii) together imply that s(t i ) is bounded in L p 2 . By taking a subsequence if necessary, s(t i ) weakly converges to s ∞ in L p 2 . In other words, h(t i ) ⇀ h ∞ = ks ∞ in the L p 2 -norm (with respect to k as before). Consequently F e h∞ ∈ L p exists as a distribution and F e h(t i ) ⇀ F e h∞ in L p . In view of (20), we now have ΛF e h∞ −cI = 0 weakly.
The standard elliptic regularity guarantees that h ∞ must be smooth, and we have the desired H ∞ -objective Einstein metric h ∞ , which is read off h ∞ = (h ∞ , H ∞ ). q.e.d.
7. An application to gauge theory: the SO(3)-moduli space We have so far worked with twisted vector bundles. For slight ease in dealing with structure groups and associated bundles, it is also useful to introduce twisted principal bundles that can be briefly laid out as follows. Fix a central extension of Lie groups 1 → Z → G → H → 1, for example take In general the multiplication maps m : Z × G → G and m : Z × Z → Z are clearly group homomorphisms. Under the group extension, a principal gerbe P = {P ij } consists of a collection of principal Z-bundles satisfying the suitable conditions as in vector bundle gerbes. Here a "tensor product" P ij ⊗ P jk by definition is the associated principal Z-bundle (P ij× P jk ) × m Z of the fiber product via the multiplication homomorphism m : Z × Z → Z. A twisted principal bundle Q ≺ P consists of a collection of G-bundles Q = {Q i } such that Q j = Q i ⊗ P ij , where the "tensor product" is the associated principal G-bundle (Q i× P ij ) × m G via the homomorphism m : Z × G → G. Its projection P(Q) via G → H is obviously a (untwisted) principal H-bundle. Of course one can also define twisted principal bundle of structure group Z (rather than G) over P, because the tensor product makes sense here. Furthermore there is not much additional difficulty in defining gerbe connections, B-fields or twisted connections as in the case of twisted vector bundles.
To get our objective version, we assume P to be n-torsion, meaning there is a principal twisted Z-bundle P ≺ P n , or equivalently, n[P] = 1 ∈Ȟ 2 (X, Z) in the sheaf cohomology of Z-valued smooth functions on X. (This is automatic under the extension (21).) Fix P and a twisted connection A on P . This brings a unique gerbe connection A on P so that A ≺ A n . Then we can look for twisted connections D on Q that obey D ≺ A. Using the curvatures of D, A together, we can simply copy Section 2 to define P -objective Chern classes c P k (Q), which are actually independent of the choices of A, D.
We now return to our main application, which will be based on twisted principal bundles under the extension (21). Take X to be a smooth Riemannian 4-manifold and S → X a principal SO(3)-bundle. Historically the moduli space M S of anti-self-dual connections on S has played an important role in the applications of gauge theory. One of the main issues is to do with the orientability and orientations of the moduli space and has been settled in Donaldson [11] for the important case that the Stiefel-Whitney class w 2 (S) has an integer lift (namely S is spin c ). Here our purpose is to handle the general case of an arbitrary w 2 (S).
More precisely let B S be space of connections on S modulo the gauge group. Each connection ∇ on S induces a connection d ∇ on the adjoint bundle g S = S × ad so(3). In turn we have the Fredholm operators Proof. This is mainly a modification of the proof in [11] for our twisted bundle set up. There are three ingredients. First, because of the exact sequence (21), by lifting S locally one has a twisted principal U(2)bundle Q = {Q i } over a principal gerbe P. (The projection P(Q) equals S and the original case with integral w 2 (S) corresponds to the trivial gerbe P.) Choose a trivialization P ≺ P 2 and let B P Q = {(D, A) : D, A are compatible connections on Q, P }/ ∼ modulo the twisted gauge transformation pairs. Here the compatibility means that A ≺ A 2 , D ≺ A for some unspecified gerbe connection A on P. When a connection A ⋆ on P is also chosen, each connection ∇ on S lifts to a unique twisted connection D ≺ A on Q in view of (21). That is to say, one has an injective map f = f : Ω 1 (g P Q ) → (Ω 0 ⊕ Ω 2 + )(g P Q ). Thus to show the theorem, it is sufficient to prove Λ P Q → B P Q is trivial. Since the choice of a trivialization P ≺ P 2 is immaterial to the discussion above, we may as well select P o = det Q = Q × det U(1) from now on. Then it remains to show that Λ Po Q → B Po Q is trivial. Next using the homomorphism λ : U(2) → SU(3), u → diag(u, det u −1 ), one introduces the twisted principal SU(3)-bundle (3). (This resembles the stablization E ; E ⊕ det E * from a U(2)-vector bundle to an SU(3)-bundle in the original proof of [11].) The canonical trivialization det Q + is a global line bundle. It follows that B Po Q can be identified with the set B Q + of twisted connections on Q + . Now that Q + has the structure group SU(3), by making suitable homotopy computations similar to [11] and [12] (especially 5.4), one sees that elements in H 1 (B Q + ) all come from [X, SU(3)] = K −1 (X)/H 1 (X) = H 3 (X). More directly the slant product over the twisted Chern class c Po 2 (Q + ) gives such a homomorphism H 3 (X) → H 1 (B Q + ). Thus each loop φ in B Po Q has the form φ γ for some loop γ in X via the Poincáre duality H 3 (X) → H 1 (X). And to show Λ Po Q is trivial, one must show the restriction Λ Po Q | φγ is so for any loop γ. The last ingredient is to use the gluing to exhibit the loop φ γ of connections in B Po Q . This is a slight generalization from the case of standard U(2)-connections in [10,11] to our twisted connection pairs. So let V → S 4 be the negative spinor bundle on the 4-sphere and J λ be an instanton on V , flattened of a small scale λ around ∞ ∈ S 4 . Write the pair J λ = (J λ , det J λ ) ∈ B det V V , where V is viewed as a twisted bundle.
Choose a twisted bundle Q ′ ≺ P with c det Q ′ 2 (Q ′ ) = c Po 2 (Q) − 1. Fix a twisted connection D ′ on Q ′ . Since g S , Λ 2 + are trivial SO(3)-bundles over the loop γ, we can choose a lifting isomorphism ρ : g S | γ → Λ 2 + | γ as our gluing parameter. Then for each pair (D ′ , A ′ ) ∈ B det Q ′ Q ′ and any point x ∈ γ, we have the glued pair [(D ′ , A ′ )# ρ(x) J λ ] ∈ B Po Q . Strictly speaking, it is better to work with the associated twisted vector bundles of Q ′ , Q and D ′ , A ′ should be flattened around x. In fact there is not much difference in gluing the twisted connections, since the gluing is done in a small neighborhood of x and essentially it is to glue component connections of D ′ , A ′ with J λ , det J λ . (As usual one should lift ρ to gluings between Q ′ and V . But the gauge classes [(D ′ , A ′ )# ρ(x) J λ ] are independent of the liftings.) Fix a pair (D ′ , A ′ ) and a uniform scale λ over the compact set γ. The result is a continuous family [(D ′ , A ′ )# ρ J λ ] in B Po Q , representing our loop φ γ . The main estimates in 3(d) of [11] gives a continuous fiberwise isomorphism j in the following diagram: where π is the obvious projection. Consequently the bundle restriction Λ Po Q | φγ is isomorphic to π * Λ det Q ′  Regardless the SO(3)-vector bundle S, it seems to be an interesting problem to investigate directly the instanton moduli space M L (E) on a twisted U(2)-bundle E ≺ L with a fixed trivialization L ≺ L 2 . Perhaps one could further extract Doanldson type invariants. In terms of algebraic geometry, this suggests the study of the moduli space of L-objective stable bundles.