A vanishing theorem for T-branes

We consider regular polystable Higgs pairs $(E, \phi)$ on compact complex manifolds. We show that a non-trivial Higgs field $\phi \in H^0 ({\rm End} (E) \otimes K_S)$ restricts the Ricci curvature of the manifold, generalising previous results in the literature. In particular $\phi$ must vanish for positive Ricci curvature, while for trivial canonical bundle it must be proportional to the identity. For K\"ahler surfaces, our results provide a new vanishing theorem for solutions to the Vafa--Witten equations. Moreover they constrain supersymmetric 7-brane configurations in F-theory, giving obstructions to the existence of T-branes, i.e. solutions with $[\phi, \phi^\dagger] \neq 0$. When non-trivial Higgs fields are allowed, we give a general characterisation of their structure in terms of vector bundle data, which we then illustrate in explicit examples.


Introduction
Vafa-Witten (VW) systems are a set of gauge-theoretic equations on a four-manifold S, introduced by Vafa and Witten [1] in the study of the S-duality conjecture for N = 4 supersymmetric YangMills theory. A key role in this analysis is played by a vanishing theorem, thanks to which for some choices of S the partition function of topologically twisted N = 4 YangMills simplifies, and can be computed in terms of the Euler characteristic of the moduli space of instantons on S. In general such a partition function contains valuable information on four-manifold invariants, as studied in cases where the theorem does not apply, see e.g. [2,3] for recent progress in this direction.
The VW equations simplify considerably when S is a Kähler surface, and can be written in terms of a Higgs pair (E, φ), with E a Hermitian vector bundle and φ a Higgs field. In this sense, they can be seen as a higher-dimensional analogue of the Hitchin equations for Higgs bundles on compact Riemann surfaces [4]. However, unlike in the generalisation made in [5], in the VW case φ is twisted by the canonical bundle of S.
Interestingly, the same VW equations for Kähler surfaces describe supersymmetric configurations of 7-branes in F-theory [6,7], which have been proposed to realise models of Grand Unification in string theory, see [8][9][10] for reviews. In this setup, the Higgs field φ contains the transverse geometrical deformations of the 7-brane embedding, and the vanishing results of [1] signal cases where a profile for φ is obstructed. Relevant for the analysis of [1] are also cases where φ = 0 but it has an Abelian profile, in the sense that the commutator [φ, φ † ] appearing in the VW equations vanishes. Remarkably the opposite case [φ, φ † ] = 0, dubbed T-brane in the string theory literature [11], is particularly interesting from the 7-brane perspective. Indeed, T-brane configurations are a key ingredient to obtain realistic Yukawas in F-theory GUT model building [11,12], and as such it is an important problem to understand for which surfaces they can be realised.
In this work we study from a general perspective manifolds where T-branes are obstructed. We consider Higgs pairs (E, φ) with the same twist as in VW systems but, for the sake of generality, we do it on complex manifolds X of any dimension, endowed with a Gauduchon metric so that Higgs polystability is well defined. We prove a vanishing theorem -Theorem 3.1 -in terms of the properties of the canonical bundle K X . If the degree of K X is negative then φ must vanish, and if K X is trivial then φ must be proportional to the identity. Restricting our analysis to Kähler metrics, we find that T-branes are only allowed when K X has positive degree.
Both our strategy and vanishing theorem are different from the ones in [1]. Instead, our proof follows the philosophy of the no-go theorem of [13], which uses the existence of φ and the stability of (E, φ) to find an obstruction in terms of K X . Compared to the vanishing results in [2,13] our present analysis is completely general, in the sense that there are no assumptions either on the manifold X or on the vector bundle E.
For those manifolds where the Higgs field is not obstructed, one may describe its different components in terms of vector bundle data. We illustrate this procedure for rank-2 bundles, namely a) an extension of two line bundles and b) its twist by an ideal sheaf. When restricting the construction to Kähler surfaces, we are able to interpret our results in terms of 7-brane physics. From this perspective, the allowed components of φ correspond to the F-flat directions of a D7-brane superpotential in case a), and of the superpotential of a D7/D3-brane bound state in case b).
The paper is organised as follows. In Section 2 we review the relation of Vafa-Witten systems, T-branes and polystable Higgs pairs, as well as the vanishing results for the Higgs field in the existing literature. We also describe a simple case to which none of these partial results apply. In Section 3 we derive the main theorem of this paper, the Vanishing Theorem 3.1, as well as some of its consequences. In those cases where Higgs fields are allowed, they should be characterised in terms of vector bundle data, as we illustrate in Section 4, both from a mathematical and a physics perspective. Section 5 describes explicit examples of these constructions, and in Section 6 we draw our conclusions.
Several technical details have been relegated to the appendices. Appendix A discusses several homological-algebra facts needed in Section 4. Appendix B discusses how the ideal-sheaf twists can be used to construct topologically non-split bundles. Appendix C provides the cohomological computations needed for the constructions of Section 5.

T-branes in Vafa-Witten systems
Let us consider a Vafa-Witten system [1] on a compact Kähler surface S. Such a system is specified by the curvature F = dA − iA ∧ A of a Hermitian vector bundle E, and by a (2,0)-form Higgs field φ, which also transforms in the adjoint of the structure group G S . 1 In terms of these two objects the Vafa-Witten equations read with ω the Kähler two-form of S and c ∈ R a constant. In the string theory literature, such a system would arise from a stack of branes wrapped around S, hosting a super-Yang-Mills theory with a symmetry group G S [6,7]. There φ contains the transverse geometrical deformations of the brane embedding, and A the gauge degrees of freedom.
Eqs.(2.1) guarantee that the system preserves supersymmetry and minimises its energy.
In general, solutions to the above equations can be specified in terms of a Higgs pair (E, φ) , φ ∈ H 0 (Hom(E, E ⊗ K S )) , (2.2) with K S the canonical bundle of S. Via a Hitchin-Kobayashi correspondence, proved for this case in [14] (see also [15]), solutions to these equations are associated to polystable Higgs pairs.
Definition 2.1. A Higgs pair (E, φ) is stable if for any non-zero coherent subsheaf S of E such that rk(S) < rk(E) and φ(S) ⊆ S × K S we have that where the slope of a torsion-free sheaf E is defined as dubbed T-branes in the string theory literature [11,16,17]. For those, as the corresponding φ is not diagonalisable, the criterion for Higgs bundle stability will be less restrictive than plain polystability for the bundle E, allowing for more general constructions.
A standard example where this happens is the original construction of Hitchin [4] generalised and adapted to the Vafa-Witten system. To construct it, let us consider a rank-2 bundle E on S of the split type We can extend these definitions to compact complex manifods X of any dimension n, by replacing S by X in (2.2) and below, and by defining the degree of a sheaf as where ω is the fundamental form of a Kähler metric on X, or Gauduchon if X is non-Kähler. 3 Note that subsheaves S of E such that rk(S) = rk(E) always have the property that µ(S) ≤ µ(E).
This is because the quotient E/S is a torsion sheaf (as it has zero rank), and the first Chern class of a torsion sheaf is dual to a positive Cartier divisor (and to the zero divisor if the support of E/S has codimension strictly higher than one). and fix its complex structure to have A (0,1) = 0. This allows to express the Higgs field φ as a two-by-two matrix such that with m ∈ H 2,0 (S, L 1 ⊗L −1 2 ). Higgs stability amounts to imposing that deg(L 1 ) < deg(L 2 ), which is easily satisfied. Explicit solutions of this sort were constructed in [13], where it was also pointed out a general obstruction to embedding them in arbitrary Kähler surfaces. Indeed, since m can be seen as a section m ∈ H 0 (S, which together with the Higgs-stability condition requires that In particular, solutions of this type cannot be realised in surfaces of positive curvature, analogously to the case of Higgs bundles on Riemann surfaces [4]. It was shown in [13] that this no-go result applies to more general T-brane configurations, with the restriction that E must be a rank-n bundle of the split type. The condition deg(K S ) ≥ 0 is simple to show for non-nilpotent Higgs fields, since then at least one Casimir of φ of degree p is non-vanishing. As such a Casimir can be seen as a section of H 0 (S, K p S ), its existence leads to the requirement deg(K S ) ≥ 0. Therefore, showing that (2.7) is a necessary condition for nilpotent T-branes leads to the following statement φ = 0 whenever deg(K S ) < 0 (2.8) An interesting result in this direction has been recently obtained in [2,3], in which the case of C * -fixed Higgs pairs in projective surfaces was studied in detail. Under those assumptions, it was shown that φ = 0 if (2.7) is not met.
The case deg(K S ) = 0 needs to be treated separately, because the above results do not exclude T-branes in general. Nevertheless, we sill show that for any choice of bundle metric [φ, φ † ] = 0 for deg(K S ) = 0 (2.9) In this work we provide a general proof that (2.7) is a necessary condition for Vafa-Witten systems with [φ, φ † ] = 0, extending the aforementioned results. More precisely, we will show the statements (2.8) and (2.9). The proof will follow a similar strategy to the one used in [13]. For the sake of clarity we will first illustrate our approach in a simple class of Higgs pairs that falls outside the cases covered by [2,3,13]. Then, in the next section we will provide the general proof.
A simple non-trivial example Let X be a complex manifold of any dimension, and let L 1 , L 2 be two line bundles over it. Consider the following non-trivial extension where i is an embedding. Let us in addition consider a map ϕ : L 2 → L 1 ⊗ K X . One may construct a Higgs pair (E, φ), by taking the Higgs field to be φ Note that since (2.10) is a complex, then φ 2 = 0. Moreover, for the same reason, φ| L 1 = φ • i = 0, which implies that the only proper subbundle of E, i.e. L 1 , is left invariant by φ. In other words, requiring that (E, φ) is a stable Higgs pair is equivalent to requiring the standard slope-stability for E.
Let us assume that E is stable, that is µ(L 1 ) < µ(E). This in particular implies that H 0 (X, End(E)) = C, which makes it impossible to undo a C * action on φ by a C * action on E. This would only be possible if the extension (2.10) is such that E = L 1 ⊕ L 2 , because then an endomorphism of the type 1 0 0 λ with λ ∈ C * would undo the action φ → λφ. In general, however, the extension (2.10) leads to nilpotent, non-C * -fixed Higgs pairs, and falls outside the analysis of [2,3].
In order to apply this construction to the Vafa-Witten equations, let us take X to be completing the proof of the vanishing theorem for this class of Higgs pairs.
The class of examples constructed by means of the non-trivial extension (2.10) gives rise to vector bundles E which, topologically, are still equivalent to E L 1 ⊕ L 2 . This means that they can be continuously connected to a split form by deforming their holomorphic structure. If dimX > 1, however, one can have vector bundles which are topologically inequivalent to a sum of line bundles, that even more so calls for a general proof of the vanishing theorem. One particular way to build such cases is to twist the sequence (2.10) by an ideal sheaf supported on a codimension-2 locus. We will give the details of this construction in Section 4, and proceed now with proving the theorem in full generality.

The vanishing Theorem
Let us now turn to the general proof of the vanishing theorem. Our strategy will be analogous to the cases discussed in the previous section. We will use the stability of the Higgs pair (E, φ) and the non-triviality of the Higgs field φ to constrain the canonical bundle K X of the complex manifold X. Our discussion extends trivially to polystable Higgs bundles, which can be seen as direct sums of stable Higgs bundles with the same slope.
Proof. Applying the definitions of degree and slope of torsion-free sheaves we have that deg(E) = deg(P) + deg(Q) and that rk(E) = rk(P) + rk(Q). Moreover, we have that Now, because P and Q are torsion-free sheaves, rk(P), rk(Q) > 0. Using (3.1), we have If on the contrary we assume that µ(P) < µ(E). Then, by (3.1), With this result, we now prove the statement (2.8) for a general complex manifold.
Proposition 3.1. The existence of a stable Higgs pair (E, φ) on a complex manifold X with φ = 0 implies that deg K X ≥ 0.
Proof. Note that the Higgs field φ : This then implies that (E, φ) is stable if and only if (E ⊗ K X , φ ) is stable. Since we are assuming that (E, φ) is stable, so is (E ⊗ K X , φ ). Moreover, by assumption ker φ is a φ-invariant subsheaf of E and Im φ is a non-vanishing φ -invariant subsheaf of E ⊗ K X .
Note that rk(E) = rk(E ⊗ K X ), and that Let us first assume that µ(Im φ) = µ(E ⊗K X ). Because of the stability of (E ⊗K X , φ ), this can only be true if rk(Im φ) = rk(E ⊗ K X ) = rk(E). Since Im φ E/ ker φ and ker φ is free of torsion, this implies that ker φ = {0}, and thus φ is injective. Hence we have that E Im φ and so µ(E) = µ(Im φ), implying by (3.2) that deg(K X ) = 0.
In fact, one can say more about the case of degree-zero canonical bundle by showing the following: Proof. If φ is an isomorphism then E E⊗K X , and by Eq. (3.2) it follows that deg K X = 0. To prove the other direction note that when deg K X = 0, then µ(Im φ) = µ(E ⊗K X ), as follows from the proof of Proposition 3.1. Therefore, by stability, rk(Im φ) = rk(E ⊗ K X ).
Nevertheless, one must still show that φ has a vanishing cokernel. For this purpose, consider the injective map between line bundles det φ : det E → det E ⊗ K X . This map is also surjective, because the torsion sheaf det E⊗K X / Im det φ has both vanishing rank and first Chern class. 5 Therefore det φ is an isomorphism, implying that φ is as well [18]. 5 We are implicitly using here that det Im φ Im det φ, but this is clearly true when φ is injective.
Indeed, on the one hand injectivity of φ implies Im φ E and hence det Im φ det E. On the other hand, injectivity of det φ implies that Im det φ det E.
The structure of the Higgs field in the case of zero-degree canonical bundle is very simple, and leads to a trivial commutator [φ, φ † ] for any choice of bundle metric. To see this, notice that in the particular case of trivial K X , a Higgs pair is given by (E, ϕ) with ϕ ∈ End(E). One may in fact consider such a pair, which we will dub untwisted Higgs pair, for a general complex surface X, and define slope stability analogously to Definition 2.1. We can then show the following result, which parallels Schur's Lemma: Proof. Suppose ϕ is not identically vanishing, so that {0} = Im ϕ ⊆ E. On the one hand, the equality holds only if rk(Im ϕ) = rk(E). On the other hand, since Im ϕ E/ ker ϕ, independently of what ker ϕ may be, we have that µ(E) ≤ µ(Im ϕ). This implies that µ(Im ϕ) = µ(E), and thus that rk(Im ϕ) = rk(E), meaning that ϕ is injective. Since the domain of ϕ coincides with its codomain, injectivity implies surjectivity, and hence ϕ is an automorphism. Moreover, being C algebraically closed, there exists λ ∈ C * such that ker(λI − ϕ) = {0}. Also, (E, ϕ) is a stable pair if and only if (E, λI − ϕ) is a stable pair, because for any subsheaf P of E, we have ϕ(P) ⊆ P if and only if (λI − ϕ)(P) ⊆ P.
Therefore, applying again the reasoning above to the map λI − ϕ : E → E, we conclude that ϕ − λI ≡ 0, since it cannot be an automorphism.
Let us now focus on Kähler manifolds. From the above result we derive the following: Proof. Corollary 3.1 implies that φ is either zero or an isomorphism. The vanishing of the commutator for the case where X is a Calabi-Yau CY (i.e. K X O CY ) follows from Lemma 3.2. All other cases with zero-degree canonical bundle are such that c 1 (K X ) is a torsion class of H 2 (X, Z), and hence that K r X O X for some integer r > 1. We may then define a r : 1 covering map π : CY → X, with π * K X = O CY . Just like for bundles [19], by the Hitchin-Kobayashi correspondence [14] π * preserves polystability of Higgs pairs and sends (E, φ) to a slope-polystable Higgs pair (π * E, π * φ) in CY. 6 Assuming that φ = 0 in X and applying our above result for CY we find that π * φ must be block-diagonal, with each block proportional to the identity. Therefore π * ([φ, pointwise, and so we must necessarily have that [φ, φ † ] = 0.
Notice that when K X is trivial this result applies to any complex manifold X. In fact, we would expect Proposition 3.2 to also hold for non-Kähler, complex manifolds with a Gauduchon metric and deg K X = 0. This would be guaranteed by the appropriate generalisation of the Hitchin-Kobayashi correspondence to Gauduchon metrics [20].
Putting together Propositions 3.1 and 3.2, we obtain the vanishing theorem: Finally, restricting to the case of T-branes in Vafa-Witten systems we obtain: In particular, S has to be properly elliptic or of general type.

Topological restrictions on Higgs pairs
Throughout the proof of Proposition 3.1 we not only have obtained that deg K S ≥ 0 for stable Higgs pairs (E, φ), but also that the inequalities must always be true for φ = 0. Here the first equality becomes strict whenever φ is not injective -like in the case of nilpotent Higgs fields -and the second one whenever φ is not surjective -like for any sort of T-brane. Even in a surface S with deg K S > 0, the condition (3.4) will restrict which Higgs pairs can be constructed on it.
Indeed, let us for simplicity consider the case where det E O. Then these inequalities are equivalent to This is reminiscent of the inequalities obtained in (2.11) for the simple T-brane example based on the extension (2.10), or to those obtained in [13] for the case of split bundles.
However, they are not fully equivalent. Indeed, taking such a nilpotent example and considering the case where deg(L 1 ) + deg(L 2 ) = 0 one obtains that (3.5) becomes This condition should be supplemented with the inequality 2 deg(L 2 ) ≤ deg(K S ) in order to arrive to (2.11) applied to this case. This extra condition describes in a more precise manner which line bundles L 2 are allowed given deg(K S ). In this sense (3.6) only contains partial information, although it is still stronger than the stability condition 0 < deg(L 2 ): strong enough to require that deg(K S ) > 0.

Structure of the Higgs field
Theorem 3.1 gives a necessary condition for a complex manifold X to allow for a non-trivial Higgs field, assuming Higgs bundle polystability. But even if deg K X ≥ 0 is satisfied, it is in general not clear whether φ can actually be non-trivial, except for the components proportional to the identity, which are in one-to-one correspondence with sections of K X . In the following we would like to give sufficient conditions for the existence of nontrivial Higgs fields, with particular emphasis on the traceless data of φ. We will focus our discussion on the particular rank-2 example analysed in section 2 and on a simple generalisation thereof, for which we will show how to construct φ in terms of the bundle data. We will then give a physical interpretation of our results, which permits to extend this picture to other settings.

Simple rank-2 example
Let us consider again the extension of two line bundles L 1 , L 2 over a complex manifold X, as discussed in section 2, namely 7 To make formulas shorter, in this section we are going to omit the zeroes in the exact sequences.
where i is an embedding. The extension class defining E is an element of the group . Following a standard construction (see e.g. [21]), we may build the following grid of long exact sequences in cohomology: whereĩ ≡ i ⊗ Id K X ,q ≡ q ⊗ Id K X , and the maps Λ 3 , Λ + will be discussed below. Since all the sheaves are locally free here, we simply have Moreover, for all α ∈ Ext 0 (E, Similar definitions apply for q * andq * . In the following we would like to show how the global sections φ ∈ Ext 0 (E, can be constructed in terms of the other elements of the grid. We will use the identification Hom(E 1 , E 2 ) E * 1 ⊗ E 2 and the notation H i (E) := H i (X, E) throughout, for all sheaves E, E 1 and E 2 on X.

The traceless Higgs fields are sections of End
where O X corresponds to multiples of Id E . Consequently with elements of H 0 (K X ) being of the form Id E ⊗s for some global section s of K X .
for every 0 = α ∈ H 0 (Hom(E, L 1 ⊗ K X )). As a consequence we have the following: , then E admits non-zero traceless Higgs fields.
Let us describe H 0 (Hom(E, L 1 ⊗ K X )) in detail, in terms of the first row of the grid in figure 1: (4.7) We can describe any element of H 0 (Hom(E, L 1 ⊗ K X )) as , where the latter maps injectively into H 0 (K X ) under i * . This in turn gives rise to a Higgs fieldĩ * (α) ∈ End(E) ⊗ K X , whose trace is: where the first equality follows by direct local computation (see Lemma A.1), and in the last one we have used that q • i = 0.
We end our analysis of H 0 (Hom(E, L 1 ⊗ K X )) by discussing sufficient conditions for the existence of traceless Higgs fields with m = 0 orv = 0. By injectivity ofĩ * and q * , To see if there is a non-trivial element of this form we again consider the first row of the grid, given by (4.7), but now the part involving the map Λ 3 . We have that if then Λ 3 has a non-trivial kernel, or equivalently dim (i * (H 0 (Hom(E, L 1 ⊗ K X )))) ≥ 1. In this case there exist non-vanishing traceless Higgs fields of the formĩ * (v).
Finally, let us consider the third row of the grid in figure 1. We have that any element ). In other words, the Higgs fields that are not in the image ofĩ * are of the type where the traceless part ψ gets injectively mapped byq * tõ ) . (4.14) From the above,ψ is the counterimage under i * of an element p ∈ H 0 (L −1 1 ⊗ L 2 ⊗ K X ). Elements of this form exist iff the matrix Λ + has a non-trivial kernel. Hence, similarly to before, a sufficient condition for non-zero traceless Higgs fields ψ ∈ H 0 (End(E) ⊗ Summary. Putting all this together, we see that any traceless Higgs field φ ∈ H 0 (End 0 (E)⊗ K X ) can be written as for some α ∈ H 0 (Hom(E, L 1 ⊗ K X )) and ψ ∈ H 0 (End 0 (E) ⊗ K X ) such that This means that: • L 1 is φ-invariant whenever ψ = 0, and hence, in this case, the pair (E, φ) is Higgs stable iff E is stable. Equivalently, only when ψ = 0 can we construct with this method a stable Higgs pair (E, φ) where the underlying bundle E is unstable.
We also have that: , there exist non-zero traceless Higgs fields of the form , there exist non-zero traceless Higgs field of the form φ = ψ.

Twisting by an ideal sheaf
The above discussion can be adapted to include an ideal sheaf twist of the extension (4.1): where I Z is the ideal sheaf of a codimension-2 locus Z ⊂ X. For simplicity, and in view of the physics applications, let us work in the case where X = S is a Kähler surface, so that Z is a finite set of distinct points. As mentioned, this twist can be used to engineer vector bundles that cannot be continuously deformed to a split form, see Appendix B.
To begin with, note that now the last term of the sequence is a non-locally-free sheaf.
Correspondingly i, thought of as a map between sheaves of sections, fails to be injective on Z. Similarly to the untwisted case (4.1), the extension class defining E is an element ξ ∈ Ext 1 (L 2 ⊗ I Z , L 1 ). This group however is more complicated, but can be described using the following long exact sequence (see [26,Chapter 2] for details): where we have used that Ext 1 (L 2 ⊗ I Z , L 1 ) O Z is the structure sheaf on Z, and thus duality. Therefore the extension class can be written as where a + ∈ H 1 (L −1 2 ⊗ L 1 ), andξ ∈ Ext 1 (L 2 ⊗ I Z , L 1 )/j(H 1 (L −1 2 ⊗ L 1 )), which has the property of having a non-zero value on Z, i.e. π(ξ) = 0. Note that the existence ofξ is obstructed if the map Y is injective.
The strategy to determine the components of a traceless Higgs field compatible with (4.18) is similar to the one applied to (4.1), so in the following we will simply point out the main differences. The first row of the grid in Figure 1 is replaced by: see Appendix A for details.
Considering the elements α ∈ H 0 (Hom(E, L 1 ⊗ K S )) as in (4.8 which guarantees that the kernel of Λ 3 in (4.21) (and thus the image of i * ) is non-trivial.
Finally, the third row of the grid in Figure 1 becomes: (see Appendix A for details). This tells us that, traceless Higgs fields of the type ψ as in (4.13) must be such that p = i * (q * (ψ)) ∈ ker Λ + ⊂ H 0 (L −1 1 ⊗ L 2 ⊗ K S ⊗ I 2 Z ). In particular, p is a section of L −1 1 ⊗ L 2 ⊗ K S that vanishes quadratically on Z. However, not all such p will lift to a Higgs field, but only those in the kernel of Λ + . Hence, similarly to Eq. (4.15) for the untwisted case, a sufficient condition for the existence of Higgs fields of the type ψ is  To make the connection it is useful to first consider the case where the extension (4.1) is trivial, in the sense that the bundle E is of the split type (2.4). Then it is straightforward to describe the holomorphic traceless deformations of the bundle connection A as where correspond to specific elements of the grid in Figure 1.
Clearly, the fact that all these deformations are holomorphic sections is a consequence of the F-term equations (2.1a) and (2.1b), but it is not the only one. F-terms also impose constraints on performing more than one deformation simultaneously. This can be easily encoded in the trilinear terms involving fluctuations that arise from the superpotential where we have expanded each of the sections a + , a − , and v in a basis of the corresponding cohomology groups, as a + = a + i ψ i + , a − = a − n ψ n − and v = v α χ α 3 with a + i , a − n , v α ∈ C. In physics terms, these complex numbers represent the vacuum expectation values (vevs) of the four-dimensional fields, and therefore the local field space. Finally we have defined The F-flatness constraints derived from the Yukawa couplings (4.29) are the following [24] Λ inα and so, e.g., care must be taken when performing a simultaneous deformation along the directions a + and v. Other deformations like m and p in (4.28) are on the other hand unobstructed by the above Yukawa couplings. Now, the extension (4.1) is described by an element of H 1 (L −1 2 ⊗ L 1 ) Ext 1 (L 2 , L 1 ), and in this sense it can be understood as a deformation of the split bundle (2.4) along the direction a + . Let us denote by a + = a + i ψ i + ∈ Ext 1 (L 2 , L 1 ) the element describing a particular bundle extension. Then Λ nα 3 ≡Λ inα 3 a + i defines a linear map where we have used Serre duality. The last equality in (4.31) then translates into the statement that the Higgs field deformations along the direction v allowed by the F-terms must belong to ker Λ 3 . To sum up, given a vector bundle E defined by the extension This result nicely matches the characterisation of traceless Higgs fields compatible with the extension (4.1), as summarised by (4.16) and the statements below. There are essentially three contributions to (4.16), that must now be identified with different elements of the grid in Figure 1, and that can be put in correspondence with sections of the form (4.28). The contribution φ =ĩ * (q * (m)) is in one-to-one correspondence with elements of Ext 0 (L 2 , L 1 ⊗ K S ) and gives rise to a nilpotent Higgs field, as in the split case. The contribution φ = ĩ * (v) − 1 2 Id E ⊗ i * (v) corresponds to elements of H 0 (K S ) that belong to the kernel of the map Λ 3 in the grid. This precisely matches our F-term analysis if we identify such a map with (4.32). Finally, the contribution φ = ψ corresponds to sections of the form p ∈ Ext 0 (L 1 , L 2 ⊗ K S ) which are in the kernel of the map Λ + in the grid. As we have assumed that H 1 (O S ) = 0, this map is such that the kernel is the whole of Ext 0 (L 1 , L 2 ⊗ K S ), and so these deformations are unconstrained. If on the other hand we consider a surface such that H 1 (O S ) = 0, then new Yukawa couplings will be developed from the superpotential (4.25) leading to an obstruction analogous to the one found above.
Indeed, the Yukawa couplings for this more general case has been worked out in [24, Appendix C], and result in (4.33) Similarly to the map Λ 3 , one can construct a map Λ + from the second term in (4.33), and the element a + ∈ Ext 1 (L 2 , L 1 ) describing the bundle extension: The Higgs-field deformations allowed by the F-term analysis are those that belong to the kernel of this map. This precisely matches the result for the contribution of the form φ = ψ in our previous analysis, upon identifying (4.34) with the map Λ + in the grid.
Finally, as the third term in (4.33) does not contain a + , no further constraint arises from it, and so as expected the Higgs-field deformation coming from m remains unconstrained.
One of the advantages of this F-term analysis is that it is easily generalised to other setups, like for instance bundles of higher rank on a Kähler surface S. Indeed, if one is able to understand the bundle E as a deformation of a split bundle, then one may identify the unobstructed Higgs field deformations in terms of the Yukawa couplings of the configuration, and therefore construct the most general Higgs field. The only drawback of this approach is that it is only reliable for small deformations of the pair (E, φ) around the split solution. Nevertheless, the F-term analysis could give valuable insight on how to generalise the mathematical construction for the above rank-2 bundle to other setups.
The ideal-sheaf twisting (4.18) also has a nice interpretation in terms of D-brane physics. This time, on top of the 7-branes wrapping S one adds |Z| D3-branes (with |Z| the number of points in Z), and switches on the vevs of the fields in the 37-sector to form a bound state. 8 The 37-sector is described by two chiral multiplets at each D3-brane location, with opposite charge under the U(1) of the D3-brane, and transforming as a doublet under the 7-brane rank-2 bundle. Let us label them as with K running over the |Z| points Q K where the D3-branes are located, which we assume near the disjoint set Z = {P K } K=1,...,|Z| . The ± above indicates charge ±1 under the relative U(1) of the 7-brane stack, assigning positive charge to the first 7-brane and negative charge to the second 7-brane. The superpotential including these modes reads where p(Q K ) stands for the value of the section p at a point Q K , same for v(Q K ) and m(Q K ). The second line does not involve any integral, because the 37-sector fields are δ-function localised at the corresponding D3-brane location. In the brane system this location is not fixed, and depends on neutral complex fields φ 33,K of each D3-brane.
Therefore, one should understand terms like p(Q K ) as a Taylor expansion where φ 33,K = Q K − P K measures the separation of the K th D3-brane from P K . Additionally the section p can be decomposed on a basis {χ λ − } ∈ H 0 (L −1 1 ⊗ L 2 ⊗ K X ) as p = p λ χ λ − , with p λ ∈ C representing the four-dimensional-field vev, so that ∂ i p(P K ) = p λ ∂χ λ − (P K ), etc. Similar expansions hold for v(Q K ) and m(Q K ), from where it follows that the second line of (4.36) not only contains cubic couplings on the fields v α , m σ , p λ , φ K 37± , φ K 73± , but also higher order couplings involving the fields φ 33,K .
Taking this into account, one can understand the constraints found for the twisted bundle (4.18) as switching on vevs to the new fields in the 37-sector, and then imposing the F-flatness constraints coming from (4.36). More precisely, to match our previous results we must switch on equal non-vanishing vevs for the fields φ K 73 + and φ K 37 + ∀K, represented by φ K + ∈ C, while keeping the vevs of φ K 73 − , φ K 37 − , φ 33,K to zero. Under this assumption, one finds that the F-flatness conditions for the fields p λ read This imposes h 0 (L −1 1 ⊗ L 2 ⊗ K X ) constraints on the values for φ K + , so generically we will not be able to have φ K . This can be understood as the physical counterpart of the constraint imposed onξ by the map Y on the sequence (4.19), by identifying {φ K + } K=1,...,|Z| = π(ξ). Regarding the constraints found for constructing the Higgs field, they can also be understood in terms of (4.36). On the one hand, the F-flatness conditions corresponding Because by assumption φ K + = 0, this implies that v(P K ) = 0, which reproduces that Higgs fields are constructed from sections of the canonical that vanish on Z, i * (v) ∈ H 0 (K S ⊗I Z ).
T hese sections must be such that the F-terms for a − arising from the first line of (4.36) also vanish, which leads to i * (v) ∈ ker Λ 3 and to the sufficient condition (4.22). Notice that the fields φ K 73 − , φ K 37 − also couple to the sections m, but since they do it quadratically and their vev vanish no constraint is imposed on them, as expected from our analysis.
Conversely, the sections p ∈ H 0 (L −1 1 ⊗ L 2 ⊗ K X ) get the most stringent constraint, as they couple quadratically to the fields φ K 73 + , φ K 37 + . The F-terms for the latter vanish if which selects sections that vanish on Z. While this is similar to the constraint (4.39), now an extra one appears due to the quartic couplings coming from the expansion (4.37).
One finds that the F-flatness condition for the fields φ 33,K reads ∂p(P K ) (φ K + ) 2 = 0 , ∀K , (4.41) or in other words the sections p vanish quadratically on Z. We therefore reproduce our previous result that p ∈ H 0 (L −1 1 ⊗ L 2 ⊗ K X ⊗ I 2 Z ). Finally, due to the first line in (4.36) we have that p ∈ ker Λ + , from where the sufficient condition (4.24) is obtained.

Explicit examples
If S is a Kähler surface, by Corollary 3.2 the existence of T-branes forces it to be properly elliptic or of general type. In this section we consider the case of a very simple properly elliptic surface that is a product of two curves. The construction presented below can nonetheless be easily adapted to more general properly elliptic surfaces.
Suppose that S = C × T where C is a curve of genus ≥ 2 and T is an elliptic curve.
Then, S is a properly elliptic surface with trivial elliptic fibration given by projection onto the first factor π := pr 1 : S → C. Note that K S π * K C . We assume for now that C is not hyperelliptic, implying that g ≥ 3.
For any P 0 ∈ C, we set T 0 := {P 0 } × T and for all n ∈ Z. Consider a rank-2 vector bundle E on S that fits into the exact sequence Referring to section 4, any element φ ∈ H 0 (End 0 (E) ⊗ K S ) can be written as for some α ∈ H 0 (Hom(E, L ⊗ K S )) and ψ ∈ H 0 (End 0 (E) ⊗ K S ) such that q * (ψ) ∈ H 0 (Hom(E, L −1 ⊗ I Z ⊗ K S ))/q * (H 0 (K S )) , (5.5) whereĩ = i ⊗ Id K S andq = q ⊗ Id K S . Moreover, α can be written as andv ∈ H 0 (Hom(E, L ⊗ K S ))/q * (H 0 (L 2 ⊗ K S )) . On the other hand, since h 0 (S, K S ) = h 0 (C, K C ) = g and the canonical linear system |K C | has no base points because g ≥ 2 (see [27, IV, Lemma 5.1]). Moreover, that there is no guarantee that there exist traceless Higgs fields withv = 0 in that case.
Remark 5.1. Note that the above construction also works when the curve C is hyperelliptic and s = 1. The computations are similar and we only state the main points.
First of all, because the curve is hyperelliptic, its genus can be 2. Therefore, g ≥ 2.
Moreover, it admits a double covering of P 1 , implying that there exists a meromorphic function on C of degree 2 (although it still does not admits a meromorphic function of degree 1 since it is not isomorphic to P 1 ). In this case, we thus have and h 1 (S, K S (−2T 0 )) = g + 1 > g = h 0 (S, K S ) > h 0 (S, K S ⊗ I Z ). (5.15) This means that the stable rank-2 vector bundle given by (5.2) with s = 1 admits a non-zero traceless Higgs field φ with m = 0 so that [φ, φ † ] = 0. There is, however, again no guarantee that there exists a traceless Higgs field such thatv = 0.
Summary of examples. Suppose that S = C × T with C a curve of genus g ≥ 2 and T an elliptic curve. Let us assume that det E O S so that c 1 (E) = 0. We have two cases: 1. C is hyperelliptic: • If g ≥ 2 and c 2 ≥ 1, consider the rank-2 vector bundle E given by with Z a finite set of c 2 distinct points in the support of K S . Then, E is stable and admits a non-zero traceless Higgs field φ with [φ, φ † ] = 0.
2. C is not hyperelliptic: In this case, g ≥ 3 (because every curve of genus 2 is hyperelliptic). Then, if c 2 ≥ 1, the rank-2 vector bundle E given by with Z a finite set of c 2 distinct points in the support of K S and s a positive integer.

Conclusions
In this paper we have proven a new vanishing theorem for regular solutions to the Vafa- One should stress that our vanishing theorem is different in nature to the one derived in [1]. There, in the case of a Kähler surface S, it was found that a necessary condition for solutions with φ = 0 is that S R tr(φ u φ † u ) ≤ 0, with R the scalar curvature and φ u the Higgs field in the unitary gauge. In cases where the presence of tr(φ u φ † u ) does not change the sign of the integral, like for instance when R does not change sign along S, this matches with the first part of Theorem 3.1 applied to Kähler surfaces, because S R dvol = − deg(K S ). In general, however, these are two different conditions, whose precise connection would be interesting to understand. For surfaces such that S R dvol > 0 our vanishing result implies that there is no Higgs field solving the Vafa-Witten equations, and so necessarily S R tr(φ u φ † u ) = 0. When K S is trivial our results imply that φ u is proportional to the identity, and so again S R tr(φ u φ † u ) = 0. Finally, when S R dvol < 0 our analysis does not provide any obvious statement on S R tr(φ u φ † u ), although it leads to the inequalities (3.4). Therefore, it would be particularly interesting to develop the interplay between both vanishing results in this case. there are regions where the Higgs field fails to be holomorphic. Indeed, poles may be induced by considering defects, which in turn correspond to additional D-branes present in the compactification and intersecting the T-brane. This situation has been explored in [24] in the special case of vector bundles topologically given by sum of line bundles. It would be important to extend these setups to general meromorphic Higgs pairs, using the techniques developed in the present paper.
In Section 4 we have given a complete characterisation of rank-2 stable Higgs pairs on a projective variety, dividing the discussion in two cases: The extension by a line bundle of a) another line bundle and b) of a non-locally-free sheaf. The first case constitutes a class of topologically split vector bundles but holomorphically non-split, whereas the second class contains topologically non-split bundles, due to the presence of the ideal sheaf. In both cases, we describe the Higgs field using a grid of long exact sequences in cohomology, deriving sufficient topological conditions for its existence. We finally provided a physical interpretation of these cohomological results, in terms of superpotentials and holomorphic Yukawa couplings, which characterise the low-energy effective physics of D7branes wrapped on surfaces.
Particularly intriguing from the physics perspective is the ideal sheaf twist, which we interpreted as the result of having coupled the D7-brane system to a system of D3branes located at points on the Kähler surface. These sorts of D3/D7 bound states have been argued in [25] to be the origin of the so-called point-like matter, [11], one of the most exotic features of T-branes. While our analysis appears to be compatible with the findings of [25], it would be interesting to investigate the connection to point-like matter further, especially by including the information of D-terms. Moreover, the physical meaning of the extension class (4.20) remains rather obscure, in particular for what concerns the role ofξ, carrying the information of the localised gluing modes, and how the latter appears in the gauge connection. We hope to come back to these issues in the near future.

A Some homological algebra
In this appendix, we explain in more detail some of the constructions appearing in Section 4 and prove the various intermediate steps therein.
First, we would like to prove the following fact, which is needed to compute the trace of the Higgs field, like in Eq. (4.9): Proof. The proof consists of a local computation. First note that all the sheaves involved, namely, L 1 , E and K X , are locally free. Every point in X thus admits an open neighbour- The first terms of the extension (4.1), when restricted to U , then become 9 Now, for any α ∈ H 0 (Hom(E, L 1 ⊗ K X )), we have so that α| U (f, g) = α 1 f +α 2 g for some α 1 , α 2 ∈ O X (U ), for all f, g ∈ O X (U ). In particular, At the same time, In other words, and tr(ĩ * (α)| U ) = h 1 α 1 + h 2 α 2 = α • i| U , proving the result. In what follows, we fill in some gaps left in Section 4, when discussing bundle extensions twisted by an ideal sheaf. Let E be a rank-2 vector bundle on a Kähler surface S. Suppose that E is given by the extension where L 1 and L 2 are line bundles on S, and Z is a finite set of distinct points in S. In other words, E is a locally-free extension of L 2 ⊗ I Z by L 1 .
A first question to address is when such locally-free extensions exist. Suppose that Z = {P 1 , . . . , P n } consists of distinct (reduced) points. A locally-free extension of L 2 ⊗ I Z by L 1 then exists if and only if every section of L −1 1 ⊗ L 2 ⊗ K S that vanishes at all but one of the P i 's also vanishes at the remaining point (see [26], Theorem 12). Note that this condition is vacuously satisfied when Z is a single point. Moreover, if one takes sufficiently many points {P 1 , . . . , P n } in general position on S, then there will be no section of L −1 1 ⊗ L 2 ⊗ K S that vanishes at all but one of the P i 's. One can therefore construct many examples of locally free extensions of L 2 ⊗ I Z by L 1 .
Let us now explain how we obtained the exact sequences (4.21) and (4.23). These in fact correspond to the long exact cohomology sequences associated to the short exact sequences of sheaves given in the following: Proposition A.1. Let E be a locally-free sheaf given by an extension of the form (A.7).
We then have the following short exact sequences of sheaves: and Indeed, by taking the long exact sequence in cohomology of (A.8), we obtain (4.21), where Λ 3 denotes the connecting homomorphism. By doing the same for (A.9), we obtain Let us now derive (A.8). Applying Hom( − , L 1 ) to (A.7), we obtain the exact sequence of sheaves In light of the above discussion, we see that giving rise to the short exact sequence To understand this sequence better, recall that for any two sheaves E 1 , E 2 , we have Moreover, from the above local discussion, and where we used that the dual of a line bundle coincides with its inverse. The short exact sequence (A.17) can therefore be written as We now turn to (A.9). Applying Hom( − , L 2 ⊗I Z ) to (A.7), we get the exact sequence giving us the short exact sequence which tensored by K S yields (A.9).

B Topologically non-split bundles
In this appendix, we will describe the topological features of a rank-2 vector bundle on a Kähler surface S, defined by the twisted extension (4.18). As we will see, twisting the sequence by an ideal sheaf gives us a simple way to construct a class of rank-2 vector bundles which may be topologically disconnected from a split form of direct sum of two line bundles.
Let Z be a set of distinct points in S, 10 with j : Z → S the corresponding holomorphic embedding. The Grothendieck-Riemann-Roch formula says: where "ch" and "td" denote the total Chern character and the total Todd class respectively, O Z is the structure sheaf of Z, j * is the push-forward map (naturally acting on sheaves), and finally j # is the push-forward in cohomology, meaning that it consists of taking first a Poincaré duality, then the natural push-forward of homology classes, and then another Poincaré duality 11 . The l.h.s. of the above formula expands to whereas the r.h.s. simply reads PD S (Z) , 10 In the context of Type IIB Orientifold (or F-theory) compactifications to four dimensions, Z corresponds to a set of D3-branes extended over the external space and completely localised in the internal part of the 7-brane worldvolume. Their orientation is fixed by supersymmetry. 11 Effectively this set of operations sends cohomology classes of Z to cohomology classes of S by simply wedging them with the class dual to Z in S.  It is now easy to design an explicit example where a vector bundle E given by (4.18) is topologically obstructed to splitting into a sum of two line bundles. Consider, for instance, a surface S cut out by a generic polynomial of degree 7 in P 3 . 12 This surface is simply connected and has 20 independent holomorphic deformations. Moreover, even though S admits 147 independent harmonic (1, 1)-forms, the genericity of the degree-7 polynomial implies that all of them but one be non-integral. 13 This means that we can only play with the hyperplane class to construct line bundles on S. Take for simplicity L −1 2 L 1 ≡ L, and choose c 1 (L) = −H, where H is the hyperplane class of P 3 . This guarantees that E is stable. Using that c 1 (S) = −3H, we find that the number of points in Z is 21, and therefore, by Eq. (B.4), we have c 2 (E) = 14. However, any traceless sum of two line bundles on this surface has a negative second Chern number, thus proving that the vector bundle E so constructed does not admit a split form in its moduli space of complex structures. 12 We may physically motivate this in the context of F-theory, where P 3 plays the role of base of the elliptic fibration. 13 This is a trivial consequence of the Nöther-Lefschetz theorem, which in this case says that the restriction Pic(P 3 ) → Pic(S) is an isomorphism.

C Cohomological computations
In this appendix, we provide a few cohomological computations that are needed in Section 5, in which we construct examples of Higgs bundles over properly elliptic surfaces.
Suppose that S = C × T where C is a curve of genus ≥ 2 and T is an elliptic curve.
Then, S is a properly elliptic surface with trivial elliptic fibration given by projection onto the first factor π := pr 1 : S → C. Note that K S π * K C . Suppose that C is not hyperelliptic. This means, in particular, that g ≥ 3. In this case, we have: where M(C) is the set of meromorphic functions on C and (f ) is the divisor of the meromorphic function f . In particular, if f ∈ H 0 (C, O C (sP 0 )) is not constant, then it must have a single pole of order ≤ s at P 0 . If s = 1, this single pole must have order 1, implying that f : C → P 1 is a degree 1 map. In other words, if h 0 (C, O(P 0 )) ≥ 2, then C is isomorphic to P 1 , which is impossible since we are assuming that g ≥ 3. Finally, when s = 2, if h 0 (C, O C (2P 0 )) ≥ 2, this would imply that there is a non-constant f ∈ M(C) with (f ) ≥ −2P 0 so that f must have a pole of order ≤ 2 at P 0 . As in the previous case, this pole cannot have order 1 because C has genus ≥ 3. Therefore, f must have a pole of order 2 at P 0 , implying that f : C → P 1 has degree two. But this would mean that C is hyperelliptic, which again contradicts our assumption and proves 1..
The statement of 2. follows from the exact sequence with n ∈ Z. Indeed, if we take n = 2, we then have Taking the long cohomology sequence, we obtain We then see that 1 ≤ h 0 (C, O C (3P 0 )) ≤ 2 and g − 3 ≤ h 1 (C, O C (3P 0 )) ≤ g − 2, implying that the statement of 2. holds for s = 3. The result for s ≥ 4 follows by induction.
Using this lemma, we can now compute the cohomology groups on S.