Compact T-branes

We analyse global aspects of 7-brane backgrounds with a non-commuting profile for their worldvolume scalars, also known as T-branes. In particular, we consider configurations with no poles and globally well-defined over a compact K\"ahler surface. We find that such T-branes cannot be constructed on surfaces of positive or vanishing Ricci curvature. For the existing T-branes, we discuss their stability as we move in K\"ahler moduli space at large volume and provide examples of T-branes splitting into non-mutually-supersymmetric constituents as they cross a stability wall.


Introduction
One important feature of type II string compactifications is the amount of information on the effective, lower-dimensional theory that one obtains by analysing BPS D-branes.
In fact, knowledge on the spectrum of BPS D-branes is a requirement to build interesting type II string vacua, since they typically host the non-trivial gauge sector of the compactification [1]. The more precise this knowledge is, the better the picture on the set of vacua on a certain region of the string landscape.
While D-brane BPS conditions have been thoroughly analysed for different classes of vacua, solving them explicitly can oftentimes be challenging. In that sense, a particularly tractable set of vacua is given by type IIB Calabi-Yau orientifolds with O3/O7-planes.
Indeed, in this case the set of space-time-filling BPS D-branes at large volume is given by D3-branes and D7-branes. On the one hand, the embedding of a single D3-brane is As usual, obstructions may be found when trying to extend a local solution globally.
In our case we find that constructing compact T-brane solutions crucially depends on the Ricci curvature of the surface S, and more precisely on its cohomology class. Indeed, we find obstructions to the existence of compact T-branes over complex four-cycles of vanishing or positive-definite curvature, like K3 or del Pezzo surfaces. On surfaces of negative-definite curvature, instead, solutions can always be constructed, generalising the result of Hitchin for Riemann surfaces of genus g > 1 [13]. Finally, for surfaces of indefinite curvature the construction will depend on the particular region of the Kähler moduli space where we sit. 2 This latter case raises the question of the fate of T-branes when we move in Kähler moduli space and, in particular, when we pass from one region to another by crossing stability walls. In this respect, we find that a T-brane is either converted into a different BPS object as it crosses the wall, or it splits into non-mutually-BPS constituents.
As could be expected, the T-brane's fate will ultimately depend on its topological data, and we analyse several interesting cases in terms of them.
The paper is organised as follows. In section 2 we specify the class of T-branes that we will be studying, with special emphasis on their global description in terms of a compact four-cycle. We then turn to discuss solutions to the BPS equations, first the analogous of the original Hitchin solution and then generalisations thereof. In section 3 we prove a topological obstruction to build compact T-brane solutions: they cannot be hosted by four-cycles of vanishing or positive-definite Ricci curvature class. Finally, in section 4 we analyse the stability of the allowed T-brane constructions as we move in large volume Kähler moduli space, and in particular their fate after crossing a stability wall. We draw our conclusions in section 5.
Some technical details are relegated to the appendices. In appendix A we give a fourdimensional interpretation of the non-harmonicity of the worldvolume flux in T-brane solutions. In appendix B we construct several explicit examples of the stability-wall transitions discussed in section 4.

Global aspects of T-branes
Consider a stack of 7-branes wrapping a compact Kähler surface S. Following [14][15][16][17], the 7-brane configuration and degrees of freedom can be characterised in terms of a eightdimensional action on R 1,3 ×S with a non-Abelian symmetry group G. In particular, such data are encoded in terms of two two-forms on S: the field strength F = dA − iA ∧ A of the 7-branes gauge boson A, and the (2,0)-form Higgs field Φ, whose eigenvalues describe the 7-brane transverse geometrical deformations. Both A and Φ transform in the adjoint of the initial gauge group G, which is nevertheless broken to a subgroup due to their nontrivial profile. Finally, such profiles need to satisfy certain equations of motion, which in the case of supersymmetric configurations are given bȳ where J is the Kähler two-form of S. These equations are a generalisation of the celebrated Hitchin system [13] to a four-manifold. Upon dimensional reduction to four dimensions, the first two equations ensure the vanishing of the F-terms, while the third equation ensures the vanishing of the D-terms.
In this paper we will analyse 7-brane backgrounds with non-commuting expectation values for the worldvolume scalar Φ, namely such that [Φ, Φ † ] = 0, also known as Tbranes in the string theory literature. We will restrict to those T-brane configurations that are globally well-defined over a compact Kähler surface S and such that the Higgs field profile is absent of poles. 3 We dub such T-brane configurations as compact T-branes, in the sense that the spectral equation for Φ describes a compact surface. Notice that poles are naturally associated to field-theory defects originating from additional 7-branes intersecting the stack, so we may interpret a compact T-brane as stack of 7-branes in isolation from the others. In other words, we may see them as basic building blocks of BPS 7-brane configurations in type IIB/F-theory compactifications. We will moreover focus on solutions of equations (2.1) involving an Abelian profile for the gauge field. In other words, in our backgrounds the source of non-commutativity of the 7-brane system will come entirely from Φ.
In order to describe the essential features of compact T-branes, in this section we will focus on the simplest possible example, namely a stack of two D7-branes. This case allows to generalise the original example of Hitchin on a Riemann surface [13] to a compact complex four-cycle. From there one may generalise the T-brane Ansatz in a number of ways, finding backgrounds with a non-harmonic worldvolume flux. As we will see, the departure from harmonicity is governed by certain non-linear differential equations, and this will allow to connect our constructions with the literature of T-brane solutions in flat space.

T-branes and non-harmonic fluxes
Let us focus on a stack of two 7-branes wrapping S, and therefore on a super-Yang-Mills theory on R 1,3 × S with symmetry group G = SU (2). We will always assume that S is simply-connected, i.e. π 1 (S) = 0. This will simplify our analysis considerably because it implies, in particular, that holomorphic line bundles on S have their topology completely specified by the first Chern class. As mentioned, we will also restrict attention to a rank-two gauge bundle V on S of split type, i.e.
where L is a line bundle whose curvature we denote by F . The F-term (2.1b) of the eight-dimensional super-Yang-Mills theory forces F to be a differential form of Hodgetype (1, 1), which gives L a holomorphic structure. Moreover, since F is closed, using the Hodge decomposition, we can uniquely write it as where the superscript h denotes the harmonic representative and α is a globally welldefined one-form. Note that the absence of non-trivial first-cohomology classes on S, following from its simply-connectedness, forbids harmonic representatives for α. We can thus always choose (globally) a gauge that kills the exact part of α, such that we can where g(x,x) is a globally well-defined real function on S (with local complex coordinates collectively denoted by x) such that S g dvol S = 0, and d c = i(∂ − ∂). Using that S is Kähler, it is easy to see that the co-differential operator δ = − * d * annihilates the expression (2.4), and hence α is co-closed. In this way, the gauge field strength becomes The function g, or equivalently α, will play a key rôle in the sequel. It will be the unknown of the non-linear partial differential equation governing T-brane backgrounds, which arises from the equation (2.1c) of the eight-dimensional super-Yang-Mills theory.
In an ordinary intersecting-brane background, where Φ is diagonalisable, this equation forces F to be primitive. By a standard result in Kähler geometry (see e.g. [36]), every primitive (1,1)-form on a Kähler two-fold is anti-self-dual with respect to the Hodgestar operator. Since F is closed, this implies then that F is also co-closed, and hence harmonic. Now, reversing the argument, a T-brane supersymmetric configuration will involve a gauge field strength which is closed but not anti-self-dual, and therefore F will not necessarily be given by the harmonic representative of a certain cohomology class.
This departure from harmonicity is described by g.
As we will see, the information that g encodes is lost in the four-dimensional effective theory. It can only be recovered when we include the D7-brane Kaluza-Klein modes into the four-dimensional description, as we discuss in appendix A. In other words, g determines the microscopic details of the T-brane background, to which only the eightdimensional theory is sensitive to.
In order to determine g let us for convenience define the global real function where, compatibly with our choice of gauge bundle V, we restrict our attention to commutators proportional to the third Pauli matrix σ 3 . Then one can see that ϕ ≥ 0 all over S and that equation (2.1c) reads Using the Lefschetz decomposition of harmonic forms, we can write where c is a constant, F h p is primitive and the numerical factor is for later convenience. Of course this splitting depends on the Kähler moduli of our string compactification, and the periods of the two summands are generally real (moduli-dependent) numbers which must add up to (half-)integer numbers to satisfy the quantization condition for F . 4 Using that S is Kähler, one can show that 2i∂∂g ∧ J = * ∆g, where ∆ is the Laplace operator in real coordinates. This leads us to an elegant rewriting of equation (2.7): (2.9) At this point, one fixes an hermitian metric on S, and solves equation (2.9) for g, or equivalently for the unitary connection A on L. Notice that a necessary requirement to solve this equation is that its r.h.s. integrates to zero, i.e.
which is nothing but the condition for vanishing D-term potential in the four-dimensional low-energy effective theory.
Practically, equation (2.9) can only be solved analytically in few situations, because in general ϕ will depend non-linearly on g. Nevertheless this equation is always of elliptic type [13] and, as such, on a compact manifold it admits a unique smooth solution if the input function ϕ is smooth and provided that (2.10) is satisfied [37].
The most convenient and adopted [11,12] approach to formulate the problem is to fix the holomorphic structure of L such that A 0,1 = 0, which turns the anti-holomorphic where m ∈ H 2,0 (S, L 2 ). Using Serre duality, we can also see m as a scalar holomorphic section of the line bundle M ≡ L 2 ⊗ K S , with K S the canonical bundle of S. By a slight abuse of notation, in the following we will describe both kind of objects with the same symbol, being clear from the context to which one we are referring. As it stands, this profile is a solution of equation (2.1a) in the holomorphic gauge. However, equation (2.1c) contains the adjoint Φ † , which depends on the metric as where the superscript + indicates complex conjugation and matrix transposition, and . This brings a non-linearity in the partial differential equation (2.9), which can now be written as where h S , the determinant of the fixed hermitian metric on S, appears because of applying the Hodge-star operator on a four-form. This is a rather non-trivial equation that reduces to a Liouville-like equation when m is constant and h S is the flat metric [11]. Nevertheless, there is a particularly nice setup in which (2.13) simplifies even further, as we discuss explicitly in the next subsection.
As a side remark, note that for the split-type configurations (2.2) we consider in this paper the stability-based algebro-geometric criterion [38] for existence and unicity of solutions of the non-Abelian BPS equations (2.1) is trivially satisfied. For instance, it is immediate to see that the only sub-bundle of V preserved by the Higgs field (2.11) (i.e. L) has negative J-slope, as enforced by the D-term equation (2.10).

The Hitchin Ansatz
The most emblematic class of Higgs-bundle configurations is probably the one originally studied by Hitchin in the case of Riemann surfaces [13]. One can straightforwardly extend this Ansatz to the present context of complex surfaces, as first suggested in [37]. This would correspond to taking the nilpotent Higgs field (2.11) such that the line bundle M is the trivial one, which amounts to demand that 5 (2.14) Since S is compact, this choice implies that the quantity m in (2.11) can only be a constant. Notice also that equation (2.14) only fixes the cohomology class of the gauge curvature in terms of that of S, but not its actual representative. Therefore, let us write the Ricci form of S as where s is another globally well-defined smooth real function on S such that S s dvol S = 0, and the factor of 2 is for later convenience. Then, eq.(2.14) states that F h = ρ h /2, or equivalently using (2.5) that 6 Loosely speaking, e g−s is the conformal factor needed to rescale the hermitian metric on the surface S to get the hermitian metric on the line bundle L. More precisely we have (2.17) Using the above relation, our partial differential equation (2.13) becomes ∆g = c + |m| 2 e 2(g−s) (2.18) where, as said, in this Hitchin set of solutions m is a complex number. Let us now analyse two possible sub-cases of this setup.

Kähler-Einstein metric
The easiest possible situation is analogous to the one originally considered by Hitchin in the case of Riemann surfaces [13]. This arises when g = s. Taking into account the D-term condition (2.10), which now simply says that c = −|m| 2 , equation (2.18) reads

19)
5 At weak coupling this is made compatible with cancellation of the Freed-Witten anomalies of the individual branes by considering a suitably-quantised primitive flux associated to the center-of-mass U (1). 6 Recall that in cohomology 1 whose unique solution on S is g(x,x) = 0. This, in turn, means that also s = 0, and thus that both the gauge flux F and the Ricci form ρ are harmonic. If in particular h 1,1 (S) = 1, then F h p = 0 in equation (2.8) and therefore we have Thus the metric on our surface S is Kähler-Einstein with Einstein constant −|m| 2 /2, that is it has constant negative Ricci curvature.
We can reverse the above argument and get a more useful statement. If we fix the metric on S to be Kähler-Einstein, then ρ = kJ with k a real constant, which in particular where we substituted the value of c fixed by the D-term (2.10). The above equation automatically implies that g(x,x) = 0, because it admits a unique smooth solution.
Therefore we conclude that, if we fix a (negatively curved) Kähler-Einstein metric on S, the vacuum solution for a constant nilpotent Higgs field involves a non-primitive, but still harmonic gauge flux.

Beyond Kähler-Einstein
If instead we consider a non-Kähler-Einstein metric on S, the vacuum profile of the gauge flux will necessarily depart from the harmonic representative, and will be uniquely fixed by the equation As before, there will be unique smooth solution for g. Note that this extension beyond Kähler-Einstein is also possible in the case of Riemann surfaces, thus directly generalising the type of solution discussed in [13].

Generalising the Ansatz
There are a few ways of generalising the above simple set of solutions, namely by considering Higgs field profiles that are non-nilpotent and by considering line bundles L that do not meet the topological condition (2.14). In the following we will consider and combine both generalisations, comparing the resulting equations for the function g with the local T-brane solutions in the literature.

Non-nilpotent Higgs field
Let us first consider the case of four-cycles where the condition (2.14) is met, but now we have a non-nilpotent profile for the Higgs field. Namely we consider it to be of the form where p ∈ H 2,0 (S, L −2 ), or equivalently a scalar holomorphic section of the line bundle P ≡ L −2 ⊗ K S . Notice that due to (2.14) we have that P K 2 S . Such a bundle will have sections in many four-cycles of negative curvature, like for instance in those where K S also does. In this case eq.(2.13) generalises to and so, using eq.(2.17) we arrive to As before, |m| 2 is a constant, while h −4 0 |p| 2 is a globally well-defined smooth function in S. Finally, enforcing the 4d D-term condition implies that c is given by so that eq.(2.25) has a (unique) solution.
Notice that now g will not vanish in the Kähler-Einstein case s = 0. Instead, eq.(2.25) will become a complicated non-linear equation for g. Near the locus where p = 0 we can Taylor expand the function h −4 0 |p| 2 , and recover an equation very similar to that obtained in the local T-brane Z 2 background of [11]. As pointed out in there, such an equation can be rewritten as a Painlevé III differential equation. Hence one would expect that, at least in a local patch near p = 0, the profile for g can be expressed in terms of solutions to that equation. Finally, one may depart from a Kähler-Einstein metric by considering s = 0. This will modify the (unique) solution for g, which will depend on the profiles of the functions |m|e −s and h −2 0 |p|e −s .

Non-trivial bundle M
Let us now consider relaxing the topological condition (2.14), or in other words assume that M ≡ L 2 ⊗ K S is a non-trivial bundle with sections. Given its definition we can express the hermitian metric in M as where h M,0 corresponds to the metric with curvature 2F h − ρ h and s is again defined by (2.15). We can then express (2.13) as with m M a globally-well defined, smooth function in S that vanishes over the same locus as m. This corresponds to an obvious generalisation of eq.(2.18), where now the input function that determines g is given by e −s m M . Since m M is non-constant, g will be non-trivial even in the Kähler-Einstein case s = 0, and so the gauge flux F will depart from harmonicity.
Finally, one may combine a non-trivial bundle M with a non-nilpotent Higgs field (2.23), again assuming that P ≡ L −2 ⊗ K S has sections. In that case, we may express the metric for this bundle as with h P,0 the metric of curvature −2F h − ρ h . We then consider the globally-well defined, vanishing smooth function in S given by p 2 P ≡ h P,0 |p| 2 . Together with the above definition for m 2 M we obtain an equation for g of the form While arising from a more general setup, this new differential equation is in fact very similar to (2.25), with the new functions that determine g now given by e −s m M and e −s p P .

A no-go theorem
The simple examples discussed in the previous section suggest that it is relatively easy to construct global T-brane configurations on four-manifolds with negative curvature. While it may seem that this preference comes from imposing the Hitchin Ansatz or generalisations, there is in fact a deeper reason behind. Indeed, in the following we will see that compact T-brane configurations with Abelian gauge bundles cannot be implemented in four-manifolds of vanishing or positive curvature. We will first show this no-go result for the configuration with symmetry group G = SU (2) and split gauge bundle of the type (2.2), and then generalise it to groups of higher rank.
The case of SU (2) In order to investigate the possible obstructions to the construction of compact T-branes, let us first consider the stack of two D7-branes wrapping a simply-connected Kähler surface S, and with split gauge bundle V = L ⊕ L −1 . As before, we may start considering the T-brane background given by the nilpotent Higgs vev where m ∈ H 0 (S, M). Now, the very fact that an holomorphic section m exists implies that the divisor associated to M ≡ L 2 ⊗ K S is effective. That is, for J in the Kähler cone with the equality holding if and only if M is trivial. 7 Moreover, the 4d D-term condition (2.10), or equivalently for a Higgs field of the form (3.1) implies that where we just used that F/2π represents c 1 (L) in cohomology. Subtracting the l.h.s. of (3.4) to the middle expression in (3.2), we get the statement that we can construct such a T-brane in a region of Kähler moduli space where (3.5) 7 We will always be at large volume, so in particular well away from boundaries of the Kähler cone.
This conditions forbids S to be K3 or a manifold with positive-definite Ricci curvature.
Indeed, if it were positive definite, the canonical class, which is represented by minus the Ricci form, would necessarily have a negative volume everywhere in Kähler moduli space. Kähler surfaces with negative-definite Ricci curvature certainly satisfy the necessary requirement (3.5), but surfaces with indefinite curvature may also do so. The second inequality we get from (3.2) and (3.4) is As an example, take the case where S has only one Kähler modulus, i.e. h 1,1 (S) = 1.
Together with the fact that S is simply-connected, this implies that every gauge "line bundle" L on S is of the form L K −n/2 , for some non-zero integer n. Then, the two conditions (3.2) and (3.4) boil down to n ≤ 1 and n > 0 respectively, which are both solved only by the choice n = 1. This is nothing but the generalisation of Hitchin's class of solutions to a four-manifold, as already analysed in [37] .
Let us now consider the most general Higgs vev compatible with a split rank-two gauge bundle, namely where now m ∈ H 0 (S, M) and p ∈ H 0 (S, P), with P ≡ L −2 ⊗ K S . Suppose now, without loss of generality, that the Fayet-Iliopoulos term in (3.3) is positive, namely condition (3.4) is satisfied. Then we obtain the following inequalities among the areas of the various Incidentally, notice that the product mp transforms as a section of H 0 (S, K 2 S ), and it appears in the spectral equation for the Higgs field. Therefore for the background (3.7) one could have guessed the obstruction to realise it in del Pezzo surfaces from a more standard, spectral-surface-based reasoning, see e.g. [39]. Nevertheless, our analysis provides more detailed information about the obstruction, like for instance the inequalities

Higher rank groups
Let us now consider a general simple Lie group G, whose algebra has the Cartan subalgebra H i and the set of roots E ρ such that We choose a basis of Cartan generators that diagonalises the Cartan-Killing metric and moreover we choose their normalisation such that the ρ i are all integer numbers. Now we take the following Ansatz for our T-brane background with m ρ ∈ H 2,0 (⊗ i (L i ) ρ i ) and ρ a set of roots such that where ρ i = ρ i g ii . Hence On the other hand, the D-term condition implies that which in turn implies the following Putting these two results together we finally obtain the following inequalities among curve and in particular we recover equation (3.5).

T-branes and stability walls
Starting from a T-brane configuration, we now want to study its stability when we move

Coincident branes
Let us consider two D7-branes wrapping a simply-connected Kähler surface S, holomorphically embedded in a Calabi-Yau threefold. As in section 2 we consider a split rank-two gauge bundle of the form (2.2), specified by a line bundle L of curvature F . We moreover consider a Kähler structure compatible with a T-brane of the nilpotent type (3.1).
Because of the D-term (3.3), the size of the vev m is controlled by the FI term F ∧ J, and thus it is proportional to the distance from the wall, which is defined by the condition F ∧ J = 0. There we get a vanishing vacuum expectation value for Φ and therefore a standard system of two coincident D7-branes with a worldvolume flux along the Cartan.
We By standard results [40] (see also [41]), the full spectrum of charged massless fields is and so the vevs for such fields a ± were assumed to vanish in the T-brane configurations of section 2. We must however take them into account in the following, to study how the D-brane configuration may react as we cross a stability wall.
On top of the charged modes there are also uncharged zero modes, which however only appear as fluctuations of Φ and not of the gauge field, because we are taking S to be simply-connected. Such fields originate from open strings with endpoints on the same D7-brane and thus corresponding to its normal deformations inside the ambient Calabi-Yau manifold. Here we only focus on relative deformations of the two branes wrapping S, and ignore the movements of their centre of mass. Therefore, these deformations appear in the Higgs-field fluctuation as Note that these vevs were also set to vanish in the T-brane configurations of section 2.
Finally, the absence of modes with negative norm (ghosts) for the strings connecting the two branes [14] leads to the following important requirements These conditions are automatically satisfied if the FI term vanishes and we are inside the Kähler cone.
Given the above spectrum one may analyse how the system behaves at both sides of the wall. For simplicity, we will first consider the case where the modes (4.3) are absent.
Then, in a sufficiently small region in Kähler moduli space around the wall, and upon dimensional reduction to 4d the D-term condition (2.1c) becomes 9 that the T-brane we started with disappears as we cross the wall, by decaying into its D7-brane constituents, which are not mutually supersymmetric. 10 Interestingly, by using the index theorem we are able to formulate a practical necessary criterion for such a decay to occur. In particular, applying the index theorem to the line bundle P, we get where the symbol h i indicates the dimension of the corresponding group H i , "ch" is the total Chern character and "Td" is the Todd class. 11 In (4.7) we have used that where the symbol #(±) denotes the number of zero modes with U (1)-charge ±. Finally, from equation (4.9) we obtain the following implication  which for negative curvature cannot be taken to zero while moving inside the Kähler cone.
Let us then consider the case where the Ricci curvature of S is indefinite. This in particular implies absence of holomorphic sections for the canonical bundle (thus S is rigid) and for any power thereof (positive and negative). Therefore no p-type modes are available and, since by assumption S is simply-connected, no a − -type modes are available either. Hence, in this class of configurations, our T-brane is forced to decay into a nonsupersymmetric vacuum when the wall is crossed.
A simple instance of a Kähler surface with the above properties can be obtained as follows. Consider P 4 with homogeneous coordinates x 1 , . . . , x 5 , blown up along a four- 12 One particular case is when I = 0, which in the literature corresponds to a wall of threshold stability.
Indeed, by looking at the definition (4.9) one realises that −I corresponds to the intersection product used in [42] to classify stability walls. where E : {w = 0} corresponds to the exceptional divisor, homeomorphic to P 2 × P 1 .
In this ambient manifold, we consider the Calabi-Yau threefold CY 3 given by the zerolocus of a smooth polynomial of bi-degree (1,4), and the D7-brane stack wrapped on It is easy to show that this surface is rigid (as a consequence of the rigidity of the exceptional divisor), and moreover has indefinite Ricci curvature, because e.g.
By using the Hirzebruch-Riemann-Roch theorem, we can also easily show that this surface has no cohomologically non-trivial one-forms

Negative curvature
Let us now consider the case where the Ricci curvature of the surface S is negative definite. Note that this does not necessarily imply that S can be holomorphically deformed, a subcase to be considered momentarily. By the observation made above, in the negative curvature case we must consider a T-brane whose m-type mode transforms under a non-trivial bundle M. The fact that M is effective and non-trivial, together with the ampleness of K S due to the negative curvature, implies that  In this case, by dimensionally reducing the D7-brane superpotential one obtains Yukawa couplings of the form which generically give an F-term mass to the negative-chirality modes a − . Now, if we impose (4.16) and cross the wall at (4.18), for h 1 (S, M) > 0 there will be an F-term potential that will make (4.18) vanish and take the system to the supersymmetric configuration of coincident D7-branes with a non-Abelian bundle created by the vev of a − .
Notice that at (4.18) we have a system of two homotopic D7-branes intersecting at a curve C, with opposite worldvolume fluxes. This is nothing but a particular case of a more general configuration, made of two intersecting D7-branes with arbitrary worldvolume fluxes. As we will now see, one can formulate the T-brane wall-crossing conditions for this more interesting case as well.

Intersecting branes
where the two sums extend over zero modes with U (1) × U (1)-charges (+, −) and (−, +) respectively. They correspond to open strings stretching from brane 2 to brane 1 and to strings going the opposite way respectively. Assuming that the intersection curve C ≡ S 1 ∩ S 2 is connected, such zero modes are counted by the following sheaf extension groups [40] (see also [41]): with K C its canonical bundle, and i 1 , i 2 the embedding maps of branes 1, 2 respectively.
In this case the wall is defined by the Kähler structure slice where F 1 ∧ J = F 2 ∧ J.
There we have a system of two intersecting D7-branes, and thus the spectrum of massless fluctuations is given by equation (4.22). Notice that, unlike in the coincident case, now the spectrum of zero modes is only counted by modes of the Higgs field. We now assume that there is at least one of these two modes, say a m-type mode with charge (+, −), so that, at one side of the wall (ξ > 0), there is a supersymmetric bound state with a T-brane profile localised at C. As we cross the wall to the other side, either this T-brane turns into a different kind of T-brane or, if no p-type mode is available, the T-brane decays into the two mutually non-supersymmetric constituents. 13 Since in this case the spectrum of charged zero modes is simpler, we are able to formulate a sufficient criterion for our T-brane to decay across the wall. First, notice that the chiral index of the theory is given by Let us for now assume that the surfaces S 1 , S 2 do not have holomorphic deformations or, if they do, that none of them will split the intersection curve into in multiple connected components. Then, calling g the genus of C and using the Riemann-Roch theorem, the existence of the m-type mode we began with implies that Finally, by the same reasoning, if the condition is satisfied, there are no p-type modes to form a T-brane on the side of the wall where the FI term is negative. Therefore, we readily see that, if the two D7-branes intersect on a sphere, the fate of our T-brane is to decay when we cross the wall. The same statement holds true when C is a two-torus and C F 1 = C F 2 . We therefore obtain a simple picture for the decay possibilities of intersecting D7-branes, summarised in figure 1. If on the other hand the surfaces S 1 , S 2 contain holomorphic deformations such that C splits into multiple components, the wall crossing picture just described may change.
Indeed, when the matter curve C = ∪ a C a is disconnected one needs to apply (4.22) separately to each individual component C a to obtain the massless spectrum. While then the relations (4.24) and (4.25) continue to hold, 14 the sufficient condition for decay (4.26) gets replaced by a significantly weaker one. This is because it is enough to find at least a p-mode localised on any of the connected components of C, in order for the two branes to bind back again into a supersymmetric system across the wall. In other words, decay will only occur when all the available holomorphic deformations of S 1 and S 2 split C in such a way that on every component C a one has I a > g a − 1.

Conclusions
In For instance, in the simplest case the following condition needs to be satisfied: Another direction would be to examine how α corrections modify the T-brane constructions considered in this paper. At moderate volumes of the compactification one may in principle apply the same strategy as in [30] to see how such corrections affect the differential equations of section 2, that govern the 7-brane background. However, as these corrections do not affect the holomorphic T-brane data and are sufficiently mild not to flip the FI-term sign, the no-go theorem of section 3 should still hold.
Finally, as the necessary conditions for the existence of compact T-branes depend on the point in the Kähler moduli space of the compactification, it would be interesting to see if our results could have any implications for Kähler moduli stabilisation.
In summary, as argued in the introduction, our findings can be seen as one further step in the classification of the full set of BPS branes in type IIB/F-theory compactifications.
As such, they should have direct consequences for the model building applications that triggered the recent study of T-branes in this context, and it would be interesting to fully explore such implications. In any event, we expect that having a good understanding of global T-brane configurations will give rise to new insights in the comprehension of string theory vacua.

Acknowledgments
We it is natural to interpret α as a set of KK modes that got a vacuum expectation value when the 4d Fayet-Iliopoulos term was switched on and the system evolved to a T-brane background. In the following we would like to give a more precise description of this intuition, in terms of the 4d effective gauge theory.
Let us begin with the D-term part of the 8d action, which is given by [15] S ⊃ where we have applied the general Ansatz of section 4.1 and in particular made use of eqs. (2.6) and (2.8). To convert this to a 4d action, we need to expand the relevant fields in eigenbasis of the Laplacian, and then perform dimensional reduction. More precisely, we denote by ψ n a real 0-form basis of the Laplacian, normalised as where V S stands for the volume of the four-cycle S. As said before, α should contain the eigenmodes of the gauge vector field A. Now, given the relation (2.4) and the fact that [∆, d c ] = 0, if the function g is an eigenmode of the Laplacian so will be α. Therefore, one naturally expands α as where a n (x) are interpreted as canonically-normalised 4d fields, which are eventually going to acquire a vev. Additionally, we can interpret the function ϕ defined in (2.6) in terms of the internal profile of the Higgs-field zero mode. More precisely, near the wall of stability we have that where m n ∈ R and φ(x) is the 4d-charged field whose vev generates a T-brane profile of the form (2.11). On the one hand, the fact that φ is canonically normalised translates into m 0 = 1. On the other hand, the fact that we obtain a finite quartic coupling for this field when we plug (A.6) into (A.1) translates to the fact that the sum n m 2 n must converge. Finally, one may easily extend this decomposition to a more general non-nilpotent-Higgsfield profile. Here for simplicity we will focus on the nilpotent case.
Plugging both expansions in the above action we obtain 4c n a n − m n |φ| 2 2 , (A.7) which is nothing but eq. (2.9) expanded in a basis of eigenmodes of the Laplacian. In other words, we have that at the wall there are cubic couplings of the form a n |φ| 2 . If now c = 0 and φ develops a vev to cancel the first term, that is the usual 4d D-term, the Kaluza-Klein modes of the gauge vector field must also do so. In particular we have that < a n > = m n 4c n |φ| 2 . (A.8) As the m n are bounded from above, these vev for the KK modes will typically decrease as their mass c n increases.

B Examples of wall-crossing for coincident branes
As a proof of existence, we will construct different examples of 4-cycles inside a compact Calabi-Yau showing the properties discussed in section 4.1. Consider the toric ambient space P 1 × P 1 × P 2 , where we label coordinates and divisor classes as given in table 1.
Using the Stanley-Reissner ideal, we can read off that the only non-vanishing intersection  product in the ambient space is given by H 1 · H 2 · H 2 3 = 1. We define a Calabi-Yau 3-fold X inside this ambient space by the zero locus of the most general polynomial in the class [X] = 2H 1 + 2H 2 + 3H 3 . One may check that X is non-singular. Using Lefshetz hyperplane theorem we know that H 1,1 (P 1 × P 1 × P 2 ) ∼ = H 1,1 (X), such that X inherits the Kähler form from the ambient space. Similarly, we have H 0,1 (X) = H 0,1 (P 1 × P 1 × P 2 ) = 0. In the following we will show different wall-crossing phenomena present on three 4-cycles inside the Calabi-Yau. we see that the Fayet-Ilioupoulos term can indeed acquire both signs depending on the position in Kähler moduli space. Notice that S c 2 1 (K S ) = 0 and I = 2, in agreement with the necessary condition of section 4.1 for a decay. And the Fayet-Ilioupoulos is given by

T-brane to T-brane crossing
which can acquire both signs depending on the position in Kähler moduli space. We read off that on one side of the wall T-branes are stable, whereas at the other side we may either have T-brane bound states, non-Abelian gauge profiles or a combination of the two.