IIFT-UAM/CSIC-16-082
T-branes and α ’-corrections Fernando Marchesano and Sebastian Schwieger
Instituto de F´ısica Te´orica UAM-CSIC, Cantoblanco, 28049 Madrid, Spain
Abstract
We study α ’-corrections in multiple D7-brane configurations with non-commutingprofiles for their transverse position fields. We focus on T-brane systems, crucialin F-theory GUT model building. There α (cid:48) -corrections modify the D-term piece ofthe BPS equations which, already at leading order, require a non-primitive Abelianworldvolume flux background. We find that α (cid:48) -corrections may either i) leave thisflux background invariant, ii) modify the Abelian non-primitive flux profile, or iii) deform it to a non-Abelian profile. The last case typically occurs when primitivefluxes, a necessary ingredient to build 4d chiral models, are added to the system.We illustrate these three cases by solving the α (cid:48) -corrected D-term equations in ex-plicit examples, and describe their appearance in more general T-brane backgrounds.Finally, we discuss implications of our findings for F-theory GUT local models. a r X i v : . [ h e p - t h ] F e b ontents α ’-corrections 43 α ’-corrections for intersecting branes 84 α ’-corrections in simple T-brane backgrounds 10 One important property of D-branes is that they greatly enhance the possibilities tobuild different 4d string theory vacua and, when present, they dictate to large extent thephenomenological interest of such vacua [1].Pictorially, we are used to think of D-branes as dynamic objects wrapped on sub-manifolds of a certain compactification manifold. While this intuition may be accuratefor a single, isolated brane, systems of multiple D-branes are known to be richer than asimple sum of submanifolds, allowing for new configurations that can be understood interms of non-Abelian bound states [2]. In this respect, attention has been given lately1o configurations which can be thought of as a non-Abelian deformation of coincident orintersecting D-brane systems [3–5], dubbed T-branes in [5]. Such T-brane configurationsare particularly interesting when they refer to systems of 7-branes, to which most of thesubsequent formal developments are related [6–13].Indeed, as follows from the scheme introduced in [14–17], in F-theory GUT modelsYukawa couplings are computed in terms of a 7-brane super Yang-Mills theory with anon-Abelian group G , a Higgs/transverse-position field Φ and a gauge vector A . Byspecifying the internal profiles for Φ and A around certain local patches, one is able tocompute the physical Yukawas of each model, see [18–24] for details. In this approach,in order to naturally achieve up-type quark masses with one family much heavier thanthe others one is forced to consider G = E n and a T-brane profile for Φ [5]. Such generalsetup can been implemented in different local models in order to achieve realistic Yukawacouplings, see [25–28].It is nevertheless important to notice that the 7-brane SYM theory described in [14–17]is subject to α (cid:48) -corrections. In the case of multiple D7-branes such α (cid:48) -corrections areencoded in the non-Abelian DBI+CS actions, and their effect can in principle be extracteddirectly from there. In practice it is however simpler to see how these correction modifythe BPS equations for multiple D7-branes, and then analyse the configurations that solvethe corrected equations. The purpose of this paper is to apply this strategy to analyse α (cid:48) -corrections in T-brane systems of D7-branes, including all those ingredients that appearin F-theory GUT model building.Since D7-branes wrapping holomorphic four-cycles are examples of B-branes, we ex-pect that α (cid:48) -corrections do not modify their F-term equations and only affect their D-termBPS equations. In other words, if we describe the corrected BPS equations as a Hitchinsystem, the holomorphic 7-brane data will remain unaffected and α (cid:48) -corrections will onlymodify the stability condition [29]. This result, which we review from the viewpointof [30, 31], allows to solve for α (cid:48) -corrected T-brane backgrounds with the same strategyused in [5]: we first define their holomorphic data and then solve the D-term equationin terms of a complexified gauge transformation acting on Φ and A . We will then seethat α (cid:48) -corrections will not only change the initial T-brane profile quantitatively, but alsoqualitatively. 2ndeed, a standard class of T-brane configurations features a Higgs field Φ along a setof non-commuting generators E i and a non-primitive worldvolume flux of the form F = − i∂∂f P (1.1)that solves the classical D-term equation. Here P is a Cartan generator of the gaugegroup G , while f is a function of the 7-brane coordinates that solves a certain differentialequation and that also enters in the profile for Φ [5]. While non-trivial, this Abelianprofile for F is relatively simple, in the sense that it could involve several, non-commutinggenerators of G . In this paper we will consider the α (cid:48) -corrected version of this class ofsystems. As a general result we find that several things can happen: i) In the most simple example of this setup, which preserves eight supercharges, thesame background is also a solution of the α (cid:48) -corrected D-term equations. ii) We may lower the amount of supersymmetry to four supercharges by a) modifying the Higgs background as Φ → Φ + ∆Φ, with [Φ , ∆Φ] = [ F, ∆Φ] = 0, b) introducing a primitive worldvolume flux H that commutes with Φ and F .Ignoring α (cid:48) -corrections a) and b) do not modify the T-brane piece of the background.However, taking α (cid:48) -corrections into account the profile for the function f is modified. iii) If we perform a) and b) simultaneously while preserving four supercharges then, ingeneral, (1.1) may not solve the α (cid:48) -corrected D-term equations and the non-primitiveflux F will have to develop new components along the non-Cartan generators E i .The T-brane profile for Φ will also become more involved.Interestingly, a) and b) are standard features that one needs to implement in localF-theory GUTs in order to engineer realistic 4d chiral models [26–28]. One may thereforeexpect that, in general, the description of T-brane systems leading to realistic F-theorymodels will be qualitatively modified when taking into account the effect of α (cid:48) -corrections,at least at the level of non-holomorphic data.The paper is organised as follows. In section 2 we derive how α (cid:48) -corrections entersystems of multiple D7-branes, and in particular how they modify their D-term equations.3n section 3 we solve such α (cid:48) -corrected D-term equations for system of intersecting D7-branes, relating the corrections to the pull-back on each individual D7-brane embedding.Then, in section 4, we turn to solve the α (cid:48) -corrected D-term equations for simple T-brane backgrounds, which already illustrate the three cases described above. In section 5we discuss how to solve α (cid:48) -corrected D-term equations in more general T-brane systemsand how the same phenomena arise in there. In section 6 we briefly comment on theimplications of our finding for some local F-theory GUT models. We draw our conclusionsin section 7.Several technical details have been relegated to the Appendices. Appendix A con-tains an alternative derivation of the α (cid:48) -corrected D-term equations by means of thenon-Abelian Chern-Simons action. Appendix B shows that α (cid:48) -corrections are trivial forcertain T-brane systems with globally nilpotent Higgs field. Appendix C shows howadding non-Cartan flux backgrounds can solve the corrected D-term equations in the T-brane backgrounds of section 4 that correspond to case iii) , at least to next-to-leadingorder in the α (cid:48) -expansion. Appendix D shows the results of the analysis of section 4applied to further SU (2) T-brane backgrounds. α ’-corrections Let us consider type IIB string theory compactified on a Calabi-Yau threefold X , and thenquotiented by an orientifold action such that the presence of O3/O7-planes is induced. Inorder to cancel the related RR charge of these orientifold content one may add differentstacks of D3-branes and D7-branes, the latter wrapping four-cycles S a ⊂ X in the internalspace and with internal worldvolume fluxes F switched on along S a .In the simplest configuration that one may consider each stack would only involve asingle D7-brane, wrapping a collection of different, isolated four-cycles {S a } . For each ofthese D7-branes one can check if the energy is minimised by looking at its BPS conditions,which amount to require that the four-cycle S is holomorphic that the worldvolume fluxthreading it is a primitive (1,1)-form in S [32–34]. These BPS conditions are captured In our conventions S is calibrated by − J and so a BPS worldvolume flux is self-dual F = ∗ S F .
4y the following functionals [35, 36] W = (cid:90) Σ P (cid:2) Ω ∧ e − B (cid:3) ∧ e λF (2.1) D = (cid:90) S P (cid:2) Im e iJ ∧ e − B (cid:3) ∧ e λF (2.2)that in 4d are respectively interpreted as a superpotential and D-term for each D7-brane.Here J is the K¨ahler form and Ω = e φ/ Ω a holomorphic (3,0)-form in X , normalisedsuch that J = − i Ω ∧ ¯Ω. In addition, B is the internal B-field, F = dA the worldvolumeflux and λ = 2 πα (cid:48) . Finally, Σ is a five-chain describing the deformations of the four-cycle S , which infinitesimally can also be parameterised by the complex position coordinatesΦ i , and P [ . . . ] stands for the pull-back on the D7-brane worldvolume, namelyP [ V µ d z µ ] α = V α + λV i ∂ α Φ i (2.3)with α a coordinate in S .More generally, one would consider configurations involving stacks of several 7-branes,with non-Abelian bundles on them and wrapping four-cycles that intersect each other. Ona given patch of the internal manifold one can describe such configurations in terms of a8d twisted super Yang-Mills theory with a given non-Abelian symmetry group G [14–17].The bosonic field content of this theory is given by a gauge field A and a Higgs-fieldΦ transforming in the adjoint of G , and whose background profiles will break G to asmaller gauge symmetry group. In this paper we are interested in configurations in whichthe profile for Φ is intrinsically non-Abelian, and more precisely in the kind of profilesconsidered in [4–6, 9–11] and dubbed T-branes in [5].Just like in the Abelian case, the non-Abelian profiles for Φ and A need to satisfycertain equations of motion that are captured by 7-brane functionals. In order to describethe non-Abelian generalisation of (2.1) and (2.2) one may proceed as follows [30,31]. Firstone uses the equations of motion of the background to locally write Ω ∧ e B = dγ , andso rewrite the integral in (2.1) as (cid:82) S P [ γ ] ∧ e λF . Then one observes that, since both W and D have both the form of the D7-brane Chern-Simons action, their non-Abeliangeneralisation should go along the same lines as described in [37]. More specifically, we See [29] for a previous, alternative derivation of these equations. W = (cid:90) S STr (cid:110) P (cid:2) e iλι Φ ι Φ γ (cid:3) ∧ e λF (cid:111) (2.4) D = (cid:90) S S (cid:110) P (cid:2) e iλι Φ ι Φ Im e iJ ∧ e − B (cid:3) ∧ e λF (cid:111) . (2.5)where ι Φ stands for the inclusion of the complex Higgs field Φ, and S for symmetrisationover gauge indices. Just like eqs.(2.1) and (2.2), these functionals describe the D-braneBPS equations whenever the approximations leading to the D-brane DBI + CS actionshold, namely internal volumes with are large and slowly varying profiles for Φ and F instring length units. In this regime the D-term functional (2.5) should take into accountall the α (cid:48) -corrections to the BPS equations for a non-Abelian system of D7-branes. In order to bring these expressions to a more familiar form let us introduce localcomplex coordinates x, y, z and take the four-cycle S along the locus { z = 0 } – that is x and y are the coordinates of S . In this local description the Higgs field is given byΦ ≡ φ ∂∂z + φ ∂∂z . (2.6)where φ is a matrix in the complexified adjoint representation of G and φ its Hermitianconjugate. Locally we may also take γ ≡ z d x ∧ d y , such that in particular we have ι Φ γ = 0.Performing a normal coordiante expansion of γ and plugging it into (2.4) then gives W = λ (cid:90) S Tr { φ dx ∧ dy ∧ F } = λ (cid:90) S Tr { ι Φ Ω ∧ F } . (2.7) That is, if we neglect higher derivative corrections of the Riemann tensor. After taking such curvaturecorrections into account one expects a non-Abelian D-term of the form [29] D = (cid:90) S P (cid:2) Im e iJ ∧ e − B (cid:3) ∧ e λF (cid:48) ∧ (cid:113) ˆ A ( T ) / ˆ A ( N )with ˆ A the A-roof genus of the tangent T and normal N bundles, and F (cid:48) = F − F N with F N the normalbundle curvature [38–42]. Here (cid:113) ˆ A ( T ) / ˆ A ( N ) = 1 − [ p ( T ) − p ( N )] + . . . with p the real four-formgiven by the first Pontryagin class. Note that this correction does not affect the Abelian D-term but it isnon-trivial in the non-Abelian case. In the following we will consider a local patch in which the K¨ahlermetric is locally flat, and therefore take p = 0 and F (cid:48) = F . It would be interesting to see if our resultscould change qualitatively when these curvature corrections become important. Crucially, the integrand doesnot depend on λ , which implies that the F-term conditions are entirely topological andreceive no α (cid:48) -corrections.We will now see that this is not the case for the D-terms (2.5), which are evaluated as D = (cid:90) S S (cid:40) λP [ J ] ∧ F − iλ ι Φ ι Φ J + iλ ι Φ ι Φ J ∧ F ∧ F − P[ J ∧ B ] − iλ ι Φ ι Φ ( J ∧ B ) ∧ F + iλ ι Φ ι Φ ( J ∧ B ) (cid:41) , (2.8)where we have kept terms of all orders in λ in this expansion. In our local patch we maytake the flat space K¨ahler form to be J = i x ∧ d x + i y ∧ d y (cid:124) (cid:123)(cid:122) (cid:125) =: ω +2 i d z ∧ d z, (2.9)decompose the background B-field as B ≡ B (cid:12)(cid:12) S + B zz d z ∧ d z and write F = λF − B (cid:12)(cid:12) S ,yielding D = (cid:90) S S (cid:40) P[ J ] ∧F + iλ ι Φ ι Φ J ) (cid:0) F − ω (cid:1) − iλ ( ι Φ ι Φ B ) ω ∧F − ω ∧ P[ B zz d z ∧ d z ] (cid:41) . Here we defined the Abelian pull-back ω to S as indicated in (2.9), such that we have ι Φ ι Φ J = 2 i [ φ, φ ] ι Φ ι Φ J = 6 i [ φ, φ ] ω . To proceed we note that 2 i [ φ, φ ] is a zero-form and secondly, that 6 i [ φ, φ ] ω has no trans-verse legs to S . That is, in both cases the pull-back P acts trivially. Lastly, one maycompute P[ J ] = ω + 2 iλ (D φ ) ∧ ( ¯ Dφ ) . (2.10)so at the end we have that the D-term equations amount to D = 0 with D = (cid:90) S S (cid:40) ω ∧F + λ D φ ∧ D φ ∧ (2 i F − B zz ω ) + λ (cid:2) φ, φ (cid:3) (cid:0) ω − F − iB zz ω ∧F (cid:1) (cid:41) . (2.11) Notice that in these references the two-form ι Φ Ω is denoted by Φ. Including curvature corrections there would be an extra term of the form iλ ι Φ ι Φ J [ p ( T ) − p ( N )]. B -field, this simplifies to D = λ (cid:90) S S (cid:40) ω ∧ F + 2 iλ D φ ∧ D φ ∧ F + (cid:2) φ, φ (cid:3) (cid:0) ω − λ F (cid:1) (cid:41) . (2.12)These expressions reproduce those found in [29], and can be recovered by analysing thenon-Abelian Chern-Simons action of a stack of D7-branes, as discussed in Appendix A.Note that both terms at leading order in λ , namely ω ∧ F + (cid:2) φ, φ (cid:3) ω , are purelyalgebra valued. Crucially, this is not the case anymore when we include higher orders,because these additional terms contain products of generators. From the original formulain (2.2) it is clear that these products have to be understood in the same way as inthe exponentiation map, which implies that for matrix algebras g ⊂ GL ( n, C ) they aresimply the matrix products in the fundamental representation of said algebra. Takinginto account the symmetrisation procedure, we end up considering terms of the formS (cid:8) T . . . T n (cid:9) = 1 (cid:88) all perm. σ T σ · · · T σ n . (2.13)Formally speaking, including higher order corrections in λ means that the D-terms arevalued in the universal enveloping algebra U ( g ) rather than g itself. α ’-corrections for intersecting branes To get some intuition on the meaning of the α ’ corrections on D-terms, let us first considerthe case where the Higgs field φ and the gauge flux F can be diagonalised, as is for thecase of intersecting D7-brane backgrounds. Then the D-term equations amount to D = λ (cid:90) S P ab [ J ] ∧ F = λ (cid:90) S (cid:0) ω + 2 iλ ∂φ ∧ ∂φ (cid:1) ∧ F, (3.1)that is to say the α (cid:48) -corrections are given entirely by the abelian pull-back of the K¨ahler-form J to S , P ab [ J ] ≡ (cid:0) ω + 2 iλ ∂φ ∧ ∂φ (cid:1) . This implies that flux needs to be primitivewith respect to this pull-back rather than with respect to ω ≡ J | S = i (d x ∧ d x + d y ∧ d y ),the difference being the α (cid:48) corrections to the D-term.Let us be more specific and consider the background φ = µ x − µ x (3.2)8nd a flux F that commutes with φ . Namely we have F = F x ¯ x dx ∧ d ¯ x + F y ¯ y dy ∧ d ¯ y + F x ¯ y dx ∧ d ¯ y + F y ¯ x dy ∧ d ¯ x (3.3)where F = F † imposes F x ¯ y = F y ¯ x and a reality condition for F x ¯ x , F y ¯ y . In particular, dueto our Ansatz these components must be of the form i ( aσ + b ), with a , b real functions.Imposing that dF = 0 and the leading order D-term condition ω ∧ F = 0 sets thesefunctions to be constant and such that F xx = − F yy , while F x ¯ y is constant but otherwiseunconstrained. The latter is also true for the α (cid:48) -corrected D-term constraint, while therelation between F xx and F yy is modified to F xx = − (1 + 4 λ | µ | ) F yy , (3.4)Notice that this condition reduces to the naive primitivity condition F xx + F yy = 0 in thelimit λ →
0, while for finite λ it gives a correction that grows with the complex parameter µ ∈ C , [ µ ] = L − .Physically, the α (cid:48) -corrected D-term condition is quite easy to understand. Indeed,notice that the Higgs-field vev in (3.4) describes an SU (2) gauge theory which is brokencompletely over generic loci, and in particular there is no D7-brane on the naive gaugetheory locus { z = 0 } . Instead we may compute the D7-brane loci via the discriminantdet ( z · − λ · φ ) = ( z − λµ x )( z + λµ x ), which indicates that the system contains twoD7-branes located at { z = ± λµ x } and µ is their intersection slope. A more suitabledescription can be obtained by passing to a new system of coordinates u ≡ z + λµ x (3.5) v ≡ z − λµ x (3.6) w ≡ y, (3.7)in which the branes loci are given by { u = 0 } and { v = 0 } , and then analysing each of theD7-branes individually in term of their Abelian D-terms. For instance, to have primitiveflux along the D7-brane located at { u = 0 } translates into0 = J | { u =0 } ∧ F (3.8) ⇒ F v ¯ v = − (cid:18) λ | µ | + 1 (cid:19) F w ¯ w (3.9) ⇒ F xx = − (cid:0) λ | µ | (cid:1) F yy (3.10)9nd similarly for { v = 0 } . This is precisely the result we obtained earlier in (3.4) fromthe perspective of the gauge theory on { z = 0 } . So intuitively the D-term equationsin this description just tells us that the flux should be primitive along the actual braneworld-volumes, rather than the locus S from which we describe the parent gauge theory. α ’-corrections in simple T-brane backgrounds After seeing the effect of α (cid:48) corrections for intersecting D7-branes, let us investigate whichtypes of effects we receive for T-brane backgrounds. In general, these backgrounds aresuch that [ φ, ¯ φ ] (cid:54) = 0 and so a non-primitive flux F , satisfying F ∧ ω (cid:54) = 0, is needed to solvethe D-term equations at leading order [5].In order to find BPS solutions for these backgrounds one may apply the strategyoutlined in [5]. Namely, one first defines the T-brane Higgs background in a unphysicalholomorphic gauge [21, 22] A (0 , = 0 ¯ ∂φ hol = 0 (4.1)and then rotate these fields by a complexified gauge transformation of the symmetrygroup G A (0 , → A (0 , + ig ¯ ∂g − φ → gφg − (4.2)in order to attain a unitary gauge in which the D-term condition is satisfied. In thefollowing we will apply this same strategy to solve for the α (cid:48) -corrected D-term equations.We will consider two simple examples in which the leading order non-primitive flux liesin the Cartan subalgebra of the symmetry group G , as this also simplifies the Ansatz tosolve the D-term equations at higher order in α (cid:48) . Let us first analyse a simple SU (2)-background already considered in [5] where the Higgsfield profile in the holomorphic gauge reads φ hol = m ax = − imE + + imaxE − (4.3)10here m, a ∈ C and [ m ] = [ a ] = L − , and the generators E ± are defined in Appendix C.This time the discriminant gives the D7-brane locus z = λ am x . Moreover, since wehave det φ hol = − m ax , we see that this is a reconstructible brane background accordingto the definition given in [5]. To solve the D-term equations we proceed as above and passto a unitarity gauge via a complexified gauge transformation in SU(2). More precisely wetake g = e f σ (4.4)which implies that in the unitarity gauge the D7-brane backgrounds reads φ = m e f axe − f , (4.5) F = − i∂∂f · σ . (4.6)At leading order in λ the D-term equations read (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f σ = [ φ, ¯ φ ] ⇒ (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f = | m | (cid:0) e f − | ax | e − f (cid:1) . (4.7)Finding f at this level amounts to solve a partial differential equation of Painlev´e III typeon the radial coordinate | x | , as has been already discussed in [5]. More precisely, we maysolve it by making the Ansatz f = f ( | x | ) and parametrise x ≡ re iθ , yielding (cid:18) d d r + 1 r dd r (cid:19) f = | m | (cid:0) e f − | a | r e − f (cid:1) . (4.8)Redefining e f ( r ) ≡ r | a | e j ( r ) further simplifies this to (cid:18) d d r + 1 r dd r (cid:19) j = | a || m | r sinh(2 j ) . (4.9)Finally we define s ≡ (cid:112) | a || m | r such that we are left with (cid:18) d d s + 1 s dd s (cid:19) j = sinh(2 j ) , (4.10)which is the standard expression for a particular kind of Painlev´e III equation analysedin [43]. Finally, we may directly solve (4.7) asymptotically near | x | = 0 by f = f ( x, x ) = log c + c | mx | + | m | | x | c (cid:0) | m | c − | a | (cid:1) + . . . (4.11)11ith c an arbitrary dimensionless parameter whose value should be close to 0 .
73 if wewant to avoid poles for large values of | x | [26].Let us now consider the α (cid:48) -corrected D-term equation. Applying (2.12) to this setupwe obtain the following equation (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f = | m | (cid:0) e f − | ax | e − f (cid:1) (cid:0) λ Q f (cid:1) + λ R [ f, f ] , (4.12)where Q f = ( ∂ x ∂ x f )( ∂ y ∂ y f ) − ( ∂ x ∂ y f )( ∂ y ∂ x f ) (4.13) R [ f, g ] = | m | (cid:2)(cid:0) ∂f ∧ ∂f e f + | a | e − f ( ∂x − x∂f ) ∧ ( ∂x − x∂f ) (cid:1) ∧ ∂∂g (cid:3) xxyy describe the new operators that appear due to the α (cid:48) -corrections. Notice however that bykeeping the Ansatz f ≡ f ( x, ¯ x ) both Q f and R [ f, f ] vanish identically and we are backto eq.(4.7). Therefore, the solution to the corrected D-term still amounts to f = f ( x, ¯ x )and the above T-brane background does not suffer any modification due to α (cid:48) -corrections.Notice that in this case the T-brane background preserves 1 / α (cid:48) -corrections do not modify the background.The analysis becomes more interesting if we consider a more general flux background,with a new component which will lower the amount of preserved supersymmetry. As usualwe may consider adding such fluxes along generators that commute with the T-branebackground. For instance we may add a worldvolume flux along the identity generatorof u (2), which could arise either from the D7-brane itself or form the pull-back of a bulkB-field. We first consider the case where this flux is H = Im ( κ dx ∧ d ¯ y ) (4.14)with κ ∈ C and [ κ ] = L − parameterising the local flux density. At leading order in α (cid:48) ,the vanishing D-term condition would allow for an arbitrary κ without modifying the T-brane background, as the above flux is primitive. Its α (cid:48) -corrected counterpart, however,has non-trivial components along the generators σ and , implying two independent12-term equations. Namely (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f = | m | (cid:0) e f − | ax | e − f (cid:1) (cid:0) λ Q f + λ | κ | (cid:1) + λ R [ f, f ] (4.15)0 = Re (cid:16) | a | e − f κx∂ y f (cid:0) x∂ x f − (cid:1) + 2 e f κ∂ y f ∂ x f (cid:17) with the second line corresponding to the D-term constraint along the identity generator.Such equation is automatically satisfied if we again impose the Ansatz f ≡ f ( x, ¯ x ), whilethe first one becomes (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f = | m | (cid:0) e f − | ax | e − f (cid:1) (cid:0) λ | κ | (cid:1) . (4.16)Hence, we are back to eqs.(4.7) and (4.11) with the replacement m → m (cid:48) = m (cid:112) λ | κ | . (4.17)Finally, let us consider the case where the flux background on the identity is H = H + H − i∂ ¯ ∂h (4.18)where H is again given by (4.14), and H is an different piece of primitive constant flux H = ρ i ( dx ∧ d ¯ x − dy ∧ d ¯ y ) (4.19)with ρ ∈ R and [ ρ ] = L − . In addition, we consider h ≡ h ( x, x, y, y ) to be an arbitraryfunction that we may expand around the origin as a polynomial, starting at quadraticorder. In addition, we write the gauge transformation (4.4) as the following expansion f = f ( x, ¯ x ) + ∞ (cid:88) i =1 ( λρ ) i f i ( x, x, y, y ) (4.20)with f ( x, ¯ x ) the solution found for ρ = 0, which near the origin behaves as (4.11) withthe replacement (4.17).In this case solving the D-term equations becomes more challenging, but one mayperform a perturbative expansion on the dimensionless parameter λρ and keep the termsup to O (( λρ ) ) in order to simplify them. On the one hand, for the D-term constraintalong the generator σ we find( ∂ x ∂ x + ∂ y ∂ y ) f σ = [ φ, φ ] (cid:0) λ Q H (cid:1) , (4.21)13here now Q H = (cid:0) ∂ x ∂ x h − ρ (cid:1) (cid:0) ∂ y ∂ y h + ρ (cid:1) − (cid:18) ∂ x ∂ y h − i κ (cid:19) (cid:18) ∂ y ∂ x h − i κ (cid:19) . (4.22)On the other hand, for the constraint along the identity we have( ∂ x ∂ x + ∂ y ∂ y ) h = λ (cid:104) | m | (cid:0) e f − | ax | e − f (cid:1) ∂ x ∂ x f (cid:16) ∂ y ∂ y h + ρ (cid:17) − R [ f, h + ρ | y | ] (cid:105) with R defined as in (4.13). We find the following solutions for h at lowest orders in λρ and near the origin h = λ ρ | mx | (cid:18) | m (cid:48) x | (cid:0) | a | + 2 c | m (cid:48) | (cid:1) − c (cid:0) | a | − c | m (cid:48) | (cid:1)(cid:19) + O ( λ ρ ) (4.23)while from (4.21) we find that the leading correction to f is f = 2 | mx | (cid:0) λ | m | | m (cid:48) | c − c − λ | am | − c | m (cid:48) x | (cid:1) + (4.24)+ 2 | mx | (cid:16) | am | c + λ | a | c + 48 λ | m | | m (cid:48) | c − λ | am | | m | | m (cid:48) | c (cid:17) where we have again Taylor-expanded around x = 0.To summarise we find that, if we add a primitive constant flux H that commuteswith the Higgs background and of the form (4.14), the D-terms equations can be solvedby an appropriate choice of gauge transformation (4.4), that induces a non-primitive fluxalong the su (2) generator σ . When we also include the constant primitive flux H of theform (4.19) the same is essentially true, but now we must also add a non-primitive flux ∂ ¯ ∂h along the identity generator of u (2) to solve the D-term constraints. Let us now consider a slightly more complicated SU(3) T-brane background, again pre-serving four supercharges. The Higgs field profile in the holomorphic gauge is given by φ hol = m µy ax µy
00 0 − µy ≡ − im E + + imax E − + mµy Q, (4.25)where the form of the generators E ± , Q and P ≡ [ E + , E − ] is detailed in Appendix C.14s before, we may solve for the D-terms equations by performing a gauge transforma-tion of the form (4.2). Because [ φ, ¯ φ ] ∝ P , the natural choice is now g = exp( f P ) and soin the unitary gauge we have a background given by φ = − ime f E + + imaxe − f E − + mµy Q (4.26) F = − i∂∂f P, With this Ansatz there is only one non-trivial D-term constraint, corresponding to thegenerator P . The α (cid:48) corrections complicate the form of this equation with respect to theleading order counterpart, and we obtain (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f = | m | (cid:0) e f − | ax | e − f (cid:1) (cid:0) λ Q f (cid:1) − λ R [ f, f ] − λ | m | | µ | ∂ x ∂ x f (4.27)By using the Ansatz f = f ( x, x ) this expression simplifies to ∂ x ∂ x f = | m | λ | m | | µ | (cid:0) e f − | ax | e − f (cid:1) (4.28)which is asymptotically solved by (4.11) with the replacement m → ˜ m = m (cid:112) λ | m | | µ | (4.29)Let us now add further worldvolume flux to this background. For simplicity we willadd it along generators that commute with the su (2) subalgebra generated by { E ± , P } .Namely we consider the following generators T = × (4.30) B = × , (4.31)Notice that an arbitrary combination of these generators does not belong to su (3) butrather to its central extension u (3). Indeed, only if we consider a worldvolume fluxsatisfying F B + 2 F T = 0 we will have an SU(3) background.Similarly to the SU(2) example one may first consider a flux that commutes with thegenerators of the T-brane background, namely of the form H = Im ( κ dx ∧ d ¯ y ) T (4.32) G = M (d x ∧ d x + d y ∧ d y ) B + N (d x ∧ d x − d y ∧ d y ) B + Im ( O d x ∧ d y ) B (4.33)15here M, N ∈ R and κ, O ∈ C . We may also generalise the Ansatz to f ≡ f ( x, x, y, y ).The corrected D-term equations then read:0 = 8 λ | mµ | ( M + N ) + N (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) f = | m | (cid:0) e f − | ax | e − f (cid:1) (cid:0) λ | κ | + 4 λ Q f (cid:1) − λ R [ f, f ] − λ | m | | µ | ∂ x ∂ x f λ Re (cid:16) | a | e − f κx∂ y f (cid:0) x∂ x f − (cid:1) + 2 e f κ∂ y f ∂ x f (cid:17) + (cid:0) e f − | ax | e − f (cid:1) Re (cid:16) κ∂ y ∂ x f (cid:17) λ κ | mµ | (cid:0) | a | e − f | − x∂ x f | − e f | ∂ x f | (cid:1) (4.34)Here the first equation correspond to the generator B and it is identical to the D-termconstraint found in (3.4) for the case of intersecting 7-branes. It fixes the relation between M and N and decouples from the rest of the equations, that will not depend on M, N, O .The second equation corresponds to the D-term along the generator P and it is againgiven by (4.27). The third and fourth equations are new, and correspond to the D-termconstraints along the generators T and E ± , respectively. From the last one we see thatthe only way to have a non-vanishing flux κ is to take the limit µ →
0, which wouldessentially take us to the previous SU (2) example.Despite this result, one is able to accommodate a background flux along the generator T by considering a slightly different Ansatz. Indeed, let us proceed as in the previous SU (2) example and generalise the above flux Ansatz to H = H + H − i∂ ¯ ∂h T (4.35) H = ρ i ( dx ∧ d ¯ x − dy ∧ d ¯ y ) Th ≡ h ( x, x, y, y ) . while returning to the Ansatz f ≡ f ( x, x ) for the flux along P . The corrected D-term16quations now read: 0 = 8 λ | mµ | ( M + N ) + N (cid:0) λ | mµ | (cid:1) ∂ x ∂ x f = | m | (cid:0) e f − | ax | e − f (cid:1) (1 + 4 λ Q H ) (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) h = 4 λ | m | | µ | (cid:0) ρ − ∂ x ∂ x h (cid:1) + 2 λ (cid:0) ρ + ∂ y ∂ y h (cid:1) (cid:0) | m | e f | ∂ x f | + 2[ φ, φ ] − | am | e − f | x∂ x f − | (cid:1) λ | mµ | (cid:0) ∂ x ∂ y h + κ (cid:1) (cid:0) | a | e − f | x∂ x f − | − e f | ∂ x f | (cid:1) (4.36)with Q H again given by (4.22). Notice the last equation now imposes 2 ∂ x ∂ y h + κ = 0,which essentially requires that the effective flux of the form (4.32) vanishes. Naively, thisseems to imply that α (cid:48) -corrected D-terms do impose constraints on worldvolume fluxescommuting with the Higgs field T-brane background, contrary to what happens at leadingorder in α (cid:48) . Nevertheless, one can show that a non-trivial κ is allowed if one generalisesthe gauge transformation Ansatz g = exp( f P ) to include complexified transformationsalong the non-Cartan generators E ± as well. We leave the somewhat technical proof ofthis statement to Appendix C, where such generalised transformations are studied in moredetail.If for simplicity we set κ = 0, make the Ansatz (4.20) and solve again perturbativelyin λρ we find the following asymptotic solutions around x = 0: f = log c + c | ˜ mx | + | ˜ m | | x | c (cid:0) c | m | − | a | (cid:0) λ | mµ | (cid:1)(cid:1) (4.37) f = 4 | ˜ mx | (cid:0) λ | m | (cid:0) | a | − c (cid:1) + c (cid:1) (4.38)+ | ˜ m | | x | c (cid:16) | a | | m | c + 2 (cid:0) λ (cid:0) | a | − | a | c (cid:0) c − | µ | (cid:1)(cid:1) − c (cid:1) + 8 λ | m | (cid:0) λ | µ | (cid:0) | a | − | a | c (cid:1) + 2 c λ | µ | (cid:0) | a | + 6 c (cid:1) + c (cid:1) + 4 λ | m | (cid:0) − | µ | (cid:0) c − λ | a | (cid:1) + | a | c (cid:0) λ (cid:0) | µ | − c | µ | (cid:1) + c (cid:1) + 6 c (cid:1) (cid:17) and h = 2 λ | ˜ mx | ρ (2 ( c + c | µ | ) − | a | ) c (4.39)+ λ | ˜ mx | ρc (1 + 4 λ | mµ | ) (cid:16) | a | | ˜ m | (cid:0) | m | (cid:0) c − λ | µ | (cid:1) − (cid:1) + 2 c | m | (cid:0)(cid:0) c + 4 λ | µ | (cid:1) + 1 (cid:1) (cid:17)
17o summarise, in this more complicated SU (3) background that preserves four su-percharges we also find different kind of solutions for the α (cid:48) -corrected D-term equations.One first class of corrections comes from the intersection slope µ that appears in φ hol ,and which corresponds to a generator Q commuting with the T-brane su (2) subalgebra { E ± , P } . Such corrections are relatively easy to take into account, as they only modifythe parameters of the Painlev´e III equation. Further non-trivial corrections come fromadding worldvolume fluxes commuting with the Higgs background. One the one hand,adding some of these primitive fluxes require a modification of the non-primitive flux ∂∂f along P and adding one of the form ∂∂h along T . On the other hand, adding some othercomponents requires a more drastic change: to generalise the standard gauge transfor-mation g to also include non-Cartan generators E ± . In the next section we will analysefrom a more general viewpoint when each of these two cases occurs. With the two examples of the previous section in mind, let us describe how α (cid:48) correctionsaffect the D-term equations for more general kinds of T-branes. As before we will take thesimplifying assumption that, given the gauge group G and its corresponding Lie algebra g ,the leading order D-term equations can be solved via a complexified gauge transformation(4.2) of the form g = e fi P i (5.1)where f i = f i ( x, x, y, y ) and P i belong to the Cartan subalgebra of g . We then write theHiggs field profile in the holomorphic gauge in the block diagonal form φ hol = m ψ ψ . . . ψ n hol , (5.2)with [ m ] = L − , and where the entry ψ i hol is an n × n matrix of holomorphic functionson x, y . One simple example of such structure is the SU(3) example of section 4.2, whichcontained a 1 × × α (cid:48) -corrected18-term equations do not couple one block to the other. The same statement holds forthe more general T-brane structure with the block-diagonal form (5.2): for the purposesof analysing α (cid:48) -corrections we can focus on each individual block ψ i hol at a time, an forgetabout the rest.In the case that ψ i hol is a 1 × α (cid:48) -corrections will be similar tothe ones studied in section 3. As in there, the α (cid:48) -corrections will impose primitivity withrespect to the standard pull-back of J on the spectral surface z = λ m ψ × ( x, y ) . (5.3)More interesting is the case where ψ i hol is a 2 × α (cid:48) -corrected D-term equations may become rather involved to solve, specially when we addadditional primitive worldvolume fluxes. In general, within that block we will have aholomorphic Higgs field profile of the form ψ × = u + u σ + u σ + u σ = u − iu + E + + iu − E − + u σ (5.4)where u i , u ± are complex functions on x, y , [ u i ] = [ u ± ] = L . Near the origin, we canapproximate such functions up to their linear behaviour, so each of them is characterisedby three independent complex numbers. However, we may absorb three numbers inconstant shifts of the local coordinates x, y, z . More precisely, by a shift in z we mayremove the constant term in u , rendering it a linear function in x, y . Similarly, by shiftson x and y we may remove the constant pieces in u and u − . Then we are left with onlyone function, namely u − that may contain a constant term, and therefore with essentiallytwo different possibilities ψ × (cid:12)(cid:12) x = y =0 = and ψ × (cid:12)(cid:12) x = y =0 = . (5.5)Examples of backgrounds of the first kind are those analysed in section 4, while severalof the second kind are studied in Appendix D. In both cases the holomorphic Higgsbackground is parameterised by eight dimension-full parameters, namely u = µ ,x x + µ ,y y u = µ ,x x + µ ,y yu − = µ − ,x x + µ − ,y y u + = µ + ,x x + µ + ,y y + (cid:15) (5.6)19here [ µ i,α ] = L − and (cid:15) = 0 , λ → g × = e ( fσ + h ) (5.7)for solving the 2 × ψ × , ψ × ] ∈ Cartan, which requires µ ,x = µ ,y = 0. We then have that in our setup ψ × = µ ,x x + µ ,y y µ + ,x x + µ + ,y y + (cid:15)µ − ,x x + µ − ,y y µ ,x x + µ ,y y . (5.8)One may now wonder if taking into account α (cid:48) -corrections will drastically change theform of the complexified gauge transformation (5.7) solving for the D-term equation. Forthis we observe that • If no background fluxes along are present, then the Ansatz (5.7) remains invariant(with h ≡ α (cid:48) -corrections may vary the specific form of f with respectto its leading order value. • If we switch a background flux H along then, for a generic ψ × , some componentsof H will preserve the Ansatz (5.7), while others will force to consider a gaugetransformation including non-Cartan generators E ± , as discussed in Appendix C.Let us be more precise on the last point, since adding non-Cartan generators to (5.7)implies having a non-Abelian flux background that will complicate the T-brane system.By inspection (see e.g., Appendix C) one quickly realises that the relevant D-term equa-tions for this problem are those along the non-Cartan components E ± , which may or maynot have solution for the Ansatz (5.7). If there is no solution, one needs to generalisethis Ansatz to include the generators E ± and therefore a non-Abelian gauge backgroundappears through (4.2).Due to the symmetrisation procedure, the D-term equations along E ± receive con-tributions only from the middle term in eq.(2.12). More precisely, assuming the Ansatz205.7) we have thatD ψ × ≡ (D ψ ) + (D ψ ) + E + + (D ψ ) − E − (5.9)= ( µ ,x d x + µ ,y d y ) + ( µ + ,x d x + µ + ,y d y + 2 ∂f ( µ + ,x x + µ + ,y y + (cid:15) )) e f E + + ( µ − ,x d x + µ − ,y d y − ∂f ( µ − ,x x + µ − ,y y )) e − f E − , and that the D-term equations along E ± read0 = D ± = 2 iλ (cid:16) (D ψ ) ± ∧ (D ψ ) + (D ψ ) ∧ (D ψ ) ∓ (cid:17) ∧ H . (5.10)From here we see that these equations are non-trivial only if the Higgs-vev ψ × hascomponents simultaneously along the identity and a (non-Cartan) generator of su (2),which will be generically the case. Moreover, the total background flux H along theidentity (including the piece − i∂ ¯ ∂h ) must be non-vanishing for this equation to be non-trivial. Let us discuss how this condition constrains the background flux H . Recall that H must satisfy the corrected primitivity condition0 = ω ∧ H + λ (cid:16) i (D ψ ) ∧ (D ψ ) − ψ ) + ∧ (D ψ ) − ] − Tr([ φ, φ ] F ) (cid:17) ∧ H (5.11)and satisfy the Bianchi identity dH = 0. Then we find that only some profiles for H maysatisfy the complex equations (5.10) and the real equation (5.11) simultaneously. Thoseprofiles that satisfy (5.11) but fail to satisfy (5.10) will not be compatible with the initialAnsatz (5.7) and therefore will require the presence of a non-Cartan flux background at O ( λ ).In practice one may find by inspection which profiles for H are compatible with theAbelian Ansatz (5.7), although in some simple cases one may be more specific. In partic-ular, let us consider the cases where • (D ψ ) + ∧ (D ψ ) − = 0Or equivalently (D ψ ) + = γ (D ψ ) − for some complex function γ . In this case onefinds that all fluxes H of the form H ∝ i (D ψ ) ∧ (D ψ ) (5.12) H ∝ i (D ψ ) − ∧ (D ψ ) − (5.13)21atisfy eq.(5.10). Moreover if ¯ γ ≡ γ − then both equations in (5.10) become thesame. In particular for γ ≡ η = ± H ∝ Re (cid:104) √ η (D ψ ) − ∧ (D ψ ) (cid:105) (5.14)also becomes a solution to (5.10). Any combination of these allowed componentssatisfying dH = 0 and (5.11) will not require a non-Abelian flux background, whilethe rest will. • (D ψ ) ± ∧ (D ψ ) = 0Or equivalently (D ψ ) ± = γ (D ψ ) for a complex function γ . In this case again bothequations in (5.10) becomes conjugate to each other and H ∝ i (D ψ ) ∧ (D ψ ) (5.15) H ∝ Im (cid:104) γ (D ψ ) ∓ ∧ (D ψ ) (cid:105) + i ψ ) ∓ ∧ (D ψ ) ∓ (5.16)automatically satisfy (5.10). Again, a combination of those satisfying (5.11) and dH = 0 will be compatible with an Abelian flux background. • (D ψ ) + ∧ (D ψ ) − = (D ψ ) + ∧ (D ψ ) = (D ψ ) − ∧ (D ψ ) = 0In this case we have that (5.10) will be solved by H ∝ i (D ψ ) ∧ (D ψ ) (5.17) H ∝ Im [ γ (D ψ ) ∧ η ] (5.18)for arbitrary complex function γ and one-form η ∈ Ω (1 , . Such that we have morefreedom to satisfy primitivity condition and Bianchi identity than in the previouscases.One can check that this general discussion reproduces the results found in the twosimple examples of section 4. On the one hand, for the SU (2) example of section 4.1we have that (D ψ ) = 0. Hence (5.10) is trivially satisfied and so non-Cartan fluxes areabsent in the corrected solution. On the other hand, in the SU (3) example of section 4.2,the 2 × ψ ) + , (D ψ ) − ∝ dx , (D ψ ) ∝ dy (5.19)22e are then in the case (D ψ ) + = γ (D ψ ) − , with γ a complicated function. It is then easyto see that H = ρ i ( dx ∧ d ¯ x − dy ∧ d ¯ y ) + O ( λ ) , ρ ∈ R (5.20)is a linear combination of the two-forms (5.12) and (5.13) which satisfies the Bianchi iden-tity and the primitivity condition at leading order. This is precisely the flux componentdenoted as H in section 4.2, explicitly shown to be compatible with the Abelian Ansatz(5.7) therein. On the contrary, a flux of the form (4.32) is shown to be incompatiblewith such an Ansatz, and non-Cartan flux generators need to be added as described inAppendix C. This again matches our general discussion, as for some choices of κ the flux(4.32) can be made of the form (5.14). But since in this example γ (cid:54) = ± The T-brane backgrounds that we considered in the previous section are very similar tothose used to generate phenomenological Yukawa hierarchies in F-theory GUTs [26–28],with the main difference that there Φ and F are valued in the Lie algebra of the exceptionalgroups E , E and E . Nevertheless, in order to build models of SU(5) unification theHiggs background is embedded in unitary subalgebras of these exceptional groups and, atleast naively, one may use this fact to apply our results.Let us for instance consider the E T-brane background constructed in [26] φ = m (cid:0) e f E + + mxe − f E − (cid:1) + µ ( bx − y ) Q , (6.1)where the generators E ± generate a su (2) subalgebra via [ E + , E − ] = P and Q a commut-ing u (1) subalgebra, see [26] for precise definitions. This background is quite similar tothe one considered in section 4.2, as one can see from acting with φ on the doublet sector( , ) − within the adjoint of e [26][ φ, R + E + + R − E − ] = − µ ( bx − y ) mm x − µ ( bx − y ) R + E + R − E − . (6.2)23aively, this action can be identified with a 2 × ψ × of the sort discussedin section 5. In fact, it is identical to the 2 × y → y − bx , a → m , mµ → µ . (6.3)One can now apply the analysis of the previous section to this case. As in the SU(3)example of section 4.2, we are in the case (D ψ ) + = γ (D ψ ) − for γ (cid:54) = ±
1. Therefore,primitive fluxes of the kind H nc × with a component of the form H nc ∝ Re (cid:0) (D ψ ) − ∧ (D ψ ) (cid:1) ∝ Re (cid:0) d x ∧ ( b d x − d y ) (cid:1) (6.4)are not allowed at order λ without adding further non-Cartan fluxes. Interestingly, forthe case b = 1 used in [26] to compute physical Yukawas, we have that such problematicflux reads H nc b =1 ∝ Re (cid:0) d x ∧ d y ) , (6.5)which allows for some primitive fluxes. In fact, the worldvolume primitive fluxes consid-ered in [26] were of the form F p = iQ R ( dy ∧ dy − dx ∧ dx ) + iQ S ( dx ∧ dy + dy ∧ dx ) (6.6)with Q R , Q S some Cartan generators that reduce to the identity for the sector of interest.Therefore, according to our naive analysis the presence of these primitive fluxes maymodify the non-primitive Abelian flux Ansatz given by g = exp( f P ) with f = f ( x, x ),but it will not require the presence of non-Cartan generators in the flux background. Henceit seems that the computation of physical Yukawas made in [26] may be affected by α (cid:48) corrections but not drastically, in the sense that the Ansatz for the T-brane backgroundtaken there survives at the next-to-leading order in α (cid:48) . This will change as soon as theworldvolume flux (6.6) is chosen more general or b is chosen such that Im b (cid:54) = 0. In this paper we have analysed the effect of α (cid:48) -corrections on BPS systems of multipleD7-branes, with special emphasis on T-brane configurations. Our main strategy has been24o compute how α (cid:48) -correction modify the D-term BPS condition, solve for the new back-ground profiles for Φ and A , and compare them with the previous leading-order D-termsolution. Since α (cid:48) -corrections do not enter holomorphic D7-brane data, this comparisoncan be made in terms of the complexified gauge transformation (4.2) in terms of whichwe solve the D-term equations.In D7-brane T-brane systems, solving the D-term equation is quite involved alreadyat leading order, which renders our analysis somewhat technical. Nevertheless, we havedrawn several lessons from the cases that we have analysed: • When the Higgs background takes a block-diagonal form (5.2), α (cid:48) -corrections canbe analysed block by block, as they do not couple different blocks. • For system of intersecting D7-branes α (cid:48) -corrections have a simple interpretation interms of the pull-back of the K¨ahler form on the actual D7-brane embedding. Itwould be interesting to see if T-brane systems allow for a similar interpretation. • In all the examples that preserve eight supercharges, α (cid:48) -corrections do not modifythe background. The classical solution also solves the corrected D-term equations.A trivial example of this are intersecting D7-branes without fluxes. • One may lower the amount of supersymmetry to four supercharges by modifyingthe Higgs field by a constant slope ∆Φ or by adding a constant primitive flux H ,both commuting with the group generators involved the T-brane background. Atleading order these additions do not modify the T-brane background at all. When α (cid:48) -corrections are taken into account the T-brane background is modified, but thereare several degrees of complexity at which this may happen i) In the simplest case α (cid:48) -corrections only modify the dimensionful parameterswhich enter the differential equation for the non-primitive flux background(1.1) and the related complexified gauge transformation (4.2), as in eqs.(4.17)and (4.29). Hence they can be typically absorbed into a coordinate redefinition. ii) In slightly more complicated cases we need to generalise the complexified gaugetransformation to g = e ( fP + h ) (7.1)25o absorb the effect of some primitive flux H . The corresponding non-primitiveflux is therefore still Abelian, with f being modified from the leading-orderexpression. The equations governing f and h are rather complicated, butone may solve them by performing a perturbative expansion in α (cid:48) -suppressedparameters. More precisely we have assumed the following hierarchy α (cid:48) ρ i (cid:28) α (cid:48) m j (cid:28) α (cid:48) . Here ρ i are primitive fluxdensity parameters and m j T-brane slope parameters. iii)
In the most complex case the Abelian Ansatz (7.1) is not sufficient to solve thecorrected D-term equations, which develop non-trivial components along non-Cartan generators (in particular those which the holomorphic T-brane datadepends on). One then needs to consider a complexified gauge transformationthat depends on such generators, as in Appendix C. The analysis for thesecorrected backgrounds is even more involved and one again needs to resort toa perturbative expansion to find solutions to next-to-leading order in α (cid:48) . • This last, more complicated case contains all the ingredients that are generic in theconstruction of 4d chiral local F-theory GUT models, so one may speculate that α (cid:48) -corrections could change qualitatively the description of these configurations, aswe have briefly discussed. In any event, the holomorphic data of these models willnot be affected by α (cid:48) -corrections. In particular the holomorphic Yukawa hierarchiesof [26–28], which only depend on such holomorphic data, will still be present after α (cid:48) -corrections are taken into account.Based on these results, one may conceive of several directions to pursue the analysis of α (cid:48) -corrections in T-brane systems. First, it would be interesting to extend our backgroundsolutions to higher orders in the α (cid:48) expansion and beyond the limit (7.2). Second, itwould be interesting to see if the interpretation of α (cid:48) -corrections for the intersecting D-brane case can be incorporated in some form for T-brane backgrounds. Moreover, itwould be interesting to verify our naive analysis of α (cid:48) -corrections in F-theory local modelsbased in exceptional groups, and compute how α (cid:48) -corrections modify the normalisation of26hiral mode wavefunctions in realistic models. Finally, it would be interesting to see theconsequences of our findings for the recent proposal to use T-branes in the constructionof de Sitter vacua [44]. Acknowledgments
We thank M. Montero, D. Regalado, R. Savelli, A. Uranga and G. Zoccarato for usefuldiscussions. This work has been partially supported by the grants FPA2012-32828 andFPA2015-65480-P from MINECO, SEV-2012-0249 of the “Centro de Excelencia SeveroOchoa” Programme, and the ERC Advanced Grant SPLE under contract ERC-2012-ADG-20120216-320421. S.S. is supported through the FPI grant SVP-2014-068525.
A D-terms from the Chern-Simons action
In section 2 we discussed how to derive the D-terms for non-Abelian stacks of D7-branesin IIB orientifolds with O3/7-planes via their generalised calibration conditions. As wewill now show, one can reach the same result by considering the 4d couplings that arisefrom the Chern-Simons action. Indeed, as was argued in [45], the D-terms of the fourdimensional effective action are related by supersymmetry to terms of the form (cid:82) ˜ B ∧ F ,where ˜ B is a 4d two-form dual to an axion and F the field strength of a gauge groupgenerator. As in other D-brane setups here the two-forms ˜ B arise from RR p -forms, andso such couplings will be contained in the D-brane Chern-Simons action.The non-Abelian Chern-Simons action for a stack of D7-branes is given by [37] S CS = µ p (cid:90) R , ×S STr (cid:16) P (cid:104) e iλι Φ ι Φ (cid:88) C ( n ) ∧ e − B (cid:105) ∧ e λF (cid:17) , (A.1)where we will use the same parametrisation for the Higgs-field as in the main textΦ = φ ∂∂z + φ ∂∂z . (A.2)For simplicity, let us assume that the odd cohomology groups of the compactificationmanifold H − ( X ) = H − ( X ) vanish. Then the harmonic components of the internal B-field are projected out, and the same applies to the 4d two-forms that could arise from27he dimensional reduction of the RR forms C and C . The only relevant 4d two-formsand their axion duals arise from the expansion of the orientifold-even RR forms C (4) = c a ω a + ρ a ˜ ω a + . . . (A.3) C (8) = e ω + . . . (A.4)where ω a , ˜ ω a run over the bases of integer two- and four-forms in the internal space,respectively (such that J = e φ / v a ω a ) and ω = d vol X / √ g X is the unique harmonicsix-form with unit integral over X . Plugging this into (A.1) gives S CS ⊃ λ µ p (cid:90) R , ×S STr (cid:40) F d ∧ (cid:20) e ∧ iι Φ ι Φ ω + c a ∧ (cid:18) P [ ω a ] ∧ F + iλ ι Φ ι Φ ( ω a ) F (cid:19)(cid:21) (cid:41) , (A.5)where F stands for the components of the D7-brane field strength with legs on R , , andwe have imposed the absence of internal B-field.The two-forms coupling to F have as 4d duals dc a = g ab K ∗ R , dρ b de = e φ ∗ R , dC (A.6)where τ = C + ie − φ is the type IIB axio-dilaton, K = K abc v a v b v c with K abc the tripleintersection numbers of X , and g ab is the inverse of g ab = K (cid:82) X ω a ∧ ∗ ω b . Such dualityrelations tells us how a vector multiplet coupling to c a and e enters the type IIB K¨ahlerpotential. Let us start from the usual expression K IIB = − log( S + ¯ S ) − log( K ) − log (cid:18)(cid:90) Ω ∧ ¯Ω (cid:19) (A.7)where S = − iτ . Here K should be seen as a function of Re T a , with T a = − K abc v a v b − iρ a .Then a vector multiplet V i coupling to these axions via a St¨uckelberg coupling Q iα shouldenter the K¨ahler potential (A.7) through the replacements S + ¯ S → S + ¯ S − Q i V i , T a + ¯ T a → T a + ¯ T a − Q ia V i . (A.8)Finally, the Fayet-Iliopoulos term corresponding to V i will be given by ξ i ∝ (cid:18) ∂K∂V i (cid:19) V =0 . (A.9)28his prescription has been applied in [46] to reproduce the D-terms of intersecting D6-brane models, which automatically include the α (cid:48) corrections of mirror type IIB setups.The latter have been analysed from this viewpoint in the Abelian case in [47]. In thefollowing we will see that it can also be used to reproduce the D-terms of α (cid:48) -correctednon-Abelian D7-brane systems.Indeed, we may apply the above prescription generator by generator of the non-Abeliangauge group of the D7-brane stack, extracting the St¨uckelberg charges Q iα from the cou-plings (cid:82) R , ˜ C α ∧ F i . At the end we obtain that the above prescription amounts to performthe following replacement in (A.5) e → e φ , c a → − v a K , (A.10)that is, to trade the two forms by their partners in the corresponding linear multiplet.We then finally obtain a non-Abelian D-term proportional to λ µ p (cid:90) S S (cid:40) P [ J ] ∧ F + iλ ι Φ ι Φ J ) F − i ι Φ ι Φ J (cid:41) . where we have used that J = e φ / v a ω a . Hence we precisely recover the expression asin (2.12). Finally, a similar analysis can be done for the case of non-vanishing internalB-field to recover (2.11). B Globally nilpotent T-brane backgrounds
In [13] it was recently shown that certain non-Abelian D7-brane vacuum solutions maybe described in terms of a single curved D7-brane. More specifically, these vacua arecompactifications of IIB string theory on R , × C with a globally nilpotent Higgs-vevin SU ( N ). Taking ( x, z ) to parametrise the C -factor, the D7-brane stack on { z = 0 } isdescribed byΦ = φ φ
0. . . . . .0 0 φ N − , φ a = (cid:112) a ( N − a ) e C ab f b / , (B.1)29here C ab is the Cartan matrix of SU ( N ) and the { f a } are functions of the D7-braneworld-volume coordinates ( x, x ). The flux is given as F = − ∂∂f a C a , (B.2)where the C a are the Cartan generators of SU ( N ). In this reference, explicit solutions { f a } to the D-term equations have been computed at leading order in α (cid:48) . This leadingorder solution was then used to provide a description of this system in terms of a single,curved D7-brane. The latter description is in principle valid whenever the field vevs arelarge compared to α (cid:48) , but the authors of [13] noted that their solution should also be validin regions where such vevs are small, due to the characteristic of their solution.In the following we will take a complementary viewpoint and analyse the above back-ground via the non-Abelian Hitchin system, better suited for for small field vevs. We willcompute their α (cid:48) -corrections explicitly and see that, just like in other T-brane backgroundspreserving eight supercharges, the classical solution is still valid after α (cid:48) -corrections aretaken into account. This implies that the classical analysis encodes all the information ofthe system, and that the dictionary built in [13] is not affected by α (cid:48) -corrections.Indeed, from eq.(2.12), we know that the corrected D-term equations are of the form D = D + λ D = 0, with D the leading order D-term and D given by D = (cid:90) S S (cid:40) i D φ ∧ D φ ∧ F − (cid:2) φ, φ (cid:3) F (cid:41) . (B.3)However in this background F , D φ and ¯ Dφ only have legs along d x and d x , and therefore D vanishes identically. Hence, the whole system is insensitive to α (cid:48) -corrections irrespec-tive of how large the values for (cid:104) φ (cid:105) , (cid:104) Dφ (cid:105) and (cid:104) F (cid:105) are. C Non-cartan flux backgrounds
When analysing non-Abelian D-term equations in section 4, we have always made theAnsatz that the gauge transformation g that defines the non-primitive flux lies entirelywithin the Cartan subalgebra of the gauge group G . However, when analysing α (cid:48) -correctedD-terms, the gauge derivatives generically introduce contributions to the D-terms also30long the non-Cartan generators. Hence, it is natural to wonder whether adding world-volume fluxes along non-Cartan generators may provide new solutions to the D-termequations.In general, introducing non-Cartan fluxes via a gauge transformation leads to veryinvolved BPS equations. For the setup at hand we may, however, follow a simple approach.Since we know that at leading order in λ no such flux is required to solve the D-termequations, we may assume that it is purely a λ -correction. This suggests that we capturethe relevant physics if we perform an infinitesimal gauge transformation φ −→ φ + [ δg, φ ] (C.1) A −→ A + i∂δg, (C.2)with δg proportional to some small parameter λ α , [ α ] = L − . In the following we willimplement this strategy for the two T-brane backgrounds analysed in section 4. SU(2) example
Let us consider the SU (2) background analysed in subsection 4.1, which we reproducehere for convenience φ = m e f axe − f , (C.3) F = − i∂∂f σ − i∂∂h . (C.4)On top of this background we perform a gauge transformation of the form δg ≡ λ (cid:18) α E + + α E − (cid:19) , (C.5)where E + = i , E − = − i . (C.6)Notice that the relation between the gauge parameters multiplying E ± is necessary forthe resulting flux to satisfy the Bianchi identity. Acting on the above background suchgauge transformation gives δφ = − iλ m (cid:0) αaxe − f + αe f (cid:1) σ (C.7) δF = − iλ ∂∂ (cid:0) α E + − α E − (cid:1) . (C.8)31e then plug this into the D-term equations and consider the linear terms induced bythis infinitesimal transformation ω ∧ δF + ω (cid:0) [ φ, δφ ] + [ δφ, φ ] (cid:1) = λ (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) (cid:0) α E + + α E − (cid:1) + (C.9)+ λ | m | (cid:0) αax + αe f + α | ax | e − f (cid:1) E + + λ | m | (cid:0) αax + αe f + α | ax | e − f (cid:1) E − . Interestingly the infinitesimal gauge transformation only introduces components in E ± ,which means these new contributions are entirely decoupled from the D-term equationswithin the main text and may be considered independently.From (C.9) we read off, that the parts in E + and E − are conjugate to each other, andso we only need to satisfy one new D-term equation: (cid:0) ∂ x ∂ x + ∂ y ∂ y (cid:1) α = − αax | m | − α | m | (cid:0) e f + | ax | e − f (cid:1) . (C.10)which we may solve asymptotically near the origin by plugging in the solution for f givenin (4.11) α = γ (cid:18) − c | mx | − | mx | c (cid:18) c + | a | | m | (cid:19)(cid:19) , (C.11)where γ ∈ C and [ γ ] = L − . We may interpret this one-parameter solution as a masslessdeformation to the T-brane background allowed at the infinitesimal level by the F- andD-terms. As pointed out in [5], this SU (2) background contains one zero mode preciselyalong the generators E ± . Therefore it is natural to relate the parameter γ with the vevof this zero mode. SU(3) example
Let us now apply this strategy to the SU (3) background of subsection 4.2, more preciselywe act with the infinitesimal gauge transformation δg ≡ λ (cid:18) α E + + α E − (cid:19) , (C.12)on the background (4.26). Now E + = i
00 0 00 0 0 , E − = − i , P = − (C.13)32o this transformation takes us to˜ φ = φ + λ m (cid:0) αmxe − f − αe f (cid:1) P (C.14)˜ A = A + iλ ∂ (cid:0) αE + + αE − (cid:1) , (C.15)so that we get new contributions to the D-term equations given by δD = − iλ ω ∧ ∂∂α E + + λ mx | m | α E + − λ α | m | (cid:0) e f + | mx | e − f (cid:1) E + + h.c. (C.16)again exclusively along the non-Cartan generators E ± . This time the D-term equationshave already some components along such non-Cartan generators. Recall from the discus-sion in the main text that it is precisely this equation that forced to set κ = 0. Thereforeone may wonder if these new contributions proportional to α may allow for a non-trivial κ . Indeed, one can confirm that a gauge transformation given by α = xα + | x | α + x | x | α + . . . , (C.18)where the constant coefficients α i depend intricately on κ, f, . . . is such a solution. Forinstance we have that α = 4 cκµ m ∗ (4 | mµ | λ + 1) (5 c + 4 λ ( | κ | c + ( c + 2) | mµ | ) + 2) × (cid:32) − λ | µ | | m | + 4 λ µ (cid:0) | κ | λ c + c − (cid:1) µ | m | + | m | (cid:0) λ (cid:0) λ | κ | + 9 | κ | (cid:1) c + 5 c − (cid:1) − | a | (cid:0) | κ | λ + 1 (cid:1) (cid:0) | mµ | λ + 1 (cid:1) (cid:33) α = − κµ ( m ∗ ) c . (C.19) D Further SU(2) T-brane backgrounds
We have analysed in section 4 two different cases of T-brane backgrounds, whose non-commuting Higgs field generators lie entirely within an su (2) subalgebra of the Lie group. More precisely, the fourth equation in (4.36) is a linear combination of those in the generators E + and E − — which are conjugate to each other. The equation in E + reads D + = − iλ | m | (cid:0) µe f ∂ x f (cid:0) ∂ y ∂ x h + κ (cid:1) − µae − f (cid:0) x∂ x f − (cid:1) (cid:0) ∂ x ∂ y h + κ (cid:1)(cid:1) (C.17)
33s discussed in section 5, whenever that is the case one may focus on such su (2) subalgebrawhen solving for the α (cid:48) -corrected D-term equations, as the equations corresponding toother generators decouple. In this appendix we will apply the analysis of section 4 tofurther SU (2) T-brane backgrounds, which are also examples of the 2 × α (cid:48) -corrections in T-brane systems.In general we will follow the strategy of subsection 4.1 when analysing the backgroundsbelow. First we consider an Ansatz with a gauge transformation of the form (4.4) with f ≡ f ( x, x, y, y ) and a worldvolume flux of the form (4.14). In general we find thatthe Ansatz for the gauge transformation can be reduced to f ≡ f ( x, x ). Moreover theeffect of κ can be absorbed in the parameter m (cid:48) defined in (4.17) in some cases, likein the T-brane examples 1 and 2, while others like T-brane example 3 seem to require avanishing κ or a non-Cartan gauge transformation (c.f. Appendix C). Second we generaliseour flux background to the form (4.18) and consider the expansion (4.20) for the gaugetransformation, which in practice result in functions f and h that only depend on ( x, x ),at least at lowest order in the expansion parameter λρ . As the procedure is identical forall the cases we present our results in a sketchy way, displaying the independent D-termequations for each Ansatz and the asymptotic solutions near the origin for the second one.All of the following examples satisfy [ φ, φ ] ≡ Cσ for some C depending on the Higgs-vev,which we will use to abbreviate the following expressions. We will compute the D-termequations for the same two Ans¨atze as in 4. That is, on the one hand for a flux consistingof the two components F = − i∂∂f · σ f ≡ f ( x, x, y, y ) H = Im ( κ dx ∧ d ¯ y ) , (D.1)henceforth called Ansatz 1, and on the other hand for F = − i∂∂f · σ H = Im ( κ dx ∧ d ¯ y ) + ρ i ( dx ∧ d ¯ x − dy ∧ d ¯ y ) − i∂ ¯ ∂h f ≡ f ( x, x ) h ≡ h ( x, x, y, y ) , (D.2)34alled Ansatz 2 in the following. T-brane 1 φ hol = m (D.3) Ansatz 1: ( ∂ x ∂ x + ∂ y ∂ y ) f = C (cid:0) λ | κ | + 4 λ Q f (cid:1) − λ | m | e f (cid:0) ∂ y ∂ y f ∂ x f ∂ x f − ∂ y f ∂ x f ∂ x ∂ y f + ∂ y f (cid:0) ∂ y f ∂ x ∂ x f − ∂ y ∂ x f ∂ x f (cid:1)(cid:1) Ansatz 2: ∂ x ∂ x f = C (cid:0) λ Q H (cid:1) ( ∂ x ∂ x + ∂ y ∂ y ) h = − λ | m | e f ∂ x f ∂ x f (cid:0) ∂ y ∂ y h + ρ (cid:1) + 4 Cλ ∂ x ∂ x f ( ρ + ∂ y ∂ y h ) Asymptotic solution f = log c + c | m (cid:48) x | + 12 c | m (cid:48) x | f = − | mx | (cid:0) c λ | m | | m (cid:48) | + c (cid:1) − c | m (cid:48) | | m | x | (cid:0) c λ | m (cid:48) | | m | + 1 (cid:1) h = − c λ ρ | m | | m (cid:48) x | − c λρ | m | | m (cid:48) x | T-brane 2 φ hol = m ax (D.4) Ansatz 1: ( ∂ x ∂ x + ∂ y ∂ y ) f = C (cid:0) λ | κ | + 4 λ Q f (cid:1) − λ | ma | e f (cid:16) ∂ y ∂ y f | x∂ x f + 1 | + 4 | x | | ∂ y f | ∂ x ∂ x f − (cid:0) x∂ y f (cid:0) x∂ x f + 1 (cid:1) ∂ x ∂ y f (cid:1) (cid:17) nsatz 2: ∂ x ∂ x f = C (cid:0) λ Q H (cid:1) ( ∂ x ∂ x + ∂ y ∂ y ) h = − λ | ma | e f | x∂ x f + 1 | (cid:0) ∂ y ∂ y h + ρ (cid:1) + 4 λ C∂ x ∂ x f ( ρ + ∂ y ∂ y h ) Asymptotic solution f = log c + 14 c | m (cid:48) a | | x | f = − c | am | | x | (cid:0) λ c | am | + 1 (cid:1) h = − λ ρc | amx | T-brane 3 φ hol = m by ax by (D.5) Ansatz 1: (cid:0) λ | mb | (cid:1) ∂ x ∂ x f − ∂ y ∂ y f = C (cid:0) λ | κ | + 4 λ Q f (cid:1) − λ | ma | e f (cid:16) ∂ y ∂ y f | x∂ x f + 1 | + 4 | x | | ∂ y f | ∂ x ∂ x f − (cid:0) x∂ y f (cid:0) x∂ x f + 1 (cid:1) ∂ x ∂ y f (cid:1) (cid:17) − iλ ab | m | κe f (2 x∂ x f + 1) Ansatz 2: ∂ x ∂ x f (cid:0) λ | mb | (cid:1) = C (cid:0) λ Q H (cid:1) ( ∂ x ∂ x + ∂ y ∂ y ) h = − λ | m | (cid:0) | a | e f | x∂ x f + 1 | (cid:0) ∂ y ∂ y h + ρ (cid:1) + 2 | b | (cid:0) ∂ x ∂ x h − ρ (cid:1)(cid:1) + 4 λ C∂ x ∂ x f ( ρ + ∂ y ∂ y h )0 = (2 x∂ x f + 1) (cid:0) ∂ y ∂ x h + κ (cid:1) symptotic solution f = log c + | am | | x | c λ | bm | + 4 f = − c | am | | x | (cid:0) λ c | ma | + 1 (cid:1) h = − λ ρ | mx | ( c | a | − | b | )4 λ | bm | + 1 T-brane 4 φ hol = m axby (D.6) Ansatz 1: ( ∂ x ∂ x + ∂ y ∂ y ) f = C (cid:0) λ | κ | + 4 λ Q f (cid:1) − λ | ma | e f (cid:32) ∂ y ∂ y f | x∂ x f + 1 | + 4 | x | | ∂ y f | ∂ x ∂ x f − (cid:0) x∂ y f (cid:0) x∂ x f + 1 (cid:1) ∂ x ∂ y f (cid:1) (cid:33) − λ | mb | e − f (cid:32) ∂ x ∂ x f | y∂ y f − | + 4 | y | | ∂ x f | ∂ y ∂ y f + 4Re (cid:0) y (cid:0) − y∂ y f (cid:1) ∂ y ∂ x f ∂ x f (cid:1) (cid:33) | a | e f Re (cid:0) κx∂ y f (cid:0) x∂ x f + 1 (cid:1)(cid:1) + | b | e − f Re (cid:0) κy∂ x f (2 y∂ y f − (cid:1) References [1] L. E. Ib´a˜nez and A. M. Uranga,
String Theory and Particle Physics. An Introductionto String Phenomenology , Cambridge University Press (2012).[2] R. C. Myers, “NonAbelian phenomena on D branes,”
Class. Quant. Grav. , S347(2003) [hep-th/0303072].[3] R. Donagi, S. Katz and E. Sharpe, “Spectra of D-branes with higgs vevs,” Adv. Theor.Math. Phys. , no. 5, 813 (2004) [hep-th/0309270].374] H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, “Flavor Structure in F-theoryCompactifications,” JHEP , 036 (2010) [arXiv:0910.2762 [hep-th]].[5] S. Cecotti, C. Cordova, J. J. Heckman and C. Vafa, “T-Branes and Monodromy,”
JHEP , 030 (2011) [arXiv:1010.5780 [hep-th]].[6] R. Donagi and M. Wijnholt, “Gluing Branes, I,”
JHEP , 068 (2013)[arXiv:1104.2610 [hep-th]].[7] R. Donagi and M. Wijnholt, “Gluing Branes II: Flavour Physics and String Duality,”
JHEP , 092 (2013) [arXiv:1112.4854 [hep-th]].[8] J. Marsano, N. Saulina and S. Schfer-Nameki, “Global Gluing and G -flux,” JHEP , 001 (2013) [arXiv:1211.1097 [hep-th]].[9] L. B. Anderson, J. J. Heckman and S. Katz, “T-Branes and Geometry,”
JHEP (2014) 080 [arXiv:1310.1931 [hep-th]].[10] A. Collinucci and R. Savelli, “T-branes as branes within branes,”
JHEP (2015)161 [arXiv:1410.4178 [hep-th]].[11] A. Collinucci and R. Savelli, “F-theory on singular spaces,”
JHEP , 100 (2015)[arXiv:1410.4867 [hep-th]].[12] A. Collinucci, S. Giacomelli, R. Savelli and R. Valandro, “T-branes through 3d mirrorsymmetry,”
JHEP , 093 (2016) [arXiv:1603.00062 [hep-th]].[13] I. Bena, J. Blab¨ack, R. Minasian and R. Savelli, “There and back again: A T-brane’stale,” arXiv:1608.01221 [hep-th].[14] R. Donagi and M. Wijnholt, “Model Building with F-Theory,”
Adv. Theor. Math.Phys. , no. 5, 1237 (2011) [arXiv:0802.2969 [hep-th]].[15] C. Beasley, J. J. Heckman and C. Vafa, “GUTs and Exceptional Branes in F-theory- I,” JHEP , 058 (2009) [arXiv:0802.3391 [hep-th]].[16] C. Beasley, J. J. Heckman and C. Vafa, “GUTs and Exceptional Branes in F-theory- II: Experimental Predictions,”
JHEP , 059 (2009) [arXiv:0806.0102 [hep-th]].3817] R. Donagi and M. Wijnholt, “Breaking GUT Groups in F-Theory,”
Adv. Theor.Math. Phys. , no. 6, 1523 (2011) [arXiv:0808.2223 [hep-th]].[18] J. J. Heckman and C. Vafa, “Flavor Hierarchy From F-theory,” Nucl. Phys. B ,137 (2010) [arXiv:0811.2417 [hep-th]].[19] H. Hayashi, T. Kawano, R. Tatar and T. Watari, “Codimension-3 Singularities andYukawa Couplings in F-theory,”
Nucl. Phys. B , 47 (2009) [arXiv:0901.4941 [hep-th]].[20] L. Randall and D. Simmons-Duffin, “Quark and Lepton Flavor Physics from F-Theory,” arXiv:0904.1584 [hep-ph].[21] A. Font and L. E. Ib´a˜nez, “Matter wave functions and Yukawa couplings in F-theoryGrand Unification,”
JHEP , 036 (2009) [arXiv:0907.4895 [hep-th]].[22] S. Cecotti, M. C. N. Cheng, J. J. Heckman and C. Vafa, “Yukawa Couplings inF-theory and Non-Commutative Geometry,” arXiv:0910.0477 [hep-th].[23] J. P. Conlon and E. Palti, “Aspects of Flavour and Supersymmetry in F-theoryGUTs,”
JHEP , 029 (2010) [arXiv:0910.2413 [hep-th]].[24] F. Marchesano and L. Martucci, “Non-perturbative effects on seven-brane Yukawacouplings,”
Phys. Rev. Lett. , 231601 (2010) [arXiv:0910.5496 [hep-th]].[25] C. C. Chiou, A. E. Faraggi, R. Tatar and W. Walters, “T-branes and Yukawa Cou-plings,”
JHEP , 023 (2011) [arXiv:1101.2455 [hep-th]].[26] A. Font, F. Marchesano, D. Regalado and G. Zoccarato, “Up-type quark masses inSU(5) F-theory models,”
JHEP (2013) 125 [arXiv:1307.8089 [hep-th]].[27] F. Marchesano, D. Regalado and G. Zoccarato, “Yukawa hierarchies at the point ofE in F-theory,” JHEP (2015) 179 [arXiv:1503.02683 [hep-th]].[28] F. Carta, F. Marchesano and G. Zoccarato, “Fitting fermion masses and mixings inF-theory GUTs,”
JHEP (2016) 126 [arXiv:1512.04846 [hep-th]].3929] R. Minasian and A. Tomasiello, “Variations on stability,”
Nucl. Phys. B , 43(2002) [hep-th/0104041].[30] A. Butti, D. Forcella, L. Martucci, R. Minasian, M. Petrini and A. Zaffaroni, “Onthe geometry and the moduli space of beta-deformed quiver gauge theories,”
JHEP (2008) 053 [arXiv:0712.1215 [hep-th]].[31] F. Marchesano, P. McGuirk and G. Shiu, “Chiral matter wavefunctions in warpedcompactifications,”
JHEP , 090 (2011) [arXiv:1012.2759 [hep-th]].[32] M. Marino, R. Minasian, G. W. Moore and A. Strominger, “Nonlinear instantonsfrom supersymmetric p-branes,”
JHEP , 005 (2000) [hep-th/9911206].[33] J. Gomis, F. Marchesano and D. Mateos, “An Open string landscape,”
JHEP ,021 (2005) [hep-th/0506179].[34] L. Martucci and P. Smyth, “Supersymmetric D-branes and calibrations on generalN=1 backgrounds,”
JHEP , 048 (2005) [hep-th/0507099].[35] H. Jockers and J. Louis, “D-terms and F-terms from D7-brane fluxes,”
Nucl. Phys.B , 203 (2005) [hep-th/0502059].[36] L. Martucci, “D-branes on general N=1 backgrounds: Superpotentials and D-terms,”
JHEP , 033 (2006) [hep-th/0602129].[37] R. C. Myers, “Dielectric branes,”
JHEP (1999) 022 [hep-th/9910053].[38] M. B. Green, J. A. Harvey and G. W. Moore, “I-brane inflow and anomalous cou-plings on d-branes,”
Class. Quant. Grav. (1997) 47 [hep-th/9605033].[39] Y. K. E. Cheung and Z. Yin, “Anomalies, branes, and currents,” Nucl. Phys. B ,69 (1998) [hep-th/9710206].[40] R. Minasian and G. W. Moore, “K theory and Ramond-Ramond charge,”
JHEP , 002 (1997) [hep-th/9710230].[41] J. F. Morales, C. A. Scrucca and M. Serone, “Anomalous couplings for D-branes andO-planes,”
Nucl. Phys. B (1999) 291 [hep-th/9812071].4042] M. Haack, D. Krefl, D. L¨ust, A. Van Proeyen and M. Zagermann, “Gaugino Con-densates and D-terms from D7-branes,”
JHEP , 078 (2007) [hep-th/0609211].[43] B. M. McCoy, C. A. Tracy and T. T. Wu, “Painleve Functions of the Third Kind,”
J. Math. Phys. , 1058 (1977).[44] M. Cicoli, F. Quevedo and R. Valandro, “De Sitter from T-branes,” JHEP ,141 (2016) [arXiv:1512.04558 [hep-th]].[45] M. Dine, N. Seiberg and E. Witten, “Fayet-Iliopoulos Terms in String Theory,”
Nucl.Phys. B (1987) 589.[46] D. Cremades, L. E. Ib´a˜nez and F. Marchesano, “SUSY quivers, intersecting branesand the modest hierarchy problem,”
JHEP , 009 (2002) [hep-th/0201205].[47] E. Plauschinn, “The Generalized Green-Schwarz Mechanism for Type IIB Orien-tifolds with D3- and D7-Branes,”
JHEP0905