Intersecting branes, Higgs sector, and chirality from $\mathcal{N}=4$ SYM with soft SUSY breaking

We consider $SU(N)$ $\mathcal{N}=4$ super Yang-Mills with cubic and quadratic soft SUSY breaking potential, such that the global $SU(4)_R$ is broken to $SU(3)$ or further. As shown recently, this set-up supports a rich set of non-trivial vacua with the geometry of self-intersecting $SU(3)$ branes in 6 extra dimensions. The zero modes on these branes can be interpreted as 3 generations of bosonic and chiral fermionic strings connecting the branes at their intersections. Here, we uncover a large class of exact solutions consisting of branes connected by Higgs condensates, leading to Yukawa couplings between the chiral fermionic zero modes. Under certain decoupling conditions, the backreaction of the Higgs on the branes vanishes exactly. The resulting physics is that of a spontaneously broken chiral gauge theory on branes with fluxes. In particular, we identify combined brane plus Higgs configurations which lead to gauge fields that couple to chiral fermions at low energy. This turns out to be quite close to the Standard Model and its constructions via branes in string theory. As a by-product, we construct a $G_2$-brane solution corresponding to a squashed fuzzy coadjoint orbit of $G_2$.

Among all 4-dimensional supersymmetric gauge theories, N = 4 super Yang-Mills (SYM) is the most special and rigid model. Due to this rigidity, N = 4 SYM has remarkable properties as a quantum field theory, such as conformal invariance and UV-finiteness [1,2]. It arises in various contexts including compactification of 10-dimensional SYM on T 6 , IIB string theory on D3 branes, and the AdS/CFT correspondence [3]. Its amplitudes exhibit a rich structure [4,5], and powerful tools such as integrability provided many remarkable insights [6]. On the other hand, the special structure of N = 4 SYM limits its applicability to realworld physics. The SUSY vacua of N = 4 SYM with SU (N ) gauge group have a simple structure, parametrized by the three commuting vacuum expectation values (VEVs) of the N = 1 chiral superfields. However, deformed or softly broken versions of N = 4 SYM, for example via extra terms in the potential, may lead to more interesting vacua and low-energy physics. For instance, it is well-known that certain mass-deformations to the superpotential, which preserve N = 1, have a much richer vacuum structure [7][8][9][10][11][12], including stacks of fuzzy behavior reminiscent of the Standard Model might be achieved for larger branes; however for generic branes we can only provide evidence for the existence of such Higgs solutions.
Summary. Due to the length of the article, we provide a brief summary of the key points already at this stage. Our starting point is the N = 4 SYM Lagrangian for gauge group SU (N ) with the following cubic and quadratic soft SUSY breaking terms with X − i := (X + i ) † , i = 1, 2, 3. We will focus on vacuum solutions X ± i ∈ Mat(N, C). Our first new insight is the following observation: The second order, i.e. double commutator, equations of motion (eom) for X admit first integral equations The full equations of motion with arbitrary mass term M ≡ M i ∈ [0, 1 2 ] are solved by a rescaling with R(M ) = 1 2 (1 + √ 1 − 4M 2 ) and any X + i that satisfy (1.2). As demonstrated in [25], this allows for vacua describing squashed fuzzy SU between background X and fluctuations φ. Our second new insight concerns properties of the regular zero modes: first, the set of regular zero modes, in the sense of (1.4), form a ring which is graded by the integral weights of su (3). Second, the decoupling conditions between X and φ are sufficient for a decoupling of the potential (1.5a) and the full equations of motion also decouple, eom(X + φ) = eom(X) + eom(φ) , (1.5b) for arbitrary mass values.
Our third new insight lies in the following construction: starting from a background X, we add regular zero modes φ which satisfy the equations of motion. Then X + φ is an exact solution due to (1.5b). Examples include (i) zero modes φ with maximal rank that reduce X + φ to the fuzzy 2-sphere, and (ii) zero modes φ with rank one such that X + φ is interpreted as C[µ] brane plus string modes. The latter provide Yukawa couplings for the analogous fermionic zero modes at the brane intersections.
While the spectra of O X+φ V and / D X+φ are not understood analytically, numerical studies show the existence of instabilities in the massless case. However, a surprising but simple observation shows that the full potential can be expressed in complete squares, such that it becomes positive semi-definite for M ≥ has no instabilities, and moreover numerical studies indicate that the number of zero modes is drastically reduced in comparison to O X V , and independent of N . This implies that our non-trivial (brane+Higgs) solutions are locally stable up to a compact moduli space, which is understood in several interesting cases.
These results are remarkable, because they point to the possibility to obtain interesting chiral low energy gauge theories from softly broken N = 4 SYM. The C[µ] solutions behave as self-intersecting branes in extra dimensions, with chiral fermionic zero modes located at the intersections. In [25,26] it has been argued that some Higgs moduli on the C[µ] branes need to acquire VEVs in order to stabilize the system and to give masses to undesired fermions via Yukawa couplings. Here, we established the existence of such Higgs VEVs, which give masses not only to fermions, but also to most of the bosonic moduli, as shown by the spectrum of O X+φ V .
There are two types of branes C[(N, 0)] and C[(N 1 , N 2 )] with different chirality properties. The space-filling 6-dimensional C[(N 1 , N 2 )] have a built-in separation of chiral modes, as seen through a suitable gauge boson mode of χ (4.1). We focus on the simplest C[(1, 1)] brane, for which we find an exact Higgs solution, and the more general rigged C[(N, 1)] branes. These are the basis for our approach to the Standard Model. On the 4-dimensional counterparts C[(N, 0)], a chiral gauge theory would require a configuration of Higgs modes which is not supported by our results.
There is another interesting general message: the underlying models are gauge theories with a gauge group of large rank N 1. In the trivial vacuum, such large-N gauge theories are governed by the t'Hooft coupling λ = g 2 N . In contrast, the C[(N 1 , N 2 )] vacua with large N i behave as semi-classical, large fuzzy branes. The fluctuation spectrum on these vacua consist of a tower of KK states with a finite mass gap (independent of N ), as well as the Higgs sector of zero modes. As discussed in section 2.5, this Higgs sector consists of string-like modes, which link some sheets of these branes with coupling strength g, as well as semi-classical modes, which are almost free. Moreover, most of the latter disappear as the string-like modes acquire some VEV. Hence, the large number of massless fluctuations in the original large-N theory is reduced to a small sector of string-like modes, which acquire VEVs, and leave few remaining zero modes. This mean that the original large-N gauge theory reduces to an effective lowenergy theory with few modes and an interesting geometric structure. This certainly provides strong motivation to study such scenarios in more detail.
Outline. The paper is organized as follows: We start by recalling N = 4 SYM in section 2, which is modified by cubic and quadratic soft breaking terms. The properties of the squashed coadjoint SU (3) orbits and the classification of the bosonic zero modes are reviewed, and the potential is organized accordingly. In section 3, we focus on the exact (classical) C[µ] solutions of 4-dimensional type, and study their zero mode sector. We find various exact solutions in the massless case, and show that all of these are free of instabilities in the presence of certain mass parameters. Moving on to 6-dimensional squashed orbits and their zero modes, we show in section 4 that chiral settings are simpler and more naturally obtained here, and identify the chirality observable χ, see (4.1). We find again exact solutions for the simplest 6-dimensional case C[(1, 1)], and comment on generalizations for larger branes and the issue of three generations. The fermionic zero modes and their Yukawa couplings are discussed in section 5. In section 6, we give a qualitative discussion how low-energy theories resembling the Standard Model might be obtained using the present framework. Finally, we conclude in section 7.
Two appendices complement this article, exemplifying solutions to the equations of motion in appendix A, and spelling out details of the novel combined solutions and the spectra of the vector Laplacian and the Dirac operator in appendix B.

Background, zero modes and Higgs potential
First we recall the setting from [25][26][27]. We start with the action of N = 4 SU (N ) SYM, which is organized most transparently in terms of 10-dimensional SYM reduced to 4 dimensions: (2.1) Here F µν is the field strength, D µ = ∂ µ − i[A µ , ·] the gauge covariant derivative, φ a , a ∈ {1, 2, 4, 5, 6, 7} are 6 scalar fields, Ψ is a matrix-valued Majorana-Weyl (MW) spinor of SO(9, 1) dimensionally reduced to 4-dimensions, and Γ a arise from the 10-dimensional gamma matrices. All fields transform take values in su(N ) and transform in the adjoint of the SU (N ) gauge symmetry. The global SO(6) R symmetry is manifest. It will be useful to work with the following complex linear combinations of dimensionless scalar fields with m having the dimension of a mass. For later, we also introduce the notation X ± j ≡ X ±α j with a normalization such that X α = X −α ∀α ∈ I = {±α j , j = 1, 2, 3} . (2.3) To introduce a scale and to allow non-trivial brane solutions, we add soft terms to the potential, thereby fixing the scale m. The cubic potential can be written as corresponding to a holomorphic 3-form on C 3 . Rewritten in a real basis, this is recognized as the structure constants of su (3) projected to the root generators [25]. The cubic term breaks the global SU (4) R symmetry to SU (3) R , which is in a sense the minimal breaking possible. The mass terms M i may break this further: for equal M i ≡ M ≥ 0, the SU (3) R is maintained. For M 1 = M 2 = M 3 (or permutations thereof) one has a global (SU (2) × U (1)) R , and if all masses are distinct there is only a (U (1) × U (1)) R left.

Preliminaries
In this section we perform some algebraic manipulations of the full potential, which allow to derive first integral equations. Also, we introduce notation for the treatment of perturbations.
Rewriting the potential. We reconsider the full potential To begin with, we rewrite the quartic potential by using the Jacobi identity of the commutator and cyclicity of the trace. This results in The cubic potential (2.7) can be expressed as By completing the square, we arrive at the following expression for the total potential Perturbation of the background. Let us add a perturbation φ α to a background X α , As discussed in [26], the perturbations imply further symmetry breaking and lead to interesting low-energy physics in the zero-mode sector of the background X . The complete potential can be worked out, Here can be viewed as gauge-fixing function in extra dimensions, and, following [27], we define following operators Equations of motion. The equations of motion (eom) for the background X can be written as For classical vacua, i.e. space-time independent X, the eom reduce and can be cast in the following form: with D as in (2.10). We observe that B ij of (2.10) and B ij are equivalent upon rescaling X.
Homogeneity of potential. The full potential exhibits a certain homogeneity pattern, which implies the relation 4V 4 + 3V 3 + 2V 2 = 0 for solutions. Hence the potential value at a solution X of the eom can be computed via

Squashed SU (3) brane solutions
It is well-known that the potential (2.4) with (2.6) has fuzzy sphere solutions X ± i ∼ R ± i J i where J i are generators of su(2) [7,10,13,14,[28][29][30][31]. However as shown in [25], there are also solutions with much richer structure corresponding to (stacks of) squashed fuzzy coadjoint SU (3) orbits C N [µ], obtained by the following ansatz We denote these as (squashed) SU (3) branes, and they are the focus of this paper. Here are root generators of su(3) X ⊂ su(N ), π is any representation on H ∼ = C N , and α 1 , α 2 are the simple roots with α 3 = −(α 1 + α 2 ). In these conventions 3 , the Lie algebra relations are 25) and in particular where (·, ·) denotes the Killing form of su (3). Note also that the choice of labeling of Using the Lie algebra relations, the equations of motion (2.20) become Assuming M i = 0 for simplicity, these equations have the non-trivial solution  A, we conclude that all R i must be equal (up to an irrelevant sign) if all M 2 i are equal. In addition, we observe that there are no solution if one of the mass parameters satisfies M 2 i ≥ 4 3 , in agreement with earlier findings [26].
where τ i denotes the Cartan generators of SU (3) R . This is a combination of the global (U (1) × U (1)) R ⊂ SU (3) R symmetry and the SU (3) X ⊂ SU (N ) gauge symmetry. We denote these U (1) K i -charges by for any 4 scalar field φ, and similarly also for fermionic fields. They define a su(3) weight lattice. The su(3) X ⊂ su(N ) gauge charges will be denoted as usual by λ(φ) ∈ su(3) * , so that K = 2τ − λ. They all live in the same su(3) weight lattice. Furthermore, we will need the τ -parity generator τ in U (3) R defined by which is broken by the cubic potential (2.7).

Geometric significance: self-intersecting branes
The qualitative features of the above solutions, and in particular the chirality properties of the fermionic zero modes discussed below, can be understood in terms of the semi-classical geometry of the solutions, interpreted as self-intersecting branes in R 6 . As for all quantized coadjoint orbits, the semi-classical geometry of C N [µ] can be extracted using coherent states. These are precisely the SU (3) orbits {g · |x , g ∈ SU (3)} of the extremal weight states |x of the irreps H Λ ∈ End(H µ ), where H µ is the representation space for the su(3) representation π µ . The set of coherent states |gx := g · |x forms a U (1) bundle over the coadjoint orbit O[µ], and the semi-classical base manifold embedded in R 6 (the brane) is recovered from the expectation values of these coherent states  [25][26][27]. The extremal weight states W|x , which lie on the discrete orbit of the Weyl group W through |x , are projected to the origin of weight space. To see this, it is sufficient to note that x|X α |x = 0 as any X ± j annihilates either |x or x|. The tangent space of the sub-variety C N [µ] is obtained by acting with the three SU (2) α subgroups, which correspond to the roots X α of su(3), on the set of extremal weight states W|x . For µ = (N, 0) and µ = (0, N ), one of these actions vanishes, leading to the 4-dimensional self-intersecting CP 2 branes depicted in figure 1b. For generic µ = (N 1 , N 2 ), the C N [µ] are 6-dimensional sub-varieties in R 6 , i.e. they are locally space-filling branes. Moreover, the non-degenerate Poisson structure on C N [µ] is recovered from the commutators [X α , X β ] ∼ i{x α , x β }, whose Pfaffian is measured by the operator χ, see (4.1). Therefore the generic 6-dimensional branes, which carry a rank 6 flux due to the symplectic form, consist of 3+3 locally space-filling sheets that cover the origin.

Fluctuations and zero modes
Consider the fluctuations of the scalar fields on a background brane C[µ] with representation space H. To organize the degrees of freedom End(H), which denotes the algebra of all possible functions, we note that the solutions (2.23) define an embedding SU (3) X ⊂ SU (N ), which acts via the adjoint on all the fields. Consequently, we decompose the su(N )-valued fields into harmonics, i.e. irreps of this SU where H Λ denotes the irreps with highest weight Λ appearing with multiplicity n Λ . For the case of µ = (N, 0) or µ = (0, N ), this decomposition is given by While the SU (3) X gauge transformations do not act on the indices α of the scalar fields φ α , the (U (1) × U (1)) K i symmetry (2.29) does act, and the α realize 3+3 of the 8 states in (1, 1) of the su(3) weight lattice due to the origin of C[µ] as a SU (3) coadjoint orbit. This allows to organize the various harmonics, which will be very useful. Assume first M i = 0. Then the squashed brane backgrounds X α admit a number of bosonic zero modes φ (0) α , as shown in [25]. To see this, we note that the bilinear form defined by / D mix on a background (2.23) can be simplified, for example, as follows: using (2.26a), where the φ ± j are perturbations as in (2.15). For R i ≡ R, this has the form of the quadratic contribution from the cubic potential (2.16), and the quadratic terms in the potential can be combined as follows: It has been proven in [25] that, under these conditions, O X V is positive semi-definite for all representations π. The zero modes of O X V fall into two classes, denoted as regular and exceptional zero modes. We will focus on the regular zero modes, and show that their classification is based on a decoupling condition, as discussed in the next section.

Regular zero modes and decoupling condition
Let φ α and ψ α , α ∈ I be two arbitrary matrix configurations, each consisting of three complex matrices or equivalently six hermitian matrices. We will say that φ α and ψ α satisfy the decoupling condition if [ψ α , φ β ] = 0 whenever α + β ∈ I or α + β = 0 (2.39) This condition is symmetric under ψ ↔ φ, and equivalent to We define regular zero modes φ α of O X V to be modes that satisfy the decoupling condition w.r.t. X α . As we will see momentarily, these modes are in fact zero modes of O X V for R i = R and M i = 0, and this definition is then equivalent to the one given in [25,27]. The conditions (2.39) or (2.40) amount to the requirement that φ α is annihilated by three ladder operators out of the six X ± j . For example, the condition for φ + i is Before classifying them more explicitly, we state some important consequences. For M i = 0, the regular zero modes φ α satisfy To prove the first statement of (2.42), consider where we have used the algebra relation (2.26a) and the Jacobi identity. All indicated commutators vanish due to decoupling condition (2.41). For the second statement of (2.42), consider Consequently, the regular zero modes, as defined by the decoupling condition, are indeed zero modes of the SU (3) branes if all R i are equal and all mass parameters M i vanish. We shall keep this name also in the general case of non-vanishing masses and distinct R i , which is discussed in section 3.1.3. Now we relate this to the group-theoretical classification of the regular zero modes on a squashed SU (3) brane C[µ] given in [25,26]. The decoupling conditions imply that any regular zero mode φ α for any fixed α ∈ I is an extremal weight vector with su(3) X weight λ in some irrep H Λ ⊂ End(H) of the decomposition (2.35). In view of (2.41), the arrow λ must be the extremal weight vector in the Weyl chamber opposite to the polarization α (or possibly on its wall). Recalling the unbroken U (1) K i of the background, this means that it is one of the six extremal U (1) K i weights 5 Λ = α − λ of any φ α ∈ H Λ , and we denote it by Hence the regular zero modes φ α,Λ have charge Λ = α − λ under the K i , corresponding to a point of the su(3) weight lattice in (the interior of) the Weyl chamber of α. The eigenvalue τ = ±1 determined by the parity of the Weyl chamber of α = ±α i . Clearly there is only one (extremal) state in H Λ for any such Λ . Since / D mix preserves Λ but flips the τ -parity (recalling τ / D mix = − / D mix τ from [26]), it follows again that / D X mix φ = 0, i.e. (2.42) holds. This provides another way to characterize the regular zero modes.
We now observe that the regular zero modes form a ring, in the following sense: For each α, let V α = {φ α,Λ } be the vector space of regular zero modes with polarization α. According to (2.48), this vector space is graded by the integral weights in the Weyl chamber corresponding to α. Moreover, the decoupling condition (2.39) -or equivalently the extremal weight property -implies that if φ α , φ α are regular zero modes with the same polarization α, then so is their (matrix) product φ α φ α . Hence, each V α is a graded ring. Moreover, the vector space V = ⊕ α V α forms a ring graded by the (integral) weights of su(3) (or rather of U (1) K i ), where we define the product of zero modes in different Weyl chambers to vanish 6 . This structure is respected by the Weyl group W, which relates the different V α . In particular, all ring elements are nilpotent, due to the cutoff in Λ.
The same analysis goes through for stacks of branes, where analogous zero modes arise as strings connecting different branes. As aforementioned, we will denote the space of zero modes as Higgs sector. Labeling the six-component vector φ = (φ α ) of zero modes by the dominant U (1) K i weight, we learn that the Higgs sector forms a nilpotent ring graded by integral weights of su(3).
Examples. Starting from the simplest solution to the decoupling condition one can use the ring multiplication of the regular zero modes to construct modes of the form and any linear combinations of these. A possible background with such a zero mode would then be On a single squashed CP 2 N brane C[(N, 0)], these exhausts all regular zero modes. The ring structure is given by α,α . In particular, the regular zero modes with maximal n = N on squashed CP 2 N link the 3 intersecting R 4 sheets at the origin, with polarization along the common R 2 [25]. These string-modes are given by |i−1, µ i, µ|, where {|1, µ , |2, µ , |3, µ } denote the 3 extremal weight vectors of µ = (N, 0). These are coherent states located at the origin on each of the three sheets. An artist's rendering of such a string mode underlying this solution is given in figure 1b. More generally, the regular zero modes can be interpreted as strings linking these sheets, shifted along their intersection.
Exceptional zero modes. In addition to the regular zero modes, there are certain exceptional zero modes which are described in [25,26]. For the squashed CP 2 N solutions, the only exceptional modes are the six Goldstone bosons corresponding to the spontaneously broken SU (3)/(U (1) × U (1)). These are easily understood also in the deformed settings considered below. For the more general branes, the explicit description of the exceptional zero modes is not known. This is one reason why we focus mostly on the CP 2 branes in this paper. Z 3 and generations. Note that the U (3) R label α for the scalar fields determines a 3-family structure, which reflects the Weyl group Z 3 × Z 2 = W. This coincides with the family and the τ -parity as determined by the U (1) i charges of the zero modes. The Z 3 × Z 2 structure is indicated by the field labels as in φ i± . This will be useful for selection rules etc.
Stringy versus semi-classical modes. The above characterization of regular zero modes hides the fact that they come with very different characteristics. We single out two extreme types: (i) the maximal or stringy zero modes, and (ii) the semi-classical or almost-commutative zero modes. The distinction corresponds to the separation of functions on non-squashed SU (3) coadjoint orbits C[µ] into UV and IR sector, as discussed in [32], but they are also distinguished by their coupling strength: All Higgs modes couple to all other (scalar, gauge and fermionic) fields through commutators [φ α , ·]. To quantify this strength, we first need to proper normalize the Higgs modes. This is dictated by the kinetic term 7 i.e. all modes should be normalized w.r.t. the trace tr(). Now the maximal Higgs modes φ string α,Λ are the ones with extremal su(3) K weight Λ ; these modes are given by rank one operators linking the extremal (i.e. coherent) weight states |x , |x of H Λ ⊂ End(H) whose weight difference is maximal. From the brane point of view, these are the extreme UV modes with the maximal momentum [32]. Since the expectation values vanish, x|X α |x = 0, these states are localized at the origin, and φ string should be interpreted as strings linking the sheets of squashed C[µ] at the origin. For large N , there are other almost-maximal zero modes, for example with rank 2 such as the modes considered in section 4. These almost-maximal modes have a similar properties as the maximal modes; hence, this broader class will be called stringy modes. The commutator of these zero modes is of order one, i.e. they are completely non-commutative functions on the branes. Putting back g (2.52), we see that their interaction strength is characterized by g.
In contrast, the zero modes φ low α,Λ with small weight Λ are matrices with high, typically maximal rank. These modes correspond to slowly-varying, semi-classical functions φ low α,Λ (x) on the C[µ] brane, given by polynomials of small degree in the X α , the lowest mode being The normalization c N for the minimal zero mode on an irreducible C[µ] can be obtained from the quadratic Casimir on H µ , which means that the φ a are almost-commutative functions on C[µ], where {·, ·} denotes the Poisson bracket, as usual for low-energy functions on fuzzy spaces. Hence their interaction strength is small, and they are become free fields as N → ∞.
To summarize, the vast number of scalar modes for large N becomes gapped on an irreducible C[µ] vacuum, and the zero modes consist of stringy modes with interaction strength g, as well as weakly interacting semi-classical modes. We will see below that the stringy modes form bound states which are stable at least for a special value of the mass parameter, in which case the zero modes are further reduced to a small number independent of N .

Aspects of the Higgs potential
Now consider the interacting potential for the Higgs sector, i.e. the zero modes φ α on a background solution X. The linear term in φ vanishes due to the eom for X, so that the effective potential for φ obtained from (2.16) is For the regular zero modes, the cubic interaction V 31 arising from the quartic term drops out. To see this, consider using the Jacobi identity, φ β = φ −β , and the gauge-fixing condition f = [X α , φ α ] = 0, which follows from the decoupling condition (2.39). Since either α ± β ∈ I or α ± β = 0 for any pair of roots α, β of su(3), the V 31 term (2.60) vanishes for the regular zero-modes, again due to the decoupling condition. Therefore V eff [φ] reduces for the regular zero mode sector to The second term is nothing else than 1 2 tr(φ α O X V,M i ≡0 φ α ) and vanishes since φ α are regular zero modes, see (2.42). Hence, the effective potential for the regular zero modes is given by and we arrive at We emphasize that the argument remains valid for any element of the ring of regular zero modes: suppose φ and φ are two six-component vectors of regular zero modes, then holds, with the component-wise multiplication defined in section 2.5. However, do not necessarily decompose in any linear fashion. Similarly, one can extend to discussion to Higgs modes connecting stacks of branes and the argument remains the same.
In particular, the potential for φ has the same structure as the original potential (2.6) for the model. Thus the quadratic potential V 2 [φ] vanishes again in the absence of mass terms, but the cubic term V 3 [φ] may entail some unstable directions. Some of the φ α are then expected to take a non-trivial VEV, which is then stabilized by the quartic term. We will indeed find such non-trivial minima for φ α , which will be denoted as Higgs vacua. Such a non-trivial Higgs linking different branes leads to a bound state of the branes.
As a further remark, we note that the mixed term quadratic in both X and φ can be written equivalently as assuming the decoupling condition between φ and X. The above argument for (2.60) to vanish does not apply to the exceptional zero modes. For single branes of CP 2 type, these are precisely the SU (3) R Goldstone bosons, which can be studied separately. However there exist other exceptional zero modes, e.g. Λ ∈ W(1, 0), Λ ∈ W(2, 0) (or conjugate) connecting C[(0, 1)] with C[(1, 0)] which need to be studied separately.
We will see that in the presence of positive mass terms M 2 i > 0, the above instability can be stabilized. However if the M 2 i are not all equal, some of the φ α turn out to acquire a negative mass. This will also lead to a non-trivial Higgs vacuum. Quantum corrections 8 might also play an important role, however we will assume that these are subleading if the branes are sufficiently large, so that the semi-classical description is valid.
Full equations of motion and decoupling. We have just seen that V(X + φ) = V(X) + V(φ) for a sum of backgrounds X, φ satisfying the decoupling condition (2.39). Now, we would like to know if such a composed background Y α = X α + φ α can be an exact solution of the full action. To address this, we cannot rely on the above reduced action for the Higgs sector, because we need δS = 0 for arbitrary fluctuations of Y α .
Consider combined (static) configurations of the form X a + φ a which satisfies the equation of motion To simplify this, consider the matrix Laplacian X+φ acting on an arbitrary ψ ∈ End(H). We obtain (2.67) Applying this on X + φ leads to (2.68) If X α and φ α satisfy the decoupling condition (2.39), we can replace / D X ad φ = / D X diag φ etc., and also [X β , Y β ] = 0 holds. This further simplifies the Laplacian to Furthermore, we note that the cubic term simplifies via (2.41) to Collecting all the intermediate steps, the full eom reduces to provided X and φ satisfy the decoupling conditions (2.39). Now recall that because the symmetry of the decoupling conditions implies that X and φ are regular zero modes of O φ V,M i ≡0 and O X V,M i ≡0 , respectively. As a consequence, the full eom for X + φ reduce to the sum of the individual eom for X and φ separately. Thus, X + φ is an exact solution if both X and φ satisfy their individual equations of motion and similar for X, as well as the decoupling condition (2.41). In fact, (2.71) are precisely the eom obtain using the above reduced potential, i.e. dropping the mixed V 31 terms such as φ X X and the mixed terms in V 3 . We can therefore use the reduced potential for backgrounds composed of decoupled solutions. However, note that finding a minimum within the Higgs sector V (φ) in general implies (2.73) only up to massive modes. This is the reason why we will find exact solutions only in special cases within the maximal Higgs sector.
Furthermore, from the homogeneity of the full potential in either X or φ, we can infer that 4V 4 + 3V 3 + 2V 2 = 0 holds at a minimum within the Higgs sector, which implies as in (2.22) V governing the dynamics of the bosonic fluctuations is positive semidefinite by virtue of representation theory [25]. This means that there are no instabilities, and moreover there exists a classification of all regular zero modes. (iii) The potential energy of the background can be computed by noting that B ij = 0 implies Now we would like to perturb the background X with regular zero modes φ to obtain a solution to the full eom for X + φ, keeping M i = 0 for the moment. Unfortunately, this is less accessible in general, because (i) Since zero modes φ do not satisfy the su(3) Lie algebra, it is not easy to find exact solutions. A notable exception are the maximal regular zero modes on C[(n, 0)]. (ii) The spectrum of the operator O X+φ V is not analytically understood, but numerical studies presented in sections 3-4, and appendix B show the existence of instabilities. Moreover, we do not have a classification of appearing zero modes. (iii) Assuming that both X and φ satisfy the first integral relations B ij = 0 = D, the combined potential energy can be evaluate to read Hence starting from a C N [µ] background, the extension to a combined background definitely reduces the potential energy. This strongly suggests the existence of a Higgs condensate. Let us try to circumvent the appearance of instabilities by including uniform mass parameters M i ≡ M > 0 and adjusting the radii R i ≡ R in (2.23) accordingly. Starting with X ± i satisfying Thus there are three solutions for |M | ≤ 1 2 . For these choices of R, the eom are satisfied and we can compute the potential from (2.11), which is describes the minimal energy configuration. This establishes the following statement (as stated in the introduction): any solution to B ij = 0 = D gives rise to a solution to the eom with uniform mass parameter M , provided one rescales with R(M ). Now we make the following very important observation: for the special mass value the full potential (2.11) is positive semi-definite, and the potential vanishes if and only if Upon rescaling, this is equivalent to B ij = 0 = D with a radius R = 2 3 . In particular, the spectrum of O Y V is guaranteed to be free of instabilities for any such solution. We will indeed find a number of nontrivial brane plus Higgs solution of this type, which are thereby local minima up to a compact moduli space. This statement is clearly reminiscent of the situation in supersymmetry, even though SUSY is explicitly broken by the potential.

Gauge fields and mode decomposition
Now consider the gauge fields A µ (y) ∈ End(H) on R 4 , which together with the gauginos constitute the N = 1 vector superfield in N = 4 SYM. The gauge fields take values in su(N ), and accordingly decompose into eigenmodes Y Λm of the matrix Laplacian X on squashed C N [µ] background. These modes acquire a mass due to the Higgs effect, given by [25] − In contrast to the scalar fields, there are no zero modes, and the gauge symmetry is broken completely on an irreducible brane. The lowest of these modes are given by Y Λm = 1 c N X α , and the corresponding KK mass scale is of order re-inserting the YM coupling constant g 2 and the cubic coupling m 2 in the potential (2.6). Now consider the coupling of the above gauge modes to the Higgs modes or to the fermions. We recall from section 2.5 that the Higgs modes arise in different types, in particular (i) maximal (stringy) modes φ string α , and (ii) semi-classical modes φ low α . A similar classification applies to the gauge modes and fermions. The effective coupling strength depends strongly on the type of modes. For the semi-classical gauge modes (these will include W -like bosons discussed in section 6), the coupling to a stringy Higgs mode φ string = r s |x x | (2.53) has the structure Hence such stringy Higgs give a contribution to the mass of nontrivial gauge boson modes, which is proportional to the difference of the wave function Y Λm (x) at the two ends of the string. However this contribution is suppressed 9 by the localization (due to (2.84)), and the main contribution to the mass of the gauge bosons arises from the background (2.83). These observations will apply in particular for the chiral A µ ∼ χ gauge mode (4.8) on 6-dimensional branes.
Nonabelian case. As usual, unbroken nonabelian gauge fields arise on stacks of coincident branes. E.g. on a stack of two coinciding branes, the gauge modes have the structure where σ i ∈ su(2) act on the two branes. Using Y Λm = 1 √ dim H 1 for the massless modes, the corresponding su(2) coupling constant is found to be smaller than g by a factor This reduction is somewhat related to (2.58).

4-dimensional branes
After the formal discussion of the algebraic properties of the regular zero modes, it is now time to show the existence of exact solutions of the form brane background C[µ] plus some nontrivial Higgs modes. We begin with the 4-dimensional fuzzy branes C[(N, 0)] in combination with point branes.

Single squashed CP 2 brane & Higgs
Now we discuss some of these branes in more detail. We begin with a squashed brane C[µ] with µ = (N, 0) and add some (regular) zero mode(s) where X α = Rπ µ (T α ). First we treat the massless case M 2 i = 0.

Minimal brane and instability towards S 2
For the minimal branes C[µ] with µ = (1, 0), the only non-trivial (regular) zero modes are given by Since the potential (2.61) for φ α is the same as for X α , there is a non-trivial minimum at r i = ±1, see (2.28). Then the full matrix configuration is This must be an exact solution according to the discussion in section 2.6, and indeed is nothing but a fuzzy sphere S 2 n with n = 3, explaining related observations in [25]. Its energy is using (2.33) and (2.63). This appears to be the global minimum of the classical potential for N = 3. We will see below that although squashed CP 2 is not the global minimum, it can be (locally) stabilized by adding a small positive mass term M 2 i > 0.

Non-minimal brane with maximal Higgs
The situation is similar, but more interesting for C[µ] with µ = (N, 0). Then the (regular) zero modes are given by φ (l) α ∝ (X −α ) l such that the full matrix configuration looks like As before, the l = 1 mode leads to an exact solution given by an irreducible fuzzy 2-sphere with (negative) energy equal twice that of C[µ]. This explains how the known fuzzy S 2 solutions are related to squashed CP 2 . However, there are other, more interesting solutions corresponding to small perturbations of the CP 2 branes localized at their intersections. For l 1, the φ (l) α are string-like modes connecting different sheets of C[µ], confined to the vicinity of the origin. The most interesting ones are the maximal zero modes with l = N , denoted by dropping the superscript N . Hereπ(T α ) is the fundamental representation 10 of su(3) acting on the 3 extremal weight states of (N, 0). In other words, the φ α are connecting the 3 coherent states on CP 2 located at the origin on each of the 3 sheets of squashed CP 2 , cf. [25]. More explicitly, denoting these states as {|1, µ , |2, µ , |3, µ }, the zero modes have the form The indicated dots • are the extremal weights of (N, 0), which are connected by the maximal Higgs ∈ End(H µ ).
which are depicted in figure 2. Since the potential V [φ] has the same form as that for the full brane we obtain a new exact solution for r i = ±1, given by . This follows again from the full equations of motion (2.71) and (2.33), noting that both X α and φ α satisfy the eom and the decoupling condition 11 . The signs of the r i are of course chosen such that the potential is a minimum. This solution describes a Higgs condensate in the CP 2 background localized at the intersections, as sketched in figure 2 and 1b. The general solutions for the combined system of C[µ] brane background with maximal Higgs are discussed in appendix B.1.1. Again, even though that this solution is not the global minimum, we will see in section 3.1.3 that it can be locally stabilized by adding a small mass term M > 0. We note that for 1 < l < N , the zero modes φ Backreaction and exact solution. In general, one should worry about the backreaction of the Higgs φ on the background brane X α . First, it is important to note that the (almost-) maximal Higgs φ with r α = ±1 are a small perturbation 12 located at the origin of the CP 2 background, so that the backreaction is very small for branes C[µ] with large µ = (N, 0). To see this, we compare the matrix elements of X α and φ α connecting the states |i, µ . For r α = R α = ±1, these are easily see to be Hence, the background generators are larger by a factor of √ N and the backreaction of the background becomes negligible for large N . Moreover, the backreaction vanishes exactly for the maximal Higgs modes, as discussed above.
Remark. The combined solution Y α = X α + φ a is not only a solution of the potential with cubic term, but also a solution of the basic N = 4 potential with a negative mass term, since cf. (2.71) and (2.69). The point is that ( X + 2 / D X ad )φ α = 0 follows from the decoupling condition for regular zero modes due to (2.38), and similarly ( φ + 2 / D φ ad )X α = 0 because X is a regular zero mode w.r.t. φ. This is quite remarkable, and it means that these solutions might arise even without the cubic terms in the action, if a negative mass term arises for these modes by quantum fluctuations.

Mass terms, stabilization and mass-induced Higgs
Now consider a single CP 2 brane (N, 0) in the presence of a (sufficiently small) positive mass terms M 2 i > 0. Then the radii R i change according to (2.27).
As a first observation, we note that the squashed brane background is stabilized for sufficiently small equal masses due to (A.8). The quadratic potential for such a background is using (2.37), where X and / D ... are the same operators as for M 2 = 0. For sufficiently small M 2 , the massive modes remain massive, because / D mix is clearly bounded. For the regular zero modes, we observe that the condition / D mix φ (0) α = 0 of (2.42) is independent of the R i . It follows that regular zero modes acquire a positive mass from the explicit M 2 contribution. For the exceptional zero modes, the above argument does not apply; for instance, on CP 2 the exceptional zero modes are the SU (3)/U (1) × U (1) Goldstone bosons, which remain massless even in the presence of explicit (equal) mass terms M 2 i = M 2 . As a consequence, a single CP 2 brane is stable in the presence of small equal masses M 2 i = M 2 , up to the flat directions due to Goldstone bosons.
For further verification, we performed a numerical analysis of the spectrum of the vector 10. For details, we refer to appendix B.1, and only summarize the results here.
• In the massless case M i = 0, we observe 6(N + 2) zero modes for the gauge-fixed O X V . These correpond to 6 Goldstone bosons plus 6(N + 1) regular zero modes, due to (2.36).
• In the massive case M i ≡ M > 0, we observed 6 zero modes and precisely 6(N + 1) massive modes with eigenvalue 4M 2 . See in particular figure 26a.
This confirms that small masses are sufficient to stabilize CP 2 branes up to Goldstone bosons.
If the M 2 i are positive but distinct, the situation is more interesting. The massive modes will stay massive for sufficiently small M 2 i , such that we may focus on the Higgs sector. However, some of the (would-be) zero modes now acquire a negative mass. Continuing the computation (2.47), the adjusted radii lead to an induced mass originating from the vector Laplacian without explicit mass terms, i.e.
with λ as in (2.48). This has one or two negative eigenvalues m i if the R i are different, depending on the sign of λ(H R ). In fact since λ 1 + λ 2 + λ 3 = 0 (from the Z 3 symmetry), we obtain the following sum rule: Note that m i depends on R i resp. M i through (2.27). Taking into account the bare masses (2.16), the quadratic part of the potential for φ is If the brane and Λ are sufficiently large, the m i will dominate M i such that one or two pairs of zero modes acquire a negative mass whenever the M i are different. This is independent of the cubic term in the potential, and it works even positive bare masses M 2 i > 0. Therefore at least one pair will definitely get a VEV φ ±α i = 0, which should in turn lead to a partial stabilization of the zero modes. Even though this scenario is interesting since it breaks the Z 3 generation symmetry, we will mostly focus on the case of equal masses in this paper.
For a CP 2 N brane, the weights of the zero modes are λ i = kα i with k ≤ N , see (2.50). Then switching on one M 2 3 > 0 leads to two negative induced mass terms m 2 1 = m 2 2 = −O(kM 2 3 ) < 0, see for instance (A.29).

Stability of the brane-Higgs system
Now consider the non-trivial solutions X + φ (l) found in section 3.1.2 involving maximal zero modes φ (l) α . An important question is whether these new solutions are stable, or if there are further zero modes or instabilities. Consider the brane C[µ] for µ = (N, 0) with maximal Higgs solution φ α = c(X −a ) N as above, and add first some additional zero modes φ (l) α which we assume not to be maximal. Thus According to (2.61), the effective potential for the combined perturbation is given by V (φ+φ). Since φ has the structure of a squashed CP 2 brane, we know that the masses of φ arising from this background are non-negative [25]. However, we have to admit the most general fluctuations φ α here, not only zero modes. Then the linear term in φ still vanishes identically, because X + φ is an exact solution of (2.71). The quadratic part of the potential for φ is given by 13 using [φ β , X β ] = 0. In general, it is not clear and, according to numerical studies, also not true that this is a positive semi-definite bilinear form. However in the presence of suitable masses M i = M , and notably for M = M * , it follows using the results in section 2.7 that the (brane +Higgs) solution is indeed a local minimum up to a compact moduli space.
Numerical results. We investigated the stability of C[(N, 0)] branes with maximal Higgs numerically by analyzing the spectrum of the vector Laplacian, see appendix B.1 for the details. We considered the massless case M i = 0 as well as the massive case M i = M . To summarize, we find the following for the fluctuations around (X + φ): • Therefore, uniform mass terms M i ≡ M are sufficient to stabilize the CP 2 brane plus maximal Higgs system up to 8 = 6 + 2 zero modes. Remarkably, the properties such as number of zero modes, seem to be independent of the brane size in the massive case M > 0. Hence the (brane+Higgs) system has a well-behaved scaling limit N → ∞, and is stable up to a compact moduli space (which could be lifted by lifting the degeneracy of the M i ). This is clearly a very interesting result, which will be seen throughout this paper.

Single squashed CP 2 brane with a point brane
Now consider a C[µ] brane and add a point brane D ≡ C[0]. We will show that there are non-trivial new vacua which involve a non-vanishing Higgs zero modes linking C[µ] and D. We choose µ = (N, 0) to be specific; the discussion for µ = (0, N ) would be analogous. The Higgs modes given by the regular zero modes discussed above are illustrated in figure 3. The inter-brane regular zero modes, originating from Hom(H µ , C), separate into 3+3 independent Higgs ϕ + i and mirror Higgsφ − i , given by distinguished by the τ -parity (2.31). Note that these determine their conjugate modes (ϕ + i ) † ∼ |0 i| etc. living in Hom(C, H µ ). Taking into account the maximal intra-brane Higgs φ ± i on C[µ] as discussed above, we find non-trivial solutions where such links ϕ + i are switched on with different strength, i.e. the point brane is connected to the C[µ] brane, but the energy is the same. For example, we can switch on one triangle consisting of two ϕ i ,φ i linking D with two corners of C[µ], connected by a maximal Higgs φ i of C[µ] as in figure 4. The existence of such solutions is explained by an SU (2) symmetry within the zero mode sector, rotating the point brane and one corner. This leads to a moduli space parametrized by some linking angles θ.
The binding energy for all these configuration is Numerical investigations indicate that these are indeed the global minima of this sub sector of the full zero mode sector. In particular there seems to be no solution which respects the Z 3 symmetry, except for ϕ = 0. In other words, the generation symmetry Z 3 is spontaneously broken.
This means that although the D brane is connected to C[µ] by some Higgs as in figure  4c, the energy is nevertheless degenerate to the case where only the maximal intra-brane Numerical results. We studied numerically the stability of the combined solutions consisting of C[(N, 0)] brane plus point brane D together with some Higgs modes, with respect to arbitrary fluctuations. The details are provided in appendix B.2.2. It turns out that for M = 0 there are typically a number of negative modes, i.e. they are unstable towards some of the (originally massive) deformation modes. This is similar to the situation in section 3.1.4. Again, these instabilities can be stabilized by adding a (small) mass term M to the background, see in particular figure 27. Then the above solutions, comprising the C[µ] brane plus point brane and several Higgs, are still exact solutions with adjusted radii according to (2.27). Here, we observe different qualitative behavior between the configurations of figure 4a and 4b, 4c.
• The triangle of maximal intra-brane Higgs of figure 4a can be stabilized with small masses, consistent with the results in section 3.1.4. The number of zero modes becomes 14 in the massive case, independent of the brane size.
• The interpolating state of figure 4b can only be stabilized with masses almost saturating the allowed values (and only for large enough branes N ≥ 5). In addition, the number of zero-modes in the massive case becomes 9, independent of the brane size.
• The connected configuration of figure 4c can also be stabilized only with relatively large mass values, but negativity of the eigenvalues is qualitatively different to the interpolating case. Again, the number of zero modes stabilizes at 9 in the massive case.
Consistent with the general results in section 2.7, we observe that there are no instabilities for the critical mass M * . The number of zero modes exceeds the expected 6 zero modes corresponding to the 6 Goldstone bosons of the broken SU (3) R . One may hope that these remaining zero modes are lifted by introducing different masses M i , but we did not verify this explicitly.

Two squashed CP 2 branes & Higgs
Now consider a system of two non-identical CP 2 branes, such as C[µ L ] and C[µ R ]. Then contains various types of zero modes. Besides the intra-brane zero modes discussed above, there are additional modes φ LR ∈ A LR etc. connecting C[µ L ] with C[µ R ]; those are the most interesting ones as we will see. We will explicitly find such solutions. Consider the case of parallel branes C[µ L ] and C[µ R ] with µ L = (N, 0) and µ R = (l, 0), possibly connected by some Higgs, as in figure 5. This set-up is interesting because we will find maximal Higgs connecting different branes, which are exact solutions at the non-linear level. These Higgs clearly breaks the U (1) × U (1) gauge symmetry on the two branes down to the diagonal U (1), and lead to various Yukawa couplings of the fermionic zero modes. This Figure 5: Parallel branes C[µ R ] and C[µ L ], with C[µ L ] being the outer brane. We only indicate the polarization for the outermost brane by two additional • next to each extremal weight state. The maximal intra-brane Higgs are denoted as φ + i and φ + i for C[µ L ] and C[µ R ], respectively. The maximal regular inter-brane zero modes are labeled by ϕ + i andφ − i , and we only draw the A LR sector here.
is the mechanism we are interested in, even though the present background may not yet be very interesting physically. Note that n identical branes would lead to u(n)-valued fields.
Besides the algebras of functions A LL and A RR on C[µ L ] and C[µ R ], respectively, the full algebra of functions contains the following intertwining part: and similarly A RL . There is again a special class of 3 + 3 maximal Higgs modes arising from the extremal states of (N, l), denoted bỹ which connect the corners of µ L and µ R . This is displayed in figure 5. Their conjugate modes are given by We will see below that each of these Higgsφ − i , ϕ + i give rise to precisely one Yukawa coupling between fermionic zero modes linking a point brane with either of the branes.
Additionally, there exist the maximal intra-brane Higgs φ ± i ∈ A LL and φ ± i ∈ A RR along some edge of C[µ L ] and C[µ R ], respectively. As summarized in figure 5, altogether we have the following maximal Higgs modes which connect the extremal weights and therefore form a closed algebra. Our aim is to find stable non-trivial solutions on top of C[µ L ] and C[µ R ], where some of these are switched on. Solutions. It is clear that there are non-trivial solutions within the maximal Higgs sector involving one closed triangle and we have exemplified such cases in figure 6. There exist several continuously parametrized solutions that interpolate between a maximal intra-brane Higgs on one brane and a closed triangle between the two branes. We refer to figures 6a-6c and figures 6a-6c for two representative cases.
In addition, as shown in figure 7a, there exists an exact solution of the form X + φ + φ involving the full brane background plus maximal intra-brane Higgs on C[µ L ] and C[µ R ] simultaneously. Moreover, one can non-trivially combine the triangular configurations of 6c and 6f and obtains another continuous family, see figure 7, which always corresponds to a configuration of two closed triangles.
The details of the numerical combinations of the several Higgs fields that give rise to exact solutions to the equations of motions are presented in appendix B.3. As far as the potential energy is concerned, the (degenerate) configurations of figure 6 count only as one closed triangle, whereas the (degenerate) configurations of figure 7 have two independent triangles, and consequently have lower potential energy. That being said and recalling the Z 3 symmetry, it is clear that the states of lowest energy, i.e. those equivalent to two closed triangles in figure  7, are highly degenerate. These solutions should be interpreted as 2 branes linked by some Higgs, but the binding energy is again zero due to the degeneracy.
correspond to arrows in the opposite directions.
Solutions. In this set-up, there is a novel type of Higgs solution which involve only links between different branes as indicated in figure 9, due to cubic terms like σ + i H + jφ + k and ϕ + iH + jσ + k . This leads to a potentially interesting structure of Yukawa couplings and chiral fermions, as discussed in sections 5 and 6. The two different types of such triangles have opposite "orientation". As further elaborated in appendix B.4, one can find various one-parameter solutions that interpolate between the configurations in figure 9 and other solutions with one closed triangle. We summarize these in figure 10. Starting from 9a there exist at least three continuous families of exact solutions which are displayed in 10a-10c, 10d-10f, and 10g-10i. Due to our knowledge of the two parallel brane case, see figure 6, it follows that configuration 10i can be deformed into to the maximal intra-brane Higgs configuration φ + i on C[µ R ]. Moreover, inspecting figure 4 reveals that configurations 10c and 10f can be deformed to closed maximal intra-brane Higgs triangles on C[µ L ] and C[µ R ], respectively. They all have the same energy. Repeating the analogous analysis for 9b, one concludes that all closed one triangle solutions are connected by continuous deformations, which all have the same energy.
As in previous cases, solutions with more than one closed triangle have a lower potential energy, and we immediately recognize the solution with maximal intra-brane Higgs φ + i and φ + i . Again, one can verify explicitly that solutions like the ones depicted in figure 11 exist, and we find continuous deformations that transform figure 11a into figure 11b as well as figure 11b into figure 11c. By analogous considerations, one infers that all other configurations involving two closed triangles are degenerate in their potential energy and can be deformed into each other. However, they lead to distinct patterns of symmetry breaking and Yukawa couplings.
Spectrum. Having established the existence of numerous novel solutions consisting of brane backgrounds and maximal Higgs, we need to address the spectrum of the vector Laplacian. 3 , including M * . As before, these non-trivial masses eliminate a large fraction of the zero modes, and stabilize their number to a level which is (roughly) independent of the system size.

Three squashed CP 2 branes & Higgs
Finally, consider the case where also D is not a point brane, but a CP 2 brane. Since the sector of maximal regular zero modes is a straight forward generalization of the previous cases, we refrain from depicting them in full detail.
Solutions. In this set-up, one can again find configurations with three closed Higgs triangles connecting the different branes as in figures 12a-12c. The energy of these configurations is conjectured to be minimal, and equal to the case where only the maximal intra-brane Higgs on the branes are switched on. Continuous deformations interpolating between these configurations are indicated in figure 12. As we have already seen in the previous cases, the all solutions with fixed number of closed triangles (here 1, 2, or 3) can be deformed into one another. Moreover, we know that solutions with the maximal number of closed triangles have the minimal potential energy. Hence these configurations are highly degenerate, but they lead to distinct patterns of symmetry breaking and Yukawa couplings. 3 including M * . Then the increasingly large number of zero modes in the massless case is reduced and stabilized to a level that appears to be independent of the systems size.

Coinciding branes with Higgs
Now consider a stack of n identical branes as above. Then the massless i.e. trivial gauge modes constitute an unbroken U (n) (or SU (n)) gauge group. As in the case of distinct branes, there are numerous exact Higgs solutions, which may or may not link the different branes in various patterns. It is clear that this leads to various patterns of (partial or complete) symmetry breaking, and it is straightforward in principle to work out the masses of the broken gauge bosons from the Higgs effect, cf. (4.9).
Taking into account the fermionic zero modes, the question arises how these fermions couple to the broken and unbroken gauge fields, which Yukawa couplings arise, and whether some kind of chiral gauge theory emerges. This is quite natural as we will see, and will be discussed in section 5.  Solutions. As a nice illustration, we can construct an exact solution which has the structure of the 7-dimensional irrep of G 2 . We accomplish this by combining the rank 2 inter-brane zero modes between the minimal brane and its conjugate with the inter-brane modes between the minimal branes and the point brane. Actually there exist two realizations of such a solution, as shown in figure 15. Following appendix B.6, one realizes that the coefficients of the involved zero modes are such that they realize the short roots of G 2 , while the long roots are realized by the background X α . Since G 2 is a Lie algebra, this solution suggests a vast generalization based on higher representations of G 2 . Indeed, the long and the short roots of G 2 satisfy the decoupling condition 14 . This should be elaborated in more detail elsewhere. For non-minimal branes C[ (N, 0)], it is difficult to find analogous exact solutions, because the regular zero modes no longer satisfy a closed algebra. Nevertheless, one would expect that similar solutions might exist for flipped non-minimal branes with a point brane. This 14 HS, unpublished; useful discussions with G. Zoupanos are acknowledged. construction is in a sense dual to the discussion of 6-dimensional branes in the next section.
Stability. The new combined solutions of G 2 -type have been obtained in the massless case. However, we can transfer them to massive solutions as before. This turns out to be necessary to obtain a instability-free spectrum of the vector Laplacian around this combined backgrounds. As in previous cases, a uniform mass parameter of order 0.45 M ≤ √ 2 3 is sufficient to achieve this, and details are provided in appendix B.6.

6-dimensional branes
Now consider the case of general C[(N, M )] branes. These are 6-dimensional quantized coadjoint orbits embedded in R 6 , which decompose into 3 L + 3 R chiral sheets with opposite flux measured by the gauge mode χ (4.1). Because these branes have maximal dimension (in contrast to the above CP 2 branes), this leads to 3 L + 3 R zero modes between a point brane and C[(N, M )], whose chirality is measured by χ. This is the basis for a chiral gauge theory.
Chirality generator and chiral sheets. For 6-dimensional (fuzzy) branes, the operator reduces to the Pfaffian of the Poisson tensor in the semi-classical limit, and therefore it should be a good observable to define the chiral L and R sheets, cf. [25][26][27]. It is easy to see [25] that on the six extremal weight states w|µ of C[µ] located at the origin, χ takes the values where |w| is the signature of the appropriate Weyl group element, and α i are the simple roots. In the semi-classical limit, χ can be identified with a function on the brane (more precisely a polynomial of degree 3), which takes positive or negative values corresponding to the orientation of the 3+3 sheets of C[µ]. This function is odd under W. Thus on the extremal weight states, χ has the simple structure The (1, 1) contribution can be excluded 15 , so that χ is the hermitian weight 0 combination in χ ∈ (0, 3) + (3, 0) .

(4.5)
Chiral Higgs. We have seen that the extremal states of 6-dimensional branes decompose into 3 L + 3 R sets of states with definite chirality. Now we consider the regular Higgs mode on such a brane, and single out those Higgsφ α which respect the chirality, i.e.
[φ α , χ] ∼ 0. (4.6) 15 It would have to be c 38 e T e where c ab c are the structure constants, which vanishes. Figure 16: Regular zero modes of rank 1 or 2 for C[(1, 1)] brane. The maximal Higgs φ + i of (a) are not chiral, but the rank 2 Higgs ϕ + i , σ + i of (b)-(d) are chiral.
These are easy to identify for the (1, 1) brane in figure 16. While the maximal regular zero modes depicted in figure 16a are not chiral, the next-to-maximal regular zero modesφ α ∈ H (3,0) + H (0,3) ⊂ End(H) depicted in figures 16b-16d are in fact chiral, and have a string-like structureφ relating extremal states with the same chirality; then (4.6) follows from (4.3).
For the simplest case C[(1, 1)], these chiral Higgs modes form a closed algebra and will lead to an exact solution, as discussed in more detail in section 4.1. This justifies the hypothesis that theseφ α acquire a VEV. The implications for the fermions and their Yukawa couplings will be discussed in section 5.
Chiral gauge field A µ and its mass. Among all the gauge fields on the C[(N, M )] brane, consider the χ-valued gauge field (4.8) We call it chiral, because it measures the chirality of the L and R sheets according to (4.3), and therefore couples accordingly to chiral fermions. We are going to argue that A µ may be the lightest non-trivial gauge mode in the presence of a chiral Higgs VEV as above, and describes a gauge field in a spontaneously broken U (1) L × U (1) R chiral gauge theory. With this in mind, the mass of the gauge boson A µ in the presence of some Higgs φ α arises as usual from assuming [X a , φ a ] = 0 (see [13]). Since χ is a weight zero mode in H (3,0) , the contribution form the brane X is obtained from (2.83) m 2 (3,0),0 = 2(Λ, Λ + 2ρ) = 2(5, 2) T G(3, 0) = 2 · 27 . (4.10) Assuming that a chiral Higgsφ takes a VEV (as justified below), it is natural to expect that A µ ∼ χ will be the lightest gauge boson, because the mass contribution [χ,φ a ][χ,φ a ] vanishes due to its defining property (4.6). All other non-chiral gauge modes A µ ∼ ρ acquire an extra mass tr[φ α , ρ] 2 . Explicitly, the masses of the lightest gauge modes (with zero weight) due to the background are as follows: using ρ = (1, 1). Here G = 2 1 1 2 is the metric on su(3) weight space. We restrict ourselves to gauge fields with weight zero here 16 ; note that e.g. Λ = (2, 0) contains no weight zero modes.
For the minimal C[(1, 1)] brane, we can also compute the mass contribution from the chiral Higgsφ α solution, which lives on the chiral triangles of the (1, 1) branes as in figure 17. To this end, we decompose the (1, 1) brane and its Hilbert space into the two triangles consisting of the L and R states, which can be viewed as (1, 0) and (0, 1) branes defined by the chiral Higgsφ α . Then all gauge modes with weight zero live in End(1, 0) = (1, 1) + (0, 0) and End(0, 1) = (1, 1) + (0, 0) w.r.t. these triangles. Hence the mass contribution for the chiral gauge field A µ ∼ χ with Λ = (3, 0) vanishes (because [χ,φ] = 0), while the mass contribution for the weight zero Λ = (1, 1) mode is Adding this to (4.11), we see that indeed the chiral boson A µ ∼ χ as well as the two Λ = (1, 1) gauge bosons are the lightest, degenerate gauge bosons (within weight zero), with mass m 2 = 2 · 36. Note that the contribution from the chiral Higgs to the non-chiral gauge bosons such as m 2 (1,1),0 should be larger on larger branes, while (4.11) is universal. It is then plausible that the chiral gauge mode A µ (4.8) becomes the lightest mode, which would entail a chiral gauge theory with a spontaneously broken U (1) L × U (1) R gauge field and 3 generations.
On larger branes, the details are more complicated, because there are many chiral Higgs which may contribute to the mass. However, the underlying geometrical mechanism is very clear: the 3 L + 3 R sheets at the origin lead a priori to a U (3) L × U (3) R gauge theory, which is broken not only by the global connectedness of the brane leading to (4.11), but also by the chiral Higgs modes which link the L and R sheets among themselves. These in turn break the symmetry to U (1) L × U (1) R with 3 generations, and hopefully leave the chiral gauge field A µ as lightest non-trivial gauge boson. Note also that in general, non-trivial Higgs configurations may lead to some back-reaction on the brane (cf. the discussion in section 4.2), which may lead to a relative shift between the L and R branes in target space, thus amplifying the effects on the symmetry breaking. This should be kept in mind in the discussion about approaching the Standard Model in section 6.

C[(1, 1)] brane with chiral Higgs solution
On the C[(1, 1)] brane, we have indeed an exact brane plus chiral Higgs solution. The underlying rank two regular zero modes have already been presented in figures 16b-16d. Because they form a closed algebra, we can combine these to form new exact solutions of the form X + ϕ and X + σ, which are depicted in figure 17. The details of how to arrange this to get an exact solution to the equations of motion are delegated to appendix B.7.
Having found two such equivalent solutions, we analyzed the spectrum of the vector Laplacian around these exact solutions. As it turns out, there are a number of negative modes, indicating potential instabilities. However by including a mass terms M i ≡ M , one can again eliminate all instabilities for 0.47 M ≤

Rigged CP 2 branes
Now consider (N, 1) branes. These can be viewed as a stack of two CP 2 branes linked by a minimal fuzzy sphere S 2 2 , see for instance [33]. We illustrate this set-up in figure 18. The minimal fuzzy 2-sphere plays the role of a Higgs linking the two CP 2 branes. This can be Figure 19: The rank 1 and 2 regular zero modes for the C[(2, 1)] brane, exemplifying the (N, 1) case. The maximal Higgs φ + i of (a) connect extremal weight states, while the next-to-maximal Higgs ϕ + i , σ + i of (b)-(d) also relate extremal and non-extremal weight states. Only the latter are (approximately) chiral.
understood noting that the extra S 2 fiber is embedded transversal to the CP 2 , leading to a 6-dimensional geometry. This is also similar to the flipped branes connected by Higgs in section 3.7, which is now realized as an exact solution with non-trivial intrinsic topology. A decomposition into (N, 0) branes can be obtained explicitly in a suitable basis, see [33]. This also leads to Yukawa couplings which remove certain fermions from the massless sector, resulting in the typical structure found on 6-dimensional branes.
Equations of motion. For these (N, 1) branes, the next-to maximal Higgs modes play again the role of chiral stringy Higgs modes. In view of figure (19b)-(19d), these modes have the structure Note that ϕ i relates again extremal states with the same chirality, but they no longer form a closed algebra because of the sub-leading H contribution. Hence the ϕ i do not yield an exact solution within the Higgs sector. Nevertheless they lower the energy of the brane, and we expect that there exist slightly deformed solutions with similar properties. Presumably, such solutions would involve small admixtures of other zero modes (and possibly massive modes). For large N , this deformation should be negligible. The argument for the σ j modes is completely analogous.

C[(N, N )] branes
Finally, we briefly consider C[(N, N )] branes as sketched in figure 20. Among the Higgs modes we mention the maximal zero modes φ + i in (2N, 2N ), and the (3N, 0) modesφ + i , Figure 20: The rank one, two, and three regular zero modes for the C[(2, 2)] brane, exemplifying the (N, N ) case. The maximal Higgs φ + i of (a) connect extremal weight states. The next-to-maximal Higgs ϕ + i , σ + i of (b)-(d) also relate extremal and non-extremal weight states. Neither the rank one nor the rank two zero modes are chiral. Only the rank three zero modes of (e)-(g) are chiral modes. σ + i (and similarly (0, 3N ) modes) connecting the opposite edges of (N, N ). These are again approximately chiral Higgs modes (similar to the (1, 1) case), which do not quite form a closed algebra, but clearly lower the energy of the brane. Hence we expect that there are nearby deformed solutions.

Nonabelian case: Stacks of 6-dimensional branes and pointbranes
Now consider a stack of two identical 6-dimensional branes as above, each with chiral Higgs φ switched on, and add also an extra point brane D to make it more interesting. This clearly leads to an unbroken U (2) gauge group. However, the results of the last sections lead to a more refined statement: the two branes lead to a U (2) L ×U (2) R gauge theory which is spontaneously broken to U (2) diag , and massive chiral gauge bosons A µ taking values in u(2) L − u(2) R . We will see in section 5 that there are also fermionic zero modes linking D with the 3 L + 3 R sheets on each brane, leading to 3 generations of chiral fermions transforming in the fundamental of U (2) L and U (2) R , respectively, which have opposite charges under A µ according to their chirality. This is a chiral gauge theory in a broken phase, reminiscent of the SU (2) L × SU (2) R Pati-Salam-type electroweak model. Further suitable (maximal, non-chiral) Higgs between the two 6-dimensional branes may break the symmetry to U (1) and lead to patterns quite close to the Standard Model, which will be discussed in section 6.

Fermions on branes with Higgs
The Dirac operator on a squashed background C[µ] is given by acting on End(H)⊗S, where S ∼ = C 8 accommodates the spinors. The ∆ ± j are fermionic ladder operators which satisfy , the X ± j act as ladder operators for the preserved U (1) K i charges. With this input, it was shown in [25][26][27] that the fermionic zero modes Ψ α,Λ on C[µ] correspond to the extremal weight states of each irrep H Λ appearing in End(H) = ⊕ Λ H Λ , and are in one-to-one correspondence to the regular zero modes φ α,Λ of the scalar fields discussed in section 2.5. The proof in [25] is based on the extremal weight properties, while a proof in the spirit of index theory was given in [26]. Consequently, the zero modes are again labeled by their U (1) K i quantum numbers Λ , and their chirality is determined by the parity τ = ±1 of (the Weyl chamber of) Λ . In addition, there exist two trivial gaugino zero modes on each brane.
The results can be summarized by stating that a quiver gauge theory arises on stacks of squashed branes ⊕n i C[µ i ], with gauge group U (n i ) on each node µ i and arrows corresponding to chiral superfields φ α,Λ labeled by the extremal weights Λ corresponding to the multiplets H Λ ⊂ Hom(H µ i , H µ j ). The trivial modes Λ = 0 on each node lead to N = 4 supermultiplets.
More specifically, on a given stack of squashed C[µ i ] branes, fermionic zero modes arise in two ways: first, as intra-brane fermions Ψ ∈ End(H µ i ) ⊗ S on some given brane C[µ i ]. The intra-brane fermions are uncharged under the gauge groups arising on the (stacks of) different branes, but they are chiral and charged under U (1) K i . The latter two features protect them from acquiring any mass terms on the C[µ i ] background, because opposite charges have opposite chirality. Nevertheless, these modes may acquire masses in the presence of nonvanishing Higgs modes due to Yukawa couplings discussed below. In contrast, the two trivial gaugino modes with Λ = 0 are unprotected, and are therefore expected to acquire a large mass due to the soft SUSY breaking, either at tree level or through loop corrections.
Second and more interestingly, fermionic zero modes also arise as inter-brane fermions Ψ ∈ Hom(H µ i , H µ j )⊗S linking two different (stacks of) branes. These are charged under the gauge groups arising on the (stacks of) different branes, chiral, and protected by their U (1) K i charges. Hence they can acquire a mass only through Yukawa couplings in the presence of Higgs modes linking different branes. Note that due to the 9+1-dimensional Majorana-Weyl condition, the Ψ ∈ Hom(H µ i , H µ j ) ⊗ S are related via charge conjugation to the Ψ ∈ Hom(H µ j , H µ i ) ⊗ S. This is important to avoid over-counting e.g. in (6.6), and to obtain a chiral gauge theory. This respects the U (3) R symmetry and the U (1) K i symmetry. Consequently, the non-vanishing Yukawa couplings in the zero-mode sector have the same structure as the cubic term V soft in the potential, and its conjugate. These Yukawa couplings are (non-)vanishing if and only if the corresponding cubic term tr(φ α i [φ α j , φ α k ]) (with the same U (1) K i quantum numbers) is (non-)vanishing. This requires in particular that the U (1) K i charges Λ of φ α j and Ψ α k add up to that of Ψ α i . In particular, the τ -parities of α i , α j , α k must be equal.
Fermionic zero modes on branes with Higgs. For a combined brane plus Higgs background Y = X + φ, the above classification of fermionic (and bosonic) zero modes does not apply any more. The reason is that the Y ± j are typically no longer ladder operators and do not satisfy any Lie algebra relations. As illustrated in various settings in appendix B, the spectrum of O Y V and / D Y may behave utterly different compared to their spectrum on the squashed backgrounds. Nevertheless, one can understand in many cases the fate of the fermionic zero modes, and obtain a qualitative understanding of the remaining low-energy sector. We will argue that some of the chiral fermionic zero modes of / D X are coupled by the Yukawa couplings induced by φ and acquire a mass. Hence, they disappear from the low-energy spectrum on the combined background X + φ. On the other hand, some other fermionic zero modes are protected and remain massless. Adding a point brane D to the combined background solution, the interbrane fermionic zero modes may lead to a very interesting low-energy physics, reproducing ingredients of the Standard Model. Due to the complicated setting, most of these arguments are only qualitative at this point, and we do not have a complete understanding in all cases. The detailed numerical results are given in appendix B.

Fermions on C[(N, 0)] branes with maximal Higgs
For a single C[(N, 0)] background, there are 6(N + 1) + 2 fermionic zero modes. These consist of 6(N + 1) modes from the decomposition End(H (N,0) ) = ⊕ N l=0 H (l,l) , plus two trivial gaugino modes. Turning on the maximal regular bosonic zero modes φ, the spectrum of / D X+φ contains only 14 zero modes independent of N , see figure 26f. This reduction clearly arises from the Yukawa couplings due to the maximal Higgs φ, which leaves the 6 + 2 zero modes from H (0,0) , plus 6 extra zero modes whose origin is obscure. Adding a point brane D to this X + φ, the number of fermionic zero modes is 22 according to figure 28b, which differs by 8 from the 14 modes on C[(N, 0)] with maximal Higgs. We can understand this as follows: A priori, there are 2 · 6 fermionic inter-brane zero modes between C[(N, 0)] and D, which arise from Hom(H (N,0) , C) ∼ = H (N,0) . However, upon switching on the maximal Higgs, Yukawa couplings tr( Ψ + 3 [φ + 2 , Ψ + 1 ]) arise as depicted in figure 21, which couple these zero modes and give them a mass. This leaves only the 6 + 2 intra-brane fermions on D, which remain massless. This explains the numerical findings. Figure 21: Chiral inter-brane fermions Ψ α,Λ and Ψ α,Λ linking C[(N, 0)] with a point-brane D, which is represented by . Their chirality is indicated by the sign ±, which is inherited from the Weyl chamber. Red arrows correspond to maximal Higgs φ + j .

Fermions on C[(1, 1)] brane with chiral Higgs, and point brane
Now we come to our most interesting solution: start with one of the two exact solutions X + ϕ or X + σ of figure 1)] with chiral Higgs, the numerical analysis reveals 20 additional fermionic zero modes. These are understood as follows: 2·6 zero-modes arise from inter-brane zero modes Ψ C,D corresponding to Hom (H (1,1) , C) and its conjugate. In addition, there are 8 trivial intra-brane fermions on D, which consist of 6 modes in End(H (0,0) ) ∼ = C and 2 gaugino modes.
The interesting point is that these 2 · 6 inter-brane zero modes do not acquire a mass, even in the presence of the chiral Higgs on C [(1, 1)]. The key word here is chiral Higgs, which by definition link the extremal weight states of C[(1, 1)] with the same chirality, see figure 17 and (4.7). Since the Ψ C,D linking D with these states have the same chirality, they cannot form a mass term, and remain massless. We view them as (toy-versions of) left-and right-handed leptons, since they couple with opposite charges to the chiral gauge field A µ of (4.8), and come in 3 generations. If some extra Higgs mode is switched on which links the states with opposite chirality, e.g. the maximal Higgs mode, then these left-and right-handed leptons would acquire a mass, as in the Standard Model. Such scenarios will be discussed further in section 6.

Fermions on rigged CP 2 brane
The story for the C[(N, 1)] branes from section 4.2 is very similar to the (1, 1) case, but should be even more interesting as far as physics and the scales are concerned. As before, we are mostly interested in a point brane D added to a C[(N, 1)] brane with Higgs. The drawback for this rigged CP 2 scenario is that we lack an exact solution which would reflect the configuration of figure 18. Nonetheless, some qualitative statements can be made. There are again the 3 + 3 fundamental chiral zero modes linking D to C[(N, 1)], which are attached to the 3 L + 3 R extremal weight (coherent) states, and are viewed as 3 generations of leptons. These leptons will survive in the presence of the chiral Higgs as before, and couple to the chiral gauge field A µ ∼ χ ∼ γ 5 of (5.4). Due to the large N , one may hope that this chiral A µ is now indeed the lightest non-trivial gauge boson on C[ (N, 1)], as discussed in section 4.2. The resulting physics is that of 3 generations of leptons coupled to A µ ∼ γ 5 . This demonstrates how a chiral gauge theory can arise from softly broken N = 4 SYM in a suitable vacuum corresponding to space-filling branes with fluxes.

Fermions on flipped minimal branes plus point brane -the G 2 brane
Let us consider the fermionic zero modes around the combined G 2 -type backgrounds of figure  15. Adding an extra point brane D to the G 2 -type solution, one might expect (i) inter-brane fermions between D and the G 2 -type solution, and (ii) intra-brane fermions on D. The numerical analysis of the Dirac spectrum, however, shows that the combined system has only 22 fermion zero modes, which is only 8 more than on the G 2 solution. This means that among the 2 · 6 inter-brane fermions and the 6 + 2 trivial fermionic modes from D, only 8 remain massless, while the remaining ones pair up and form massive states. The reason is that the G 2 orbits have higher dimension, and the chirality properties of the 6-dimensional solutions no longer apply. Therefore the present G 2 solution is less interesting for the application in section 6, but it may give hints how to find non-trivial Higgs solutions on several SU (3) branes. The task is trickier here, because Ψ DC L , for instance, provides two fermions connecting to the same corner of H ((N L ,0) , whose chirality is given by their τ -parity. They will be paired up by the intra-brane Higgs connecting these corners. However, by inspection, all solutions found in section 3.4 break the Z 3 symmetry, and they also typically involve intra-brane Higgs (except for the solutions in section 3.5). Hence, even though the surviving massless fermions are chiral and have different couplings to A L µ and A R µ , the generation symmetry Z 3 is not respected. This leads to a somewhat strange low-energy theory far from the Standard Model. Nevertheless, it is conceivable -and even reasonable -that the inter-brane Higgs acquire by some other mechanism a VEV which does respect Z 3 , as indicated in figure 22. Then Since we do not have a dynamical justification for such a Z 3 -invariant Higgs configuration, we will focus on the configurations in sections 5.2 and 5.3 in the following discussion towards the Standard Model.

Approaching the Standard Model
At first sight, it may seem impossible to get anything resembling the Standard Model from deformed N = 4 SYM. After all the Standard Model is chiral, while N = 4 SYM is not. In fact, any low-energy gauge theory arising in some vacuum of a deformation of N = 4, as considered here, will have index zero, cf. [34]. However, the Standard Model extended by right-handed neutrinos ν R does have index zero, and this is what we aim to approach with sterile ν R , which are uncharged under the gauge group of the SM. The scenario to be discussed will be reminiscent of (a supersymmetric extension of) the Pati-Salam model [35] in the broken phase. This is a refinement of the brane configuration proposed in [26], using the results of the previous sections.
Consider the brane configuration of figure 24, which is constructed in terms of our squashed brane solutions as follows: The D d brane is realized by a rigged brane C[(N, 1)], which decomposes into two chiral D Ld + D Rd branes, with chiral Higgsφ switched on as in section 4.2.
The D u has the same structure 17 as D d , which decomposes into two chiral D Lu + D Ru branes, again with chiral Higgsφ. Finally add 4 point branes, denoted as 3 × D c + D l for reasons which will become clear soon. The overall brane configuration D l + 3 × D c + (D u + D d ) in the Next, we assume that there exists a non-vanishing "Pati-Salam" Higgs 18 φ S linking D Ru with D l , which breaks the gauge group to This breaking might also be achieved or accompanied by a displacement of the two R branes, while the L branes remain coincident. Here Q is the electric charge, L is the lepton number, B the baryon number, and such that B is traceless. Note that Q is traceless provided dim H u = dim H d , which strongly suggests that the D u and D d have the same structure. We also introduce the weak hypercharge Q and Y will reproduce the correct charge assignments of the Standard Model, and we recover the Gell-Mann-Nishjima formula Fermions. Now consider the off-diagonal fermions linking these branes, which arise as zero modes Ψ ij ∈ End(H i , H j ) linking the extremal weight states of the branes C[µ i ] and C[µ j ].
Consider first the fermions between the point branes D l , D c and D u , D d . Since the former are point branes and D u as well as D d have the structure of C[(N, 1)] branes (by assumption), we can simply apply the results of sections 5.2 and 5.3. Recalling that the extremal weight states of D u separate into 3 L + 3 R chiral extremal weight states |µ i Lu , |µ i Ru and similarly for D d , we obtain 3 generations of chiral leptons linking D l with D u , D d , and 3 generations of chiral quarks linking D l with D u , D d . In the basis (|µ i Lu , |µ i Ld , |µ i Ru , |µ i Rd , |0 l , |0 j c ), we denote the inter-brane fermions as Here the left-handed quarks and leptons arise as SU (2) doublet acting on the D u , D d branes. This SU (2) is broken by φ S , which will be discussed in more detail below. Similarly, the right-handed quarks and leptons also arise as SU (2) doublets Note that the entries below the diagonal are not independent but related to the upper entries by charge conjugation, see section 5. The charge generators (6.3), (6.4) are given explicitly by which results in the following quantum numbers for the off-diagonal modes dropping the obvious SU (3) c assignment. All quantum numbers of the Standard Model are correctly reproduced, and three families arise automatically due to the Z 3 symmetry 19 . There are also some extra modes, including HiggsinosH u,d (as in the MSSM), the ν R which is uncharged under the SM gauge group, gauginos, Winos, and some sterile diagonal fermionic modes. Furthermore, there are also extra modes e with the same quantum numbers as e R , and u with the same quantum numbers as u R . Their fate depends on the detailed structure of e.g. φ S and will not be discussed here. This way of obtaining the correct SM charges is familiar from the context of matrix models [23,26,36,37] and from intersecting brane constructions in string theory [38][39][40].
Higgs sector. In the present background, all the above fermions are exactly massless, even though the unbroken symmetry is only (6.1). The reason is that the U (2) L × U (2) R symmetry of the two L and R sheets of the 6-dimensional branes D u and D d is broken because the sheets are connected (see section 4). This can be viewed as breaking via some background "Higgs", which however does not couple to the above fermions. On the other hand, recall that all the above fermionic zero modes have scalar superpartners, as discussed in the first part of the paper. In particular, this includes the electroweak Higgs doublets H u , H d with Y (H d ) = 1 (as in the SM) and Y (H u ) = −1 (as in the MSSM), which fit into the above matrix structure as It is reasonable to assume that these are intra-brane Higgs modes within D u and within D d , realized by bosonic zero modes ϕ u and ϕ d as discussed in section 4.2. This means that they take the following VEVs with Q = 0. More explicitly, these Higgs modes have the structure Since they are intra-brane modes, they do not induce any further symmetry breaking; nonetheless, they do induce the desired Yukawa couplings between the left-and right-handed leptons and quarks, as in sections 5.2, 5.3. Then the low-energy phenomenology should be fairly close to that of the Standard Model, extended by the various extra fields as above.
These H u,d should be viewed as part of the combined background, which gives mass in particular to the W ± bosons discussed below, as explained in section 4. The lowest fluctuations of this vacuum will involve all the constituents and may behave similar to a SM Higgs, while the detailed composition of the background will enter only via the various gauge and Yukawa couplings. We note that this is somewhat similar to a Pati-Salam model in the broken phase [35]. Whether or not such a scenario may be realistic is another issue, and perhaps there is a better background. The main point here is to show that one may come surprisingly close to SM-like low-energy physics, with only few reasonable assumptions on the VEVs of the zero modes.
Now consider the ν R in more detail. Its fate is clearly affected by the presence of the Higgs link φ S between D l and the R states of D u , which leads to extra Yukawa couplings of the ν and some intra-brane fermions. This would entail a mixture of fermionic states, which is too complicated to be fully analyzed here.
Gauge bosons. It is well-known that a background consisting of stacks of branes leads to massless U (n i ) gauge bosons within stacks of n i coinciding identical branes. This yields (6.1) in the present background.
The story becomes more interesting if we account for the lightest massive gauge bosons. Assume for the moment that there is no φ S . Then the D u + D d would define a U (2) gauge symmetry which enhances to U (2) L × U (2) R by taking the chiral A µ ∼ χ ∼ γ 5 of (5.4) on the C[(N, 1)] branes into account. Together with the SU (4) from the D l + 3 × D c , this is reminiscent of a Pati-Salam model in the broken phase. Clearly the SU (2) L along with the Y contributes the W ± and Z gauge bosons of the electroweak sector, and the U (2) R is broken in the presence of φ S , leading to (6.1). This provides the basic structure of an extended Standard Model. One may hope that the various extra fields acquire a sufficiently high mass to be negligible at low energies, but this is beyond the scope of this paper.
Discussion. Let us briefly address some of the numerous open questions. One concern is that we had to assume that the φ S and the H u,d acquire the appropriate VEVs, without having an exact solution. However, if the H u,d are realized as links along the edges of C[(N, 1)], then there are in fact non-vanishing cubic terms involving e.g. tr(H u φ S φ S ), which lower the energy. It is, therefore, plausible that there is such a solution, but it would presumably have a non-vanishing back-reaction on the brane, which we cannot compute. This is a non-trivial problem, and quantum effects might play an important role.
Another question is whether a suitable hierarchy could arise between the 3 generations. Even though the background under consideration has an exact Z 3 symmetry, this could easily be broken in the Higgs sector, or possibly by introducing different mass parameters M i .
A further issue is the extra massless U (1) B gauge field, which amounts to baryon number. Even though this protects from proton decay, there should not be any such massless gauge field, and it is not clear how to remove this in the present field-theoretic setting. However, it might disappear via a Stückelberg-type mechanism in an analogous matrix model setting with an axion, cf. [41][42][43][44] and the discussion in [26]. In fact, all results of the present paper carry over immediately to the IKKT matrix model, where noncommutative U (N ) N = 4 SYM arises on a stack of N (3 + 1)-dimensional noncommutative brane solutions, while the internal structure of the present paper are unchanged. Thus, the present paper can also be seen as a possible way to obtain interesting particle physics from the IKKT model, cf. [37,45].
Finally, we note that instead of realizing, for instance, the D u branes via C[(N, 1)], we could alternatively use separate L and R branes C[(N L , 0)] and C[(N R , 0)], linked by suitable Higgs as in section 5.5. These links would give mass to the mirror fermions, which disappear from the low-energy theory. The remaining discussion follows the logic employed above. We recall, that this has been the setup proposed in [26]. The links are essentially an integral part of the C[(N, 1)] branes linking their chiral sheets, realized in an exact solution.

Discussion and conclusion
Building on previous work [25,26], we studied vacua corresponding to squashed fuzzy coadjoint SU (3) branes in N = 4 SYM, softly broken by a SU (3)-invariant cubic potential and masses. We found a rich class of novel vacua that include non-trivial condensates of the zero modes. The condensates are interpreted as string-like Higgs modes linking the self-intersecting branes.

Summary.
On a formal level, we showed that the potential can be rewritten in terms of complete squares (2.11), so that the equations of motion follow from a set of first integral equations (1.2). These observations allow to extend solutions from the massless to the massive case, and to establish the absence of any instabilities for a preferred mass parameter M * , at the classical level. Furthermore, we give a useful new characterization of the regular zero modes (the Higgs modes) on the SU (3) branes in terms of a decoupling condition (2.41). As a consequence, the full potential and equations of motion decouple completely for a combination X + φ of brane plus Higgs mode. This is the basis for establishing a rich class of new exact solutions of this type, which lead to a spontaneously broken gauge theory with non-trivial Higgs VEVs and corresponding Yukawa couplings, realizing the ideas put forward in [26].
The decoupling conditions and the related rewriting of the potential in terms of complete squares are in many ways reminiscent of supersymmetry. It provides non-trivial moduli spaces of vacua, and establishes their stability. However there is no underlying symmetry, so that at present we cannot extend these structures to the quantum level.
At a more technical level, the first integral equations are somewhat weaker than su(3) Lie algebra relations (2.25). Nonetheless, su(3) representations provide a valuable starting point, and most -but not all -of our solutions are based on su(3) in some way. Clearly the basic branes C[µ] arise directly from su(3). Using the decoupling properties of the zero modes, we constructed numerous novel combined brane plus Higgs solutions including the following: (i) The fuzzy 2-sphere S 2 N arises from the minimal regular zero mode φ + i = −X − i on any C[µ] brane background. However, these are large perturbations, which completely change the geometry of the brane.  (3), as discussed in section 3.7. The discussion of the fermion sector in section 5 is mostly qualitative. In the presence of several Yukawa couplings the situation is complicated, and a full treatment is beyond our scope. However in many cases we are able to explain the fermionic zero modes of / D X+φ , notably for the C[(1, 1)] brane with chiral Higgs and point brane, which is used in our approach to the Standard Model. This is based on the detailed computations of appendix B.
Discussion. On a more physical level, two of the most interesting features of the new brane plus Higgs vacua are the following: (i) they have a mass gap, and (ii) typically only a small number of zero modes exists. The number of zero modes is independent of the rank N 1 of the underlying SU (N ) gauge theory and of the size dim H Λ of the brane C[Λ]. This observation is remarkable, since the starting point is a gauge theory with large N , which is typically organized in terms of a t'Hooft 1 N genus expansion with effective coupling λ = g 2 N . In contrast, the C[(N 1 , N 2 )] vacua with large N i behave as semi-classical, large branes. The fluctuation spectrum on these vacua consists of a small number of zero modes with typical coupling strength g, and a large tower of typically weakly interacting KK modes with a finite mass gap independent of N . Hence, the original large N gauge theory reduces to an effective low-energy theory with few modes and an interesting geometric structure. This should provide sufficient motivation to study them in more detail.
The most interesting aspect of these vacua is that they can lead to a chiral gauge theory at low energies, with interesting properties not far from the Standard Model. The point is that the underlying branes are locally space-filling in the 6 extra dimensions and carry a flux, so that bi-fundamental fermions are charged and expected to have chiral zero modes. This is precisely what happens, although the overall index is bound to be zero. Elaborating on a previous proposal by using our new solutions, we discussed in section 6 a brane configuration which comes fairly close to the Standard Model. The set-up is reminiscent of the Pati-Salam model in the broken phase and of intersecting brane constructions in string theory. Even though we do not claim that this is realistic, it certainly provides strong motivation for further work.

Future directions.
There are many open issues which should be addressed in future work. One task is to understand better the chiral Higgs solution (4.13) on the (N, 1) branes and the associated chiral gauge bosons, which seem to be particularly interesting for physics. Another is to justify a non-vanishing "Pati-Salam" φ S in our Standard Model approach, and to see how close to real physics one may come in this way. More generally, it would be desirable to have a more systematic understanding and perhaps a classification of the solutions of the first-order equations of motion (2.12).
At some point of course, these classical considerations will no longer suffice. We have argued that a classical treatment should be justifiable to some extent on large branes, due to the existence of a gap with few remaining low-energy modes, and the mild UV behavior of the softly broken N = 4 model. Nevertheless, quantum effects need to be taken into account at some point. Due to the global SU (3) symmetry, the relevant terms in the effective potential must preserve the form (2.6). Moreover, the critical mass M * of (2.80) marks the transition between trivial and non-trivial stable vacua; hence, it should play a special place in the full quantum theory, possibly as an RG fixed point. A direct loop computation by summing over all higher KK modes seems far too complicated. For a deformed N = 4 model, one strategy might be to invoke holography; however, this is questionable since (i) the model is not conformal and (ii) the vacuum is highly non-trivial. Fortunately, a suitable alternative technique was recently proposed in [32], which is based on string-like states on the fuzzy brane backgrounds. This geometric approach has been applied successfully in the purely fuzzy context of [32,46], but not yet in the present field-theoretic setting with fuzzy extra dimensions. Hence, this needs to be developed elsewhere.
The take-home message it that a simple deformation of N = 4 SYM with large N can reduce at low energy to an effective gauge theory with few string-like Higgs modes coupled to chiral fermions, interpreted in terms of a gauge theory on intersecting branes in 6 extra dimensions. Moreover, the possible quantum numbers include those of the Standard Model. This should provide sufficient motivation to study such scenarios in more detail and at the quantum level, and it will be interesting to see how far these solutions can reach.

A.1 Preliminaries
Potential. The full potential for an ansatz like (2.23) reads Phase degeneracy. For any configuration (R 1 , R 2 , R 3 ) which solves (2.27), we can freely change two of the three phases R i → R i e iϑ i , due to the U (1)×U (1) symmetry of the potential. The third phase is then fixed by the equations of motion, and it is easy to see that if two R i are real then so is the third. We will not spell out this trivial degeneracy in the solutions below. Note that an overall sign flip R i → −R i does in general not map solutions into solutions.

A.2 Solutions
The only solutions are up to phases. In this case, we observe that V (S 1 ) = 1 2 > V (S 0 ) = 0. Hence, S 0 is the minimum.
There are nine three types of solutions, given by up to phases. We can compute the potential and find One can verify that V (S 1 ) < V (S 2 ) for 0 < M < Also, observe that S 1 = S 2 for M = 1 2 . Hence S 1 is a (relative) minimum for 0 < M < 1 2 and the absolute minimum for 0 < M < M * , see also figure 25. Note that within each solution S i the radii are equal, |R 1 | = |R 2 | = |R 3 | =: R(S i ). We also observe (A.14) Hence the radius with the lowest energy in this regime is given by Note that the critical mass M * (2.80) marks the transition between trivial and non-trivial stable vacua.  α ) = −lα. These masses (3.14) are given by For the solution in section A.4 we can simplify this to such that so that In particular, the maximal Higgs Taking into account the cubic terms arising from the soft potential, the r i have to satisfy the following eom −N ∆R 2 r 1 + r 1 2r 2 which satisfy the Lie algebra relations (2.25), with all required normalizations.
Single brane. For a single brane, we define the background as in (2.23), where we only consider equal radii R i ≡ R(M ) = 1 2 1 + √ 1 − 4M 2 for 0 < M < 1 2 . The 6 hermitian matrices X a are then obtained by inverting (2.2). The definition of the (gauge-fixed) vector Laplacian as is implemented, with structure constants (2.7) determined by Multiple branes. Multiple branes (N l , K l ) for l = 1, . . . , d are realized as a straightforward extension of the previous paragraph, described by

.1 Solution to eom
With the notation shown in figure 2 we use the following ansatz: Here and in the following, the radius R = R(M ) (A.15) is always fixed by solving the eom for equal masses M i = M . Then we find the following family of solutions to the eom: This exhibits the above-mentioned flat direction corresponding to two U (1) phases within the Higgs sector, due to (2.63).

B.1.2 Fluctuation spectrum
For a single C[(N, 0)] brane, O X V has a positive semi-definite spectrum, but the number of zero modes (in the gauge fixed case) increases with the brane size as 6(N + 2). As shown in [25][26][27], there are six regular zero modes for each irrep appearing in the endomorphism space. Here, End(H (N,0) ) ∼ = ⊕ N l=0 H (l,l) and we expect 6(N + 1) regular zero modes. Additionally, there are six Goldstone bosons coming from SU (3)/U (1) 2 , as the background only preserves the U (1) K i symmetries. Numerically, we find indeed 6(N + 2) zero modes in the massless case. As depicted in figure 26a, turning on uniform masses M i ≡ M < 1 2 allows to lift all zero modes except the 6 modes associated to the Goldstone bosons.
Similarly, the zero modes of the Dirac operator / D X are shown in 26b and were classified in [25][26][27]. We observe indeed 6(N +1)+2 fermionic zero modes, among which 6(N +1) originate from the one-to-one correspondence with regular bosonic zero modes, and the remaining two are trivial gaugino modes. The spectrum of / D X is independent of the bosonic mass.  (a) (c) Minimal brane. The combination of minimal brane C[(1, 0)] together with its maximal Higgs reduces to the fuzzy 2-sphere. We find that O X V is positive semi-definite and O X V,fix has 11 zero modes. Including masses partially lifts them and one finds 5 remaining zero modes. These are the Goldstone bosons of SU (3)/(U (1)×SU (2)), because the fuzzy 2-sphere preserves a full SU (2). In addition, the Dirac operator has 8 zero modes. B.2 (N, 0) brane + point brane B.2.1 Solution to eom We use the notation of figure 3 and the following ansatz: for f j , r j , s j ∈ C and Y j,− = (Y j,+ ) † . The setup necessarily contains the above solutions including the trivial solution, solutions with maximal intra-brane Higgs i.e. f j only, and triangular solutions formed out of (f 1 , r 2 , s 3 ), (f 2 , r 3 , s 1 ), or (f 3 , r 1 , s 2 ). The question is whether there are more general solutions or whether two or more triangles can exist simultaneously.
Triangular solutions. Consider for instance the triangular system comprised of f 1 , r 2 , s 3 ∈ C, where all remaining coefficients vanish. Then we find the following solutions:

B.2.2 Spectrum
As special case of (B.8), for f 3 = z = √ 1 − x 2 and y = x, consider the following 1-parameter family of solutions with x ∈ R, |x| ≤ 1. We computed the spectrum of the vector Laplacian and found that a number of negative modes exist in the massless case, see figure 27. There are several observations: • In the massless case, the number of negative modes for x = 0 increases like 3 (N − 2), for x = 1 as (N − 2) + 6, and for 0 < x < 1 as 3(N − 2) + 6. Hence the number of negative modes is smallest for x = 1 for large N .
• However, the eigenvalues of the negative modes behave peculiar for x > 0. We observe from figures 27b, 27c that a small number of negative modes acquire relatively large negative eigenvalues that can only be lifted by mass parameters M i ≡ M approaching the limiting value M * = √ 2/3. Nonetheless, for all these configurations one can lift all negative modes.
• As shown in figure 27a, the negative modes for x = 0 can be lifted by relatively small masses. This is to be expected as the x = 0 configuration is the direct sum of C[(N, 0)] plus maximal intra-brane Higgs with a independent point brane D. In other words, we can compare to figure 26d.
Turning our attention to the number of zero modes of O X+φ in this combined background, we summarize our numerical findings in figures 28a and 28c. As before for the single CP 2 brane, with or without Higgs modes, the number of zero modes of O X V stabilizes upon inclusion of mass parameters M i ≡ M .
• • For the C[(N, 0)] plus maximal intra and inter brane Higgs configuration x > 0, there are 9 zero modes in the gauge fixed case.
Again, these numbers are then independent of the size of the system, i.e. the value of N . Considering the zero modes of / D X+φ , we observe the following: • For x = 0 as in figure 28b, there exist 22 fermion zero modes, which we can explain by 14 fermion zero modes from C[(N, 0)] with maximal Higgs plus 8 zero modes from the point brane D. From these 8 modes, 6 correspond to the regular bosonic zero modes on D and 2 are trivial gaugino modes.
• For x ∈ (0, 1), the situation remains qualitatively the same. The properties remain independent of the brane size N .
for f j , h j , r j , s j ∈ C and Y j,− = (Y j,+ ) † . Due to the considerable number of free complex parameters and their nonlinear appearance in the full eom, we can only probe a subset of solutions. Motivated by the solutions found in section B.1 and B.2, we assume that we can restrict to solutions with real coefficients.

B.3.1 Solution to eom
There are the following types of solutions: Maximal Higgs. Solve for f j , h j ∈ C, all others coefficients vanish. Then (f j ) and (h j ) can be any solution of (B.5) and any combination thereof is an exact solution for the two brane case.       The other x configurations behave similarly for x > 0, in the sense that the properties are largely N independent, cf. figure 30d. For the choice x = 0, we observe a marked N dependence in the fermionic zero mode spectrum, see figure 30b.
for f j , h j , p j , q j , r j , s j , u j , v j ∈ C and Y − j = (Y + j ) † . Due to the considerable number of free complex parameters and their nonlinear appearance in the full eom, we can only probe a subset of solutions. As before, we assume that we can restrict to solutions with real coefficients.
Maximal Higgs. As in the previous cases, one finds solutions to the eom which contain only the maximal Higgs, i.e. non-trivial values for f j , h j . The more involved the connection of the two CP 2 branes to the point brane becomes, i.e. 0 ≤ x < 1, the more does the fermionic zero modes spectrum exhibit a pronounced dependence on the system size N i . The bosonic zero modes exhibit this only in the massless case, and the massive case seems less N i dependent.   Consider the following parametrization Y + j = X + j + f j φ + j,3 + g j φ + j,2 + h j φ + j,1 + p j ϕ + j,3,2 + q j (ϕ − j,3,2 ) † + r j ϕ + j,3,1 + s j (ϕ − j,3,1 ) † + u j ϕ + j,2,1 + v j (ϕ − j,2,1 ) † (B.20) with f j , g j , h j , p j , q j , r j , s j , u j , v j ∈ C. The subscript in φ j,1 means maximal intra-brane Higgs on C[(N 1 , 0)], while ϕ + j,3,2 means maximal inter-brane Higgs from C[(N 3 , 0)] to C[(N 2 , 0)] and so forth.
Maximal Higgs. Consider the configuration with only maximal Higgs, i.e. solve f j , g j , h j ∈ C and all other coefficients vanish. Then we find that all possible combinations of (f j ), (g j ), (h j ) independently taking the form of (B.5) are in fact solutions in the three brane case. Note that the potential energy for the case f j , g j , and h j simultaneously non-zero is the smallest.
Single triangles. We solve the eom for p 1 , u 2 , s 3 ∈ R and all other coefficients vanish. We find the following solutions: Note that this is not the usual family obtained from 2 phases. Similarly, there are analogous solutions for p 2 , u 3 , s 1 ∈ R or p 3 , u 1 , s 2 ∈ R. Additionally, one can consider the triangles going the other way around, i.e. q 1 , v 3 , r 2 ∈ R, q 2 , v 1 , r 3 ∈ R, or q 3 , v 2 , r 1 ∈ R.
Multiple triangular subsystem of type I. Consider the configuration in figure 12a such that we solve the eom for p j , u j , s j ∈ R and all other coefficients vanish. As solutions we find all possible combinations of (B.21), i.e. we find 5 3 = 125 real solutions. Note that these configurations have the same potential energy as the configuration with all maximal Higgs non-vanishing.
Multiple triangular subsystem of type II. Consider the configuration in figure 12b; i.e. we solve the eom for g 2 , h 3 , p 1 , p 3 , q 3 , r 2 , s 1 , s 2 , u 1 ∈ R and all other coefficients vanish. Since the solutions is comprised of three independent triangles, we find 5 3 = 125 cases, again. Note that the configurations with all triangles non-trivial have the same potential energy as the configuration with three non-vanishing maximal Higgs configurations.

B.5.2 Spectrum
Analogous to the other set-ups, we can evaluate the spectrum of the vector Laplacian and the Dirac operator around a combined background. We restrict ourselves to two configurations: (i) the solution of figure 12a, and (ii) the maximal intra-brane solution of figure 12e. The number of negative modes is shown in figure 33a and 33b, respectively. Again, we observe that for large enough mass values 0.47 M ≤ M * all negative modes can be lifted consistently.
For configuration 12e, we observe from figure 33d that a large number of zero modes disappears after inclusion of uniform mass values and the resulting number depends slightly on the system size. The fermionic zero modes exhibit that the system roughly corresponds to three independent copies of C[(N i , 0)] plus maximal intra-brane Higgs. In detail, we observe from figure 33f that for the system C[ (N 1 , 0) The more intricate configuration 12a shows a qualitatively similar behavior. Note, however, that the number of bosonic and fermionic zero modes, see figure 33c and 33e respectively, is much lower compared to the other configuration. B.6 (1, 0) brane + (0, 1) brane + point brane B.6.1 Solution to eom With the notation introduced in figures 13-14 we employ the following ansatz for f j , h j , p j , q j , r j , s j , u j , v j , x j , y j ∈ C. Inspecting the various Higgs modes there are various exact solutions.
Maximal intra-brane Higgs. As seen in previous cases, the brane background together with the maximal regular zero modes φ j and / orφ j leads to an exact solution of the equations of motion.

B.7.1 Solution to eom
Following the conventions of figure 16, we employ the ansatz: with g j , h j ∈ C and Y − j = (Y + j ) † . There are two classes of solutions (restricting to real coefficient does not exclude any non-trivial solution) which are Moreover, and similar to all previous cases, the number of bosonic zero modes is reduced by non-trivial mass values and is found to be 8. We can understand these as 6 Goldstone bosons plus the two phases in the g j resp. h j , as in section B.1.2. In addition, there are 20 fermionic zero modes.