An extended standard model and its Higgs geometry from the matrix model

We find a simple brane configuration in the IKKT matrix model which resembles the standard model at low energies, with a second Higgs doublet and right-handed neutrinos. The electroweak sector is realized geometrically in terms of two minimal fuzzy ellipsoids, which can be interpreted in terms of four point-branes in the extra dimensions. The electroweak Higgs connects these branes and is an indispensable part of the geometry. Fermionic would-be zero modes arise at the intersections with two larger branes, leading precisely to the correct chiral matter fields at low energy, along with right-handed neutrinos which can acquire a Majorana mass due to a Higgs singlet. The larger branes give rise to $SU(3)_c$, extended by $U(1)_B$ and another $U(1)$ which are anomalous at low energies and expected to disappear. At higher energies, mirror fermions and additional fields arise, completing the full ${\cal N}=4$ supersymmetry. The brane configuration is a solution of the model, assuming a suitable effective potential and a non-linear stabilization of the singlet Higgs. The basic results can be carried over to ${\cal N}=4$ $SU(N)$ super-Yang-Mills on ordinary Minkowski space with sufficiently large $N$.


Introduction
The main result of this paper is to establish a background of the IKKT or IIB model [1] with low-energy physics close to that of the standard model. This is part of the programme of using matrix models as basis for a theory of fundamental interactions and matter, which has been pursued for many years from various points of view [2][3][4][5][6][7][8]. We focus here on the relation with particle physics, restricting ourselves to the case of flat 4-dimensional space-time. Indeed it is well-known that flat Minkowski space arises as "brane" solution of the IKKT model, realized as noncommutative plane R 4 θ . It is also known that the fluctuations of the (bosonic and fermionic) matrices around a background consisting of N coincident such R 4 θ branes give rise to non-commutative maximally supersymmetric N = 4 U (N ) super-Yang-Mills (SYM) on R 4 θ , cf. [3,6]. Accordingly, our results can be interpreted as statements within noncommutative In particular, we can compute the mass spectrum given by the eigenvalues of the internal Dirac operator / D int on our background, as well as the approximate localization and chirality of the corresponding fermionic modes. The results are consistent with the expectations. In particular, we clearly see near-zero modes which are localized as predicted on the intersecting branes, with the expected chiralities. Their eigenvalues approach zero for increasing N , with a clear gap to the next eigenvalues corresponding to mirror fermions. For a range of parameters we even find good quantitative agreement with our estimates for the Yukawa couplings, including the first series of mirror fermions.
Our brane configuration is a solution of the bare matrix model action, supplemented by a simple SO(6)-invariant term in the potential. Although we add such a term by hand here (thus explicitly breaking supersymmetry), it seems plausible that (a more complicated form of) such a potential arises in the quantum effective action of the original model. This reflects the interaction of the branes extended in the extra dimensions, due to the conjectured -and to some extent verified [1,5,[11][12][13][14] -relation with supergravity. The singlet Higgs S is a solution at the linearized approximation.
Other ways to obtain chiral fermions in the IKKT model and similar models have been proposed in the literature. This includes warped extra dimensions [15], allowing to circumvent the index theorem [16] which applies to product spaces R 4 θ × K. However, no such warped solution of matrix models is known at present. Chiral fermions can be obtained in unitary matrix models [17], which in a sense have a built-in toroidal compactification. However these models are not supersymmetric, which may lead to problems upon quantization. In string theory, there are many ways to obtain chiral fermions, however this entails the vast landscape of string compactifications with its inherent lack of predictivity. Avoiding this is in fact one of the main motivations for the IKKT model. Nevertheless, many of the present ideas related to brane constructions of the standard model originate from string theory, cf. [18]. Finally, it seems likely that a somewhat adapted brane configuration can be found in the BFSS model [2].
We should also state the potential problems and pitfalls of our proposal. At some scale above the electroweak scale, mirror fermions come into play, which couple to the standard model gauge bosons, and may decay into standard model fermions via extra massive gauge bosons. In order to be at least near-realistic, there should be a sufficiently large gap between the electroweak scale and the scale of the mirror fermions. Unfortunately at tree level, it turns out that this gap is not very large. However we argue that quantum corrections should increase this gap, since a tower of massive Kaluza-Klein gauge bosons couples to the mirror fermions (as well as to the ordinary fermions) but not to the electroweak gauge bosons or the Higgs. Proton decay is prevented by baryon number conservation, which is violated only by a quantum anomaly.
The solution presented here a priori leads to two generations, which arise from two widely separated intersection regions of the underlying branes, with the same structure and chiralities. It seems straightforward to extend them to any even number of generations, by introducing multiple branes. To get an odd number of generations is less clear; one possibility is that the singlet Higgs S leads to a deformation of the background and removes one intersection region.
At this point, it is perhaps a bit optimistic to hope that the backgrounds proposed here -with some adjustments -can be phenomenologically viable. On the other hand it seems at least conceivable, and the fact that we can get so close in this maximally (super)symmetric matrix model is certainly very remarkable. This should provide motivation to investigate these observations in more detail.
This paper is organized as follows. In section 2, we collect the required facts about the matrix model, and recall the relation with noncommutative N = 4 SYM. From that point on the paper may be read and interpreted by anyone familiar with N = 4 SYM. In section 3, the organization of the fermions and their quantum numbers is recalled from [9]. The central idea of the Higgs realized as intrinsic part of a minimal brane is explained in section 3.3. In section 3.4 we give the brane solution (3.18) of the matrix model, which is the centerpiece of the paper. It is also spelled out with all branes in (5.1). The rest of the paper is devoted to establishing the low-energy physics on this background. The chiral fermions at the brane intersections are established in section 4 in the flat limit N → ∞, where they become exactly massless. The case of finite N is then discussed in section 4.3 using an ansatz motivated by the previous section, which allows to estimate the corresponding Yukawa couplings. These are compared with numerical computations. The symmetry breaking and the resulting lowenergy effective field theory is elaborated in section 5, which allows to make contact with the standard model. In the appendix, we elaborate the reduction of the fermions to 4 dimensions.

The matrix model
Our starting point is the IKKT or IIB model [1], which is given by the action The indices A, B run from 0 to 9, and are raised or lowered with the invariant tensor η AB of SO (9,1). The X A are Hermitian matrices, i.e. operators acting on a separable Hilbert space H, and Ψ is a matrix-valued Majorana Weyl spinor of SO(9, 1), with Clifford generators Γ A . We also introduced a scale parameter Λ 0 with dim Λ 0 = L −1 . This model enjoys the fundamental gauge symmetry as well as the 10-dimensional Poincaré symmetry and a N = 2 matrix supersymmetry [1]. The tilde indicates the corresponding spin group. Defining the matrix Laplacian as the equations of motion of the model take the form for all A, assuming Ψ = 0.

Noncommutative branes and gauge theory
We focus on matrix configurations (in fact solutions, ultimately) which describe embedded noncommutative (NC) branes. This means that the X A can be interpreted as quantized embedding functions [6] of a 2n-dimensional manifold embedded in R 10 . More precisely, there should be a quantization map Q : C(M) → A ⊂ L(H) which maps functions on M to a noncommutative (matrix) algebra, such that commutators can be interpreted as quantized Poisson brackets, and A as quantized algebra of functions on M. In the semi-classical limit indicated by ∼, matrices are identified with functions via Q, in particular, X A = Q(x A ) ∼ x A , and commutators are replaced by Poisson brackets. For a more extensive introduction see e.g. [6]. Then the commutators encode a quantized Poisson structure on (M 2n , θ αβ ). This Poisson structure sets a typical scale of noncommutativity Λ NC . We will assume that θ αβ is non-degenerate, so that the inverse matrix θ −1 αβ defines a symplectic form on M 2n ⊂ R 10 . The prototype of such a noncommutative brane solution is given by the 4-dimensional quantum plane R 4 θ , defined by [X µ ,X ν ] = iθ µν 1 where θ µν has rank 4. It obviously satisfies X A = 0. We can assume that this plane is embedded along µ = 0, ..., 3, withX a = 0 for a = 4, ..., 9. The (well-known) key observation is that fluctuations of the matrices around this background describe (non-commutative) N = 4 gauge theory on R 4 θ . Interpreting the fluctuations A A as u(N )-valued functions in R 4 θ , the matrix model reduces to (cf. [3,6]) Since |G| = |η| = 1 in four dimensions, we will drop √ G from now on. All fields take values in u(N ). In particular, we can read off the u(N ) coupling constant, Although the action (2.9) is written in a way that looks like the standard N = 4 SYM, it is in fact noncommutative N = 4 SYM on R 4 θ . In the present paper, we will focus on those aspects where this distinction becomes (almost) irrelevant, emphasizing that the basic results also apply to standard N = 4 SYM on commutative R 4 .
To describe the internal or "extra-dimensional" sector described by the Φ a or X 4...9 , we need to consider more general branes M 2n . Being embedded in R 10 , they are equipped with the induced metric which is the pull-back of η AB . However, this is not the effective metric on M 2n . It turns out that the effective action for fields and matter on such NC branes is governed by a universal effective metric G αβ given by [6] G αβ = ρθ αα θ ββ g α β , for n > 1. This can be seen using the semi-classical form of the matrix Laplace operator 5 acting on scalar fields Φ ∼ φ. Then the matrix equations of motion (2.5) take the simple form hence the embedding x A ∼ X A is given by harmonic functions on M with respect to G αβ . The prime example of a compact noncommutative brane is the fuzzy sphere S 2 N [20,21]. Its embedding in R 3 is given in terms of 3 matrices Y i = cL i , where L i is the generator of the N -dimensional irreducible representation of su (2). Then In this paper, we will give such a compactification in terms of stacks of suitable K, resulting in a 4-dimensional gauge theory on R 4 that resembles the standard model at low energy. The constructions of this paper also apply to SU (N ) N = 4 SYM theory on ordinary R 4 . Then the brane configurations become backgrounds of the 6 scalar fields, and our results state that the low-energy physics of such a background resembles that of the standard model. 3 The standard model from branes in the matrix model 3

.1 Fields and symmetries
In order to recover the standard model from the matrix model, all fields must be realized as matrices in the adjoint of some big U (N ) gauge group. The SU (3) c × SU (2) w × U (1) Y gauge group must arise at low energies from the fundamental SU (N ) gauge group by some symmetry breaking mechanism, and the standard model matter fields must transform in the appropriate way. It is quite remarkable that this is possible at all. Such an embedding of the standard model fields was given in [9] (cf. [22]), which we take as starting point here. The fermionic matrices (including a right-handed neutrino) are realized as follows The electric charge Q and the weak hypercharge Y are then realized by the adjoint action of the following SU (N ) generators In particular, the Gell-Mann Nishima relation is satisfied. Furthermore, we need a mechanism which breaks the U (N ) gauge group down to the standard model gauge group SU (3) c × SU (2) w × U (1) (possibly extended by additional U (1) factors), such that Q and Y arise as above. This can be achieved naturally by a suitable arrangement of stacks of compact branes in the extra dimensions, analogous to brane constructions in string theory [18]. In the matrix model, such a collection of coincident branes K i can be realized by block matrix configurations X a (i) acting on H (i) ∼ = C N i , cf. [23]. This suggests a brane configuration [9] with 2 + 1 + 1 "electroweak" branes 2 × D w ⊕ D a ⊕ D b leading to SU (2) w × U (1) 3 , along with a "leptonic" brane D l which carries a U (1) L gauge group (corresponding to lepton number), and three coincident "baryonic" branes D B which carry the SU (3) c × U (1) B gauge group: Here k coincident branes are described by (...) ⊗ 1 k . This background breaks 6 the U (N ) gauge symmetry down to the product of U (k i ) as follows A priori, fermions on such noncommutative branes are not chiral, and thus cannot realize the standard model. Remarkably, chiral fermions do arise on intersections of such (noncommutative!) branes as shown in [9], provided they locally span the internal space R 6 . Thus for suitable arrangements of the above branes, the required chiral fermions (3.1) may indeed arise on the corresponding intersections of D l and D B with the electroweak branes D w , D a , D b . However due to the trivial topology of R 10 , there are always additional intersections, leading to additional fermions with the opposite chiralities. This is quite unavoidable for branes with product geometry 7 R 4 × K ⊂ R 10 , as can be shown by an index theorem [16]. We propose a simple solution to this problem here, which at the same time provides a compelling mechanism for the electroweak Higgs. First, we note that the Higgs doublets with Y (H d ) = 1 (as in the standard model) and Y (H u ) = −1 (as in the MSSM) fit into the above matrix structure as This indeed leads to the desired pattern of electroweak symmetry breaking. We also added a "sterile" Higgs S, which is a singlet under the standard model gauge group, occupying the same slot as ν R . This leads to a modified matrix background of the form which however still does not resolve the problem of chirality doubling. The solution comes from replacing the two branes connected by the Higgs with a single noncommutative brane, recognizing the Higgs as intrinsic part of the geometry. This is explained in the next section.

Higgs from deconstructing compact branes
For two branes connected by an off-diagonal Higgs as above, the embedding matrices generate an irreducible algebra which contains the original branes as sub-algebras, and should therefore be interpreted geometrically as a single compact space K. Conversely, a single compact brane K can be considered as a 2-brane system glued together at the boundary by some Higgs. For example, S 2 N can be interpreted as 2 disks in the xy direction near the north and south poles connected by an equatorial strip, which realizes the Higgs. In mathematical terms, we split the Hilbert space of a fuzzy sphere H N = C N = C N/2 ⊕ C N/2 ∼ = H (1) ⊕ H (2) into two halves interpreted as D (1) and D (2) , and write the embedding matrices as We can then interpret the two diagonal blocks as 2 a priori separate branes, linked by the Higgs field φ. Note that D (1) and D (2) have opposite Poisson structure near the origin, and are transversally separated by the diameter. The two groups U (1) = U (H (1) ) and U (2) = U (H (2) ) corresponding to the diagonal blocks can be viewed as gauge groups on the two half-branes 8 . Then φ intertwines these gauge groups, and plays the role of a Higgs. Indeed, the 4-dimensional gauge fields corresponding to U (1) , U (2) will acquire a mass due the Higgs effect. One problem with this idea is that the Kaluza-Klein gauge modes on K would not respect in general these upper or lower half-branes, but spread over the entire compact quantum space. Moreover, they would not respect the localized fermions arising on intersections of branes in a clear-cut way. These problems are avoided for fuzzy spaces with N = 2 represented on C 2 = H (1) ⊕ H (2) with dim H (1) , H (2) = 1; we call them minimal N = 2 quantum spaces. This leads to a simple set of gauge modes arising from a short KK tower.

Minimal electroweak branes with Higgs
Applying this idea to the above brane scheme, we interpret the Higgs φ u as compactification of (the "half-branes" defined by) the first and fourth line in (3.8) into a single compact brane denoted as D u , and φ d as compactification of the second and third line into another compact brane D d . If these two branes touch each other at some point, an approximate (i.e. spontaneously broken) U (2) gauge group arises at the intersection, corresponding to Although D u intersects D l at two points leading to fermions with opposite chirality (as implied by the index theorem), these two localized fermions transform in different representations of the spontaneously broken electroweak gauge group, as in the standard model. Thus the chirality problem is resolved by identifying the Higgs as intrinsic part of the brane geometry.
The simplest example of such a configuration is given by two fuzzy spheres embedded as follows where λ i are su(2) generators in the N -dimensional representation. For r u = r d they touch at the south pole p − (and also at the north pole p + ), which leads to an approximate (spontaneously broken) SU (2) at p − as shown 9 in section 5. The corresponding Higgs φ will be identified shortly. These two fuzzy spheres realize the branes D u and D d touching each other.
Similarly, D d ∩ D l intersecting at p + and p − gives e R and e L , while D u ∩ D l at p + and p − gives ν R and ν L , respectively. The electroweak gauge fields should arise from the low-energy gauge fields on D u and D d . However, these gauge fields will in general not respect the decomposition into upper and lower halves, and will not couple to the chiral fermions at the various intersections in a clean way. This problem is resolved if these two fuzzy spheres are realized by S 2 N =2 , with minimal Hilbert Figure 3: minimal SM brane scheme spaces H ∼ = C 2 . This leads to the following "minimal" electroweak matrix configuration visualized in figure 3. Note that X 6 has four eigenvalues, two of which coincide if r u = r d .
In the absence of φ, the unbroken gauge group given by the commutant (stabilizer) of this background is therefore U (2) L × U (2) R if r d = r u . The gauge modes which do not commute with X 6 acquire a mass m 2 ∼ [X 6 , [X 6 , .]] given by the difference of the X 6 eigenvalues. The U (2) L is broken in the presence of φ, which will play the role of the electroweak Higgs. The U (2) R will also be broken by the scalar Higgs S, as elaborated in section 5. Furthermore, it turns out that chiral fermionic zero modes arise at the intersections even for this very fuzzy geometry, realized by coherent states |± u,d on the branes located at p ± . These fermions couple to the low-energy gauge group as required, and are connected by the φ u,d . Although these φ u,d clearly correspond to the electroweak Higgs sector, the precise role of their fluctuations and the relevance of the other geometrical moduli in the above configuration remains to be clarified.
Discussion. Before establishing these claims in more detail, we briefly discuss some of the further issues arising in this scenario. We need to specify the dimensions and type of the various branes. First, the above remark suggests that all four electroweak D0 branes corresponding to |± u,d should be located on both branes D B and D l , in order to obtain chiral near-zero modes which couple appropriately to the electroweak gauge fields. This suggests that D B and D l should be (nearly) coincident.
To get chiral fermions, the branes must span the internal space R 6 at the intersections. Thus we have two possibilities: either the electroweak branes D u,d are two-dimensional and D l,B are four-dimensional, or conversely. This choice affects the effective 4-dimensional gauge couplings, via the volume or trace over the extra dimensions. It turns out that the first possibility leads to a pattern of the electroweak gauge coupling constants (in particular the Weinberg angle) which seems unrealistic. We therefore take D l,B to be 2-dimensional fuzzy branes The extra K N 1 does not significantly change the above picture of the electroweak symmetry breaking, and merely introduces a multiplicative factor to the low-energy gauge groups. This allows a reasonable pattern of low-energy coupling constants, as discussed in section 5.
An important question is the fate of the extra U (1) gauge fields, which always arise in similar brane constructions [18]. Each brane comes with an associated U (1) acting with 1 i on H (i) , which do not acquire any mass term from a Higgs mechanism. The trace-U (1) decouples completely in the commutative limit (i.e. for ordinary N = 4 SYM), and can be identified with a gravitational mode on non-commutative space-time [6,7]; we will therefore ignore it in the present paper. Furthermore a U (1) B ∼ 1 3 1 B corresponding to baryon number B arises on the baryonic brane D B , and a U (1) L ∼ 1 l corresponding to lepton number L arises on the leptonic brane D l . Some of these will be affected by quantum anomalies, as discussed later. Most importantly, the electric charge also arises in this way, which is of course anomaly-free. Finally, quantum effects are expected to play an important role. Besides introducing (benign) anomalies, they will also mediate the interaction between the branes, which is expected to play an essential role in selecting and stabilizing the appropriate brane configuration. This should be a central theme for future work in this context.
Singlet Higgs S. To avoid an exactly massless U (1) B−L gauge field and to break SU (2) R for r u = r d , we assume that there is an extra singlet Higgs S connecting D u with D l at the location of ν R . S can be seen as superpartner of ν R , and it is a singlet of the standard model gauge group (cf. (3.3)). In the presence of a VEV S = 0, the D l and D u branes are unified into a single compact brane, which is natural in view of t Q = 1 2 (1 u +1 l −...). Clearly S breaks U (1) B−L , leaving only one extra U (1) 5 gauge field besides the standard model gauge group at low energies. That U (1) 5 acquires a mass by the electroweak Higgs, and is anomalous at low energies. Such anomalous U (1) gauge fields are expected to disappear from the low-energy spectrum by some variant of the Stückelberg mechanism, as discussed e.g. in [24][25][26]. The symmetry breaking will be discussed in more detail in section 5.
Finally, S = 0 allows to write down a Majorana mass term for the right-handed neutrino, such as Such a term is compatible with the full SU (N ) gauge symmetry, and could therefore arise in the quantum effective action even at high scales.

Intersecting brane solutions
In general, compact quantum spaces in Euclidean signature are never solutions of the classical matrix equation of motion X a = 0. However, quantum effects will contribute to the effective action. It is generally expected that this can be related to some sort of effective (super-) gravity in higher dimensions; for some partial results from the matrix model point of view see e.g. [1,5,[11][12][13][14]. In particular, this should lead to an attractive interaction between nearly-coincident branes, and it is plausible that suitable compact brane configurations may be stabilized in this way. Lacking more specific results, we will model these quantum contributions to the effective potential on a 4-dimensional space-time R 4 θ by a SO (6)-invariant function f tr N 9 a=4 X a X a : Note that the non-commutativity scale ρ allows to write down a dimensionless invariant radius operator. This leads to the equations of motion 10 which will have non-trivial brane solutions (reflecting the above discussion) provided f < 0 in some range. In particular, we give a solution with the properties discussed above, where D l and D B are realized as fuzzy spheres S 2 N l and a stack of three S 2 N B , while the electroweak branes D u and D d are realized as S 2 Recall that a fuzzy sphere S 2 N is the matrix algebra Mat(N, C) generated by the spin N −1 Denote the generators of S 2 Nu by L i and those of S 2 N l by K i . The generators of S 2 2 are σ i , which are the Pauli matrices σ i divided by 2. We also use the notation which implies and similarly for the other branes. Nevertheless, since the above effective action is certainly oversimplified, we will keep the different geometrical moduli r i , φ i , R i henceforth, assuming only that they have the same scale. The above brane configuration can alternatively be obtained as solution of the bare N = 4 SYM equations of motion on R 4 without quantum corrections, by letting them rotate in the extra dimensions as X a = R a b (t)X b , whereX a is given by (3.18), cf. [27,28]. This is indeed a solution for suitable rotations in the 4 − 5, 6 − 7 and 8 − 9 planes. However the rotation may distort the low-energy effective field theory in 4 dimensions, and we will not pursue this possibility here.
Intersections. Now consider the intersections of these branes. If r, φ RN u,l , then D u and There are hence two widely separated intersection regions located in target space approximately at ±R 2 N l (1, 0, 0, 0, 0, 0). Since the spheres are oriented, the helicity of the would-be zero modes is the same in the two intersection regions, as discussed in section 4. These two intersection regions could therefore be interpreted in terms of two generations. Alternatively, a deformation of D l (e.g. by the singlet Higgs S) might remove one of these intersection regions, leaving only one generation at this stage 11 .
For the intersections of the other brane pairs, we need similarly We will also impose N u = N d , so that SU (2) L can act naturally on the Hilbert spaces of H u and H d , as discussed below. To satisfy all these conditions 12 , it follows that N l ≈ N B , and R u ≈ R d , so that D u ≈ D d and D l ≈ D B from a geometric point of view.

Flat limit
To understand the intersections discussed above for small but finite r, φ in a simple way, we want to approximate the large fuzzy spheres near these intersections by tangential quantum planes. We will thus replace D u by R 2 89 × S 2 2 and D l by R 2 67 . In the limit of large N , the tangent space to a "point" on the fuzzy sphere generated by RL i tends indeed to the quantum plane R 2 θ , if accompanied by a suitable scaling of R. As the number of Planck cells is N and the area is proportional to R 2 N 2 , R should scale as R ∼ N − 1 2 in order to have a constant Planck cell size and thus a well-defined flat limit. However, in the above configuration, a scaling of R ( ) u,l would have to be accompanied by a scaling of φ, r. Hence, in order to keep φ, r constant, we keep R ( ) u,l constant, and thus obtain a quantum plane with noncommutativity 11 For example, this is achieved by slightly shifting one end of D l . 12 As discussed later, one way to introduce additional generations is via additional branes D (i) u,d . Their parameters R (i) and their quantum numbers N (i) should be very close to each other to ensure that they all intersect with the same D l,B branes. This leads to further constraints, and different N (i) are possible only if R u,d ≤ φ. θ ∼ N . Specifically, we can replace the tangent spaces of the large spheres by quantum planes Here the sign depends on the sign of L 3 and K 3 respectively, and determines the chiralities of the would-be zero modes. Then the effective noncommutativity of the tangential generators x a = R √ N y a is given by θ ∼ R 2 N , and e.g. the equation of motion (3.20) for the 8 − 9 components becomes We can now describe the intersections of D u with D l in more detail, assuming r, φ N R. In the limit of large N , we can write Assuming R u N u = R 2 N l to have perpendicular intersections, this reduces to the intersections of a minimal ellipsoid with a quantum plane, R 2 θ(89) × S 2 2 ∩ R 2 θ(67) . The picture of intersecting branes makes sense even for minimal fuzzy spheres S 2 N =2 , since their coherent states are located at the corresponding classical ellipsoid Taking into account the curvature of D l near y = 0, the intersection is determined by the 456 coordinates where ϕ is the angle on the normalized minimal fuzzy sphere, and ϑ on the large circle of D l . This suggests the following ansatz for the would-be zero modes ψ = |ϕ+ u ϑ| l ⊗ |s i (3.27) in terms of coherent states located at their classical intersection; here |ϕ+ u is the product of coherent states 13 located at the angle ϕ on S 2 2 and at the north pole L 3 = + Nu−1 2 of S 2 Nu , ϑ| l is a coherent state on S 2 N l located at the angle ϑ, and |s i indicates a suitable spinor state. It is not hard to see that this leads to approximate zero modes, consistent with the picture expected from the flat limit. However, we largely restrict ourselves to the flat limit in this paper, as elaborated below.

Equations of motion for the singlet Higgs S.
To complete the background, we should solve the equations of motion for the singlet Higgs S, linking D u with D l at ν R . Such a link between two branes will be localized at one (or both) of their intersections, suggesting an ansatz Here |p+ u denotes the tensor product of a state p on S 2 2 and the coherent state L 3 |+ = Nu−1 2 |+ on S 2 Nu , and q is a state on S 2 N l . To obtain a solution, we let S rotate in the 8 − 9 plane, via the ansatz Here t = c µ X µ is a vector of the four-dimensional Minkowski space, time-like and normalized w.r.t. the effective metric on R 4 θ : G µν c µ c ν = ρη µ ν θ µ µ θ ν ν c µ c ν = −1.
Now we confine ourselves to the linearized equation of motion for the perturbation H a , which reads Similarly, we compute for a ∈ {4, 5, 6, 7} It follows that for a ∈ {8, 9}, the equation of motion (3.31) is fulfilled. For a ∈ {8, 9}, we note that 13 Coherent states on fuzzy spheres are obtained by SO(3) rotations of the highest weight states, cf. [30].
Hence, it suffices to choose the state n p n ⊗ q n such that The double commutator on the lhs is a hermitian operator on C 2 ⊗C N l , which can be diagonalized. One would expect that the two lowest eigenstates λ are localized near the intersections (0, 0, ±1) on S 2 2 and S 2 N l . This expectation can be verified numerically with very high precision. Choosing one of these corresponds to either coupling the right-or the left-handed neutrino to the scalar Higgs. Then (3.28) is a solution of the equations of motion (3.31) for We have thus established that the linearized equation (3.31) indeed has a solution with the desired properties. This completes our brane configuration of the matrix model. To estimate λ, we use the natural ansatz S = |ϕ+ u ϑ| l in terms of coherent states on the branes (cf. (3.27)). Then for large N , consistent with numerical results. The seemingly ad-hoc coupling of S to D u rather than D d can be interpreted as spontaneous breaking of the SU (2) L × SU (2) R gauge symmetry down to SU (2) L × U (1). This is discussed further in section 5. Furthermore, the back-reaction of S to D u,l might lead to a shift of the branes, possibly removing the other intersection regions of the branes (e.g. at X 4 ≈ −RN ). Then a single generation would arise for the above background.
A rotating scalar Higgs carries a background U (1) current, which may be useful to set the scale for the branes. We will verify later that the rotation of S does not affect the Dirac equations governing the fermionic would-be zero modes, at least in the above linearized approximation. Nevertheless (e.g. to avoid undesired backreactions), it may be desirable to have a static background. It is not clear if a simultaneous solution of both (3.20) and (3.33) without free parameter ω S can be achieved; however this could be possible by taking into account the non-linear contributions 14 from S, as well as quantum contributions. Alternatively, S might not be needed at all if the two minimal ellipsoids in D u,d touch only at one point, so that no SU (2) R arises. We leave these issues for future work.

Chiral fermions in the flat limit N → ∞
In this section, we consider the limit N → ∞, where the fuzzy spheres S 2 N become much larger than the minimal electroweak branes S 2 N =2 . We can then replace S 2 N by a quantum plane R 2 θ near the intersection with S 2 2 , and obtain exact results for the (would-be) chiral fermions. This allows to understand the resulting low energy physics in a simple way.

S 2 intersecting R 2
We want to understand the origin of massless chiral fermions arising on the intersection of the above minimal ellipsoids embedded in the 456 directions with a flat brane R 2 θ in the 67 plane, dropping the 89 directions for now. Thus consider D u realized by M at(2, C) acting on H (u) ∼ = C 2 , and D l realized by an operator algebra acting on H (l) . We should therefore find the (near-) zero modes of the "internal" Dirac operator in the 4567 direction (using the conventions of appendix A) for the off-diagonal fermions Ψ ∈ H (u) ⊗ H * (l) for a background of two branes .

(4.2)
Here ∆ a are the SO(6) Gamma matrices. As a warm-up, consider first the intersection of a single D0-brane with R 2 θ (cf. [31]). The D0 brane is given by a projector X a (u) = p a |p u p| u located at p a = (p 4 , p 5 , p 6 , 0). Then the fermions linking the state |p u with R 2 θ have the form for some state ψ on R 2 θ . Then we can write i.e. p is located in D l . Furthermore, it is easy to see following [9] that / D (2) Ψ = 0 if and only if |ψ l is a coherent state on R 2 θ localized at p ∈ D l , with definite chirality associated with D l ∼ = R 2 . This can be seen by introducing the shifted creation-and anihilation operators for R 2 Heren = a † a is the number operator. We also introduce a fermionic oscillator representation for the Gamma matrices ∆ a , Hence the chirality operator on R 2 θ is given by acting on the spin-1 2 irreducible representation. Moreover, it is straightforward to show that With these tools, we can write From either equation it follows that / D (2) Ψ = 0 if and only if Ψn = 0 = (1 + χ)Ψ, which means that ψ (l) is a coherent state on R 2 θ localized at p ∈ R 2 , with definite helicity χ = −1. Putting these results together, it follows that / D has zero modes linking the D0 brane with R 2 It is remarkable that this is optimally localized at p ∈ R 2 . However, there are two degenerate states with both chiralities s i = ±1, corresponding to a vanishing index [16]. If p is located at some finite distance from R 2 θ , then these states are massive. Now we switch on a Higgs field realized by the non-commutative X a (u) given by the background (3.12). We denote the basis of H (u) with |± u , so that (4.14) We claim that now / D int has precisely one chiral zero mode located at each p ± . To see this, we write down again the Dirac operator for off-diagonal fermions acting on the above states as Noting that {X 6 , X 7 (l) } satisfy the same algebra R 2 θ as {X 6 (l) , X 7 (l) }, we can introduce modified ladder operatorsã and therefore Therefore / D (2) Ψ = 0 if and only ifãã † Ψ = 0 = (1 + χ)Ψ. This is equivalent tõ which means that Ψ consists of coherent states localized at X 6 = r 2 σ 3 and X 7 = 0. More explicitly, expanding Ψ in the appropriate basis {|+ u n + | l , |− u n − | l } (dropping helicities) where n ± | l denotes an oscillator basis with origin p ± , it follows that the zero modes of / D (2) have the form Imposing in addition / D (1) Ψ = 0 using (4.16) and noting that ∆ + |+, ↑ u = 0 and ∆ − |−, ↓ u = 0 leaves the following two zero modes of / D int with definite chirality L, R in R 4 . These are the two chiral zero modes located at p ± expected from the picture of intersecting branes. As a check, it is straightforward to verify using (4.19) that the states (4.23) are indeed zero modes of / D int . If D l is described by some curved brane, then these would-be zero modes acquire some small mass. The associated Yukawa coupling will be proportional to the Higgs φ, as discussed in the next section.

R 2 × S 2 intersecting R 2
Finally we add the missing R 2 θ(89) to D u = R 2 89 × S 2 . This is achieved simply by adding ∆ 8 X 8 + ∆ 9 X 9 to / D (int) and / D (1) , leading to an additional term where b, b † form the oscillator representation of R 2 θ(89) and The additional contribution vanishes if and only if bΨ = 0 = βΨ, so that the above results generalize immediately. We obtain the following two zero modes of / D int with definite chirality L, R in R 6 . These are the two chiral zero modes located at p ± expected from the picture of intersecting branes. Finally, note that the Dirac equation for these fermionic would-be zero modes are not affected by H a (S) due to the coherent state property h a X a |0 u = 0, except for ν R , which could acquire a Majorana mass term via the gauge singlet tr N (ν R S).
Mirror fermions. Besides these zero modes, there are additional pairs of massive fermions ("mirror fermions") with opposite chirality at the same intersections, coupling to the same gauge fields. The lowest ones arise from the opposite helicity of the coherent state |+ on the minimal S 2 2 , with eigenvalue of / D (1) of order 2φ. We denote those by ψ. Additional sets of ultra-massive fermions with mass of order θ arise from other helicity and oscillator states on R 2 θ . In this way, a chiral model emerges at low energies from the non-chiral underlying N = 4 theory, with a large hierarchy between the low-energy chiral fermions and their massive mirror partners. Such a mirror model could be phenomenologically viable provided the hierarchy is sufficiently large.
One potential problem is the fact that the tree level mass of these lowest mirror fermions is only by a factor √ 2 higher than the tree-level W mass, both being determined by φ (see section 5.2). However, this is also the scale of the KK modes on the large branes, which couple to the fermions but not to the electroweak gauge fields. It then seems reasonable that quantum effects increase the mass of the mirror fermions sufficiently high above the electroweak scale.

Deformations, would-be zero modes and Yukawa couplings 4.3.1 Analytical expectation
Armed with these results for the flat case, we would like to understand the fermions arising at the intersections of the compact branes D u ∩ D l given by large and small fuzzy spheres (3.18). We expect that the qualitative features of the flat limit survive: there should be pairs of near-zero eigenmodes of / D int called would-be zero modes henceforth, which due to 3)) decompose into states Ψ L , Ψ R of definite chirality, which in turn are approximately localized at the intersections p ± of the branes. However, the helicities should be determined by the local tangent planes at the intersections. We therefore expect that the following ansatz in terms of coherent states should be appropriate at least if the intersection 15 is perpendicular. Here the coherent states are located at the intersection of the branes, with slightly modified helicity orientation reflecting the local geometry. The incompatible spin orientations of the pair of would-be zero modes leads to non-vanishing Yukawa couplings and eigenvalues. To gain some analytic insights, let us compute these 15 Here the classical orbit of the loci of the coherent states on the fuzzy branes is relevant. For the fuzzy sphere, this has radius R N −1 Yukawa couplings explicitly for the above ansatz (4.27). Consider first In the last step, we observed that only the second term in (σ + ∆ − + σ − ∆ + ) can give nonvanishing matrix elements between −, | u and |+, u , and evaluated the action of σ − . The contribution from the coherent states on D l can be approximated by since the two spin directions should be appropriately aligned, as long as D l is much larger than S 2 2 . In the flat limit this inner product would be exponentially suppressed with the distance of the two coherent states on D l , although this factor is typically very close to 1 for the compact branes under consideration. On the other hand, the contribution from the coherent states on D u can be approximated by the spin contribution only, This is non-vanishing only due to the non-alignment of the two helicities at the two intersections. Assuming that the spinor wavefunctions factorize (as in the flat case), the spinor associated with S 2 2 is given by where ∼ sin( 1 2 ϑ) and ϑ measures the angle of the two distinct tangent planes T p ± M relative to the flat limit. This gives using (3.26) for the present geometry. Combining these results, we obtain the desired Yukawa couplings This is clearly small for the would-be zero modes under consideration, while for the mirror fermions ψ with reversed spin associated with the S 2 2 we would get due to |∆ + | u ≈ 1. In particular, there is naturally a large hierarchy O( 2 ) between the lowest chiral sector corresponding to the standard model, and the first series of massive mirror fermions. Moreover these quantities are accessible, both to (refined) analytical considerations and to numerical methods. Of course they will also be subject to quantum corrections, which are out of the scope of the present paper. One may hope that these quantum corrections help to increase the separation of the mirror fermions from the electroweak W, Z bosons, as discussed below.

Numerical results
Some aspects of the theoretical expectations derived in the previous subsection can be verified numerically. We compute the eigenvalues of the internal Dirac operator for the off-diagonal spinors connecting the two branes, and identify them with the Yukawa couplings of their chiral components. Restricting to a regime where the Planck cells of the larger fuzzy spheres are greater than r, we can neglect the Gaussian factor in (4.34), and obtain for the lowest eigenvalue and for the next eigenvalue, i.e., the mirror fermions, In Figures 4, 5, and 6, we see that these estimates agree quite well with numerical results (the red lines show the expectations from (4.36) and (4.37)). Also the next eigenvalue λ 2 is shown. Furthermore, we see that one can produce large hierarchies for moderate N .  However, varying φ while keeping r fixed leads to a dramatic deviation from the expectation (4.36), as shown in Figure 7 (note that the scale is logarithmic): There is a very pronounced minimum of λ 0 at φ = r/2 (which is excluded by the equation of motion in Section 3.4). This is also seen for other choices of N, R, so it seems to be a universal behavior. From geometrical considerations, one would rather expect φ ≈ 1 √ 2 r to be special, as then the branes intersect orthogonally. Hence, a complete understanding of the Yukawa couplings is missing, but the generation of a large gap between the lowest and the next eigenvalue of / D int is certainly possible.
For our arguments, it is crucial that the lowest eigenstates, when projected to a definite 6dimensional chirality, are very well localized at ±(0, 0, 1) on S 2 2 , and are essentially eigenvectors of χ 45 = 2Σ 45 , the chirality operator corresponding to the 4−5 plane. 16 From (4.32), we expect the expectation value of 1 − Σ 45 in the lowest eigenstate ψ 0 to be roughly (4.38) Figure 8 shows the expectation value of 1−σ 3 and 1−Σ 45 in the lowest eigenstate as a function of R = R l,u = R l,u . Also the expectation (4.38) is plotted. 17 We see that the deviation from being an eigenstate indeed decreases for increasing R. 16 Note that they are then also essentially eigenvectors of χ 67 χ 89 . Projecting on the eigenspaces of, say, χ 89 , one obtains a vector which is essentially an eigenvector of Σ 45 , Σ 67 , and Σ 89 . 17 Note that our ansatz was that ψ 0 is an eigenstate of σ 3 corresponding to S 2 2 , so the above discussion does not give a prediction for the expectation value of 1 − σ 3 .

Gauginos
Besides the fermions arising at the interactions of the various branes, fermions also arise in the diagonal blocks, as functions on the corresponding branes. They can be viewed as (generalized) gauginos, i.e. superpartners of the gauge bosons or scalar fields. All these fermions are nonchiral, i.e. both chiral sectors couple identically to the gauge and scalar fields. This includes the gluinos, binos, winos, Higgsinos, etc. Note that the gauginos corresponding to U (1) B and U (1) L are neutral under the full standard model gauge group, and therefore decouple at low energies. As usual they may be considered as dark matter candidates. The gluinos and other gauginos are expected to get radiative mass. Furthermore, there are towers of higher KK modes for all these fermions. However, note that the present backgrounds are far from supersymmetric, since e.g. the scalar superpartners of the standard model fermions have tree level mass of order θ ∼ R 2 N .

Fermion masses and Yukawas
In this section, we show that the 4-dimensional masses of the would-be zero modes are given by the corresponding Yukawa couplings. As recalled in Appendix A, cf. also (2.9), the Dirac operator can be written as Here is the massless 4-dimensional Dirac operator on R 4 θ withγ µ = ρ 1 2 θ νµ γ ν , and / D int is defined in terms of the X a . Now consider a pair of eigenspinors ψ ± of the internal Dirac operator in terms of two chirality eigenstates (such as our would-be zero modes), with Γ (int) ψ = ±ψ to be specific. Then ρ 1 2 / D int ψ ↑ = mψ ↓ , corresponding to a Yukawa coupling m. Now consider a 32-component spinor whose internal components consist of the above two internal helicity states, recalling the definition of β in section 4.2. This vanishes due to βψ = 0 for the would-be zero modes, using the explicit form of S.

Symmetry breaking and 4-dimensional fields
Spelling out all the branes, the background (3.18) is given by We also included the Higgs singlet S, which connects the branes D u and D l . We want to understand the bosonic modes which arise as fluctuations X a → X a +A a about the above background. In general, such fluctuations on a fuzzy brane K N can be written as a finite sum where Y lm ∈ M at(N, C) stands symbolically for the harmonics of = [X a , [X a , .]] on K N . This applies both to scalar fields, gauge fields and the gauginos. It provides a geometric interpretation of the matrix-valued fields on R 4 in terms of towers of massive Kaluza-Klein modes on K N . On the fuzzy sphere S 2 N , this KK tower arises at roughly equidistant masses determined by the eigenvalues of , with lowest non-trivial eigenvalue ∼ R 2 . Therefore at low energies, it suffices to keep only the massless modes ∼ 1 on the S 2 N . Then the A(x, y) can be viewed as functions on R 4 taking values in the above space of 8 × 8 matrices. In particular, the stack of three coincident D B branes gives rise to the massless gluons with unbroken U (3) = SU (3) c × U (1) B gauge symmetry, as well as an associated finite tower of massive KK modes. On the other hand, the KK tower on the minimal branes is very short, and contains in particular the electroweak Higgs, the Z boson, and the B 5 and C µ bosons as discussed below.

Hierarchical symmetry breaking
To determine the masses of the low-energy gauge bosons explicitly, it is useful to first replace the two minimal fuzzy spheres S 2 N =2 by a stack of 4 coincident D0 branes X a (1) = p a 4 i=1 |i i| located at some point p a on the coincident D l ∼ = D B branes described by 1 4 ⊗ X a (2) . This background admits a U (4) × U (4) symmetry. Now we switch on a non-vanishing singlet Higgs S ∼ |i p| l , where |p l is a coherent state on D (l) located at the D0 branes p a . Since S has rank one, it breaks the symmetry to the commutant U (3) B × U (3) × U (1) S , where U (1) S acts diagonally on |i ⊕ |p l . This U (1) S can be traded for U (1) tr , which has a clear geometric interpretation. We assume here that this breaking happens at a high scale, and restrict ourselves to the commutant of S from now on. The fermionic modes on such a background are still non-chiral.
Next, we introduce the long axis along X 6 of the electroweak ellipsoids by turning on r > 0. This breaks the above symmetry further to the commutant of X 6 (in U (3) B × U (3) × U (1) tr ). The bosons C µ associated with this breaking will be discussed below. Using the explicit form (5.1), this commutant is given by Here we assume that D l,B is represented on C N 2 , and λ α ∈ u(3) = su(3) ⊕ u(1) B . Note that t 5 acts as on the fermionic zero modes. It is therefore anomalous and expected to disappear from the low-energy spectrum, along with U (1) B . This leaves exactly the gauge group of the standard model SU (3) c × SU (2) w × U (1) Y , extended by the anomalous U (1) B , U (1) 5 , and the geometric U (1) tr . The U (1) 5 is also broken by the electroweak Higgs, as elaborated below. Finally, we switch on the electroweak Higgs φ u,d , so that the 4 D0 branes expand to form two fuzzy ellipsoids S 2 2 . Then the symmetry breaks down as desired to SU (3) c × U (1) Q × U (1) B × U (1) tr , with charge generator Here U (1) B is anomalous, and U (1) tr is a geometric mode associated to gravity. As a check, the unbroken gauge group of the above brane configuration can alternatively be obtained as follows. Consider first the background without S, given by two coincident branes D u D d and 4 coincident branes D l D B . This background has an unbroken U (2) × U (4) gauge symmetry. Now we switch on the scalar Higgs S. This breaks the symmetry to its

Four-dimensional gauge bosons and masses
We recall the 4-dimensional form of the effective action (2.9) of the matrix model in our brane background. To obtain the proper coupling constants for the corresponding gauge fields, we 18 Here I N1 indicates the identification C Nu ∼ = C N d of the Hilbert spaces of S 2 Nu and S 2 N d , assuming N u = N d = N 1 .
introduce canonically normalized generators with tr N (t itj ) = 1 2 δ ij , via t ±3 = c Lt±,3 , The point is that thet α act on the full Hilbert space of the matrix model and satisfy rescaled commutation relations, while the t α act on the reduced Hilbert space of physical fermionic states as in the standard model. To identify the low-energy gauge couplings, we write the gauge fields in two ways using (5.3) The effective standard model coupling constants in the second line are identified from the covariant derivative on the fermions on the fermionic would-be zero modes. Since the relevant fermionic (would-be) zero modes are made of one-dimensional (coherent) states in the internal Hilbert space, the term tr Nψγ µ iD µ ψ in (2.9) reduces to the appropriate Lagrangian for the 4-dimensional fermions in the standard model, without any extra factors coming from tr N . We can therefore identify the gauge fields W i , B, etc. with those of the the standard model, where g is the SU (2) L coupling constant, g is the U (1) Y coupling constant, g S is the strong coupling constant, and g 5 the one associated with U (1) 5 . These tree-level couplings apply at very high energies. The kinetic terms of these gauge fields have the standard normalization, and by gauge invariance their full action must be Here tr red is the trace in the adjoint representation of the reduced low-energy gauge group ⊕ i g i generated by the t i , with gauge fields A, W, etc. corresponding to the (extended) standard model; for example, the contributions of the gluons is Finally consider the action for the scalar fields Φ a , which describe the internal branes and contain in particular the Higgs. Their kinetic term leads as usual to SSB of the gauge fields. The most interesting part is the electroweak symmetry breaking, induced by the minimal fuzzy ellipsoids S 2 N . Let us elaborate their effect on the low-energy fields. These terms arise from In view of (5.1), it is natural to organize the non-vanishing entries of X + = X 4 + iX 5 in terms of "effective" Higgs doublets with Here The Z boson is identified as the combination of W 3 and B which acquires a mass, On the other hand, the last form of (5.5) guarantees that t Q does not couple to the Higgs. The masses are obtained from 19 We can then read off the W and Z bosons masses in the high energy regime, taking into account a factor N 1 from tr N : The U (1) 5 is anomalous at low energies, hence it is expected to disappear from the low-energy spectrum by some Stückelberg-type mechanism, cf. [24][25][26]. The photon and the Z-boson are then identified as usual .
This gives the Weinberg angle and sin 2 θ W = 1 (5.23) For N 1 = N 2 this gives sin 2 θ W = 3/8 and g S = g, as in the SU (5) GUT. Similarly, we can compute the mass of the gauge bosons C µ associated with the breaking of SU (3) → SU (2) L × U (1) due to r > 0. A typical generator t C ∼ |+ d −| u,d 1 N 1 links the standard model fermions to the first massive mirror fermions, such asd R ↔ d L , u L or e R ↔ ν L , e L . Since t C relates different eigenvalues of X 6 , we have [X 6 , t C ] = ±r t C . There will also be a mass contribution from the Higgs φ in the same way as the W bosons, leading to a mass term This is larger than the W ± mass, assuming r > φ 2 . Next consider the contribution to the gauge boson masses from the singlet Higgs S via X a (S) = h(e 8 + ie 9 )e iω S t S + h.c.. Recalling (3.28) and (3.29), the relevant terms in the action are dropping possible fluctuations of S here. Even though this goes beyond the linear approximation that we used for S, it is clear that this gives a mass to every gauge field coupling to S. It breaks the lepton number U (1) L , and in the absence of r, φ it breaks the electroweak U (4) to U (3) (subsequently broken to SU (2) L by (5.24)). In particular S breaks U (1) B−L , which is anomaly free and would otherwise lead to an unphysical massless gauge boson, whereas U (1) L is anomalous and expected to disappear from the low-energy spectrum anyway. Finally, the fermion masses for the off-diagonal fermions arise from taking into account the factor 2 from (A.13), which also enters the kinetic term. Here f ψ is the Yukawa coupling for the fermion under consideration, as discussed in section 4.3. The trace tr N gives no extra factor since the fermions are made from coherent states. Therefore the fermion mass is given by where f ψ is the corresponding Yukawa coupling. For the first series of mirror fermions we found f ψ ≈ 1, so that their tree level (!) mass is about √ 2 times the W mass. In contrast, the standard model fermions have much smaller Yukawas.
At first sight, the low scale of the mirror fermions seems very bad. However, keep in mind that we merely computed the tree level masses here 20 , valid at high energies in the N = 4 regime. At lower energies, the Yukawa couplings will be subject to quantum corrections. For example, an effective factor α > 1 in front of the internal Dirac operator α / D (int) rises the fermion masses without affecting the boson masses, thus increasing the gap between the electroweak scale and the first mirror fermions. More specifically, since the fermions are given by localized (coherent) states on the large branes S 2 N , they couple to all the massive KK gauge and scalar fields arising on these branes. These KK modes start at a scale set by R, which is comparable to the scale of the first mirror fermions ψ by (3.20). Therefore they will contribute significantly to the Yukawa couplings. In contrast, these KK modes do not contribute to the mass of the W, Z gauge bosons, because these are ∝ 1 on the large branes. This should magnify the gap between the electroweak scale and the first mirror fermions, and one may hope that the model can become phenomenologically viable in this way.
In any case, the model clearly predicts mirror fermions with opposite chirality at not very high energies. These mirror fermions interact with the standard model gauge bosons, and can decay into the standard model fermions via the heavy gauge bosons C µ . More quantitative statements would require computing quantum effects.

Moduli and the Higgs potential
The action for the geometrical moduli φ, r, R i is obtained from the modified matrix model action (3.15) as 20 The term V quant (3.15) in the effective potential does not affect the mass of the fermions and gauge bosons.
We have The coefficient of the term of order δφ vanishes, by the equation of motion (3.20). The remainder can be simplified, and comparison with the kinetic term yields the following mass squared for the fluctuations δφ (here we introduced physical units): This is positive and somewhat larger than m 2 W , unless f is too negative. There is another interesting set of low-energy perturbations, given by Goldstone bosons of the global SO(6) symmetry acting uniformly on all matrices, corresponding to local rotations of the matrix background. This affects only the trace-U (1) sector of the model and leads to metric perturbations related to the effective or "emergent" gravity on R 4 θ , as elaborated in [29] for a similar type of background.

Further aspects
Anomalies and massive gauge fields. In the present type of background (as in analogous brane-configurations in string theory [18]), a U (1) gauge symmetry arises on each brane, some of which are anomalous at low energies. This does not signal an inconsistency, since the fundamental U (N ) gauge symmetry is anomaly-free. Rather, it indicates that the corresponding anomalous gauge bosons acquire a mass and disappear from the low-energy physics. This topic has been discussed extensively in the literature, see e.g. [26], or [24,25] in a closely related context, based on a type of Stückelberg mechanism with an axion.
In fact, axion-like fields appear in non-commutative gauge theory via the term η(x)F ∧F , where the "axion" is realized by the geometric field η(x) = Gg in the picture of emergent gravity [6,7,32,33]. This should be related to the Chern-Simons-terms arising in the Dbrane action in string theory. The precise origin of such mass terms in the present context should be clarified.
In particular, baryon number U (1) B is such an anomalous gauge symmetry. It should still provide protection from proton decay, in contrast to many grand-unified models. This is important in view of the highly populated spectrum of fields at intermediate energies.
Generations. Additional generations can arise if the large fuzzy spheres S 2 N in either D u,d or D l,c are replaced by stacks of spheres with slightly different parameters. On the other hand, we have seen that there are in fact two separated intersection regions contributing to e.g. D u ∩D l . This would also manifest itself as doubling of generations, which is actually unwelcome as it would imply an even number of generations. However, one of these intersection regions could be removed in principle.
Right-handed neutrinos. One clear prediction of our solution is the presence of righthanded neutrinos ν R , which acquire a Dirac mass term determined by the corresponding Yukawa coupling. In addition, it seems plausible that a Majorana mass term (3.14) is induced by quantum effects. This aspect should be studied in more detail. For a survey on the phenomenological aspects of right-handed neutrinos we refer to the recent review [34].

Discussion and conclusion
We have shown that the IKKT model can behave very similar to the standard model at low energies, for suitable backgrounds. We provided such backgrounds consisting of branes in the internal space, which are solutions of the matrix model (with linearized singlet Higgs S), assuming a suitable stabilizing term in the effective potential. Our results also apply to N = 4 SU (N ) SYM with sufficiently large N , challenging the standard lore that N = 4 SYM can only be a "spherical cow" approximation to realistic gauge theories. We recover the chiral fermions of the standard model with the correct quantum numbers coupling appropriately to the electroweak gauge fields. Right-handed neutrinos arise, as well as towers of massive Kaluza-Klein modes of other fields, ultimately completing the full N = 4 spectrum at very high energies. Our results are supported by numerical computations of the spectrum of low eigenvalues of the internal Dirac operator / D int , verifying also the chirality and localization properties of the corresponding fermionic modes.
One clear prediction is the existence of mirror fermions at intermediate energies, which can decay to standard model fermions via massive gauge bosons. The mass of the lowest mirror fermions is rather low at tree level (about √ 2 times the W mass, which obviously would not be realistic), however it seems likely that quantum effects raise their scale to higher energies; this should be studied in detail elsewhere.
The Higgs sector is found to be more intricate than in the standard model, consisting of two doublets, which form an intrinsic part of the internal rotating branes. This should lead to some protection from quantum corrections. The electroweak scale is set by the geometrical scale of the internal compact branes. Another important parameters is the rank N of the internal matrices, which determines the size of the "large" internal fuzzy spheres, and in particular the hierarchy of the Yukawa couplings for the standard model fermions versus the mirror fermions. The standard model Yukawas can be made arbitrarily small for N → ∞, while those of the mirror sector remain fixed at tree level. However, to assess the viability of the resulting model, quantum corrections due to the towers of massive modes must be taken into account.
There are many open questions and issues raised by this work. One important issue is the scale of the mirror fermions. A reliable computation of this and other physical parameters requires computing the quantum corrections due to integrating out the Kaluza-Klein tower of massive fields. Due to the rich spectrum this is a formidable task even at one loop, which however should be feasible. Understanding the quantum contributions to the effective potential is also essential to clarify the stability of the background, and to clarify whether it is necessary to add a stabilizing term such as (3.15) by hand. Ultimately, this should also allow to select preferred backgrounds among the mini-landscape of matrix model configurations.
Although we have intentionally hidden any noncommutative aspects, it should be clear that our solution really defines a fully noncommutative version of the (extended) standard model on quantized Minkowski space R 4 θ . If the required stabilizing potential indeed arises through quantum effects in N = 4 SYM, the model can be expected to be perturbatively finite and free of pathological UV/IR mixing, in contrast to previous proposals [35]. Moreover, gravity is expected to be included automatically in the matrix model, encoded in the trace-U (1) sector [6,7], [36]. However, this is not yet fully understood.
Another open issue is the non-linear back-reaction effects of the singlet Higgs S. It is not clear if a rotating S can be incorporated in a fully consistent way, or if S should be seen as an approximation to some more complicated static background. As discussed in the text, the breaking of SU (2) R might also be achieved by deforming the background, which is also related to the question of generations. A related aspect is the possible Majorana mass term for ν R , which is expected to arise due to S.
If the above issues can be resolved in a satisfactory way, many interesting physical issues could be addressed, including in particular the physical properties of the Higgs. In any case, we have certainly demonstrated that there is no fundamental obstacle for obtaining nearstandard-model physics from the matrix model. Forschung under grant P24713. Useful discussions with J. Nishimura and A. Tsuchyia are gratefully acknowledged.

A Clifford algebra and reduction to 4 dimensions
The ten-dimensional Clifford algebra, generated by Γ A , naturally separates into a fourdimensional and a six-dimensional one as follows, Here the γ µ define the four-dimensional Clifford algebra, and are chosen to be real corresponding to the Majorana representation in four dimensions for η µν = diag(−1, 1, ..., 1). Then γ 0 = −γ † 0 = −γ T 0 and γ i = γ † i = γ T i . The ∆ a define the six-dimensional Euclidean Clifford algebra, and are chosen to be real and antisymmetric. The ten-dimensional chirality operator separates into four-and six-dimensional chirality operators This means that the lower-diagonal matrices ψ * 12 are nothing but "anti-particles" of the upperdiagonal "particles", so that there is no further doubling of the chiral fermions ψ 12 stretching from brane 1 to brane 2 as identified in the text. Spelling out this block-matrix structure of the fermions, the Yukawa couplings are where X c = diag(X c (1) , X c (2) ). Two of these terms coincide Tr ψ * T 12 γ 0 γ 5 X c (1) ∆ c ψ 12 = − Tr X c (1) ψ T 12 ∆ c γ 0 γ 5 ψ * 12 (A.12) using the Grassmann nature of ψ, so that Tr Ψγ 5 / D int Ψ = Tr ψ T 11 γ 0 γ 5 / D 6 ψ 11 + ψ T 22 γ 0 γ 5 / D int ψ 22 + 2ψ * T 12 γ 0 γ 5 / D int ψ 12 (A. 13) This gives rise to the Yukawa couplings computed in section 4.3. 21 Note that T transposes only the spinor.