On Multimatrix Models Motivated by Random Noncommutative Geometry II: A Yang-Mills-Higgs Matrix Model

We continue the study of fuzzy geometries inside Connes’ spectral formalism and their relation to multimatrix models. In this companion paper to Pérez-Sánchez (Ann Henri Poincaré 22:3095–3148, 2021, arXiv:2007.10914), we propose a gauge theory setting based on noncommutative geometry, which—just as the traditional formulation in terms of almost-commutative manifolds—has the ability to also accommodate a Higgs field. However, in contrast to ‘almost-commutative manifolds’, the present framework, which we call gauge matrix spectral triples, employs only finite-dimensional algebras. In a path-integral quantization approach to the Spectral Action, this allows to state Yang–Mills–Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields exactly mirror those of the Yang–Mills–Higgs theory on a smooth manifold.


Introduction
The approximation of smooth manifolds by finite geometries (or geometries described by finite-dimensional algebras) has been treated in noncommutative geometry (NCG) some time ago [36] and often experiences a regain of interest; in [20,25], for instance, these arise from truncations of space to a finite resolution. In an ideologically similar vein but from a technically different viewpoint, this paper addresses gauge theories derived from the Spectral Formalism of NCG, using exclusively finite-dimensional algebras, also for the description of the space(time). This allows one to make precise sense of path integrals over noncommutative geometries. Although this formulation is valid at the moment only for a small class of geometries, the present method might shed light on the general problem of quantization of NCG, already tackled using von Neumann's information theoretic entropy in [15,24], by fermionic and bosonic-fermionic second quantization, respectively. Traditionally, in the NCG parlance, the term 'finite geometry' is employed for an extension of the spacetime or base manifold (a spin geometry or equivalently [18,48] a commutative spectral triple) by what is known in physics as 'inner space' and boils down to a choice of a Lie group (or Lie algebra) in the principal bundle approach to gauge theory. In contrast, in the NCG framework via the Spectral Action [13], this inner space-called finite geometry and denoted by F -is determined by a choice of certain finite-dimensional algebra whose purpose is to encode particle interactions; by doing so, NCG automatically rewards us with the Higgs field. Of course, the exploration of the right structure of the inner space F is also approached using other structures, e.g., non-associative algebras [7,9,27,54] for either the Standard Model or unified theories, but in this paper we restrict ourselves to (associative) NCG-structures.
Still in the traditional approach via almost-commutative geometries M × F [14,52,57], the finite geometry F plays the role of discrete extra dimensions or 'points with structure' extending the (commutative) geometry M , hence the name. What is different in this paper is the replacement of smooth spin geometries M by a model of spacetime based on finite-dimensional geometries ('finite spectral triples') known as matrix geometry or fuzzy geometry [3]. Already at the level of the classical action, these geometries have some disposition to the quantum theory, as it is known from well-studied 'fuzzy spaces' [23,37,49- At this point, it is pertinent to clarify the different roles of the sundry finite-dimensional algebras that will appear. Figure 1 might be useful to illustrate why matrix algebras that differ only in size are given different physical nature. In this cube, pictorially similar to Okun's 'cube for natural units' [28,40], classical Riemannian geometry sits at the origin (0, 0, 0). Several NCG-based theories of physical interest may have, nevertheless, the three more general coordinates ( , 1/N, F ) described now: • The F -direction in Fig. 1 describes (bosonic) matter fields. Mathematically the possible values for F correspond to a 'finite geometry.' These were classified by Paschke and Sitarz [43] and diagrammatically by Krajewski [35]. Particle physics models based on NCG and the Connes-Chamseddine spectral action [2,6,14,19,21,25] 'sit along the F -axis'. From those spectral triples F , only their algebra appears in Fig. 1.
• A finite second coordinate, 1/N > 0, means that the smooth base manifold that encodes space(time) has been replaced by a 'matrix geometry,' which in the setting [3] is a spectral triple based on an algebra of matrices of size N (and albeit finite-dimensional, escaping Krajewski's classification).
• The remaining coordinate denotes quantization when = 0. In the path integral formalism, the partition function is a weighted integral Z = dξ e iS(ξ)/ over the space of certain class of geometries ξ, the aim being the quantization of space itself, having quantum gravity as motivation.
Here S is the classical action. Accordingly, the planes orthogonal to the axis just described are: • The plane ( , 1/N, 0) of base geometries. On the marked plane orthogonal to F lie 'spacetimes' or 2 'base manifolds' and, when these are not flat, they can model gravity degrees of freedom. If F = 0, no gauge fields live on such space.

•
The plane ( , 0, F ) = lim N →∞ ( , 1/N, F ). On the plane orthogonal to the 'matrix geometry' axis, one has the quantum, smooth geometries (meaning, their algebra is or contains a C ∞ (M ) as factor). The longterm aim is to get to the 'quantum smooth geometry plane' as matrix algebras become large-dimensional, which is something that, at least for the sphere, is based on sound statements [44][45][46][47] in terms of Gromov-Hausdorff convergence. Additional to such large-N , one might require to adjust the couplings to criticality [8,29,34]. This can also be addressed using doubly scaling limits together with the Functional Renormalization Group to find candidates for phase transition; for models still without matter, see [42]. • The plane (0, 1/N, F ) = lim →0 ( , 1/N, F ) of classical geometries. By 'classical geometry' we mean a single geometrical object (e.g., a Lorentzian or Riemannian manifold, a SU(n)-principal bundle with connection, etc.), which can be determined by, say, the least-action principle (Einstein Equations, SU(n)-Yang-Mills Equations, etc.). This is in contrast to the quantization of space, which implies a multi-geometry paradigm, at least in the path integral approach. The program started here is not as ambitious as to yield physically meaningful results in this very article, but it has the initiative to apply three small steps-one in each of the independent directions away from classical Riemannian geometry-and presents a model in which the three aforementioned features coexist. This paves the way for NCG-models of quantum gravity coupled to the rest of the fundamental interactions (it is convenient to consider the Figure 2. Depicting the organization of this article, following the path P QR. Here, F ym-h = (M n (C), M n (C), D F ) corresponds to the spectral triple for the Yang-Mills-Higgs theory and G f is a fuzzy four-dimensional geometry. As outlook (dashed), to reach a smooth geometry at the point S one needs a sensible limit (e.g., large-N and possibly tuning some parameters to criticality) in order to achieve phase transition theory as a whole, due to the mutual feedback between matter and gravity sectors in the renormalization group flow; cf. [22] for an asymptotic safety picture). For this purpose, we need the next simplifications, as illustrated in Fig. 2: • Our choice for the finite geometry F is based on the algebra A F = M n (C) (n ≥ 2). This is the first input, aiming at a SU(n) Yang-Mills theory.
• Instead of the function algebra on a manifold, we take a simple matrix algebra M N (C). This is an input too. (Also N is large and n need not be.) • We use random geometries instead of honest quantum geometries; this corresponds with a Wick rotation from e iS(ξ)/ , in the partition function, toward the Boltzmann factor e −S(ξ)/ . This setting is often referred to as random noncommutative geometry [5,29].
Random NCG was introduced in [8]. While aiming at numerical simulations for the Dirac operators, Barrett-Glaser stated the low-dimensional geometries as a random matrix model. The Spectral Action of these theories was later systematically computed for general dimensions and signatures in [41]. Also, in the first part of this companion paper, the Functional Renormalization Group to multimatrix models [42] inspired by random noncommutative geometry was addressed for some two-dimensional models obtained in [41]. Solution of the matrix-models corresponding to one-dimensional geometries was addressed in [1], using Topological Recursion [26] (due to the presence of multitraces, in its blobbed [10] version).
The organization of the article is as follows. Next section introduces fuzzy geometries as spectral triples and gives Barrett's characterization of their Dirac operators in terms of finite matrices. Section 3 interprets these as variables of a 'matrix spin geometry' for the (0, 4)-signature. Section 4 introduces the main object of this article, gauge matrix spectral triples, for which the spectral action is identified with Yang-Mills theory, if the piece D F of Dirac operator along the 'inner space spectral triple' 3 vanishes, and with Yang-Mills-Higgs theory, if this is non-zero, D F = 0 (see Sect. 5). Our cutoff function f appearing in the Spectral Action Tr H f (D) is a polynomial f (instead of a bump function 4 ). In Sect. 6, we make the parallel of the result with ordinary gauge theory on smooth manifolds. Finally, Sect. 7 gives the conclusion and Sect. 8 the outlook, while also stating the explicit Yang-Mills-Higgs matrix model for further study.
This article is self-contained, but some familiarity with spectral triples helps. Favoring a particle physics viewpoint, we kept the terminology and notation compatible with [57].

Spectral Triples and Fuzzy Geometries
Let us start with Barrett's definition of fuzzy geometries that makes them fit into Connes' spectral formalism.

Definition 2.1.
A fuzzy geometry is determined by • a signature (p, q) ∈ Z ≥0 , or equivalently, by for all T, W ∈ M N (C) and v, w ∈ V . To wit γ : V → V is self-adjoint with respect to the Hermitian form (v, w) = av a w a on V ∼ = C k and satisfying γ 2 = 1. This k is so chosen as to make V irreducible for even s. Only the ±1-eigenspaces of V with the grading γ are supposed to be irreducible, if s is odd The representation is often implicit • an anti-linear isometry, called real structure, J f := C ⊗ * : H f → H f given in terms of the involution * (in physics represented by †) on the matrix algebra and C : V → V an anti-linear operator satisfying, for each gamma matrix, • a self-adjoint operator D on H satisfying the order-one condition • the condition 5 Dγ f = −γ f D for even s. Moreover, the three signs above impose: Notice that, in this setting, the square of J f is obtained from C as specified above, but we added the redundant Eq. (2.3a), as this equation appears so for general real, even spectral triples. which is typically an axiom for spectral triples, is not assumed in our setting. However, one can show that it is a consequence of those in Definition 2.1. The axiom states that the right A-action ψb : commutes with the left A-action for each a, b ∈ A. Since J = C ⊗ * , and the algebra acts trivially on V , The focus of this paper is dimension four, but we still proceed in general dimension. We impose on the gamma matrices γ μ the following conditions: (γ μ ) 2 = +1 V , and γ μ Hermitian for μ = 1, . . . , p, Since it will be convenient to treat several signatures simultaneously, we let (γ μ ) 2 =: e μ 1 V for each μ = 1, . . . , d. According to Eq. (2.6), one thus obtains the unitarity of all gamma-matrices: without implicit sum, and for each v, w ∈ V . Let these matrices generate Ω := γ 1 , . . . , γ d R as algebra, for which one obtains a splitting Ω = Ω + ⊕ Ω − where Ω ± is contains products of even/odd number of gamma-matrices. According to [3,Eq. 64], the Dirac operator D f solves the axioms of an even-dimensional fuzzy geometry whenever it has the next form:

7)
{A, B} ± := AB ± BA, where T ∈ M N (C) and the sum is over increasingly ordered multi-indices I = (μ 1 , . . . , μ 2r−1 ) of odd length. With such multi-indices I, the following product γ I := γ μ1 · · · γ γ2r−1 ∈ Ω − is associated (the sum terminates after finitely many terms, since gamma-matrices square to a sign times 1 V ). Moreover, still as part of the characterization of D f , e I denotes a sign chosen according to the following rules: • Type (0,4), s = 4, Riemannian. Since the triple product of anti-Hermitian gamma matrices is self-adjoint, (γ α γ μ γ ν ) * = (−) 3 γ ν γ μ γ α = γ α γ μ γ ν , so are the operator-coefficients, which have then the form {H αμν , • } for (H αμν ) * = H αμν : where γρ means the product of gamma matrices with indices different from ρ, multiplied in ascending order; see the restriction in the sum in the expression for D In the sequel, we use K I generically for either H I or L I , whose adjointnesstype is then specified by the signature and by I. We also define the sign e I by K * I := e I K I , or equivalently by (γ I ) * = e I γ I , for a multi-index I. In four dimensions, one has for triple indices I =μ [41, App. A] In summary, a fuzzy geometry of signature (p, q) has following objects: Although next equation is well-known, we recall it due of its recurrent usefulness later. In any dimension and signature, it holds: (2.10) Each inscribed segment in the chord diagrams denotes an index-pairing between two indices labeling their ends, say λ and θ, which leads to η λθ ; all the pairings of each diagram are then multiplied bearing a total sign corresponding to (−1) to the number of simple chord crossings. This picture is helpful to compute traces of more gamma-matrices, but is not essential here; see [41] to see how the spectral action for fuzzy geometries was computed by associating with these chord diagrams noncommutative polynomials in the different matrix blocks K I composing the Dirac operator. Incidentally, notice that so far this chord diagram expansion is classical, unlike that treated by Yeats [58, §9], which appears in the context of Dyson-Schwinger equations.

Toward a 'Matrix Spin Geometry'
We restrict the discussion from now on to dimension four, leaving the geometry type (KO-dimension) unspecified. Next, we elaborate on the similarity of the fuzzy Dirac operator and the spin-connection part spanned by multiindices, which has been sketched in [3, Sec. V §A] for d = 4. The identification works only in dimensions four and, if 'unreduced' (cf. Footnote 6 above) also three. For higher dimensions, quintuple products appear; for lower ones, triple products are absent. Although it would be interesting to address each dimensionality separately, since the physically most interesting case is dimension 4, we stick to it.
Remark 3.1. Since some generality might be useful for the future, or elsewhere (e.g., in a pure Clifford algebra context), even though we identify the geometric meaning only for the objects in Riemannian signature, we prove most results in general signature.
The coefficients ω i = 1 2 ω μν i γ μν of the spin connection ∇ S (the lift of Levi-Civita connection) are here expressed with respect to a base γ μν = 1 4 [γ μ , γ ν ] that satisfies the o(4) Lie algebra in the spin representation (see, e.g., [17, §11.4]). The gamma matrices with Greek indices (or 'flat') γ μ relate to the above Γ i (x) = e i μ γ μ by means of tetrads e i μ (x). The coefficients e i μ ∈ C ∞ (U ), by definition, make of the set of fields (E μ ) μ=0,1,2,3 = (e i μ · ∂ i ) μ=0,1,2,3 an orthonormal basis of X(U ) with respect to the metric g of M , which is to say In contrast to the commutation relations that the elements of the coordinate base ∂ i = ∂/∂x i satisfy, one generally has [E μ , E ν ] = 0 for the non-coordinate base E 0 , . . . , E 3 , also sometimes called non-holonomic [55, §4]. Notice that in the fuzzy setting only Greek indices appear.
This, together with the fact that rather η μν instead of g ij appears in the Clifford algebra, should not be interpreted at this stage as flatness. Instead, for fuzzy geometries the equivalent of a metric is encoded in the signature η = diag(e 0 , . . . , e 3 ) and in the matrices parametrizing the Dirac operator.
In Riemannian signature, we rewrite 7 (cf. Ex. 2) Simultaneously (up to the trivial factor 1 V ), we identify the commutators [L μ , • ] with iE μ = ie j μ ∂ j and the coefficients of the triple gamma products {H μσν , • } eμσν with the full anti-symmetrization i 4 ω [μ|ik e i |σ e k ν] of the spin connection coefficients in the three Greek indices. The triple products of gammamatrices present in the Dirac operator (3.2) are the analogue of those in the spin connection appearing in D M = iγ μ (E μ + e i μ ω i ), here in the 'flat' (nonholonomic or non-coordinate) basis E 0 , . . . , E 3 . Altogether, ∇ S f can be understood as the matrix spin connection.
We let Δ 4 = {0, 1, 2, 3} and denote by δ μνασ the fully symmetric symbol with indices in Δ 4 , which is non-vanishing (and then equal to 1) if and only if the four indices are all different; equivalently, δ μνασ = | μνασ |, in terms of the (flat) Levi-Civita symbol . Remark on notation. Specially when dealing with fuzzy geometries, we sometimes do not use Einstein's summation (traditional in differential geometry). We avoid raising and lowering indices as well, e.g., gamma matrices are presented only with upper indices. We set This lemma is proven in "Appendix A." Notice that in Eq. (3.3) the repeated indices μ, ν in the RHS are not summed (therefore the index-symmetry of δ μνλρ with the antisymmetry of γ μ γ ν does annihilate that term).

Lemma 3.3. The square of the Dirac operator of a fuzzy geometry
is We defined also the (whenever non-vanishing) signs For the first term, one obtains To get the first two terms in the RHS of Eq.
We now see that the only Gothic letter left unmatched, c, is precisely the fifth term. Indeed, due to Lemma 3.2, where in the last step we exploited the skew-symmetry of the gammas with different indices to annul the restriction α < σ on the sum by introducing sgn(σ − α). The term in square brackets is s μνασ .
Notice that the analogy in Table 1 goes further, since in the case of a smooth manifold spin manifold (M, g), the fields ∂ 0 , . . . , ∂ 3 , or equivalently

Gauge Matrix Spectral Triples
We restrict the discussion to even KO-dimensions ( = 1) and define the main spectral triples for the rest of the article. Their terminology is inspired by the results. The reader might want to see Table 2, which will be hopefully helpful to grasp the organization of the objects introduced this section. But first, we recall that the spectral triple product G 1 × G 2 of two real, even spectral triples Definition 4.1. We define a gauge matrix spectral triples as the spectral triple We should denote these geometries by G (N ) f × F (n) , but for sake of a compact notation, we leave those integers implicit and write G f × F .

Yang-Mills Theory From Gauge Matrix Spectral Triples
In order to derive the SU(n)-Yang-Mills theory on a fuzzy base, we choose the following inner space algebra: A F = M n (C). This algebra acts on the Hilbert space H F = M n (C) by multiplication. The Connes' 1-forms Ω 1 D (A)

and its fluctuations
Notation Given by Lies in Dirac op. Type where the sums are finite. The latter algebra is the fuzzy analogue of the algebra In order to compute the fluctuated Dirac operator, we start in this section with the fluctuations along the fuzzy geometry (labeled with f) and leave those along the F direction for the Sect. 5. Thus, turning off the 'finite part' D F = 0, one obtains for ω f of the form (4.1), with respect to the 'purely fuzzy' Dirac operator On the Yang-Mills matrix spectral triple over a four-dimensional fuzzy geometry of type (p, q), i.e., of signature η = diag(+ p , − q ), the fluctuated Here, the curly brackets are a generalized commutator {A, Proof. The theorem follows by combination of Lemma 4.3 with Lemma 4.4, both proven below.
Proof of Lemma 4.3. We set X = 0 globally in this proof. Pick a homogeneous vector in the full Hilbert space and a, c ∈ A F , the action of ω on Ψ yields The second part of the inner fluctuations is, for each the next expression: where the last step is a consequence of Eq. (4.7) .
to be multiplied by the right. Hence, As a result, the fully-fluctuated operator acting on . The triviality of the part of the Dirac operator along the finite geometry F implies that , and the significance of each factor can be obtained by comparison with the smooth case. There, the inner fluctuations of a Dirac operator on an almost-commutative geometry are given by Recall that in the smooth case it is customary to treat only Riemannian signature together with self-adjointness (which we do not assume) for each gamma-matrix Γ i = c(dx i ), c being Clifford multiplication. For each point x of the base manifold M , one has A i (x) ∈ i su(n) = i Lie SU(n). Since Eq. (4.8b) represents the fuzzy analogue, that equation can be further reduced We now have to add the fluctuations resulting from the triple products of gamma matrices.
Lemma 4.4 (Fluctuations with respect to the X μ -matrices). With the same notation of Theorem 4.2 and additionally setting K μ = 0, the innerly fluctuated Dirac operator D gauge is given by (4.10) Proof. See "Appendix A." From the last subsections, the rules for S μ and A μ lead to the manifest self-adjointness of D ω f .

Field Strength and the Square of the Fluctuated Dirac Operator
We introduce now the main object of the gauge theory. To this end, let 8 Definition 4.5. We abbreviate the following (anti)commutators Proposition 4.6. The square of the fluctuated Dirac operator of a Yang-Mills gauge matrix spectral triple that is flat (X = 0, x = s = 0) is given by The first summand is known from Lemma 3.3. One obtains the last summand by Eq. (4.6) and using the Clifford algebra relations just as in the proof of that lemma. The result reads We renamed indices and rewrote the last summand in (4.16) as 1 2 γ μ γ ν ⊗[a μ , a ν ] To make the notation lighter, we also mean by a μ a ν the composition a μ • a ν from now on (also for k μ ). Using Eq. (4.6), one can obtain for the two summands in the middle of Eq. (4.15); further abbreviating Again, we used the Clifford relations for the gamma matrices and renamed indices. Equations (4.16) and (4.17) imply Expanding Eq. (4.12) Proof. According to Eq. (4.4) so D 2 ω has the same structure already observed in the 'fuzzy Lichnerowicz formula' above (Lemma 3.3). To be precise, notice that one can compute the square of the present Dirac operator by replacing the in D f the following operators: k → k + a and x → x + s.

Gauge Group and Gauge Transformations
For any even spectral triple, the Hilbert space H is an A-bimodule. The right action of A on the Hilbert space H Ψ is implemented by the real structure J, Ψa := a o Ψ := Ja * J −1 Ψ. Both actions define the adjoint action Ad(u)Ψ := uΨu * of the unitarities u ∈ U(A) on H. We want to determine the action of the unitarities U(A) = {u ∈ A | u * u = 1 = uu * } of the algebra A on the Dirac operator, The commutant property is essential for the subalgebra to be also a subalgebra of the center Z(A), as we will see later. The gauge group G of a real spectral triple is defined via the adjoint action Ad u (a) = uau * of the unitary group U(A) on H as follows: (4.24) Before proceeding to compute it for a case concerning our study, we do the notation more symmetric, setting n 1 = N and n 2 = n for the rest of this section. We assume n 1 > n 2 ≥ 2. The next statement is not surprising, but due to the presence of the tensor product, some care is needed. Before proving this proposition, broken down in some lemmata below, we recall the characterization of the gauge group that will be used. Namely, the next short sequence is exact, according to [ We now verify that they do, so that after computing U(A) and U(Z(A)), we can finally obtain the gauge group by this isomorphism (4.26).
where * i is the involution of A i and Ψ an arbitrary vector in the Hilbert space described above. Therefore, z 1 ⊗ z 2 ∈ A J . One verifies that this proof leads equally to Z(A) ⊂ A J by taking other representing element z 1 λ ⊗ z 2 λ −1 (λ ∈ C × ) the same conclusion Z(A) ⊂ A J is reached, which restricted to the unitarities gives U(Z(A)) ⊂ U(A J ).
Proof. The embedding C × → C × × | det | C × is given by λ → (λ, λ −1 ) and the next map C × × | det | C × → U{Z(A 1 ⊗ A 2 )} by (z 1 , z 2 ) → z 1 ⊗ z 2 . Being the rest an easier case than that the proof of Lemma 4.10, the details on exactness can be deduced from there.
We are now in position to give the missing proof.
Proof of Proposition 4.8. According to Eq. (4.26), where one passes to the second line by Lemma A.1. By Lemma 4.11 for the group in the 'numerator' and Lemma 4.10 for the one in the 'denominator,' By the third group isomorphism theorem, one can 'cancel out' the C × , and get Notice that in each group | det | only constrains the real parts, while it respects the U(n 1 ) and U(n 2 ) in the numerator and the two factors U(1) of each C × in the denominator. We conclude that

Unimodularity and the Gauge Group
It turns out that for real algebras the gauge group does not automatically include the unimodularity condition, and this property needs to be added by hand. Since this is relevant for the algebra that one uses as input to derive the Standard Model (cf. discussion in [57, Ch. 8.1.1, Ch. 11.2]) we address also the unimodularity of the gauge group, when the base itself is noncommutative. Given a matrix representation of a unital * -algebra A, the special unitary group of A is defined by

SU(A) := {m ∈ U(A) | det[ (m)] = 1}.
We now define the following morphisms δ i : GL(n i ) → C × , which shall be useful in the description of the special unitary group we care about (notice that both morphisms depend on the pair (n 1 , n 2 ) and the different signs in the exponents).

Lemma 4.12. The special unitary group of
fits in a short exact sequence of groups: where U(n 1 ) × det U(n 2 ) is the (categorical) pullback of any of the two morphisms (4.28) along the remaining one.
On the other hand, if g 1 ⊗ g 2 ∈ ker(det) ⊂ SU(A 1 ⊗ A 2 ) then each g i ∈ GL(n i ) (otherwise its determinant vanishes and by assumption it is 1) so we can write them in matrix polar form g i = p i u i with u i ∈ U(n i ) and p i = p * i positive definite. Since, in particular, Being both p i 's positive definite Hermitian matrices, they can be written as Solving each equation leads to Λ 1 = r 1/2 1 n2 and Λ 2 = r −1/2 1 n2 , so we can forget the v i 's, since Λ i is central.
By way of contrast, an important one conceptually, we stress that for SU(n)-Yang-Mills(-Higgs) finite geometries where one has A 1 = M N (C) and 10 Here the fact that the unitary quaternions {q ∈ H : q * q = 1 = qq * } are unimodular (i.e., their determinant is 1 in the embedding of H into 2 × 2 matrices), U (H) ∼ = SU(2), causes that unimodularity has influence on the H summand. That is why μ 3 appears as fiber instead of μ 3×2 . A 2 = M n (C) (so n 1 = N and n 2 = n above), n is the 'color' analogue, the two (special) unitary factors in Proposition 4.8 or the unimodular analogue above, have a different nature. The PU(N ) [resp. SU(N )] describes the symmetry of the base (and could be understood as the finite-dimensional analogue of diffeomorphisms of a manifold) and PU(n) [resp. SU(n)] along the fibers.

Yang-Mills-Higgs Theory with Finite-Dimensional Algebras
The Higgs field being considered at the same footing with the gauge bosons is one of the appealing characteristics that is offered by the gauge theory treatment with NCG. We now recompute the results of Sect. 4, revoking the restriction D F = 0. The aim is a formula informed by Weitzenböck's. The Weitzenböck formula, D 2 ω = Δ S⊗E + E, includes the Higgs Φ and extends Lichnerowicz's formula, to the product of the spinor bundle S with a vector bundle E. It is given in terms of an endomorphism E in Γ(End(S ⊗ E)): where (∇ S⊗E ) j and F ij are locally the connection on S ⊗ E and the curvature on E, respectively. Further, γ M is the chirality element or γ 5 in physicists' speak. (See e.g., [57,Prop. 8.6] for a proof.)

The Higgs Matrix Field
We now turn off the fuzzy-gauge part of the spectral triple in order to compute the fluctuations along the finite geometry F . These fluctuations are namely generated by the second summand in the original (in the sense, 'unfluctuated'). Dirac operator of the product spectral triple . Also φ * = φ holds. Proof. As before, one computes the corresponding Connes' 1-forms a[γ f ⊗ D F , c] in terms of a = 1 V ⊗ W ⊗ a and c = 1 V ⊗ T ⊗ c, being W, T ∈ M N (C) and a, c ∈ M n (C). Namely, We rename φ := X ⊗ a[D F , c], since W, T are arbitrary and their product can be replaced by any matrix X ∈ M N (C). Thus, DF (M n (C)) as claimed. Since from the onset γ is self-adjoint, so must be φ, since ω * F = ω F is required. The remaining part of the fluctuations is self-adjoint, as argued before.
In Eq. (5.3) of the proof, one could also have computed directly, using the explicit formula (3.4) for the chirality: The complex conjugate in the last line appears since C is anti-linear. The sign is chosen as follows: notice that σ(η) is purely imaginary for the (1,3) and (3,1) signatures (and otherwise it is a sign). This means that the sign ± in last equation is (−1) #number of minus signs in η = (−1) q . This different way to compute leads to the same result as the one given in the proof. Indeed, for four-dimensional geometries (−1) q is precisely , according to the sign table in Definition 2.1, namely = −1 for KO-dimensions 2 and 6 and = +1 for KO-dimensions 0 and 4. From Proposition 5.1 and Theorem 4.2, the form of the most general fluctuated Dirac operator follows: the Higgs field, since in the smooth Riemannian case (where the analogous relation reads Φ (C ∞ ) = D F + J F φ (C ∞ ) J −1 F ) its analogue in the context of almostcommutative geometries leads to the Standard Model Higgs field, when the finite algebra A F is correctly chosen (cf. [14,57]).
Corollary 5.2. The fluctuated Dirac operator D ω on the 'flat' (x = 0) fuzzy space factor G f of a gauge matrix geometry G f × F satisfies Notice also that γμ = γ α γ ρ γ σ anti-commutes with γ, for Since the matrices γ μ and γμ, μ ∈ Δ 4 , span (the projection to V of) D gauge , the anti-commutator {D gauge , D Higgs } is traceless also if the fuzzy space is 'curved,' X = 0.

Transformations of the Matrix Gauge and Higgs Fields
Throughout this section, we always assume the Riemannian signature. We now compute the effect of the gauge transformations, already explicitly known for the Dirac operator, on the field strength F μν and on the Higgs field. For the former, this requires to know how the matrices A μ transform under G(A; J) = U(A)/U(A J ). We can pick a representing element of G(A; J) in u ∈ U(A) directly, since the apparent ambiguity up to an element z ∈ U(A J ) leads to 11 The last line is obtained since U(A J ) = U(Z(A)), so z is central (and thus z * too). Hence, [D, z * ] = 0 (Table 3).
Next, observe that, by definition, and also by Jacobi identity on M N ⊗n (C), with analogous expressions for [l ν , a μ ] • and [a μ , a ν ] • . This allows to write the field strength as the commutator with another quantity F μν ∈ M N ⊗n (C) that we call field strength matrix, 8a) Table 3. Notation for the matrices parametrizing the Dirac operator of Riemannian four-dimensional Yang-Mills-Higgs matrix spectral triples and its fluctuations along The accompanying gamma-matrices in the former case are omitted. The rows in gray will not be used below (X is set to zero, and this implies the vanishing of the rest of operators in gray rows). See Eq. (5.5) for more details We now find the way the field strength transforms under the gauge group. By definition, the transformed field strength is given by the expression F μν evaluated in the transformed potential a u μ , this latter being dictated by the way the Dirac operator transforms under G(A; J). Specifically, Concerning the Higgs, we come back to Eq. (5.5). We deduce from there and from (4.20), that the matrix field φ, which parametrizes by (5.5e) the Higgs field, transforms like The transformation of the field strength is more interesting: In Riemannian signature, the field strength of a Yang-Mills (-Higgs) finite geometry transforms under the gauge group as follows: which is completely determined by the next transformation rule on the field strength matrix Proof. Observe that for the pair l μ , a u ν the same argument given about Eq. (5.7) for the pair l μ , a ν holds, and so does for the other pair of composition commutators appearing in the u-transformed field strength. Therefore, we can indeed write it in terms of the matrix F u μν : We now compute the transformed field strength matrix and all those terms that imply A (namely the transformation under u of T μν := F μν − [L μ , L ν ] → T u μν ) and infer from that those the gauge transformations on the field strength matrix F μν .
The contributions to T u μν split into three: LA-terms (i.e., containing L μ , A ν or L ν , A μ ), AA-terms, and LL-terms. We compute them separately: • AA-terms: uA μ u * , uA ν u * = u[A μ , A ν ]u * , clearly • LL-terms: When the commutators are expanded, the next LL-terms yield the quantity in bracelets, which can be neatly rewritten: This can be also obtained expanding the commutators as above; the last two commutators yield u{[L μ , A ν ]−[L ν , A μ ]}u * +r(L, A). The excess terms r(L, A) are actually cancelled out with the two first propagators, yielding for the final expression of the LA-terms: In view of the last equalities, we can conclude that which, re-expressed in terms of F, yields F μν → F u μν = Ad u (F μν ). Remark 5.4. Notice that L μ being the fuzzy analogue of the derivatives, the 'surprising term' [L μ , L ν ] is the analogue 12 of [∂ μ , ∂ ν ], which is identically zero on the algebra C ∞ (M ). This seems to (but, as we will see, does not) imply the freedom of choice as to whether we take the field strength matrix as defined above by F μν , or ratherF μν = F μν − [L μ , L ν ] (called T μν above). According to Eq. (5.14),F μν transforms then as Although for quadratic actions the last two terms add up to a traceless quantity, higher powers of the Dirac operator would mix the gauge sector with others. This confirms that the definitions in Eqs. (4.12) and (5.8b) are correct. For only then, the pure gauge sector (i.e., powers of F ) obtained from Tr H (D 2m ω f ) would be expressible (see [41], and for m = 2, Eq. (2.10) above) as a sum over chord diagrams ξ, with μ = (μ 1 , . . . , μ m ), ν = (ν 1 , . . . , ν m ), The scalars ξ μ1ν1μ2ν2...μmνm are expressed as sums of m-fold products of the bilinear form η ασ (signature) and are irrelevant for the discussion. The important conclusion is that, due to Proposition 5.3, the traced quantity is gauge invariant, since the transformation rule ignores the 'space-time indices' μ i and ν i . The quartic computation is explicitly given below.

Traces of Powers of D
The next statement is obvious: one gets the result by Eq. (4.14).
for some sign ± in last line, which is in fact irrelevant since Tr V (γ μ γ ν γ ρ γ) = 0 for any choice of indices. The line before the last is also traceless. Further, using Tr V (γ μ γ ν γ α γ ρ ) given in Eq. (2.10), (5.19) By symmetry of η and skew-symmetry of F , the first chord diagram vanishes, and by the same token, also the second line in Eq. (5.19). The second chord diagram comes with a minus sign and, using the skew-symmetry F , one can see that the third diagram yields the same contribution, namely μ,ν,ρ,σ (−η νρ η μα ) Tr M C N ⊗n (F μν F αρ ). For the second to last line, Tr V (γ μ γγ ν γ) = −η μν Tr V 1 V . Dividing the whole Eq. (5.19) by dim V = 4 and get the claim.

The Spectral Action for Yang-Mills-Higgs Matrix Spectral Triples: Toward the Continuum Limit
We now give the main statement and, after its proof, we compare it with [13, §2], which derives from NCG the Yang-Mills-Higgs theory over a smooth manifold. Since in differential geometry the Einstein summation convention is common, we restore it here (also in the fuzzy context) together with the raising and lowering of indices with the constant signature η μν = (η −1 μν ) and η μν . Using the lemmata of previous sections, we can give a short proof to the main result: Tr where each sector is defined as follows:

2)
and the rest terms in the ellipsis represents operators Tr M C N ⊗n [P ] being P ∈ C l μ , a μ | μ = 0, . . . , 3 of order ≥ 5. Further, g e is the even part of the polynomial g truncated to degree < 5.Moreover, one obtains positivity for each of the following functionals, independently: Proof. Recall D = D gauge +D Higgs . It is obvious that Tr H (D) = 0. The possible crossed-products contributions to Tr(D 3 ) are Tr(D 2 gauge D Higgs ) and Tr(D gauge D 2 Higgs ). The former vanishes because in spinor space V we have to trace over γ μ γ ν γ, which vanishes. Similarly, D 3 Higgs is traceless since γ 3 = γ is, and and D 3 gauge vanishes by Tr V (γ μ γ ν γ ρ ) = 0. Thus odd powers of D are traceless, at least for degrees < 5.
Hence inside the trace over H, f can be replaced by its even part f e . Notice that by Lemmas 5.5 and 5.6, then Observe that by definition, Eq. (4.14), ϑΦ 2 = d μ d μ Φ 2 , so the last term yields by expansion of the commutators and cyclicity, The result follows by inserting the definitions from Eq. (6.2) and by observing that the trace of D 6 is a noncommutative polynomial (which we do not determine) of homogeneous degree 6 in the eight letters a and l; this is, in the worse case, the rest term in (6.1).
Regarding positivity: First, notice that a μ a μ = a μ (e μ a μ ) = a μ (a μ ) * is a positive operator, and that so is ϑ ∈ End(M N (C) ⊗ H F ) by the same token, is well-defined and its trace positive, since f e is by definition an even polynomial.
Further relations like [k μ , a ν ] * = −e μ e ν [k μ , a ν ], and similar ones for all the commutators defining the field strength, lead to F * μν = −e μ e ν F μν . Since η = diag(e 0 , . . . , e 3 ), one obtains the positivity of the operator Therefore, also the positivity holds summing over μ, ν, which is a positive multiple of S f ym , whose positivity also follows. Similarly, since Φ is self-adjoint, even powers of it are positive, thus so is S f H . We now comment on the interpretation of this result. For fuzzy geometries, the equivalent of integration over the manifold is tracing operators M N (C) → M N (C). (At the risk of being redundant, notice that the unit matrix in that space has trace N 2 .) First, recall that Φ is self-adjoint. We identify the Higgs field H on a smooth, closed manifold M with Φ, so the quartic part M |H| 4 vol of the potential for the Higgs is Tr M C N ⊗n (Φ 4 ). In the Riemannian case, in order to address the gauge-Higgs sector 13 , notice that since Φ = Φ * , if a 4 = 1, This interpretation of d μ = l μ + a μ as the covariant derivative D μ = ∂ μ + A μ for Yang-Mills connection, with the local gauge potential A μ absorbing the coupling constant (cf. Def. 4.5 and Remark 5.4). Next, notice that F μν is a matrix-version of the SU(n)-Yang-Mills (local) curvature F μν for the action S ym . If a 4 = 1, one has the exact correspondence For the time being, the previous identifications hold only the Riemannian signature, since for (p, q) = (0, 4) anti-commutators appear; these, unlike commutators, are no longer derivations in the algebraic sense. Nevertheless, keeping this caveat in mind, we extend the previously defined functionals to any signature (there, each l μ is replaced by k μ ). It holds then in general signature.
of the smooth case is further supported by the fact that [l μ , a ν ] • generalizes the multiplication operator ∂ μ A ν , on top of the reason already given in Remark 5.4. The alternative to this definition, using only l μ • a ν in place of [l μ , a ν ] • (and similar replacements), yields instead (∂ μ • A ν )ψ = (∂ μ A ν ) · ψ + A ν ∂ μ (ψ) on sections ψ (fermions). Notice also that for the smooth field strength one gets the positivity of the type of Eq. (6.4), namely − Tr su(n) (F μν F μν ) ≥ 0, due to F μν F μν = −F μν F μν [17, below Eq. 1.597]. We summarize this section in Table 4.
Remark 6.2. Notice that in the expression for the Yang-Mills action, when the model is fully expanded in terms of the fields k and a, the next tetrahedral action appears (6.7) as well as the same type of action, , in the variable a. The reference to a tetrahedron is justified when one writes that action in full, where the faint (blue) lines correspond to contractions of Greek indices and black lines to matrix-indices i, j, m, l. Modulo the restriction μ = ν present in the sum, this kind of action is an example of the 'matrix-tensor model' class [4].

Conclusions
We introduced gauge matrix spectral triples, computed their spectral action and interpreted it as Yang-Mills-Higgs theory, if the inner-space Dirac operator is non-trivial (and as Yang-Mills theory if it is trivial), for the fourdimensional geometry of Riemannian signature. We justified this terminology based on Remark 5.4 and Sect. 6; in particular see Table 4 for the summary.
Higgs lagrangian The partition function of the Yang-Mills-Higgs theory is an integral over gauge potentials A μ and a Higgs field Φ in (subspaces of the) following matrix spaces where Ω 1 f and Ω 1 F are the Connes' 1-forms along the fuzzy and the finite geometry, respectively, both parametrized by (finite) matrices, see Sect. 8. Additionally, the partition function for the spectral action implies an integration over four copies of su(N ); each of these matrix variables L μ appears as the adjoint l μ = ad Lμ = [L μ , • ]. These operators l μ are interpreted as degrees of freedom solely of the fuzzy geometry, in concordance with the identification of Der(M N (C)) with a finite version of the derivations on C ∞ (M ), that is, vector fields.
As in the almost-commutative setting M × F , with M a smooth manifold, the Higgs field arises from fluctuations along the finite geometry F and the Yang-Mills gauge fields from those along the smooth manifold M . This is apparent in the parametrizing matrix subspaces (see Eq. (8.3)) for the matrix Higgs field and the matrix gauge potentials, which are swapped if one simultaneously 14 exchanges n ↔ N and F ↔ f. The Yang-Mills-Higgs matrix theory has a projective gauge group G = PU(N ) × PU(n). The left factor corresponds with the symmetries of the fuzzy spacetime and the right one with those of the 'inner space' of the gauge theory (a similar interpretation holds for the unimodular gauge groups in Lemmas 4.12 and 4.13), so the whole group G could be understood as C ∞ (M, SU(n)) after a truncation has been imposed on M . A rigorous interpretation, e.g., in terms of spectral truncations [20], is still needed.
Another approach to reach a continuum limit resembling smooth spin manifolds is the Functional Renormalization Group, which could be helpful in searching the fixed points (cf. the companion paper [42] for the application of this idea to general multimatrix models).

Outlook
Aiming at a model with room for gravitational degrees of freedom, the careful construction of a Matrix Spin Geometry needs a separate study (in particular requiring X μ = 0 and thus also a more general treatment than that of Sect. 5). If that is concluded, one could identity for signature (0, 4) In order to give a more structured appearance to the partition function for Riemannian (p = 0), flat Yang-Mills-Higgs spectral triples, we recall the dependence of our functionals on the fundamental matrix fields L μ , A μ and φ. The L's are functioning as derivatives L μ ∈ su(N ) ⊗ 1 n , and l μ = ad Lμ = [L μ , • ] is the derivation defined by the adjoint action, l μ ∈ Der M N (C) ⊗ 1 n , for each μ. One arrives at a similar situation with the matrix gauge potentials anti-Herm. ⊗ M n (C) s.a. ⊂ M N (C) ⊗ M n (C), where the subindex in the curly brackets restricts to anti-Hermitian 1-forms. In terms of these (a 0 , a 1 , a 2 , a 3 ) = a = a(A μ ) is defined, again, via derivations: a μ = ad Aμ = [A μ , • ], which already bear a non-trivial factor in the inner space. 15 This yields dependences l = l(L), a = a(A). Further, by Eq. (5.5e), also Φ = Φ(φ). All in all, this yields for each sector A). The partition function, using a polynomial g(x), reads • the measure dD = dL dA dφ is the product of Lebesgue measures on the three factors of (8.2). While writing down the path integral does not solve the general problem of how to quantize noncommutative geometries, this finite-dimensional setting might pave one of the possible ways there, for instance, also by addressing these via computer simulations (Barrett-Glaser's aim). However, it should be stressed that the treatment of this path integral is not yet complete, due to the gauge redundancy to be still taken care of. A suitable approach is the BV-formalism 16 (after Batalin and Vilkovisky [12]), all the more considering that it has been explored for U(2)-matrix models in [33], and lately also given in a spectral triple description [32].
En passant, notice that since the main algebra here is M N (A) with A a noncommutative algebra, the Dyson-Schwinger equations of these multimatrix models would be 'quantum' (in the sense of Mingo-Speicher [38, §4]; this is work in progress).
For Eq. (3.3c), one notices that γ μ has to jump, in order to pass to the other side, three matrices (one of which is γ μ itself), which yields the sign (−1) 2 .
If ν = μ, we first determine the corresponding RHS term up to a sign, and thereafter correct it. First, it is clear that γμγν is a product of γ μ (which appears in γν), with γ ν (which appears in γν) and, additionally, with the other two gamma-matrices whose indices {λ, ρ} that are neither μ nor ν. But each one of the latter appears twice, once in γν and once in γμ. The matrix is then proportional to γ μ γ ν , which with the squared matrices γ λ and γ ρ yield ς μν e λ e ρ γ μ γ ν for λ, ρ ∈ Δ 4 \ {μ, ν} and λ = ρ, for a sign ς μν = ± that we now determine. To enforce the inequality of all the indices, we introduce δ μνλρ , but since e λ e ρ δ μνλρ is symmetric in λ and ρ, we have to divide the sum over those indices by 1/2. To find the correct sign ς μν , by explicit computation one sees that ς μν = −1 if and only if (μ, ν) is (0, 2), (2, 0), (3,1) or (1,3) and ς μν = +1 in all the other cases. That is, ς μν = −1 if and only if |μ − ν| is even. But this is precisely equivalent to ς μν = (−1) |μ−ν|+1 . Proof. Notice that for (so far, arbitrary) a i , b i ∈ A i (i = 1, 2), one has by adding and subtracting a 1 b 1 ⊗ b 2 a 2 and rearranging, Conversely, notice that if a 1 ⊗ a 2 = 0, then we are done, so we suppose a 1 ⊗ a 2 ∈ Z(A 1 ⊗ A 2 ) \ {0}. If the LHS of the previous equation vanishes for each b 1 ⊗ b 2 ∈ A 1 ⊗ A 2 , so does for b 1 = 1; in which case, one gets a 1 ⊗ [a 2 , b 2 ] = 0 for each b 2 ∈ A 2 , so a 2 ∈ Z(A 2 ), since a 1 = 0 by assumption. Repeating the argument now taking b 2 = 1 instead, one gets Z(A 1 ) ⊗ Z(A 2 ) ⊃ Z(A 1 ⊗ A 2 ).