On Multimatrix Models Motivated by Random Noncommutative Geometry Ii: a Yang-Mills–Higgs Matrix Model

ON MULTIMATRIX MODELS MOTIVATED BY RANDOM NONCOMMUTATIVE GEOMETRY II: A YANG-MILLS–HIGGS MATRIX MODEL

CARLOS I. PEREZ-SANCHEZ

Abstract. We continue the study of fuzzy geometries inside Connes’ spectral formalism and their relation to multimatrix models. In this companion paper to [arXiv:2007:10914, Ann. Henri Poincaré, 2021] 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 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.

1. 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 [LLS01] and often experiences a regain of interest; in [DLM14, CvS20], 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 [CCvS19] and [DKvS19], 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 [Con13, RV06] 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 arXiv:2105.01025v2 [math-ph] 2 Aug 2021 bundle approach to gauge theory. In contrast, in the NCG framework via the Spectral Action [CC97], 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 [BH10,

Key words and phrases. Noncommutative geometry, random matrices, spectral action, spectral triples, gauge theory, random geometry, fuzzy spaces, multimatrix models, quantum spacetime, Yang-Mills theory, Cliﬀord algebras, almost-commutative manifolds. 1 2 CARLOS I. PEREZ-SANCHEZ

Fur12, BF20, Tod19] 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 ˆF [Ste06, CCM07, vS15], 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 [Bar15]. 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’ [Mad92, DHMO08, SS13, SS18, ŠT20, Ste21], which are not always based on Connes’ formalism1. This article lies in the intersection and treats ‘fuzzy spaces’ inside the Spectral Formalism. At this point it is pertinent to clarify the different roles of the sundry finite- dimensional algebras that will appear. Figure1 might be useful to illustrate our terminology. Sitting at the origin, starting from classical Riemannian geometry, one can change, deform or enhance along each of the three independent axis described next:

Figure 1. Three axis representing independent theories departing from spin Riemannian geometry, all inside NCG. Abbreviations and terminology: YM=Yang-Mills; SM=Standard Model; “Base geometry” refers to structures that describe purely gravitational models.

‚ The F -direction in Figure1 represents what is usually known as ‘ﬁnite geometries’ and stands for the addition of matter content. On the marked

1Also, other proposals related to discretizations or truncations [DLM14, GS20, GS21, BSZ20] are (closer to) spectral triples. FINITE ALMOST-COMMUTATIVE GEOMETRIES 3

plane orthogonal to F lie ‘spacetimes’ or ‘base manifolds’. These geometries are used as gravitational models implying no matter content. Particle physics models based on NCG and the Connes-Chamseddine spectral action [CCM07, Bar07, DLM14, DDS18, Bes21, CvS19] encode the particle interactions on ‘points of the F -axis’, so to say. These are classiﬁed by Krajewski’s diagrams [Kra98]; see also [PS98].

‚ Displacement along the ‘matrix geometries’ direction signifies the replacement of the smooth base manifold with a finite algebra, as well as adapting the rest of the objects of a spectral triple to this setting, introduced in [Bar15]. On the shaded (in the online version, green) plane orthogonal to this axis, one has the smooth geometries; the eventual aim is to get to the ‘smooth geometry plane’ as the algebras become large-dimensional. Addi- tional to such large-N one might require to adjust the couplings to criticality [BG16, Gla17, KP21]. This can also be addressed using the Functional Renormalization Group to find candidates for phase transition; for models still without matter, see [Pér21].

‚ The remaining axis denotes quantization, here in the path integral formalism. The partition function is a weighted integral Z “ dξ eiSpξq{~ over the space of certain class of geometries ξ, the aim being the quantization of space ş itself, having in mind quantum gravity as motivation. Here S is a classical action of a model that lies on the ‘classical plane’ (~ Ñ 0). By ‘classical geometry’ we mean a single geometrical object (e.g. a Lorentzian or Rie- mannian manifold, an SUpnq-principal bundle with connection, etc.), which can be determined by, say, the least-action principle (Einstein Equations, SUpnq-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 theory as a whole, due to the mutual feedback between matter and gravity sectors in the renormalization group ﬂow; cf. [DEP14] for an asymptotic safety picture). For this purpose we need the next simpliﬁcations, illustrated in Figure2:

‚ Our choice for the ﬁnite geometry F is based on the algebra AF “ MnpCq (n ě 2). This is the ﬁrst input, aiming at a SUpnq Yang-Mills theory.

‚ Instead of the function algebra on a manifold, we take a simple matrix algebra MN pCq. 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 eiSpξq{~, in the partition function, towards the Boltzmann factor e´Spξq{~. This setting is called random noncommutative geometry [Gla17, BDG19]. 4 CARLOS I. PEREZ-SANCHEZ

Figure 2. Depicting the organization of this article, following the path P QR. Here, FYM-H “ pMnpCq,MnpCq,DF q corresponds to the spectral triple for the Yang-Mills–Higgs theory and Gf is a fuzzy 4-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

Random NCG was introduced in [BG16]. 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 systemati- cally computed for general dimensions and signatures in [Pér19]. Also, in the first part of this companion paper, the Functional Renormalization Group to multimatrix models [Pér21] inspired by random noncommutative geometry was addressed for some two-dimensional models obtained in [Pér19]. Solution of the matrix-models corresponding to one-dimensional geometries was addressed in [AK19], using Topo- logical Recursion [EO07] (due to the presence of multitraces, in its blobbed [Bor15] 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. Section3 interprets these as variables of a ‘matrix spin geometry’ for the p0, 4q-signature. Section4 introduces the main object of this article, finite almost-commutative geometries, for which the spectral action is iden- tified with Yang-Mills theory, if the piece DF of Dirac operator along the ‘inner space spectral triple’2 vanishes, and with Yang-Mills–Higgs theory, if this is non- zero, DF ‰ 0 (see Sec.5). Our cutoff function f appearing in the Spectral Action

2This is usually referred to as ‘ﬁnite spectral triple’ but in this paper all spectral triples are ﬁnite dimensional. FINITE ALMOST-COMMUTATIVE GEOMETRIES 5

TrH fpDq is quartic-quadratic; this is not the ﬁrst time a polynomial f is used (e.g., see the approach by [MvS14] in the spin network context). In Section6 we continue making the parallel with ordinary gauge theory on smooth manifolds, and also give the explicit Yang-Mills–Higgs matrix model. Finally, Section7 gives the conclusion and the outlook. This article is self-contained, but some familiarity with spectral triples helps. Favoring a particle physics viewpoint, we kept the terminology and notation com- patible with [vS15].

Contents 1. Introduction1 2. Spectral Triples and Fuzzy Geometries5 3. Towards a ‘Matrix Spin Geometry’9 4. Finite almost-commutative geometries 13 4.1. Yang-Mills theory from finite almost-commutative geometries 13 4.2. Field Strength and the square of the fluctuated Dirac operator 16 4.3. Gauge group and gauge transformations 19 4.4. Unimodularity and the gauge group 23 5. Yang-Mills–Higgs theory with finite-dimensional algebras 25 5.1. The Higgs matrix field 25 5.2. Transformations of the matrix gauge and Higgs fields 27 5.3. The quadratic-quartic Spectral Action 31 6. Towards the continuum limit 31 6.1. The Spectral Action of a Yang-Mills–Higgs finite geometry 32 6.2. The Yang-Mills–Higgs matrix model 34 7. Conclusion and Outlook 35 Appendix A. A lemma and a proof 37 References 38

2. 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 pp, qq P Zě0, or equivalently, by

η “ diagp`,..., `, ´,..., ´q “ diagp`p, ´qq p q 1 2 ‚ three signs , , P t´looomooon1, `1u ﬁxedlooomooon through s by the following table:

s ” q ´ p mod 8 0 1 2 3 4 5 6 7 ` ` ´ ´ ´ ´ ` ` 1 ` ´ ` ` ` ´ ` ` 2 ` + ´ + ` + ´ + 6 CARLOS I. PEREZ-SANCHEZ

‚ a matrix algebra Af “ MN pCq ‚ a Cliﬀord C`pp, qq-module V or spinor space

‚ a chirality γf “ γ b 1A : Hf Ñ Hf for the vector space Hf “ V b MN pCq with inner product

˚ hv b T, w b W i “ pv, wq TrN pT W q for all T,W P MN pCq and v, w P V To wit γ : V Ñ V is self-adjoint with respect to the Hermitian form pv, wq “ k 2 a v¯awa on V – C and satisfying γ “ 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 ř supposed to be irreducible, if s is odd

‚ a left-Af representation on Hf , %paqpv b W q “ v b paW q, for a P Af and W P MN pCq. The representation % is often implicit

‚ an anti-linear isometry, called real structure, Jf :“ Cb˚ : Hf Ñ Hf given in terms of the involution ˚ (in physics represented by :) on the matrix algebra and C : V Ñ V an anti-linear operator satisfying, for each gamma matrix, C2 “ and γµC “ 1Cγµ (2.1)

‚ a self-adjoint operator D on H satisfying the order-one condition

´1 rDf ,%paqs ,Jf %pbqJf “ 0 for all a, b P A (2.2) 3 ‚ the condition“ Dγf “ ´γf D for‰ even s. Moreover, the three signs above impose: 2 Jf “ , (2.3a) 1 Jf Df “ Df Jf , (2.3b) 2 Jf γf “ γf Jf . (2.3c)

Notice that, in this setting, the square of Jf 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. For s odd, γf can be trivial γf “ 1H. The number d :“ p ` q is the dimension and s :“ q ´ p (mod 8) is the KO-dimension. Remark 2.2. It will be useful later to stress that the ‘commutant property’ (cf. for instance [vS15, eq. 4.3.1]) ra, Jb˚J ´1s “ 0 , for all a, b P A , (2.4) which is typically an axiom for spectral triples, is not assumed in our setting. How- ever, one can show that it is a consequence of those in Deﬁnition 2.1. The axiom states that the right A-action ψb :“ boψ “ Jb˚J ´1ψ, for b P A, ψ P H, commutes with the left A-action % for each a, b P A. Since J “ C b ˚, and the algebra acts trivially on V , abopv b mq “ aJpv b b˚m˚q “ v b pambq (2.5) o “ b apv b mq , v P V, m P MN pCq .

3This condition does not appear in list given by Barrett and in fact follows from the construction of the explicit γ matrices, so it is tautological but useful to emphasize, as it also appears in the smooth case [vS15]. Barrett also allows algebras A over R, C and H; for quaternion coeﬃcients, MN{2pHq Ă MN pCq. FINITE ALMOST-COMMUTATIVE GEOMETRIES 7

The focus of this paper is dimension four, but we still proceed in general dimension. We impose on the gamma matrices γµ the following conditions: µ 2 µ pγ q “ `1V , and γ is Hermitian for µ “ 1, . . . , p, (2.6a) µ 2 µ pγ q “ ´1V , and γ is anti-Hermitian for µ “ p ` 1, . . . , d “ p ` q. (2.6b) Since it will be convenient to treat several signatures simultaneously, we let pγµq2 “: eµ1V for each µ “ 1, . . . , d. According to Eqs. (2.6), one thus obtains the unitarity of all gamma-matrices: µ µ µ ˚ µ µ µ 2 pγ v, γ wq “ ppγ q γ v, wq “ peµγ γ v, wq “ peµq pv, wq “ pv, wq pno sumq 1 d for each v, w P V . Let these matrices generate Ω:“ hγ , . . . , γ iR 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 [Bar15, Eq. 64], the Dirac operator Df solves the axioms of an even-dimensional fuzzy geometry whenever it has the next form: I Df pv b T q “ γ v b tKI ,T ueI and eI P t`1, ´1u , (2.7) I ÿ tA, Bu˘ :“ AB ˘ BA, where T P MN pCq and the sum is over increasingly ordered multi-indices I “ pµ1, . . . , µ2r´1q of odd length. With such multi-indices I the following product γI :“ γµ1 ¨ ¨ ¨ γγ2r´1 P Ω´ is associated (the sum terminates after ﬁnitely many terms, since gamma-matrices square to a sign times 1V ). Moreover, still as part of the characterization of Df , eI denotes a sign chosen according to the following rules:

‚ I ‚ if γ is anti-Hermitian (so eI “ ´1), then tKI ,T ueI “ rLI ,T s, i.e. tKI , ueI is a commutator of the anti-Hermitian matrix KI (denoted by LI ); and I ‚ if γ is Hermitian, so must be KI , which will be denoted by HI . Then

‚ eI “ `1, and tKI ,T ueI “ tHI ,T u, so tKI , ueI is an anti-commutator with a Hermtian matrix HI . Example 2.3. Some Dirac operators of fuzzy d-dimensional geometries, d “ 2, 3, 4. 1 ‚ Type (0,2). Then s “ d “ 2, so “ 1. The gamma matrices are anti- Hermitian and satisfy pγiq2 “ ´1. The Dirac operator is p0,2q 1 2 Df “ γ b rL1, ‚ s ` γ b rL2, ‚ s

‚ Type (0,3), s “ 3. In this signature, the gamma matrices can be replaced for the quaternion units {, | and k to express the p0, 3q-geometry Dirac operator as4 p0,3q Df “ tH, ‚ u ` {rL1, ‚ s ` |rL2, ‚ s ` krL3, ‚ s

‚ Type (0,4), s “ 4, Riemannian. Since the triple product of anti-Hermitian gamma matrices is self-adjoint, pγαγµγνq˚ “ p´q3γνγµγα “ γαγµγν, so are

4This formula diﬀers from the most general p0, 3q-geometry Dirac operator [Bar15, Eq. 73] spanned by eight gamma matrices, since ours corresponds to a simpliﬁcation (also addressed in §V. A of op. cit.) byproduct of V being irreducible and the product of all gamma matrices being a scalar multiple of the identity. 8 CARLOS I. PEREZ-SANCHEZ

˚ the operator-coeﬃcients, which have then the form tHαµν, ‚ u for pHαµνq “ Hαµν:

p0,4q α κ λ µ Df “ γ b rLα, ‚ s ` γ γ γ bt Hκλµ , ‚ u α κăλăµ ρˆ ÿ ÿ γ Hρˆ ρ ρˆ “ γ b rLρ, ‚ s ` γ b tHlooomooonρˆ, ‚ u (loomoontρ, κ, λ, µu “ t0, 1, 2, 3u) ρ ÿ 0 ‚ Type (1,3), s “ 6, Lorentzian. Let γ be the time-like gamma matrix, i.e. the only one squaring to `1. Then p1,3q 0 i ‚ ‚ Df “ γ b tH0, u ` γ b rLi, s i 0ˆ ÿ γ 0 i j 1 2 3 ` γ γ γ b rLij, ‚ s ` γ γ γ btH0ˆ, ‚ u (2.8) iăj hkkikkj ÿ In the sequel we use KI generically for either HI or LI , whose adjointness-type is ˚ then speciﬁed by the signature and by I. We also deﬁne the sign eI by KI :“ eI KI , I ˚ I or equivalently by pγ q “ eI γ , for a multi-index I. In four dimensions, one has for triple indices I “ µˆ [Pér19, App. A] q`1 eµˆ “ eµp´1q 1 ď µ ď d “ p ` q “ 4, for signature pp, qq . (2.9) In summary, a fuzzy geometry of signature (p, q) has following objects:

‚ Af “ MN pCq ‚ Hf “ V b MN pCq, Hilbert-Schmidt inner product on MN pCq

‚ a representation of Af on Hf , %paqpv b T q “ v b aT

‚ Df given by Eq. (2.7)

‚ Jf “ C b ˚ with C anti-linear satisfying Eq. (2.1)

‚ γ γ 1 , with γ constructed from all γ-matrices; see Eq. (3.5) for f “ b MN pCq d “ 4 Although next equation is well-known, we recall it due of its recurrent usefulness later. In any dimension and signature, it holds:

µ µ µ µ ν α ρ TrV pγ γ γ γ q “ dim V ¨ ρ ν + ρ ν + ρ ν (2.10) ˆ α α α ˙ “ dim V ¨ pηµνηαρ ´ ηµαηνρ ` ηµρηναq Each inscribed segment in the chord diagrams denotes an index-pairing between two indices labeling their ends, say λ and θ, which leads to ηλθ; all the pairings of each diagram are then multiplied bearing a total sign corresponding to p´1q to the number of simple chord crossings. This picture is helpful to compute traces of more gamma-matrices, but is not essential here; see [Pér19] to see how the spectral action for fuzzy geometries was computed by associating with these chord diagrams noncommutative polynomials in the diﬀerent matrix blocks KI composing the Dirac operator. Incidentally, notice that so far this chord diagram expansion is classical, unlike that treated by Yeats in [Yea17, §9], which appears in the context of Dyson- Schwinger equations. FINITE ALMOST-COMMUTATIVE GEOMETRIES 9

3. Towards 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 multi-indices, which has been sketched in [Bar15, Sec. V §A] for d “ 4. The identification works only in dimensions four and, if unreduced, also three (cf. Footnote4 above). For higher dimensions, quintuple products appear; for lower ones, triple products are absent. Although it would be interesting to address each dimensionality separately, being dimension four the physically most interesting case, we stick to it. 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 Rie- mannian signature, we prove most results in general signature. For a Riemannian spin manifold M, recall the local expression (on an open U Ă M) of the canonical Dirac operator on the spinor bundle S Ñ M for each section ψ there, j S 8 pDM ψqpxq “ iΓ pxq∇j ψpxq , for x P U and ψ P Γ pU, Sq, (3.1a) S ∇i “Bi ` ωi . (3.1b) 1 µν S The coefficients ωi “ 2 ωi γµν of the spin connection ∇ (the lift of Levi-Civita 1 connection) are here expressed with respect to a base γµν “ 4 rγµ, γνs that satisfies the op4q Lie algebra in the spin representation (see e.g. [CM07, §11.4]). The gamma µ i i µ matrices with Greek indices (or ‘flat’) γ relate to the above Γ pxq “ eµγ by means i i 8 of tetrads eµpxq. The coefficients eµ P C pUq, by definition, make of the set of fields i pEµqµ“0,1,2,3 “ peµ ¨ Biqµ“0,1,2,3 an orthonormal basis of XpUq with respect to the i j ij metric g of M, which is to say gpEµ,Eνq “ ηµν. Thus tΓ pxq, Γ pxqu “ 2g pxq “ ´1 µ ν µν 2pg qijpxq for x P U, but tγ , γ u “ 2η . In contrast to the commutation rela- i tions that the elements of the coordinate base Bi “ B{Bx satisfy, one generally has rEµ,Eνs ‰ 0 for the non-coordinate base E0,...,E3, also sometimes called non- holonomic [Tor20, §4]. Notice that in the fuzzy setting only Greek indices appear. This, together with the fact that rather ηµν instead of gij 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 the signature η “ diagpe0, . . . , e3q and in the matrices parametrizing the Dirac operator. In Riemannian signature, we rewrite5 (cf. Ex. 2.3) µ S Df “ pγ b 1N qp∇f qµ , (3.2a) µ S ÿ σ ν ‚ ‚ p∇f qµ “ 1V b rLµ, s ` γ γ b tHµσν, ueµσν . (3.2b) 0ăσď3 1ăνď3 pµÿăσq pσÿăνq

5The restrictions 0 ă σ ď 3 and 1 ă ν ď 3 account for the appearance in Eq. (3.2b) of exactly three gamma matrices whose indices are increasingly ordered, as in the characterization (2.7) of fuzzy Dirac operators. To match the canonical Dirac operator on a spinor bundle, one could redeﬁne H, fully anti-symmetrize, and compare expressions. 10 CARLOS I. PEREZ-SANCHEZ

Simultaneously (up to the trivial factor 1V ), we identify the commutators rLµ, ‚ s

j ‚ with iEµ “ ieµBj and the coefficients of the triple gamma products tHµσν, ueµσν i i k with the full anti-symmetrization 4 ωrµ|ike|σeνs of the spin connection coefficients in the three Greek indices. The triple products of gamma-matrices present in the Dirac operator (3.2) are the analogue of those in the spin connection appearing in µ i DM “ iγ pEµ ` eµωiq, here in the ‘flat’ (non-holonomic or non-coordinate) basis S E0,...,E3. Altogether, ∇f can be understood as the matrix spin connection. We let ∆4 “ t0, 1, 2, 3u and denote by δµνασ the fully symmetric symbol with indices in ∆4, which is non-vanishing (and then equal to 1) if and only if the four indices are all different; equivalently, δµνασ “ |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 rising and lowering indices as well, e.g. gamma matrices are presented only with upper indices. We set k “ pkµqµP∆4 ,K “ pKµqµP∆4 , x “ pxµqµP∆4 . Similarly, previously undefined index-free objects refer to all entries of those already defined.

Lemma 3.1. For any µ, ν P ∆4 “ t0, 1, 2, 3u the following relations are satisﬁed:

µ νˆ µ µ 0 1 2 3 α σ γ γ “ p´1q δν γ γ γ γ ` sgnpν ´ µq δµνασeµγ γ , (3.3a) αăσ γµˆγµ “ ´γµγµˆ´, ÿ ¯ (3.3b)

γνˆγµ “ `γµγνˆ pν ‰ µq . (3.3c)

Proof. For the ﬁrst equation: If µ “ ν, then

γµγµˆ “ γµγ0 ¨ ¨ ¨ γµ ¨ ¨ ¨ γ3 “ p´1qµγ0 ¨ ¨ ¨ γµ ¨ ¨ ¨ γ3 , since the ﬁrst γµ has to ‘jump’ µxgamma-matrices in order to form γ0γ1γ2γ3. If µ ‰ ν, precisely the two gamma matrices with indices diﬀerent from µ and ν survive, µ 2 which explains δµνασ. The LHS of Eq. (3.3a) contains also pγ q “ eµ1V . To justify the sign p´1qµsgnpν ´ µq, notice, by explicit computation, that if µ ă ν then the pµ, νq-pairs p0, νq and p2, νq yield positive sign, whereas p1, νq negative. Thus the sign is p´1qµ (in no case the sign depends on ν, as far as µ ă ν). The situation is inverted if µ ą ν, where the pairs p1, νq and p3, νq yield positive sign and p2, νq negative. Thus the sign is p´1qµ´1. Notice that in γµˆγµ the last matrix has to move 4´µ´1 places to the right to ar- rive at the µ-th factor, which says that γµˆγµ “ p´1q3´µγ0γ1γ2γ3 “ ´p´1qµγ0γ1γ2γ3 and, by Eq. (3.3a), Eq. (3.3b) follows. For Eq. (3.3c) one notices that γµ has to jump, in order to pass to the other side, µ 2 three matrices (one of which is γ itself), which yields the sign p´1q .

Lemma 3.2. For µ, ν P ∆4 one has for any signature η in four dimensions 1 γµˆγνˆ “ p´1q1`|µ´ν| δ e e γµγν ´ 1 δµ ¨ e ¨ detpηq . (3.4) 2 µνλρ λ ρ V ν µ ρ,λ ÿ FINITE ALMOST-COMMUTATIVE GEOMETRIES 11

Notice that in Eq. (3.4) the repeated indices µ, ν in the RHS are not summed µ ν (therefore the index-symmetry of δµνλρ with the antisymmetry of γ γ does anni- hilate that term).

Proof. To obtain the second summand, notice that if µ “ ν, then the LHS is of the form pγιγλγτ q2 with pairwise different indices (i.e. anti-commuting gamma- µˆ µˆ λ τ λ τ λ λ τ τ matrices). Therefore γ γ “ eιγ γ γ γ “ éιγ γ γ γ “ éιeλeτ 1V . Since each 2 e ‚ is a sign and tι, λ, τ, µu “ ∆4, by multiplying the last expression by eµ “ 1 one µˆ µˆ arrives to γ γ “ éµ ¨ pe0e1e2e3q1V “ éµ detpηq1V . If ν ‰ µ, we first determine the corresponding RHS term up to a sign, and thereafter correct it. First, it is clear that γµˆγνˆ is a product of γµ (which appears in γνˆ), with γν (which appears in γνˆ) and, additionally, with the other two gamma- matrices whose indices tλ, ρu that are neither µ nor ν. But each one of the latter appears twice, once in γνˆ and once in γµˆ. The matrix is then proportional to γµγν, λ ρ µ ν which with the squared matrices γ and γ yield ςµνeλeργ γ for λ, ρ P ∆4ztµ, νu and λ ‰ ρ, for a sign ςµν “ ˘ that we now determine. To enforce the inequality of all the indices we introduce δµνλρ, but since eλeρδµνλρ is symmetric in λ and ρ, we have to divide the sum over those indices by 1{2. To find the correct sign ςµν, by explicit computation one sees that ςµν “ ´1 if and only if pµ, νq is p0, 2q, p2, 0q, p3, 1q or p1, 3q and ςµν “ `1 in all the other cases. That is, ςµν “ ´1 if and only if |µ ´ ν| |µ´ν|`1 is even. But this is precisely equivalent to ςµν “ p´1q . We now need the explicit form of the chirality γ γ 1 , given by f “ b MN pCq 1 pq´pqpq´p`1q 0 1 2 3 0 1 2 3 γ “ píq 2 γ γ γ γ “: σpηqγ γ γ γ . (3.5) This factor σpηq in γ in front of the matrices is ´1, ì, `1, í, for the signatures pp, qq “ p0, 4q, p1, 3q, p2, 2q, p3, 1q, respectively, corresponding to KO-dimensions s “ 4, 2, 0, 6.

Lemma 3.3. The square of the Dirac operator of a fuzzy geometry Gf of signature η “ diagpe0, . . . , e3q is

2 µν 1 µ ν Df “ 1V b η kµ ˝ kν ` γ γ b rkµ, kνs˝ ´ detpηqeµ1V b xµ ˝ xµ µ,ν 2 µ ÿ ÿ µ ν 1 α σ ` tµνγ γ b rxµ, xνs˝ ` sµνασ ¨ γ γ b txν, kµu˝ (3.6) µăν 2 µ,ν,σ,α ÿ ÿ 1 µ ` p´1q γ b rxµ, kµs˝ , σpηq µ ÿ with the ‘commutator’ rf, gs˝ given by f ˝ g ´ g ˝ f in terms of the next operators (which are themselves commutators or anti-commutators)

‚ ‚ kµ :“ tKµ, ueµ and xµ :“ tKµˆ, ueµˆ . (3.7) We deﬁned also the (whenever non-vanishing) signs µ sµνασ :“ eµp´1q ¨ sgnpν ´ µq ¨ sgnpσ ´ αq ¨ δµνασ P t´1, 0, `1u , (3.8) 1`|µ´ν| tµν :“ p´1q δµνλρeλeρ P t´1, 0, `1u . (3.9) λăρ ÿ 12 CARLOS I. PEREZ-SANCHEZ

2 Proof. One straightforwardly ﬁnds Df “ pa ` b ` c ` d ` eqpk, xq with

µ ν apk, xq “ γ γ b pkµ ˝ kνq , (3.10a) µ,ν ÿ µˆ µ µ µˆ bpk, xq “ γ γ b pxµ ˝ kµq ` γ γ b pkµ ˝ xµq , (3.10b) µ ÿ µˆ ν µ νˆ cpk, xq “ γ γ b pxµ ˝ kνq ` γ γ b pkµ ˝ xνq , (3.10c) µ‰ν ÿ µˆ µˆ dpk, xq “ γ γ b pxµ ˝ xµq , (3.10d) µ ÿ µˆ νˆ epk, xq “ γ γ b pxµ ˝ xνq . (3.10e) µ‰ν ÿ For the ﬁrst term one obtains

µ ν apk, xq “ γ γ b kµ ˝ kν µ,ν ÿ µ ν 1 “ γ γ b kµ ˝ kν ` kν ˝ kµ ` rkµ, kνs˝ µ,ν 2 ÿ ´ ¯ µ ν 1 µν 1 ν µ “ γ γ b pkµ ˝ kνq ` η 1V ´ γ γ b kν ˝ kµ ` rkµ, kνs µ,ν 2 2 ÿ ´ ¯ ´ ¯ µν 1 ν µ “ 1V b η kµ ˝ kν ´ γ γ b rkµ, kνs˝ . µ,ν 2 ÿ To get the ﬁrst two terms in the RHS of Eq. (3.6) one renames indices in the last term. The third summand is precisely d after applying Lemma 4.4 with µ “ ν. The fourth term is e, also by Lemma 4.4. The sixth and last term in Eq. (3.6) come µˆ µ from b; if one uses tγ , γ u “ 0 and Eq. (3.3a)|µ“ν, after using introducing the chirality element:

µ µˆ bpk, xq “ γ γ b pkµ ˝ xµ ´ xµ ˝ kµq µ ÿ µ “ p´1q γ b rkµ, xµs pvia Eq. 3.5q . µ ÿ We now see that the only Gothic letter left unmatched, c, is precisely the ﬁfth term. Indeed, due to Lemma 3.1,

ν µˆ µ νˆ cpk, xq “ γ γ b pxµ ˝ kνq ` γ γ b pkµ ˝ xνq (by Eq. 3.3c) µ‰ν ÿ µ νˆ µ νˆ “ γ γ b pxν ˝ kµq ` γ γ b pkµ ˝ xνq (index renaming) µ‰ν ÿ µ νˆ “ γ γ b txν, kµu µ‰ν ÿ µ α σ “ p´1q eµ pδµνασsgnpν ´ µqqγ γ b txν, kµu (by Lemma 3.1) µ‰ν αăσ ÿ ÿ 1 µ α σ “ p´1q eµsgnpν ´ µqsgnpα ´ σqδµνασ γ γ b txν, kµu 2 µ,ν,α,σ ÿ “ ‰ FINITE ALMOST-COMMUTATIVE GEOMETRIES 13

Riemannian Concept Smooth Geometry Fuzzy Geometry

i Base of XpUq Eµ “ eµpxqBi lµ “ rLµ, ‚ s 1 i k ‚ Spin connection i,k 4 ωrα|ike|σeνs hασν “ tHασν , u Table 1. Analogies betweenř smooth spin geometry and Riemannian fuzzy geometries. Local expressions in a chart U of M are given. Here, XpUq are the vector fields on U, whose non-coordinate base is tEµu where in the last step we exploited the skew-symmetry of the gammas with different indices to annul the restriction α ă σ on the sum by introducing sgnpσ ´ αq. The term in square brackets is sµνασ. Notice that the analogy in Table1 goes further, since in the case of a smooth manifold spin manifold pM, gq, the fields B0,..., B3, or equivalently E0,...,E3, (locally) span the space of vector fields XpMq on M, which consists of derivations in 8 C pMq. The analogue of Bj is here (after the base change to Eµ) the derivation in

‚ DerpMN pCqq that corresponds to lµ “ adLµ “ rLµ, s.

4. Finite almost-commutative geometries We restrict the discussion to even KO-dimensions (1 “ 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 Table2, which will be hopefully helpful to grasp the organization of the objects introduced this section. But ﬁrst, we recall that the spectral triple product G1 ˆ G2 of two real, even spectral triples Gi “ pAi, Hi,Di,Ji, γiq is

pA1 b A2, H1 b H2,D1 b 1H2 ` γ1 b D2,J1 b J2, γ1 b γ2q . Definition 4.1. We define a finite almost-commutative geometry6 as the spectral triple product Gf ˆ F of a fuzzy geometry Gf with a finite geometry F “ pAF , HF ,DF ,JF , γF q, dim AF ă 8. If F is a finite geometry with AF “ MnpCq and HF “ MnpCq with 2 ď n, we say that Gf ˆ F is a Yang-Mills–Higgs finite (almost-commutative) geometry. If moreover DF “ 0 above holds, then Gf ˆ F is called Yang-Mills finite (almost-commutative) geometry. pNq pnq We should denote these geometries by Gf ˆ F , but for sake of a compact notation, we leave those integers implicit and write Gf ˆ F . 4.1. Yang-Mills theory from finite almost-commutative geometries. In order to derive the SUpnq-Yang-Mills theory on a fuzzy base we choose the following inner space algebra: AF “ MnpCq. This algebra acts on the Hilbert space HF “

6The reader will notice that such spectral triple is even ‘doubly noncommutative’, and that ‘finite almost- commutative’ is from the onset a misnomer, but it is also the terminology that will most likely evoke the definition. The main reason is that ‘doubly noncommutative’ could just as well describe the product of two finite (in the ordinary sense) spectral triples, but this does not represent the fact that one factor has a spacetime meaning, while the other describes the gauge interactions—this is what ‘almost-commutativity’ suggests. Respecting a common NCG-terminology, we shorten ‘finite dimensional (...) geometry’ by only ‘finite (...) geometry’. 14 CARLOS I. PEREZ-SANCHEZ

1 MnpCq by multiplication. The Connes’ 1-forms ΩDpAq for A “ MN pCq b MnpCq are then elements of the form

ω “ arD, cs with a “ W b a, c “ T b c P MN pCq b MnpCq , (4.1) ÿ ÿ ÿ where the sums are finite. The latter algebra is the fuzzy analogue of the algebra 8 8 C pM, AF q “ C pMq b AF of an (8-dimensional, smooth) almost-commutative geometry. 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 Section5. Thus, turning off the ‘finite part’ DF “ 0, one obtains

´1 Dgauge :“ Dωf “ Df b 1F ` ωf ` Jωf J (4.2) for ωf of the form (4.1), with respect to the ‘purely fuzzy’ Dirac operator

µ µˆ ‚ ‚ Df b 1F “ γ b tKµ, ueµ ` γ b tXµ, ueµˆ , (4.3a) µ ÿ Kµ “ Kµ b 1F and Xµ :“ Xµ b 1F . (4.3b)

Theorem 4.2. On the Yang-Mills ﬁnite (almost-commutative) geometry over a four-dimensional fuzzy geometry of type pp, qq, i.e. of signature η “ diagp`p, ´qq, the ﬂuctuated Dirac operator D “ Df b 1F reads

µ µˆ ‚ ‚ Dgauge :“ Dωf “ γ b tKµ ` Aµ, ueµ ` γ b tXµ ` Sµ, ueµˆ , (4.4) µ ÿ 1 in terms of matrices Aµ, Sµ P ΩDpAf b AF q satisfying

˚ ˚ q`1 pAµq “ eµAµ, and pSµq “ p´1q eµSµ . (4.5)

Here, the curly brackets are a generalized commutator tA, Bu˘ “ AB ˘ BA depend- ing on eµ, eµˆ P t`1, ´1u. Proof. The theorem follows by combination of Lemma 4.3 with Lemma 4.4, both proven below.

Lemma 4.3 (Fluctuations with respect to the Kµ-matrices). With the same notation of Theorem 4.2 and setting Xµ “ Kµˆ “ 0—cf. Eq. (3.2) and Eq. (3.7)—the innerly

ﬂuctuated Dirac operator Dgauge is given by

µ ˚ 1 Dgauge|X 0 “ γ b Kµ ` Aµ, ‚ where eµpAµq “ Aµ P Ω pAq . (4.6) “ eµ D µ ÿ ( Proof of Lemma 4.3. We set X “ 0 globally in this proof. Pick a homogeneous vector in the full Hilbert space Ψ “ v b Y b ψ P H “ V b MN pCq b HF . For 1 a “ 1V b W b a and a “ 1V b T b c parametrized by T,W P MN pCq and a, c P AF , FINITE ALMOST-COMMUTATIVE GEOMETRIES 15 the action of ω on Ψ yields

1 ωf pΨq “ arDf b 1F , a spΨq µ ‚ “ p1V b W b aq γ b tKµ, ueµ b 1, 1V b T b c pΨq µ µ “ ÿ ‰ ‚ ‚ “ γ v b W tKµ, ueµ T ´ T tKµ, ueµ pY b acψq µ ÿ ´ ¯ µ “ γ v b W tKµ,TY ueµ ´ T tKµ,Y ueµ b acψ µ ÿ ´ ¯ µ “ γ v b W KµTY ` eµTYKµ ´ T pKµY ` eµYKµq b acψ µ ´ ¯ ÿ µ “ γ v b W rKµ,T s Y b acψ µ ÿ µ ` ˘ “ γ b W rKµ,T s b ac Ψ µ ÿ ` ˘ µ so ωf “ µ γ b Aµ b b, relabeling b “ ac P AF and Aµ :“ W rKµ,T s. Notice that since ř µ ˚ µ ˚ ˚ pγ b Aµ b bq “ eµγ b Aµ b b (4.7)

˚ ˚ the self-adjointness condition ωf “ ωf is achieved if and only if pAµ b bq “ eµpAµ b bq for each µ. The second part of the inner ﬂuctuations is, for each

Ψ “ v b Y b ψ P V b MN pCq b MnpCq , the next expression:

´1 1 ´1 pJωf J qpΨq “ JarDf b 1F , a sJ pΨq

` µ ˘ ´1 ˚ ˚ “ pC b ˚N b ˚nqpγ C v b AµY b bψ q µ ÿ µ ´1 ˚ ˚ ˚ ˚ “ pCγ C v b pAµY q b pbψ q (cf. Eq. 2.1) µ ÿ γµ µ ˚ ˚ “ γlooomooonv b YAµ b ψb µ ÿ γµ 1 1 Ψ 1 A b ˚ “ p b MN pCq b nq p V b µ b q µ ÿ e γµ 1 1 Ψ 1 A b , “ p µ b MN pCq b nq p V b µ b q µ ÿ ´1 µ where the last step is a consequence of Eq. (4.7). Thus Jωf J “ µ eµγ b p ‚ qpAµ b bq where the bullet stands for the argument in MN p q b Mnp q Ă H to C řC be multiplied by the right. Hence

´1 µ ωf ` Jωf J “ γ b Aµ b b ` eµp ‚ qpAµ b bq , with (4.8a) µ ´ ¯ ˚ ÿ 1 eµpAµ b bq “ Aµ b b P ΩDpAq for each µ. (4.8b) 16 CARLOS I. PEREZ-SANCHEZ

As a result, the fully-ﬂuctuated operator acting on Ψ “ v b Y b ψ P H is

pDf b1F qΨ

µ µ Dωf Ψ “ γ v b tKµ,Y ueµ b ψ ` γ v b AµY b bψ ` eµYAµ b ψb . hkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkjµ µ ÿ ÿ ` ˘ or defining Kµ :“ Kµ b 1n and Aµ :“ Aµ b b P MN pCq b MnpCq, one has µ ˚ 1 Dω “ γ b Kµ ` Aµ, ‚ , pAµq “ eµAµ P Ω pAq . f eµ D µ ÿ ( The triviality of the part of the Dirac operator along the finite geometry F implies that Ω1 A Ω1 M Ω1 M M , Dp q “ Df b1F NbnpCq “ Df p N pCqq b npCq where MNbnpCq abbreviates MN` pCqbMnp˘Cq (in sub-indices, later further shortened C as MNbn too), 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 µ 8 Γ b pAµ ´ JF AµJF q, with Aµ “ íaBµb P C pMq b AF . µ ÿ 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 Γi “ cpdxiq, c being Clifford multiplication. For each point x of the base manifold M one has Aipxq P i supnq “ i Lie SUpnq. Since Eq. (4.8b) represents the fuzzy ˚ analogue, that equation can be further reduced to b “ b P MnpCqs.a. “ i upnq and ˚ pAµq “ eµAµ, that is

µ ˚ µ i upNq if eµ “ `1 iff pγ q “ `γ , A P (4.9) µ µ ˚ µ #upNq if eµ “ ´1 iff pγ q “ ´γ . We now have to add the fluctuations resulting from the triple products of gamma matrices.

Lemma 4.4 (Fluctuations with respect to the Xµ-matrices). With the same notation of Theorem 4.2 and additionally setting Kµ “ 0, the innerly ﬂuctuated Dirac operator Dgauge is given by µˆ q`1 ˚ 1 Dgauge|K 0 “ γ b Xµ ` Sµ, ‚ p´1q eµpSµq “ Sµ P Ω pAq . (4.10) “ eµˆ D µ ÿ ( Proof. See AppendixA.

From the last subsections, the rules for Sµ and Aµ lead to the manifest self- adjointness of Dωf . 4.2. Field Strength and the square of the ﬂuctuated Dirac operator. We 7 introduce now the main object of the gauge theory. To this end, let rf, gs˝ “ pf ˝ gq ´ pg ˝ fq for any endomorphisms f, g of the same vector space. Similarly, we deﬁne tf, gu˝ “ pf ˝ gq ` pg ˝ fq.

7Here we emphasize the composition to avoid potential confusion arising from the objects inside commutator already being (anti-)commutators themselves. Should no confusion arise, so we drop the ˝. FINITE ALMOST-COMMUTATIVE GEOMETRIES 17

Notation Given by Lies in Dirac op. type

Kµ (fundamental) MN pCq ˚ Kµ “ eµKµ , / Xµ (fundamental) MN pCq / /Pure fuzzy ˚ 1`q / Xµ “ Kµˆ Xµ “ eµp´1q Xµ / ./ ‚ kµ tKµ, ueµ EndpMN pCqq / ‚ / xµ tXµ, ueµˆ EndpMN pCqq / / / -/ Kµ Kµ b 1F MN pC ¨ 1F q

Xµ Xµ b 1F MN pC ¨ 1F q , , /Df b 1F / kµ tKµ, ‚ ue End MN p ¨ 1F q / / µ C / / . / X , ‚ End M 1 / xµ t µ ueµˆ ` N pC ¨ F q˘ / / / / /Finite ` ˘ / / - / A A b, Ω1 M A /almost µ µ b D b1F N p F q / f / ˚ . Aµ “ eµÀµ, b P pA˘F qs.a , commu- / 1 / / Sµ Sµ b b Ω MN pAF q / /tative Df b1F /Fluctua- / / / S˚ “ e p´1qq`1S , b P pA q / / µ µ ` µ˘ F s.a ./ tions / / a tA , ‚ u End M pA q / µ µ eµ N F / / / / ‚ / / sµ tSµ, ueµˆ End`MN pAF q˘ / / / / / -/ ` ˘ - Table 2. Notation for different matrices and operators appearing in the Dirac operator D “ Df b1F of finite almost commutative geometries Gf ˆF (case with DF “ 0) and its fluctuations. In the table MN pC¨1F q “ MN pCqb pC ¨ 1F q

Definition 4.5. We abbreviate the following (anti)commutators

‚ ‚ ‚ kµ :“ tKµ, ueµ , xµ :“ tXµ, ueµˆ “ tKµˆ, ueµˆ , (4.11a)

‚ ‚ aµ :“ tAµ, ueµ , sµ :“ tSµ, ueµˆ . (4.11b)

It follows in particular that kµ b 1F “ kµ. The field strength Fµν P EndpAf b AF q of a finite almost-commutative geometry Gf ˆ F is defined as

Fµν :“ rkµ ` aµ, kν ` aνs˝ . (4.12)

Proposition 4.6. The square of the fluctuated Dirac operator of a Yang-Mills finite almost-commutative geometry that is flat (X “ 0, x “ s “ 0) is given by

2 1 µ ν Dgauge|X“0 “ γ γ b Fµν ` 1V b ϑ , (4.13) 2 µ,ν ÿ where µν ϑ :“ η paµ ` kµq ˝ paν ` kνq . (4.14) µ,ν ÿ 18 CARLOS I. PEREZ-SANCHEZ

´1 Proof. Squaring Dgauge “ Df b 1F ` ωf ` Jωf J one gets 2 2 ´1 Dgauge “ pDf q b 1F ` pDf b 1F qpω ` JωJ q (4.15) ´1 ´1 2 ` pω ` JωJ qpDf b 1F q ` pω ` JωJ q . 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 ´1 2 µ ν 1 pω ` JωJ q |X“0 “ γ γ b paµ ˝ aνq (4.16) µ,ν 2 ÿ µν 1 ν µ ` η 1V ´ γ γ b aν ˝ aµ ` raµ, aνs µ,ν 2 ÿ ´ ¯ ´ ¯ µν 1 ν µ “ 1V b η aµ ˝ aν ` γ γ b raµ, aνs . µ,ν 2 ÿ being rf, gs “ f ˝ g ´ g ˝ f a simplified notation for the composition-commutator. 1 µ ν We renamed indices and rewrote the last summand in (4.16) as 2 γ γ b raµ, aνs To make the notation lighter, we also mean by aµaν the composition aµ ˝ aν from now on (also for kµ). Using Eq. (4.6) one can obtain for the two summands in the

‚ middle of Eq. (4.15); further abbreviating kµ “ tKµ b 1F , ueµ one obtains

´1 ´1 pDf b 1F qpω ` JωJ q ` pω ` JωJ qpDf b 1F q |X“0

ν µ µ ( ν ‚ ‚ ‚ ‚ “ γ b tKν, ueν γ b tAµ, ueµ ` γ b tAµ, ueµ γ b tKν, ueν µ,ν ÿ ` ˘` ˘ ` ˘` ˘ ν µ µ ν “ pγ b kνqpγ b aµq ` pγ b aµqpγ b kνq µ,ν ÿ ν µ µ ν “ pγ γ b kνaµq ` pγ γ b aµkνq µ,ν ÿ 1 µν ν µ 1 µ ν “ 2η 1V ´ γ γ b pkµaν ` aµkνq ` γ γ b pkµaν ` aµkνq µ,ν 2 2 ÿ ´ ¯ µ µ 1 µ ν “ 1V b pkµa ` a kµq ` γ γ b ´ kνaµ ´ aνkµ ` kµaν ` aµkν µ 2 µ,ν ÿ ÿ ´ ¯ µ 1 µ ν “ 1V b tkµ, a u ` γ γ b rkµ, aνs ´ rkν, aµs . (4.17) µ 2 µ,ν ÿ ÿ ´ ¯ Again, we used the Cliﬀord relations for the gamma matrices and renamed indices. Equations (4.16) and (4.17) imply

2 2 1 µ ν pDgauge|x“s“0q “ Df b 1F ` γ γ b rkµ, aνs ´ rkν, aµs ` raµ, aνs 2 µ,ν ÿ ´ ¯ µ µ ` 1V b a aµ ` tk , aµu . µ ÿ Expanding Eq. (4.12)

Fµν “ rkµ, aνs˝ ´ rkν, aµs˝ ` raµ, aνs˝ ` rkµ, kνs , (4.18) µ µ µ and using Lemma 3.3|X“0 together with ϑ “ k ˝ kµ ` tk , aµu˝ ` a ˝ aµ one gets the result. FINITE ALMOST-COMMUTATIVE GEOMETRIES 19

Proposition 4.7. The fluctuated Dirac operator of a finite Yang-Mills geometry satisfies

2 µν 1 µ ν Dgauge “ 1V b η pkµ ` aµq ˝ pkν ` aνq ` γ γ b rkµ ` aµ, kν ` aνs˝ µ,ν 2 ÿ ` detpηqpéµq1V b pxµ ` sµq ˝ pxµ ` sµq µ ÿ µ ν ` tµνγ γ b rxµ ` sµ, xν ` sνs˝ (4.19) µăν ÿ 1 α σ ` sµνασ ¨ γ γ b txν ` sν, kµ ` aµu˝ 2 µ,ν,σ,α ÿ µ ` p´1q γ b rxµ ` sµ, kµ ` aµs˝ . µ ÿ Proof. According to Eq. (4.4) µ µˆ Dω “ γ b pkµ ` aµq ` γ b pxµ ` sµq µ ÿ 2 so Dω has the same structure already observed in the ‘fuzzy Lichnerowicz formula’ above (Lem. 3.3). To be precise, notice that one can compute the square of the present Dirac operator by replacing the in Df the following operators: k Ñ k ` a and x Ñ x ` s. 4.3. 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 Q Ψ is implemented by the real structure J, Ψa :“ aoΨ:“ Ja˚J ´1Ψ. Both actions define the adjoint action AdpuqΨ:“ uΨu˚ of the unitarities u P UpAq on H. We want to determine the action of the unitarities UpAq “ tu P A | u˚u “ 1 “ uu˚u of the algebra A on the Dirac operator, 1 ´1 ˚ 1 ´1 U D ` ω ` JωJ U “ D ` ωu ` JωuJ , (4.20a) ` ˘U :“ Adu, u P UpAq , (4.20b) which namely leads to the transformation rule ˚ ˚ ω ÞÑ ωu “ uωu ` urD, u s (4.20c) for the inner fluctuations. It is instructive to present a variation of the original proof given in [CM07, Prop. 1.141] for the analogous property of general spectral triples. Verifying this again is important, since the axiom ra, bos “ 0 that appears in op. cit., does not appear in the present axioms. However, according to Remark 2.2 above, it is a consequence of these in the fuzzy setting. So one can see that not only there, but also for finite almost-commutative geometries, the commutant o property rA, A s “ 0 (elsewhere an axiom) holds. Indeed, since J “ C b ˚N b ˚n, for a, b P A, abopv b mq “ aJpv b b˚m˚q “ v b pambq (4.21) o “ b apv b mq , v P V, m P MNbnpCq . The commutant property is essential for the subalgebra ˚ AJ :“ ta P A | aJ “ Ja u Ă A (4.22) 20 CARLOS I. PEREZ-SANCHEZ to be also a subalgebra of the center ZpAq, as we will see later. Proof of Eq. (4.20); adapted from [CM07, §10] to fuzzy geometries. We split the adjoint action into the right action by u˚, z :“ pu˚qo “ JuJ ´1, and the left action by u, U “ uz. ˚ ˚ ‚ Transformation of D: Applying wDw “ D ` wrD, w s consecutively for w “ z, u, one gets UDU ˚ “ upD ` zrD, zsqu˚ “ D ` urD, u˚s ` zrD, z˚s . (4.23) 1 ‚ Transformation of ω: since ω P ΩDpAq, ω “ arD, bs (or sum of this 1-forms), one also has (2.2) (2.5) ωz˚ “ ωuo “ arD, bsuo “ auorD, bs “ uoarD, bs “ z˚ω , so UωU ˚ “ upzωz˚qu “ uωu˚, since zz˚ “ 1. Also the term urD, u˚s is absorbed from the pure Dirac operator, then ω ÞÑ uωu˚ ` urD, u˚s . ´1 ´1 ˚ ˚ ‚ Transformation of JωJ : Similarly one obtains UJωJ U “ Jpuωu qJ. But actually zrD, z˚s from Eq. (4.23) can be taken from the transformation of the pure Dirac operator and passed to that of JωJ ´1, contributing, by the axioms (2.3) of the fuzzy geometry, since one can rearrange it as ˚ ´1 ˚ ´1 ˚ ´1 1 ˚ ´1 zrD, z s “ JuJ DJu J ´ Ju J D “ JurD, u sJ . The gauge group G of a` real spectral triple is˘ defined via the adjoint action ˚ Adupaq “ uau of the unitary group UpAq on H as follows: ´1 GpA,Jq “ tAdu | u P UpAqu “ tuJuJ | u P UpAqu . (4.24) Before proceeding to compute it for a case concerning our study, we do the notation more symmetric, setting n1 “ N and n2 “ n for the rest of this section. We assume n1 ą n2 ě 2. The next statement is not surprising, but due to the presence of the tensor product, some care is needed.

Proposition 4.8. Let G1 ˆG2 “ Gf ˆF be a ﬁnite almost-commutative geometry, with algebra A “ A1 bA2, A1 “ Mn1 pCq and A2 “ Mn2 pCq, and reality J “ J1 bJ2. The gauge group is given by the product of unitary projective groups GpA,Jq “ PUpn1q ˆ PUpn2q. Before proving this proposition, broken down in some lemmas below, we recall the characterization of the gauge group that will be used. Namely, the next short sequence is exact, according to [vS15, Prop. 6.5]:

1 Ñ UpAJ q Ñ UpAq Ñ GpA,Jq Ñ 1 . (4.25)

Thus, if the groups UpZpAqq and UpAJ q coincide, then GpA,Jq – UpAq{UpZpAqq . (4.26) We now verify that they do, so that after computing UpAq and UpZpAqq, we can ﬁnally obtain the gauge group by this isomorphism (4.26).

Lemma 4.9. For A and J as in Proposition 4.8, UpZpAqq “ UpAJ q. FINITE ALMOST-COMMUTATIVE GEOMETRIES 21

Proof. First, observe that if a P AJ and b P A, then ab “ Ja˚J ´1b “ a0b “ ba0 “ bJa˚J “ ba , where one gets the equalities at the very left or very right by the deﬁning property (4.22) of AJ , and the third equality by the commutant property (4.21). Hence AJ Ă ZpAq, and thus UpAJ q Ă UpZpAqq. We only have to prove the containment UpZpAqq Ă UpAJ q. According to Lemma A.1 (proven in AppendixA), ZpAq “ ZpA1 b A2q “ ZpA1q b ZpA2q. Since the representation % of A1 b A2 on H1 b H2 “ V b A1 b A2 is the fundamental on each factor (except the trivial action on spinor space factor V ) by Schur’s Lemma, each ZpAiq consists of multiples of the identity. Then, for any z1 b z2 P ZpA1q b ZpA2q one has ˚ %rpz1 b z2q sJΨ “ p1V b z¯1 b z¯2qpC b ˚1 b ˚2qΨ

“ pC b ˚1 b ˚2qp1V b z1 b z2qΨ “ J%pz1 b z2qΨ where ˚i is the involution of Ai and Ψ an arbitrary vector in the Hilbert space described above. Therefore z1 b z2 P AJ . One verifies that this proof leads equally ´1 ˆ to ZpAq Ă AJ by taking other representing element z1λ b z2λ pλ P C q the same conclusion ZpAq Ă AJ is reached, which restricted to the unitarities gives UpZpAqq Ă UpAJ q. Lemma 4.10. The following is a short exact sequence of groups: ˆ ` ` α 1 Ñ C Ñ tR ˆ Upn1qu ˆ| det | tR ˆ Upn2qu Ñ UpA1 b A2q Ñ 1 , where8 ` ` tR ˆ Upn1qu ˆ| det | tR ˆ Upn2qu (4.27) ` ` :“ pρ1, u1, ρ2, u2q P R ˆ Upn1q ˆ R ˆ Upn2q | ρ1ρ2 “ 1 . ` Proof. Let us abbreviate the group in the middle as follows G “ tR Ûpn1quˆ( | det | ` tR ˆ Upn2qu and define α : G Ñ UpA1 b A2q by pρ1, u1, ρ2, u2q ÞÑ ρ1u1 b ρ2u2.

Suppose pρ1, u1, ρ2, u2q P ker α, so that αpρ1, u1, ρ2, u2q “ ρ1u1 b ρ2u2 “ 1n1 b 1n2 . Since ker α Ă G one has ρ1ρ2 “ 1. Thus, the previous equation yields u1 b u2 “

1n1 b 1n2 , which says that in the lhs u1 and u2 are a scalar multiples and mutual inverses. Then ker α – tρ, λ, ρ´1, λ´1u, and if one embeds Cˆ ãÑ G as follows (which will be the definition of the leftmost map) z “ |z| ¨ eiθ “ r ¨ eiθ ÞÑ pr, eiθ, r´1, eíθq, one gets exactness at G. The rightmost map UpA1 b A2q Ñ 1 is the determinant in absolute value. Its kernel has elements g1 bg2 P UpA1 bA2q satisfying | detpg1 bg2q| “ 1. But this condition is satisfied by all elements gi “ ρiui, as far as pρ1, u1, ρ2, u2q P G. Conversely, if iψ ˆ g1 bg2 P UpA1 bA2q satisfies | detpg1 bg2q| “ 1, then there exists a λ “ |λ|ë P C ´1 iψ ´1 iψ with g1 “ λ ¨ u1 and g2 “ λ ¨ u2. Then αp|λ|, e ¨ u1, |λ| , e ¨ u2q “ g1 b g2. Hence kerp| detp ‚ q|q Ă im α too, and the sequence is exact also at UpA1 b A2q.

Lemma 4.11. The following group sequence is exact: ˆ ˆ ˆ 1 Ñ C Ñ C ˆ| det | C Ñ UtZpA1 b A2qu Ñ 1 .

8 ` That group is isomorphic to R ˆ Upn1q ˆ Upn2q, but we will keep the full notation and the embedding for later convenience. 22 CARLOS I. PEREZ-SANCHEZ

(ρ, u j )

) i 1 , u − (ρ ui uj ρ 1/ρ

Figure 3. Picture of the group G implied in the description of the group ˆ UpA1 bA2q – G{C . There the indices refer to each Upniq-factor, i ‰ j, and the ρ and ρ´1 might lie outside the unit circle (thick line)

ˆ ˆ ˆ 2 The group C ˆ| det | C in the middle is the subgroup of pC q whose entries pz1, z2q ´1 satisfy |z1| “ |z2| .

ˆ ˆ ˆ ´1 Proof. The embedding C ãÑ C ˆ| det | C is given by λ ÞÑ pλ, λ q and the next ˆ ˆ map C ˆ| det | C Ñ UtZpA1 b A2qu by pz1, z2q ÞÑ z1 b z2. 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),

GpA,Jq – UpA1 b A2q UtZpA1 b A2qu

– UpA1 b A2qLUtZpA1q b ZpA2qu . where one passes to the second line by LemmaL A.1. By Lemmas 4.11 for the group in the ‘numerator’ and Lemma 4.10 for the one in the ‘denominator’, ` ` ˆ tR ˆ Upn1qu ˆ| det | tR ˆ Upn2qu C G A,J . p q – ˆ ˆ ˆ “ C ˆ| det | C {C ‰L By the third group isomorphism theorem,` one can˘ ‘cancel out’ the Cˆ, and get ` ` tR ˆ Upn1qu ˆ| det | tR ˆ Upn2qu G A,J . p q – ˆ ˆ “ C ˆ| det | C ‰ Notice that in each group | det | only` constrains the˘ real parts, while it respects ˆ the Upn1q and Upn2q in the numerator and the two factors Up1q of each C in the denominator. We conclude that

Upn1q ˆ Upn2q GpA,Jq – – PUpn1q ˆ PUpn2q . Up1q ˆ Up1q FINITE ALMOST-COMMUTATIVE GEOMETRIES 23

4.4. 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 [vS15, 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 SUpAq :“ tm P UpAq | detr%pmqs “ 1u . ˆ We now deﬁne the following morphisms δi : GLpniq Ñ C ,

n2 ´n1 δ1pg1q “ rdetn1 pg1qs and δ2pg2q “ rdetn2 pg2qs , (4.28) which shall be useful in the description of the special unitary group we care about (notice that both morphisms depend on the pair pn1, n2q and the diﬀerent signs in the exponents).

Lemma 4.12. The special unitary group of A1 b A2,

SUpA1 b A2q “ tu1 b u2 P UpA1 b A2q | detpu1 b u2q “ 1u ﬁts in a short exact sequence of groups: κ 1 Ñ Up1q ãÑ Upn1q ˆdet Upn2q Ñ SUpA1 b A2q Ñ 1 , where Upn1q ˆdet Upn2q is the (categorical) pullback of anyone of the two morphisms (4.28) along the remaining one.

Proof. Define the homomorphism κ by pu1, u2q ÞÑ u1 b u2. Suppose that u1 b u2 P ˆ ker κ, so κpu1, u2q “ u1 bu2 “ 1n1 b1n2 . This means that there exists a λ P C with ´1 u1 “ λ1n1 and u2 “ λ 1n2 , but by assumption ui P Upniq, so λ P Up1q. Thus the ´1 image of the inclusion Up1q ãÑ Upn1q ˆdet Upn2q λ ÞÑ pλ1n1 , λ 1n2 q is the kernel of κ. The last map to the right is the determinant. If u1 b u2 P im κ, then by definition of the fibered group Upn1qˆdet Upn2q, δ1pu1q “ δ2pu2q holds. But this happens if and n2 n1 only if 1 “ rdetn1 pu1qs ¨ rdetn2 pu2qs “ detpu1 b u2q “ pdet ˝κqpu1, u2q. Therefore the image of κ is in the kernel of the determinant. On the other hand, if g1 b g2 P kerpdetq Ă SUpA1 b A2q then each gi P GLpniq (otherwise its determinant vanishes and by assumption it is 1) so we can write them ˚ in matrix polar form gi “ piui with ui P Upniq and pi “ pi positive definite. Since, in particular, p1u1 b p2u2 P UpA1 b A2q, one obtains ˚ ˚ ˚ ˚ 2 2 1n1 b 1n2 “ p1u1u1 p1 b p2u2u2 p2 “ p1 b p2 . (4.29)

Being both pi’s positive deﬁnite Hermitian matrices, they can be written as pi “ ˚ viΛivi for Λi “ diagpλi,1, . . . , λi,nq with λi,m ě 0 and vi P Upniq. But then Eq. ` 2 ˚ (4.29) means the existence of certain r P R for which v1pΛ1q v1 “ r ¨ 1n1 and 2 ˚ ´1 1{2 ´1{2 v2pΛ2q v2 “ r ¨1n2 . Solving each equation leads to Λ1 “ r 1n2 and Λ2 “ r 1n2 , so we can forget the vi’s, since Λi is central.

In summary, there exist scalars ρi such that pi “ ρi1ni with ρi ą 0 and ρ1 “ 1{ρ2. This relation shows that g1 b g2 “ u1 b u2 “ κpu1, u2q, since in the tensor product ´1 ˆ g1 b g2 “ pλ g1q b pλg2q for any λ P C (here, in particular, choosing λ “ ρ1). 24 CARLOS I. PEREZ-SANCHEZ

By construction, ui are unitarities, which, by assumption, moreover satisfy 1 “ detpg1 b g2q “ detpu1 b u2q “ δ1pu1q{δ2pu2q. Hence pu1, u2q P Upn1q ˆdet Upn2q and g1 b g2 “ κpu1, u2q, which concludes the proof of exactness at SUpA1 b A2q. Lemma 4.13. There exists an exact sequence of groups ι ξ ζ 1 Ñ µmcdpn1,n2q Ñ Up1q ˆ SUpn1q ˆ SUpn2q Ñ Upn1q ˆdet Upn2q Ñ µn1¨n2 Ñ 1 where mcdpn1, n2q is the maximum common divisor of n1 and n2. Proof. From left to right we start defining the maps and checking exactness along ´1 ni the way. The first map is ιpλq “ pλ, λ ¨ 1n1 , λ ¨ 1n2 q. Since detni pz ¨ 1ni q “ z “ 1 for z P µmcdpn1,n2q, the map is well-defined, and clearly is also injective. ´1 The next map is given by ξpz, m1, m2q “ pzm1, z m2q. Since mi have unit ´1 determinant, the condition δ1pzm1q “ z “ 1{δ2pz m2q is satisfied (cf. Eq. (4.28) ´1 above). The pair pzm1, z m2q is thus in the fibered product Upn1q ˆdet Upn2q, by its definition and ξ is thus well-defined. To verify the exactness, notice that if pz, m1, m2q is such that ξpz, m1, m2q “ ´1 ´1 pzm1, z m2q “ p1n1 , 1n2 q, since each mi P SUpniq, one has detn1 pz 1n1 q “ 1 and ´1 detn2 pz1n2 q “ 1. Hence z P µn1 X µn2 . Thus pz, m1, m2q “ pz, z ¨ 1n1 , z ¨ 1n1 q and therefore ker ξ Ă im ι since the group that generates this intersection z P µn1 X µn2 is µmcdpn1,n2q. The other containment holds also, since ´1 ´1 ´1 ξ ˝ ιpλq “ ξpλ, λ ¨ 1n1 , λ ¨ 1n1 q “ pλ ¨ rλ ¨ 1n1 s, λ ¨ rλ ¨ 1n2 s q “ p1n1 , 1n2 q for each λ P µmcdpn1,n2q. Hence ker ξ “ im ι and the sequence is exact at the node having the triple product. 1{n1 1{n2 The last map is given by ζpu1, u2q “ rdet1pu1qs ¨ rdet2pu2qs , which for pu1, u2q in the fibered group product satisfies, by definition,

1{n1 1{n2 n1n2 n1n2 1 “ δ1pu1q{δ2pu2q “ trdet1pu1qs ¨ rdet2pu2qs u “ rζpu1, u2qs .

(well-deﬁnedness). To see that ker ζ Ă im ξ, take pu1, u2q in the ﬁbered product 1{n1 1{n2 group satisfying ζpu1, u2q “ rdet1pu1qs ¨ rdet2pu2qs “ 1. This means that

1{n1 ´1 1{n2 λ0 :“ rdetn1 pu1qs and λ0 “ rdetn2 pu2qs (4.30) ´1 are consistent. Due to Eq. (4.30), conveniently used, both matrices λ0 ¨ u1 and ´1 λ0 ¨ u2 are special unitary, and we also obtain pu1, u2q “ ξpλ0, λ0 ¨ u1, λ0 ¨ u2q. Finally, on the other hand,

´1 1{n1 1{n2 pζ ˝ ξqpλ, m1, m2q “ ζpλm1, λ m2q “ rdetn1 pλ1n1 qs rdetn2 pλ1n2 qs “ 1 so the inverted injection holds ker ζ Ą im ξ too. The last result extracts the Lie group part of the ﬁbered group in terms of which we computed the unimodular gauge group in Lemma 4.12. This proof was inspired by a proof by Chamseddine-Connes-Marcolli but is diﬀerent from it due to the presence of the tensor product of algebras, whilst [CCM07, Prop. 2.16] or [CM07, Prop. 1.185] focus on unitarities of semi-simple algebras, A1 ‘ A2 ‘ ... ‘ Ak. In particular for the Standard Model [CM07, Prop. 1.199], the unimodular gauge group is the well-known tUp1q ˆ SUp2q ˆ SUp3qu{µ6 Standard Model gauge group (cf. also [vdDvS12, §6.2]). The embedding of the group µ6 of sixth roots of unit in the Lie group is given by λ ÞÑ pλ, λ3, λ2q, as pointed out in [vS15, §11.2.1]. FINITE ALMOST-COMMUTATIVE GEOMETRIES 25

Our embedding of the roots of unit appearing in the above Lemma is diﬀerent, since the determinant for tensor products of algebras is governed by another rule: n2 n1 detpa1 b a2q “ rdetn1 pa1qs ˆ rdetn2 pa1qs for each a1 b a2 P A1 b A2. On the physical side, the origin of the two roots of unit groups in the exact sequence

1 Ñ µ3 Ñ Up1q ˆ SUp2q ˆ SUp3q Ñ SUpAF q Ñ µ12 Ñ 1 [CM07, Seq. 1.661] characterizing the unimodular gauge group for the algebra of the Standard Model 9 AF “ C ‘ H ‘ M3pCq is quite diﬀerent: on the one hand, the group µ3 comes from M3pCq; and on the other µ12 does depend also on the number of generations and the representation of fermions.

By way of contrast, an important one conceptually, we stress that for SUpnq-Yang- Mills(–Higgs) finite geometries where one has A1 “ MN pCq and A2 “ MnpCq (so n1 “ N and n2 “ 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 PUpNq [resp. SUpNq] describes the symmetry of the base (and could be understood as the finite dimensional analogue of diffeomorphisms of a manifold) and PUpnq [resp. SUpnq] along the fibers.

5. 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 offered by the gauge theory treatment with NCG. We now recompute the results of Section4, revoking the restriction DF “ 0. The aim is a formula informed by Weitzenböck’s. The Weitzenböck 2 SbE formula, Dω “ ∆ ` 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 ΓpEndpS b Eqq: 1 1 2 i j j SbE E “ R b 1 ` 1 b Φ ´ iΓ Γ b Fij ` iγM Γ b adp∇j qΦ , (5.1) 4 i,j 2 j ÿ ÿ bE where p∇S qj and Fij are locally the connection on S b E and the curvature on E, respectively. Further, γM is the chirality element or γ5 in physicists’ speak. (See e.g. [vS15, Prop. 8.6] for a proof.) 5.1. 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 D “ Df b 1F ` γ D D 1 γ 1 D where D D˚ M is the Dirac f b F “ f b F ` b MN pCq b F F “ F P npCqs.a operator of the finite geometry F .

Claim 5.1. The inner ﬂuctuations of the Dirac operator along the ﬁnite geometry F are ω Jω J ´1 Ψ γ φ Ψ 2 γ 1 1 Ψ 1 φ , (5.2) p F ` F qp q “ p b qp q ` p b MN pCq b MnpCqq p V b q

9 Here the fact that the unitary quaternions tq P H : q˚q “ 1 “ qq˚u are unimodular (i.e. their determinant is 1 in the embedding of H into 2 ˆ 2 matrices), UpHq – SUp2q, causes that unimodularity has inﬂuence on the H summand. That is why µ3 appears as ﬁber instead of µ3ˆ2. 26 CARLOS I. PEREZ-SANCHEZ for each Ψ P H “ V b MN pCq b AF . These are parametrized by φ P MN pCq b Ω1 M . Also φ˚ φ holds. DF p npCqq “

Proof. As before, one computes the corresponding Connes’ 1-forms arγf b DF , cs in terms of a “ 1V bW ba and c “ 1V bT bc, being W, T P MN pCq and a, c P MnpCq. Namely,

ωF “ arγf b DF , cs a γ 1 D , c “ r b MN pCq b F s 1 W a γ 1 D , 1 T c “ p V b b q b MN pCq b F V b b “ γ b WT b arD“ F , cs ‰

We rename φ :“ X b arDF , cs, since is clear that W, T are arbitrary and their product can replaced by any matrix X M . Thus ω γ φ Ω1 A P N pCq F “ b P γf bDF p q “ M Ω1 M as claimed. Since from the onset γ is self-adjoint, so must N pCq b DF p npCqq ˚ be φ, since ωF “ ωF is required. The remaining part of the ﬂuctuations acting on v b Y b m P V b MN pCq b MnpCq are

´1 ´1 pJωF J qpv b Y b mq “ pC b ˚N b ˚nqpγ b φqpC b ˚N b ˚nq pv b Y b mq ´1 ˚ ˚ “ p`C b ˚N b ˚nqpγC v b XY b arDF , c˘sm q ´1 ˚ ˚ ˚ “ CγC v b YX b parDF , csm q (5.3) 2 ˚ ˚ “ γv b YX b mparDF , csq , since Cγ “ 2γC (cf. table of Def. 2.1). Therefore,

Jω J ´1 Ψ 2 γ 1 1 Ψ 1 X˚ a D , c ˚ p F qp q “ t b MN pCq b MnpCqup qtp V b b p r F sq u 2 γ 1 1 Ψ 1 φ , (5.4) “ t b MN pCq b MnpCqup qp V b q since φ “ X b arDF , cs is self-adjoint, as argued before.

In Eq. (5.3) of the proof one could also have computed directly, using the explicit formula (3.5) for the chirality:

CγC´1 “ pCσpηqγ0C´1qpCγ1C´1qpCγ2C´1qpCγ3C´1q “ pCσpηqγ0C´1qγ1γ2γ3 “ σpηqγ0γ1γ2γ3 “ ˘γ .

The complex conjugate in the last line appears since C is anti-linear. The sign is chosen as follows: notice that σpηq 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 p´1q#number of minus signs in η “ p´1qq. This diﬀerent way to compute leads to the same result as the one given in the proof. Indeed, for 4-dimensional geometries p´1qq is precisely 2, according to the sign table in Deﬁnition 2.1, namely 2 “ ´1 for KO-dimensions 2 and 6 and and 2 “ `1 for KO-dimensions 0 and 4.

From Claim 5.1 and Theorem 4.2, the form of the most general ﬂuctuated Dirac operator follows: FINITE ALMOST-COMMUTATIVE GEOMETRIES 27

DHiggs Dgauge µ µˆ Dω “ γ b Φ ` γ b pkµ ` aµq ` γ b pxµ ` sµq , (5.5a) hkkikkj µ hkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkj ÿ µ µˆ D “ γ b DF ` γ b kµ ` γ b xµ , (5.5b) µ ÿ µ µˆ ω “ γ b φ ` γ b Aµ ` γ b Sµ , (5.5c) µ ÿ ´1 2 µ q µˆ JωJ “ γ b p ‚ qφ ` eµγ b p ‚ qAµ ` p´1q eµγ b p ‚ qSµ , (5.5d) µ ÿ 2 with Φ: 1 D φ ‚ φ 1 D φ, ‚ 2 . (5.5e) “ MN pCq b F ` ` p q “ MN pCq b F ` t u

We will call Φ M Ω1 M M Ω1 M the Higgs P N pCqs.a b t DF r npCqsus.a Ă N r DF p npCqqs s.a. field, since in the smooth Riemannian case (where the analogous relation reads pC8q pC8q ´1 ( Φ “ DF ` JF φ JF ) its analogue in the context of almost-commutative geometries leads to the Standard Model Higgs field, when the finite algebra AF is correctly chosen (cf. [CCM07, vS15]).

Corollary 5.2. The fluctuated Dirac operator Dω on ‘flat’ (x “ 0) fuzzy space on the geometry Gf ˆ F satisfies

2 1 µ ν 2 Dω|X“0 “ γ γ b Fµν ` 1V b pϑ ` Φ q . (5.6) 2 µ,ν ÿ Proof. Since γµγ “ ´γγµ one has also that γµˆ “ γαγργσ anti-commutes with γ, for γαγργσγ “ ´γαγργγσ “ `γαγγργσ “ ´γγαγργσ . µ µˆ Since the matrices γ and γ , µ P ∆4, span (the projection to V of) Dgauge, the anti- 2 2 2 commutator tDgauge,DHiggsu vanishes so pDωq “ Dgauge ` DHiggs. The gauge part 2 2 2 2 Dgauge is known from Proposition 4.6; on the other hand, DHiggs “ pγ bΦq “ 1V bΦ from the axiom in Deﬁnition 2.1 for the chirality γ.

5.2. 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 Fµν and on the Higgs field. For the former, this requires to know how the matrices Aµ transform under GpA; Jq “ UpAq{UpAJ q. We can pick a representing element of GpA; Jq in u P UpAq directly, since the apparent ambiguity 10 up to an element z P UpAJ q leads to ωuz “ puzqωpuzq˚ ` uzrD, puzq˚s “ uωu˚ ` uz rD, z˚su˚ ` z˚rD, u˚s ˚ ˚ ˚ u “ uωu ` uzz rD, u s “ ω . ( ˚ The last line is obtained since UpAJ q “ UpZpAqq, so z is central (and thus z too). Hence rD, z˚s “ 0.

10The next equation is based on van Suijlekom’s [vS15, §8.2.1] 28 CARLOS I. PEREZ-SANCHEZ

Is given by Notation or satisfies Lies in Dirac op. type

˚ Lµ (fundamental) supNq pLµ “ ´Lµq ˚ Riemannian Xµ (fundamental) MN pCqs.a. (Xµ “ Xµ) , /Pure fuzzy l rL , ‚ s Der M p q / µ µ N C ./ xµ tXµ, ‚ u End`pMN pCqq˘ / / -/ DF (fundamental) MnpCqs.a Finite geometry

L L 1 su N 1 µ µ b MnpCq p q b MnpCq X X 1 M 1 , µ µ b MnpCq N pCqs.a. b MnpCq , D b 1 / / f MnpCq / l rL , ‚ s Der M p q b 1 / / µ µ N C MnpCq / / . / x tX , ‚ u End M p ¨ 1 q / µ µ ` N C MnpCq ˘ / / / / /Finite ` ˘ -/ / 1 / Aµ Aµ b b ΩD 1 pMN b MnqpCq /almost f b F / 1 / Sµ Sµ b b ΩD 1 `pMN b MnqpCq˘ , /commu- f b F ./ /Fluctuations aµ rAµ, ‚ s End pM`N b MnqpCq ˘ / tative ./(gauge) / / sµ tSµ, ‚ u End`pMN b MnqpCq˘ / / / / / ` ˘ / / ˚ 1 - / φ φ “ φ MN p q b Ω Mnp q / C DF C / / tφ, ‚ u Fluctuations / Φ ` ˘ , / End pMN b MnqpCq /(Higgs) / `1MN p q b DF . / C / / ` ˘ / -/ Table 3. Notation for the matrices parametrizing- the Dirac operator of Riemannian four-dimensional Yang-Mills–Higgs ﬁnite almost commutative geometry and its ﬂuctuations along D “ Df b 1F ` γf b DF , which are split into blocks along the gauge (Df b 1F ) and Higgs parts (γf b DF ). 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.

Next, observe that, by deﬁnition, and also by Jacobi identity on MNbnpCq,

rlµ, aνs˝ “ Lµ, rAν, ‚ s ´ Aν, rLµ, ‚ s (5.7) “ “rLµ, Aνs, ‚ ‰ “ ‰ (5.8) with analogous expressions for rl“ν, aµs˝ and‰ raµ, aνs˝. This allows to write the ﬁeld strength as the commutator with another quantity Fµν P MNbnpCq that we call ﬁeld strength matrix,

Fµν “ rFµν, ‚ s , (5.9a)

Fµν :“ rLµ, Lνs ` rLµ, Aνs ´ rLν, Aνs ` rAµ, Aνs . (5.9b) We now find the way the field strength transforms under the gauge group. By definition, the transformed field strength is given by the expression Fµν evaluated FINITE ALMOST-COMMUTATIVE GEOMETRIES 29

u in the transformed potential aµ, this latter being dictated by the way the Dirac operator transforms under GpA; Jq. Specifically, u u u u u Fµν “ rlµ, lνs˝ ` rlµ, aν s˝ ´ rlν, aµs˝ ` raµ, aν s˝ , (5.10a) u u aµ “ rAµ, ‚ s , (5.10b) u ˚ ˚ ˚ Aµ “ AdupAµq ` urLµ, u s “ upAµqu ` urLµ, u s , (5.10c) u u u u u Fµν :“ rLµ, Lνs ` rLµ, Aν s ´ rLν, Aν s ` rAµ, Aν s . (5.10d) Concerning the Higgs, we come back to Eqs. (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

u ˚ ˚ φ ÞÑ φ “ uφu ` urDF , u s , u P GpA; Jq . (5.10e) The transformation of the ﬁeld strength is more interesting:

Proposition 5.3. In Riemannian signature, the field strength of a Yang-Mills(– Higgs) finite geometry transforms under the gauge group as follows: u u Fµν “ rFµν, ‚ s ÞÑ Fµν “ rFµν, ‚ s u P GpA; Jq , (5.11) which is completely determined by the next transformation rule on the field strength matrix

u ˚ Fµν ÞÑ Fµν “ uFµνu “ AdupFµνq . (5.12) u Proof. Observe that for the pair lµ, aν the same argument given about Eq. (5.7) for the pair lµ, aν 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 u u u u u terms of the matrix Fµν :“ rLµ, Lνs ` rLµ, Aν s ´ rLν, Aν s ` rAµ, Aν s as follows: u u u u u u Fµν “ rLµ, Lνs ` rLµ, Aν s ´ rLν, Aµs ` rAµ, Aν s, ‚ “ rFµν, ‚ s . (5.13) We now compute“ the transformed field strength matrix‰ and all those terms that u imply A (namely the transformation under u of Tµν :“ Fµν ´ rLµ, Lνs Ñ Tµν) and infer from that those the gauge transformations on the field strength matrix Fµν. u ˚ ˚ Tµν “ ` Lµ, AdupAνq ` urLν, u s ´ Lν, AdupAµq ` urLµ, u s ˚ ˚ `“ AdupAµq ` urLµ, u s, Ad‰ upA“νq ` urLν, u s ‰ ˚ ˚ “ `”Lµ, uAνu ` Lµ, urLν, u s ı ˚ ˚ ´“Lν, uAµu ‰ ´“Lν, urLµ, u s‰ ˚ ˚ ˚ ˚ ` “uAµu , uA‰νu “ ` urLµ, u s‰, uAνu ˚ ˚ ˚ ˚ ` “urLµ, u s, urLµ‰, u “s ` uAµu , urLν‰, u s u The contributions to“ Tµν split into three:‰ LA“-terms (i.e. containing‰ Lµ, Aν or Lν, Aµ), AA-terms, and LL-terms. We compute them separately: ˚ ˚ ˚ ‚ AA-terms: uAµu , uAνu “ urAµ, Aνsu , clearly ‚ LL-terms: When the commutators are expanded, the next LL-terms “ ‰ ˚ ˚ ˚ ˚ Lµ, urLν, u s ´ Lν, urLµ, u s ` urLµ, u s, urLµ, u s “ ‰ “ ‰ “ ‰ 30 CARLOS I. PEREZ-SANCHEZ

yield the quantity in bracelets, which can be neatly rewritten:

˚ ˚ LµuLνu ´ LµLν ´ uLνu Lµ ` LνLν ˚ ˚ `LνLµ ´ LνuLµu ` uLµu Lν ´ LµLν ˚ $ ˚ ˚ , “ urLµ, Lνsu ´ rLµ, Lνs ’ `uLµLνu ´ uLνLµu ` LµLν ´ LνLµ / &’ ˚ ˚ ˚ ˚ ./ `uLνu Lµ ´ LµuLνu ` LνuLµu ´ uLµu Lν

’ ˚ ˚ /˚ ˚ ˚ ˚ ‚% LA-terms: Lµ, uAνu ´ Lν, uAµu ` urLµ, u-s, uAνu ` uAµu , urLν, u s . This can be also obtained expanding the commutators as above; the last two “ ‰ “ ‰ “ ˚ ‰ “ ‰ commutators yield utrLµ, Aνs´rLν, Aµsuu `rpL, Aq. The excess terms rpL, Aq are actually cancelled out with the two ﬁrst propagators, yielding for the ﬁnal expression of the LA-terms:

˚ u rLµ, Aνs ´ rLν, Aµs u (5.14) In view of the last equalities, we` can conclude that˘

u ˚ ˚ Tµν “ uTµνu ` urLµ, Lνsu ´ rLµ, Lνs (5.15)

“ AdupTµνq ` Adu rLµ, Lνs ´ rLµ, Lνs , ` u ˘ which, re-expressed in terms of F, yields Fµν Ñ Fµν “ AdupFµνq.

Remark 5.4. Notice that Lµ being the fuzzy analogue of the derivatives, the ‘sur- 11 prising term’ rLµ, Lνs is the analogue of rBµ, Bνs, which is identically zero on the algebra C8pMq. 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 Fµν, ˜ ˜ or rather Fµν “ Fµν ´ rLµ, Lνs (called Tµν above). According to Eq. (5.15), Fµν transforms then as ˜ ˜u Fµν ÞÑ Fµν “ AdupFµνq ` Adu rLµ, Lνs ´ rLµ, Lνs . (5.16) ` traceless˘ Although for quadratic actions the last twoloooooooooooooomoooooooooooooon 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 Eq. (4.12) and Eq. (5.9b) are correct. For only then, the pure gauge sector (i.e. powers of ) obtained from Tr D2m would be F Hp ωf q expressible (see [Pér19], and for m “ 2, Eq. (2.10) above) as a sum over chord diagrams ξ, with µ “ pµ1, . . . , µmq, ν “ pν1, . . . , νmq,

µ1ν1µ2ν2...µmνm u u ξ Tr C tF ¨ ¨ ¨ F u . (5.17) MNbn µ1ν1 µmνm ξ µ,ν m-chordÿ diag. ÿ

The scalars ξµ1ν1µ2ν2...µmνm are expressed as sums of m-fold products of the bilinear form ηασ (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’ µi and νi. The quartic computation is explicitly given below.

11 The precise statement is that adrLµ,Lν s is the analogue of rBµ, Bν s, but Jacobi identity allows one to state this in terms of rLµ, Lν s only. FINITE ALMOST-COMMUTATIVE GEOMETRIES 31

5.3. The quadratic-quartic Spectral Action. The next statement is obvious:

Lemma 5.5. The fully fluctuated Dirac operator on the Yang-Mills–Higgs finite geometry satisfies in ‘flat space’ (i.e. X “ 0),

1 2 2 TrH Dω “ TrM C pϑ ` Φ q (5.18) 4 X“0 Nbn ` ˇ ˘ Proof. From Eq. (5.6), since tracingˇ the ﬁrst summand yields, due to index symmetry of η and index skew-symmetry of the ﬁeld strength,

µ ν µν TrHpγ γ b Fµνq “ η dim V Tr C Fµν “ 0 MNbn µ,ν µ,ν ÿ ÿ one gets the result by Eq. (4.14). Lemma 5.6. The fully fluctuated Dirac operator of the Yang-Mills-Higgs finite geometry satisfies in ‘flat space’ (i.e. X “ 0),

1 4 1 µν 2 2 TrH D “ ´ Tr C pFµνF q ` Tr C pϑ ` Φ q . (5.19) ω X“0 MNbn MNbn 4 2 µ,ν ` ˇ ˘ ÿ ` ˘ ˇ 2 Proof. Squaring the expression for pDω|X“0q given by Lemma 5.2.

2 2 1 µ ν ρ σ pDω|X“0q “ γ γ γ γ b FµνFρσ 4 µ,ν,ρ,σ ÿ 1 µ ν 2 2 2 2 ` γ γ b Fµνpϑ ` Φ q ` pϑ ` Φ qFµν ` 1V b pϑ ` Φ q . 2 µ,ν ÿ ` ˘ The expression pa ` bq2 “ a2 ` ta, bu ` b2 can be simpliﬁed, when ta, bu is replaced µ ν α ρ by 2ab (or 2ba). Using TrV pγ γ γ γ q as given in Eq. (2.10),

µ µ µ 2 2 dim V ρ ν + ρ ν + ρ ν TrHpDω|X“0q “ ¨ TrM C FµνFαρ 4 Nbn µ,ν,ρ,σ ˆ α α α ˙ ÿµν 2 ` ˘ ` dim V η Tr C pFµνpϑ ` Φ qq (5.20) MNbn µ,ν ÿ 2 2 ` dim V Tr C pϑ ` Φ q . MNbn 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. (5.20). The second chord diagram comes with a minus sign and, using the skew-symmetry F , one can see that the third νρ µα diagram yields the same contribution, namely p´η η q Tr C pFµνFαρq. µ,ν,ρ,σ MNbn We can divide the expression by dim V “ 4 and readily get the claim. ř

6. Towards the continuum limit We now give the main statement and, after its proof, we compare it with [CC97, §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). 32 CARLOS I. PEREZ-SANCHEZ

6.1. The Spectral Action of a Yang-Mills–Higgs ﬁnite geometry. Using the lemmas of previous sections, it is immediate to prove the main result:

Theorem 6.1. For a four dimensional Yang-Mills–Higgs ﬁnite almost-commutative 1 2 4 geometry the quartic-quadratic Spectral Action with fpxq “ 2 px ` x q reads 1 Tr fpDq “ Sf ` Sf ` Sf ` Sf, (6.1) 4 H YM H g-H ϑ where we deﬁne each sector as follows:

f 1 µν SYMpl, aq :“ ´ TrM C pFµνF q , (6.2a) 4 Nbn f S pΦq :“ Tr C fpΦq, (6.2b) H MNbn f 2 S pl, a, Φq :“ Tr C ϑ Φ , (6.2c) g-H MNbn f ` 1{2˘ S pl, aq :“ Tr C f ϑ . (6.2d) ϑ MNbn Moreover, one obtains positivity for each of the following` ˘ functionals, independently: f f f f Sϑ ,SYM,Sg-H,SH ě 0 .

Proof. Notice that by Lemmas 5.5 and 5.6,

1 1 1 2 1 4 TrH fpDq “ TrV b TrM C D ` D 4 4 Nbn 2 2 1 !2 1 ) 2 2 1 µν “ TrM C pϑ ` Φ q ` ppϑ ` Φ qq ´ FµνF Nbn 2 2 4 !1 2 2 4 2 2 1) µν “ TrM C ϑ ` ϑ ` Φ ` Φ ` Φ ϑ ` ϑΦ ´ FµνF . Nbn 2 4 The result follows by inserting! ` the definitions from Eq. (6.2). ˘ ) µ ˚ Regarding positivity: First, notice that aµa “ aµpeµaµq “ aµpaµq is a positive operator, and that so is ϑ P EndpMN pCq b HF?q—cf. Eq. (4.14)—by the ˚ same token, ϑ “ µpa ` kqµpa ` kqµ ě 0 . Thus fp ϑq is well-defined and its trace positive, since f is an even function. ř ˚ Further relations like rkµ, aνs “ éµeνrkµ, aνs, and similar ones for all the ˚ commutators defining the field strength, lead to Fµν “ éµeνFµν. Since η “ diagpe0, . . . , e3q, one obtains the positivity of the operator µν ˚ ´FµνF “ FµνpéµeνFµνq “ FµνpFµνq ě 0 , (no sum). (6.3) Therefore, also the positivity holds summing over µ, ν, which is a positive multiple of f SYM, whose positivity follows too. Similarly, since Φ is self-adjoint, even powers of it f ˚ µ are positive, thus so is SH. Finally, if βµ “ Φpkµ àµq, due to pa`kqµ “ pa`kq one has µ µ 2 µ µ ˚ Tr C pkµ ` aµqpk ` a qΦ “ Tr C pk ` a qΦ Φpkµ ` aµq MNbn MNbn ˚ ( “ Tr C pβ βµq , ( MNbn µ µ ÿ hence Sg-H is positive. We now comment on the interpretation of this result. For fuzzy geometries, the equivalent of integration over the manifold is tracing operators MN pCq Ñ MN pCq. FINITE ALMOST-COMMUTATIVE GEOMETRIES 33

(At the risk of being redundant, notice that the unit matrix in that space has trace N 2.) We identify the Higgs ﬁeld H on a smooth, closed manifold M with Φ, so 12 4 4 the quartic part |H| vol of the potential for the Higgs is Tr C pΦ q. In the M MNbn Riemannian case, in order to address the gauge-Higgs sector, notice that integration ş by parts yields the following identiﬁcation:

µ 2 µ 2 µ Tr C lµl Φ Ø BµB p|H| qvol “ ´ pBµHqpB Hqvol . MNbn żM żM ` ˘ This is consistent with the interpretation of lµ`aµ as the covariant derivative Dµ “ Bµ ` Aµ for Yang-Mills connection, with the local gauge potential Aµ absorbing the coupling constant (cf. Remark 5.4). Using Eq. (4.14) one gets the correspondence between the fuzzy and the smooth gauge Higgs sectors:

f 2 S pl, a, Φq :“ Tr C ϑ Φ (6.4) g-H MNbn

` ˘ µ µ 2 µ “ Tr C plµ ` aµqpl ` a qΦ Ø ´ DµHpD Hqvol . MNbn żM ` ˘ Next, notice that Fµν is a matrix-version of the SUpnq-Yang-Mills curvature, and for the action SYM,

f 1 µν SYMpl, aq :“ ´ TrM C pFµνF q (6.5a) 4 Nbn Ù 1 S p q “ ´ Tr p µνqvol . (6.5b) YM A 4 supnq FµνF żM For the time being, the previous identifications hold only the Riemannian signature, since for pp, qq ‰ p0, 4q 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 lµ is replaced by kµ). It holds then in general signature. The identification of Fµν “ rlµ, lνs˝`rlµ, aνs˝´rlν, aµs˝`raµ, aνs˝ in the Rie- mannian case (and its extension Fµν “ rkµ, kνs˝ `rkµ, aνs˝ ´rkν, aµs˝ `raµ, aνs˝ to the general signature) with the curvature Fµν “BµAν ´ BνAµ ` rAµ, Aνs of the smooth case is supported by the fact that rlµ, aνs˝ generalizes the multiplication operator BµAν, on top of the reason already given in Remark 5.4. The alternative to this definition, using only lµ ˝ aν in place of rlµ, aνs˝ (and similar replacements), yields instead pBµ ˝ Aνqψ “ pBµAνq ¨ ψ ` AνBµpψq on sections ψ (fermions). Notice also that for the smooth field strength one gets the positivity of the type of Eq. µν µν (6.3), namely ´ TrsupnqpFµνF q ě 0, due to FµνF “ ´FµνFµν [CM07, below Eq. 1.597]. We summarize this section in Table4, before computing, in next section, the matrix model corresponding to the path-integral quantization of this action.

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

12Notice that Φ˚ “ Φ so the analogy of Φ with the modulus of H is justiﬁed. 34 CARLOS I. PEREZ-SANCHEZ appears

f 1 µ ν S pkq :“ ´ Tr C kµkνk k (6.6) MNbn 2 µ‰ν ÿ ` ˘ as well as the same type of action, Sfpaq, in the variable a. The reference to a tetrahedron is justiﬁed when one writes that action in full,

µ ν v pk q pk q pk q pk q „ v0 v2 3 µ ij ν jm ml li ∼ v0 v1

v3 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 Sf is an example of the ‘matrix-tensor model’ class [BCTV20].

Meaning Random matrix case Smooth operator (Riemannian signature)

Derivation lµ Bi

Gauge potential aµ Ai Higgs ﬁeld Φ H

Covariant Derivative dµ “ lµ ` aµ Di “Bi ` Ai

”{ 0 ” 0

Field strength rdµ, dν s “ rlµ, lν s ` rDi, Djs “ rBi, Bjs ` rlµ, aν s ´ rlν , ahkkkikkkjµs ` raµ, aν s BiAj ´ BjAihkkikkj` rAi, Ajs

2 4 2 4 Higgs potential TrpΦ ` λΦ q M |H| ` λ|H| vol 2 2 Gauge-Higgs coupling Trpϑ Φ qş ´` M |DiH| vol˘ 1 µν 1 ij Yang-Mills action ´ 4 TrpFµν F q ´ 4 M şTrsupnqpFijF qvol ş Table 4. In this table Tr P denotes the trace of operators P : MNbnpCq Ñ MNbnpCq; gauge potential means the local expression for the connection; ﬁnally, λ stands for a coupling constant one can integrate in the quartic part of the Spectral Action. The analogies implying lµ ØBi hold only for the Riemannian signature

6.2. The Yang-Mills–Higgs matrix model. We now focus on the matrix model in the Riemannian case (p “ 0). In order to give a more structured appearance to the partition function, we recall the dependence of our functionals on the fundamental matrix ﬁelds Lµ, Aµ and φ. The L’s are functioning as derivatives Lµ P supNq b 1n,

‚ and lµ “ ad Lµ “ rLµ, s is the derivation deﬁned by the adjoint action, lµ P FINITE ALMOST-COMMUTATIVE GEOMETRIES 35

Der MN pCq b 1n, for each µ. One arrives at a similar situation with the matrix gauge potentials ` ˘ A Ω1 M M M M , µ P Df r N pCqs anti-Herm. b npCqs.a. Ă N pCq b npCq where the subindex in the curly( brackets restricts to anti-Hermitian 1-forms. In terms of these pa0, a1, a2, a3q “ a “ apAµq is deﬁned, again, via derivations:

‚ 13 aµ “ ad Aµ “ rAµ, s, which already bear a non-trivial factor in the inner space . This yields dependences l “ lpLq, a “ apAq. Further, by Eq. (5.5e), also Φ “ Φpφq. All in all, this yields each sector f f f f SYM “ SYMpL, Aq ,Sg-H “ Sg-HpL, A, φq , f f f f SH “ SHpφq ,Sϑ “ Sϑ pL, Aq . 1 2 4 The partition function, using the function fpxq “ 2 px ` x q, reads

f 1 Z “ exp ´ TrH fpDq dD (6.7) N 4 ż ´ ¯ where

‚ the Spectral Action is given by Theorem 6.1 f f,f ‚ the partition function Z “ ZN,n implies integrating over the matrix space N that depends on the parameters N and n via p“0,q“4 ˆ4 gauge ˆ4 Higgs pLµ, Aµ, φq P N “ NN,n “ supNq ˆ NN,n ˆ NN,n , (6.8) with the Higgs and gauge ﬁelds“ matrix‰ spaces“ deﬁned‰ by Higgs : iu N Ω1 M M Ω1 M (6.9a) NN,n “ p q b r DF p npCqqss.a. Ă N DF npCq s.a. gauge : iΩ1 M iu n M Ω1 M (6.9b) NN,n “ Df p N pCqqs.a. b p q Ă n “Df ` N pCq ˘‰(anti-Herm. ‚ the measure dD “ dL dA dφ is the product “ Lebesgue` measures˘‰( on the three factors of (6.8).

7. Conclusion and Outlook We introduced finite almost-commutative geometries, 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 four dimensional geometry of Riemannian signature. We justified this terminology based on Remark 5.4 and Section6; in particular see Table4 for the summary. The partition function of the Yang-Mills-Higgs theory is an integral over gauge potentials Aµ and a Higgs field Φ in (subspaces of the) following matrix spaces 1 1 Aµ P MnpΩf q and φ P MN pΩF q 1 1 where Ωf and ΩF are the Connes’ 1-forms along the fuzzy and the finite geometry, respectively, both parametrized by (finite) matrices. Additionally, the partition function for the spectral action implies an integration over four copies of supNq; each

‚ of these matrix variables Lµ appears as the adjoint lµ “ ad Lµ “ rLµ, s. These operators lµ are interpreted as degrees of freedom solely of the fuzzy geometry,

13 We recall that a is actually dependent on l, since the operators aµ “ rAµ, ‚ s parametrize the inner fluctuations of the Dirac operator Df b1F , itself parametrized by l, but this dependence is not made explicit. 36 CARLOS I. PEREZ-SANCHEZ in concordance with the identification of DerpMN pCqq with a finite version of the derivations on C8pMq, that is, vector fields on M. As in the ordinary almost-commutative setting MˆF , with M a smooth manifold, the Higgs field arises from fluctuation 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 (6.9) for the matrix Higgs field and the matrix gauge potentials, which are swapped if one simultaneously14 exchanges n Ø N and F Ø f. The Yang-Mills–Higgs theory has a projective gauge group G “ PUpNq ˆ PUpnq. 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 C8pM, SUpnqq after a truncation has been imposed on M.A rigorous interpretation, e.g. in terms of spectral truncations [CvS20], 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 [Pér21] for the application of this idea to general multimatrix models). 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 re- quiring Xµ ‰ 0 and thus also a more general treatment than that of Section5). If that is concluded, one could identity for signature p0, 4q

‚ Lemma 3.3 with ‘Fuzzy Lichnerowicz formula’,

‚ Proposition 4.6 with ‘Fuzzy ﬂat Weitzenböck formula’, and

‚ Proposition 4.7 with ‘Fuzzy Weitzenböck formula’. Whereas stating 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. 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-formalism15 (after Batalin-Vilkovisky [BV83]), all the more considering that it has been explored for Up2q-matrix models in [IvS17], and lately also given in a spectral triple description [Ise21]. En passant, notice that since the main algebra here is MN pAq with A a noncommutative algebra, the Dyson-Schwinger equations of these multimatrix models would be ‘quantum’ (in the sense of Mingo-Speicher [MS13, §4]; this is work in progress).

Acknowledgements. I thank the Faculty of Physics, Astronomy and Applied Computer Science, Jagielloniain University for hospitality (in 2018). I am also indebted to Andrzej Sitarz and Alexander Schenkel for useful comments during the preparation. This work was supported by the TEAM programme of the Foundation for Polish Science co-ﬁnanced by the European Union under the European Regional Development Fund (POIR.04.04.00-00-5C55/17-00).

14To be strict, one has to swap also the anti-Hermiticity by the Hermiticity in the both lines of (6.9), but this is clearly fixed by an imaginary factor and is ignored here. 15 Tangentially, a discussion on gauge theories and the BV-formalism in the modern language of L8-algebras appears in [CGRS21], in a noncommutative field theory (but also different) context. FINITE ALMOST-COMMUTATIVE GEOMETRIES 37

Appendix A. A lemma and a proof

Lemma A.1. Let Ai be unital, associative algebras, and let ZpAiq be the center of Ai. Then ZpA1 b A2q “ ZpA1q b ZpA2q.

Proof. Notice that for (so far, arbitrary) ai, bi P Ai (i “ 1, 2), one has by adding and subtracting a1b1 b b2a2 and rearranging,

ra1 b a2, b1 b b2s “ ra1, b1s b pb2a2q ` pa1b1q b ra2, b2s . (A.1)

Clearly, if ai P ZpAiq for i “ 1, 2, then the RHS vanishes for each bi P Ai, that is, a1 b a2 P ZpA1 b A2q. Therefore ZpA1q b ZpA2q Ă ZpA1 b A2q. Conversely, notice that if a1 b a2 “ 0, then we are done, so we suppose a1 b a2 P ZpA1 b A2qzt0u. If the LHS of the previous equation vanishes for each b1 b b2 P A1 bA2, so does for b1 “ 1; in which case, one gets a1 bra2, b2s “ 0 for each b2 P A2, so a2 P ZpA2q, since a1 ‰ 0 by assumption. Repeating the argument now taking b2 “ 1 instead, one gets ZpA1q b ZpA2q Ą ZpA1 b A2q. Proof of Lemma 5.6. Again, in the whole proof we set K “ 0, even though the notation will not reﬂect it. This can be obtained by small modiﬁcations from the previous lemma: if Kµ “ 0 for each µ, then

1 µˆ ωf pΨq “ arDf b 1F , a spΨq “ γ v b W rXµ,T sY b acψ µ µˆ ÿ “ γ b W rXµ,T s b ac Ψ µ ÿ ` µˆ ˘ “ γ b Sµ b ac Ψ µ ÿ ` ˘ and Sµ :“ W rXµ,T s. Since a and c are arbitrary matrices, we rename b “ ac. Again, since the

µˆ ˚ µˆ ˚ ˚ pγ b Sµ b bq “ eµˆγ b Sµ b b (A.2) ˚ and since we had set already b P i upnq we conclude that Sµ “ eµˆSµ. This sign is q`1 eµˆ “ p´1q eµ, according to [Pér19, App. A]. It follows from the deﬁnition, that the anti-linear operator C : V Ñ V satisﬁes CpγαγργσqC´1 “ CγαC´1 ¨ CγρC´1 ¨ σ ´1 α ρ σ µˆ ´1 µˆ Cγ C “ γ γ γ for each triple of indices α, ρ, σ P ∆4. Therefore Cγ C “ γ ´1 and the operator Jωf J can thus readily be computed: for Ψ “ v b Y b ψ P V b MN pCq b MnpCq, ´1 1 ´1 pJωf J qpΨq “ JarDf b 1F , a sJ pΨq

` µˆ ˘ ´1 ˚ ˚ “ pC b ˚N b ˚nqpγ C v b SµY b bψ q µ ÿ µˆ ´1 ˚ ˚ ˚ ˚ “ pCγ C vq b pSµY q b pbψ q µ ÿ γµˆ γlooomooonµˆ 1 1 Ψ 1 e S b “ p b MN pCq b MnpCqq p V b µˆ µ b q µ ÿ 1 q`1e γµˆ 1 1 Ψ 1 S b . “ pp´ q µ b MN pCq b MnpCqq p V b µ b q µ ÿ 38 CARLOS I. PEREZ-SANCHEZ

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