A new algebraic structure in the standard model of particle physics

We introduce a new formulation of the real-spectral-triple formalism in non-commutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent $A$ (the algebra of differential forms on some possibly-noncommutative space) on $H$ (the Hilbert space of spinors on that space), and to reinterpret this representation as a simple super-algebra $B=A\oplus H$ with even part $A$ and odd part $H$. $B$ is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional real-spectral-triple formalism of NCG are elegantly recovered from the simple requirement that $B$ should be a differential graded $\ast$-algebra (or"$\ast$-DGA"). Moreover, this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. In particular, these new constraints provide a novel interpretation of electroweak symmetry breaking that is geometric rather than dynamical. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.


Introduction
At low energies, the laws of physics appear to be accurately described by a particular effective field theory (EFT) -the so-called standard model of particle physics (coupled to Einstein gravity). But the EFT framework leaves certain basic questions about the standard model unanswered, and gives us less guidance than we would like about what might come beyond the standard model. In the EFT construction of the standard model, one specifies as input the basic symmetries of the theory, as well as the list of fundamental fields and how they transform under those symmetries. The dynamics are then described by the most general Lagrangian that is built from those fields and invariant under those symmetries. But what determines the symmetries -e.g. why is the standard model gauge group SU (3) × SU (2) × U (1)? What determines the basic list of fermions fields, and why do they transform in the particular representations (and with the particular charges) that they do? Similarly, what determines the basic list of scalar fields, their representations and charges? It is natural to look for a mathematical framework that is compatible with the EFT description of the standard model, but goes further in addressing some of these questions. A framework that is more restrictive than EFT -that only permits a subset of the theories that would seem valid from the EFT standpoint -could explain more about the standard model, and give more guidance about beyond-the-standard-model physics. This paper builds on earlier work on non-commutative geometry (NCG) [1][2][3][4] and its relationship [5][6][7][8][9][10][11][12][13][14][15][16][17] to the structure of the standard model (for a pedagogical introduction, see [18][19][20][21]). The idea, in brief, is that the observed structure of the standard model (coupled to Einstein gravity) may be reinterpreted as arising from the fact that the underlying spacetime is non-commutative (i.e. it is described by a certain kind of NCG, consisting of a 10=4+6 dimensional space, with four continuous/commutative "ordinary" dimensions, and six discrete/non-commutative "extra" dimensions of a certain type). And the claim is that this perspective captures or explains something important about the structure of the standard model that is missed in ordinary EFT. Much as Euclidean geometry allows one, given the first two angles of a triangle, to infer the third angle, the NCG approach allows one, given certain features of the particle content of a gauge theory, to infer other features of the particle content -features that would be completely independent inputs from the EFT standpoint. For example, in the NCG approach, once one chooses: (i) the gauge symmetry, (ii) the basic list of fermion fields in the theory, and (iii) the representations under which they transform, then (iv) the scalar fields in the theory and the representations under which they transform are determined (i.e. they are output, whereas in EFT they must be specified as additional independent input). Fundamentally this is because, in the NCG approach, the Higgs fields have a geometric meaning that puts them on the same footing as the gauge fields: the gauge and Higgs fields are two different pieces of the connection on the non-commutative space. Moreover, the fermion fields and their representations are much more restricted in NCG than in EFT [16,17,22]. Fundamentally this is because in EFT the fermions are governed by the representations of finite-dimensional Lie groups, while in NCG they are governed by the representations of finite-dimensional associative *algebras (which are much more restricted: e.g. the Lie group SU (N ) has an infinite number of different finite-dimensional irreducible representations, while the associative algebra of N ×N complex matrices M N (C) has only one or two irreducible representations, depending on whether we regard it as an algebra over C or over R -see e.g. [22]). As a simple example, in the NCG approach, the observed fact that all the fermions in the standard model transform in either the trivial or fundamental representation [of SU (2) and SU (3)] is an explained output, whereas in EFT this is an unexplained input.
In this paper, we introduce a new formulation of non-commutative geometry. We explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. Our approach, in brief, is as follows: (i) we start with a * -algebrâ A (the "algebra of coordinates"); (ii) we use it to define a related algebra A (the univer-sal * -algebra of forms overÂ); (iii) we take the simplest non-trivial graded representation of A on a Hilbert space H (i.e. we take H to be a graded space with just two non-zero components, H R and H L ); and (iv) we note that this representation may be reinterpreted as a super-algebra B = A ⊕ H, with even part A and (square-zero) odd part H. This super-algebra B is the fundamental object in our approach: we note that (nearly) all of the basic axioms and assumptions of the traditional ("spectral triple") formulation of noncommutative geometry may be elegantly recovered from the simple requirement that B should be a differential graded * -algebra (or " * -DGA"). But this requirement also yields other, new, geometrical constraints. When we apply our formalism to the spectral triple traditionally used to describe the geometry of the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. Thus, our proposal shares and even extends the key advantage of the traditional NCG reformulation of the standard model: that it is more restrictive and explanatory than the EFT framework (as sketched in the previous paragraph).
In a sense, our new proposal only differs from our earlier one [23,24] by a small change -namely, the fact that the representation of A on H is now appropriately graded -but the improvement is dramatic. Although (as reviewed in Section 6 below) our earlier proposal already had some of the nice features of the new one, it also had two key drawbacks (mentioned in [23], but emphasized and clarified in [25]). The first drawback was that, although A was a differential graded algebra, its extension to B was not (due to the presence of junk forms [2,3,26]). The second drawback was that, although our new "second-order condition" seemed to mesh very nicely with the six-dimensional discrete/non-commutative part of the standard model geometry, it was incompatible with the ordinary four-dimensional continuous/commutative part. In their paper [25], Brouder et al had an important insight -they pointed out that these two problems could be simultaneously resolved by taking an appropriately graded representation of A on H. Although the formulation we suggest here is very different than the one proposed by Brouder et al, their basic insight -that the representation of A on H should be appropriately graded -is still the key. In our proposal, we point out a different (more minimal) graded representation; and we find that many other nice results also fall neatly into place as a result.
The outline of this paper is as follows. In Section 2 we introduce the idea of a differential-graded * -algebra (or " * -DGA"). Although * -algebras and (non-commutative) DGAs have been widely studied on their own, the combined object (a non-commutative * -DGA) seems to have been studied less, and has some novel features that will play a key role in our analysis. We particularly direct the reader's attention to remarks (ii) , (v) and (v) , which do not seem widely known, and will be crucial in what follows. The goal of Section 3 is: first, to introduce Eilenberg's idea that the represention of an algebra A may be regarded as a new "Eilenberg algebra" B = A ⊕ H (a "square-zero extension" of A, and a particularly simple type of super-algebra); second, to explain how this idea naturally generalizes to representing * -algebras, DGAs, and * -DGAs; and third, to define the tensor product of two Eilenberg * -DGAs. In Section 4, we introduce a simple type of Eilenberg * -DGA B = A ⊕ H and explain its relevance to NCG. First (in Subsection 4.1), given a * -algebraÂ (the "algebra of coordinates"), we define a corresponding algebra A (the universal * -algebra of differential forms overÂ). Second (in Subsection 4.2), we take the simplest non-trivial graded representation of A on a Hilbert space H (i.e. we take H to be a graded space with just two non-zero components H = H L ⊕ H R ). Third (in Subsection 4.3), we show that the simple and natural requirement that B is an associative * -DGA elegantly unifies (nearly) all of the axioms and assumptions underlying the traditional (spectral triple) formulation of NCG; and, in addition, we show that this same requirement also yields other, new, geometric constraints (which do not appear in the traditional formulation of NCG). In Section 5, we apply our formalism to the particular geometric data traditionally used to describe the standard model in NCG; and we show that our new geometric constraints correspond to physically meaningful and phenomenologically correct constraints (which thereby represent an improvement over the traditional spectral triple formulation of the standard model). We also discuss how our formalism connects to the issues of quantization and lorentzianversus-euclidean signature. In Section 6, we mention some phenomenological consequences of our previous formulation that carry over to the new formulation, and we mention some interesting directions for future work.
Although * -algebras and (non-commutative) DGAs have been widely studied on their own, the combined object (a non-commutative * -DGA) seems to have been studied less, and has some novel features that will play a key role in our analysis. We particularly direct the reader's attention to remarks (ii) , (v) and (v) , which do not seem widely known, and will be crucial in what follows.
(i) An algebra A (over a field F) is a vector space (over F) equipped with an F-bilinear product aa ∈ A (for a, a ∈ A). A is called "commutative" if the "commutator" [a, a ] ≡ aa −a a vanishes for all a, a ∈ A; and "associative" if the "associator" [a, a , a ] ≡ (aa )a − a(a a ) vanishes for all a, a , a ∈ A. (Note that we do not assume either commutativity or associativity of A in this section.) (ii) A * -algebra is an algebra A (over F) that is equipped with an additional structure: a * operation. By a * operation, we mean an anti-automorphism from A to A -i.e. an invertible F-anti-linear map from A to A that is an anti-homomorphism: (2.1) and also has the property that (a * ) * is proportional to a: For example: the complex numbers C are a commutative * -algebra (over R), where the * operation is complex conjugation z →z; and n × n complex matrices M n (C) are a non-commutative * -algebra (over C), where the * operation is the conjugate transpose ( †) operation m → m † .
(ii) Ordinarily, we can then use the argument aa = ((aa ) * ) * = (a * a * ) * = (a * ) * (a * ) * = 2 aa to determine that = +1, so that * becomes an involution, (a * ) * = a; but, as we will see in the Subsection 3.2, this argument fails in the nilpotent sector of an Eilenberg algebra, and the more general possibility = ±1 is allowed. 1 In the general case where (a * ) * = a, we should be careful to distinguish between the operation " * " and its inverse " * ".
(iii) A graded algebra A is an algebra that decomposes into subspaces, A = ⊕ m A m , where the product respects the decomposition: a m ∈ A m , a n ∈ A n ⇒ a m a n ∈ A m+n .
(iv) A differential graded algebra (or "DGA") is a graded algebra A that is also equipped with a left-differential d L : a linear map from A m to A m+1 that is nilpotent and satisfies the left-Leibniz rule (see Appendix A.1) d L [a m a n ] = d L [a m ]a n + (−1) m a m d L [a n ] (a m ∈ A m , a n ∈ A n ). (2.3b) An example is the exterior algebra of differential forms: it is graded, since an m-form wedged with an n-form is an (m + n)-form, and it is equipped with a differential: the usual exterior derivative d on differential forms.
(v) A differential graded * -algebra (or " * -DGA") is a DGA that is also a * -algebra (with the properties listed above). In particular, Eq. (2.1) becomes (see Appendix A.1) (a m a n ) * = a * n a * m . (v) A * -DGA comes equipped with a left-differential d L and a * -operation; and from these, it is natural to also construct the right-differential d R : (i.e. " * " composed with d L composed with "inverse * "). Note that, since d L is nilpotent (2.3a) and satisfies the left-Leibniz rule (2.3b), it follows that d R is also nilpotent: 1 We have chosen the name for this ± sign because, as we will see in Section 4, it is linked to the ± sign that appears in the standard definition of a real spectral triple in NCG: J 2 = (see [20,21]). 2 We have chosen the name for this ± sign because, as we will see in Section 4, for the models we consider it is linked to the sign that appears in the standard definition of a real even spectral triple in NCG: Jγ = γJ (see [20,21]). Care should be taken however, as this correspondence is not true in general, and only holds for NCG models in which the Z2 grading of an even spectral triple is identified with the differential grading of its corresponding Eilenberg * -DGA (as explained below in Sections 3 and 4). and satisfies the right-Leibniz rule (see Appendix A.1) d R [a m a n ] = a m d R [a n ] + (−1) n d R [a m ]a n . (2.6b) Note that d R maps m-forms to (m + ) forms: in other words, when = +1, it is a right-differential in the usual sense, but when = −1, it is a right-differential with respect to the inverted grading. Also note the following identities which follow from the definition and which say that a * operation which appears inside the argument of a differential may be "pulled outside" at the cost of swapping d L ↔ d R . (a m ⊗ a n )(a p ⊗ a q ) = (−1) np a m a p ⊗ a n a q , (2.8) the * operation on A is given by (see Appendix A.2): (a m ⊗ a n ) * = (−1) mn a m * ⊗ a n * , (2.9) and the differential d is given by: where 1 and 1 denote the identity operators on A and A , respectively.

Representing * -DGAs
The goal of this section is to introduce Eilenberg's idea that the represention of an algebra A may be regarded as a new "Eilenberg algebra" (a particular type of super-algebra with A as its even part); to explain how this idea naturally generalizes to representing * -algebras, DGAs, and * -DGAs; and to define the tensor product of two Eilenberg * -DGAs.

Representing algebras (with Eilenberg algebras)
In this subsection, we introduce Eilenberg's perspective on representing an algebra A (via an associated super-algebra B). Let A be an algebra (over F), and let H be a vector space (over F); following Eilenberg and Schafer [27,28], we define a bi-representation R of A on H (or, equivalently, a bi-module H over A) as a pair of F-bilinear products ah ∈ H and ha ∈ H (a ∈ A, h ∈ H). Now notice that this definition of a bi-representation of A on H (or, equivalently, of a bi-module H over A) is equivalent to the definition of a new algebra (over F) with the product between elements of B (b = a + h and b = a + h ) given by where aa ∈ A is the product inherited from A, while ah ∈ H and ha ∈ H are the products inherited from R, and hh = 0. We will call such an algebra an "Eilenberg algebra." 3 Also notice that the algebra B defined this way is automatically a superalgebra -i.e. a Z 2 -graded algebra, with "even" and "odd" subspaces A and H, respectively.
We stress that, so far, we have not assumed anything about the associativity of A or B. On the one hand, if we now assume that B is associative, then (as explained in the following paragraph) we precisely recover the traditional associative definition of the (left-, right-, or bi-)representation of A on H. But, on the other hand, we need not necessarily assume that B is associative: for example, if A is a Jordan algebra (an important type of non-associative algebra), then it is natural to define its representation on H by taking B to also be a Jordan algebra [27,28,30]. In fact, this is what originally led us to adopt Eilenberg's perspective in [23,31]: it is a way of defining the representation of A on H that naturally generalizes from non-commutative geometry (where the algebra of coordinates, may be non-commutative) to non-associative geometry (where the algebra of coordinates may also be non-associative).
Let us now explain our assertion (from the previous paragraph) that if we assume B is associative, then we precisely recover the traditional definition of an associative representation of A on H. If B is associative, all the associators [b, b , b ] must vanish. This implies four non-trivial constraints: [a, a , a ] = 0, while the remaining associators (in which two or three arguments are from H) vanish trivially because hh = 0. Note that (3.3a) is simply the requirement that A itself is associative; (3.3b) says that ah is a traditional associative left-representation of A on H; (3.3c) says that ha is a traditional associative right-representation of A on H; and (3.3d) says that the left-and right-representations commute with each other. In other words, we recover the traditional associative definition of a left-right bi-representation of A on H (or, equivalently, the traditional associative definition of a left-right bi-module H over A); and the special cases of a left-representation (left-module) or right-representation (rightmodule) are recovered, respectively, when either the right action ha or the left action ah vanishes identically.

Representing * -algebras (with Eilenberg * -algebras)
In this subsection, we explain how to extend Eilenberg's idea from algebras to * -algebras: one simply requires that B itself is a * -algebra. We will see how many of the traditional axioms/assumptions of NCG follow from this requirement.
To extend Eilenberg's construction from algebras to * -algebras, we must promote B from an algebra to a * -algebra. The * operation on B should have all the properties explained in Section 2, plus one more: compatibility with the * operation on the subalgebra A ⊂ B. This together with (2.2) implies compatibility with the intrinsic Z 2 grading on B (i.e. b * ∈ A when b ∈ A, and b * ∈ H when b ∈ H), and fixes the * operation to be for a ∈ A, h ∈ H, and where a * is the * -operation on A, while J is an invertible anti-linear operator on H. We will call such an algebra an "Eilenberg * -algebra." In Section 2, we took the * -algebra A to satisfy (a * ) * = a (and then derived = 1); but for an Eilenberg super-algebra we should allow a different constant in the even and odd sectors: (a * ) * = 0 a and (h * ) * = 1 h. Then, as in Section 2, the argument 0 aa = ((aa ) . If the * operation still has some finite period (i.e. * n = 1 for some finite n): it follows that 1 is a root of unity; but then, the From now on, to match standard NCG notation, let us drop the subscript "1" and simply refer to 1 as " ." We thus recover the standard NCG axiom J 2 = , where = ±1.
The fact that B is a * -algebra thus implies and unifies four traditionally-assumed facts about NCG, including: (i) that A is a * -algebra; (ii) that H is equipped with an invertible anti-linear operator J; and (iii) that J 2 = where = ±1. In addition, (iv) the antihomomorphism property (bb ) * = b * b * implies (ah) * = h * a * and (ha) * = a * h * , which then implies that A is not just left-represented or right-represented on H, but left-right bi-represented on H, with the left and right representations related by Finally, in the NCG context, H will be a Hilbert space, so compatibility with the inner product on H will also require J to be anti-unitary (3.6)

Representing DGAs (with Eilenberg DGAs)
In this subsection, we explain how to extend Eilenberg's idea from algebras to DGAs: one simply requires that B itself is a DGA. Let us proceed in two steps: (i) first, we define graded bi-representations (or, equivalently, graded bi-modules), and (ii) second we define differential graded bi-representations (or, equivalently, differential graded bi-modules).
(i) Suppose A is a graded algebra (over F) with grading A = ⊕ m A m , and H is a vector space (over F) with grading H = ⊕ m H m . As the natural extension of Eilenberg's preceding definition of a bi-representation (or bi-module), let us define a graded bi-representation R of A on H (or, equivalently, a graded bi-module H over A) as a pair of F-bilinear products ah ∈ H and ha ∈ H that respect the grading: (ii) Next let A be a DGA, with grading A = ⊕ m A m and differential d; and let H be a vector space with grading H = ⊕ m H m . Then we will say that a graded bi-representation R of A on H is a differential graded bi-representation of A on H (or, equivalently, a differential graded bi-module H over A) if H is also equipped with its own differential d: i.e. a linear operator from H m to H m+1 that is nilpotent and satisfies the graded Leibniz conditions for a m ∈ A m and h n ∈ H n . With these definitions, we now notice that the definition of a graded bi-representation of the graded algebra A on H (or, equivalently, the definition of a graded bi-module H over A) is simply equivalent to the definition of the algebra B given above, together with the condition that B is graded as follows: [where in the applications to NCG in Section 4, A m will only be non-zero for integer values of m or some subset thereof, while H m will only be non-zero for m = {0, 1} (when = 1) or for m = ± 1 2 (when = −1)]. And the definition of a differential graded bi-representation of the DGA A on H (or, equivalently, the definition of a differential graded bi-module H over A) is simply equivalent to the requirement that the algebra B is itself a DGA with respect to the grading (3.10). We will call such an algebra B an "Eilenberg DGA. " We also notice that the algebra B is now graded in two different ways or, more precisely, it is graded over the ring Z × Z 2 . In other words, it has its new grading m (A m ⊕ H m ) over Z; but, in addition, it still has another independent Z 2 grading that splits it into an even part A and an odd part H, thereby making it a super-algebra (or, in this case, a super-DGA).
We again stress that, so far in this subsection, we have not assumed anything about the associativity of A or B. If we now assume that B is associative, then we precisely recover the traditional associative definition of a (differential) graded representation of the (differential) graded algebra A on H. But, as discussed above, we need not necessarily assume that B is associative; and for some purposes, the fact that this approach to representing a (differential) graded algebra naturally generalizes to the non-associative case may be crucial. 4 3.4 Representing * -DGAs (with Eilenberg * -DGAs) Finally, in this subsection, we explain how to extend Eilenberg's idea from algebras to * -DGAs: one simply requires that B itself is a * -DGA.
Having laid all the groundwork in the previous three subsections, there is little to be added here. A differential graded * -representation of A on H (or, equivalently, a differential graded * -module H over A) is simultaneously a * -representation ( * -module) in the sense of Subsection 3.2, and a differential graded representation (differential graded module) in the sense of Subsection 3.3. And, as before, these conditions may be succinctly summarized by saying that the algebra B defined above is a * -DGA (and, moreover, a super- * -algebra, because of its intrinsic Z 2 grading). We will call such an algebra an "Eilenberg * -DGA."

Tensoring two Eilenberg * -DGAs
We have seen how a * -DGA A may be represented by an Eilenberg * -DGA B = A ⊕ H; and we would now like to define the tensor product of two such Eilenberg * -DGAs. At the end of Section 2, we explained how to take the tensor product of two generic * -DGAs A and A to obtain a new * -DGA A. If we directly apply this construction to two Eilenberg DGAs

Application to NCG
In the previous section we explained, in general terms, how to extend Eilenberg's approach: from representing algebras to representing * -DGAs. In this section, our goal is to describe a simple type of Eilenberg * -DGA B = A ⊕ H: first, given a * -algebraÂ (the "algebra of coordinates"), we define A as the universal * -algebra of differential forms overÂ; and then, to complete the definition of B, we take H to be the simplest possible non-trivially graded space -a space with just two components H = H L ⊕ H R -and follow this idea where it leads. We find that this construction does a remarkably good job of unifying and explaining many aspects of the traditional NCG formalism 5 , on the one hand, and resolving key problems/puzzles in the traditional NCG construction of the standard model of particle physics, on the other. So far in this paper, we have not assumed associativity, since we want to emphasize that one of the features of our formalism is that it retains the advantage of Eilenberg's perspective -i.e. it naturally lends itself to generalization, from the associative to the non-associative case. This is surely an interesting direction for future work. But for the rest of this paper, we will restrict to the associative case -i.e. the case where B is associative (and hence A andÂ are also associative -since our main goal in the remainder of the paper will be to show how we can thereby neatly unify and illuminate the traditional axioms of traditional (associative) NCG and also fix several problems in the traditional NCG construction of the standard model.

The * -DGA A
As in the traditional formulation of NCG, we start by choosing a (possibly non-commutative) * -algebraÂ (over F): roughly, this may be thought of as the algebra of coordinates. We can then define A, the universal * -algebra of forms overÂ, as follows: For every elementâ ∈Â, let us introduce a corresponding formal symbol d[â] which has the following familiar linearity and Leibniz properties: We can regardâ as a zeroform, d[â] as a one-form, and d[â] * as an -form (i.e. as a one-form or a minus-one-form, depending on the sign of -see Section 2). Next consider an arbitrary term constructed by taking a product ofâ's, d[â]'s and d[â] * 's: to take a rather complicated example, consider (5) ]â (6) , whereâ (1) , . . . ,â (6) ∈Â: note that this example is a 3-form if = +1, or a 1-form if = −1. We can take the product of two such terms by simply juxtaposing them, and using the product inherited fromÂ: e.g., the product of the 1-form a 1 = d[â (1) ]â (2) and the one-form a 1 =â (3) d[â (4) ] is the 2-form a 2 = a 1 a 1 = d[â (1) ]â (5) d[â (4) ], where we have used the productâ (2)â(3) =â (5) inÂ. The algebra A is obtained by considering all such terms (and all linear combinations of such terms over F), with the product just defined. Note that this algebra is automatically graded, A = ⊕ m A m , where A m is the space of m-forms. The * operation onÂ extends to a * operation on A in the natural way (by recursive application of the rules described in Section 2): so, for example, (λ d[â (1) ]â (2) * d[â (3) ] * â(4) ) * =λâ (4) * d[â (3) ]â (2) d[â (1) ] * . We can also extend d[. . .] to a differential on A in the natural way by recursive application of the graded Leibniz rule d[a m a n ] = d[a m ]a n + (−1) m a m d[a n ] (a m ∈ A m , a n ∈ A n ), along with the definitions (4) ] * . In short: A, the universal * -algebra of differential forms overÂ, is the algebra generated byÂ, d[Â ] and d[Â ] * , modulo the various relations described above.
Note that this "universal * -algebra of differential forms overÂ" is similar to the more familiar "universal algebra of differential forms overÂ" (presented e.g. in [2]). But, as indicated by the differerence in name, there is an important difference between these two algebras, coming from the way the * operation is handled: note, in particular, that we do not add the extra relation d

The * -DGA B = A ⊕ H
Now let us take H to be the simplest possible graded space, with just two non-vanishing components which we can call H R and H L (for "right" and "left"): (4.1) Without loss of generality, we can take H R to be the component of lower grading while H L is the component of higher grading. Note that, when = +1, we should take H R and H L to have ordinary integer gradings (0 and 1, respectively, like the two states in a fermionic fock space), but when = −1, we should take H R and H L to have half-integer gradings (−1/2 and +1/2, respectively, like the two states of a spin-1/2 particle).
As explained in Subsection 3.3, since B is a DGA, its differential d must map m-forms in H to (m + 1)-forms in H and so, in particular, must map H R to H L , and H L to 0. Thus, for h ∈ H, we can write where d H is a linear operator on H that, in the {H L , H R } basis, is only non-zero in its upper off-diagonal block: In the previous section, we saw that d 2 vanishes on A; and now, since H has only two components, it immediately follows that d 2 also vanishes on H, and hence on the whole of B (as is required in order for B to be a DGA). So far we have specified A and H; but to specify the algebra B = A ⊕ H, we must also specify the product between A and H. Actually, we only need to specify the product ah -i.e. the left-action of a zero-formâ ∈Â on an element h ∈ H -since (as we shall see) the remaining products a m h and ha m (the left-action or right-action of an arbitrary m-form a m ∈ A m on an element h ∈ H) are then determined by the general structure of an  [Note that, in the NCG literature, the operators Lâ and Râ (the left and right representations ofâ on H) are sometimes denoted π(â) and π(â) 0 , respectively.] As explained in Subsection 3.2, the fact that B is a * -algebra then implies that H is equipped with an invertible anti-linear operator J (h * = Jh) and, moreover, that the right-representation Râ is determined by the left-representation Lâ via the familiar NCG formula Râ = JLâ * J −1 . (4.5) (ii) Next we consider the left-and right-action of d[â] on H: where by |h| we mean the order of the Hilbert space element h, according to the grading: i.e. |h| is 0 or 1 when = +1, and |h| is ±1/2 when = −1. Note that (−1) |h| is an operator, not just a number, so care must be taken with regards to its position in Eq. (4.7b).
(iii) Finally we consider the left-and right-action of d[â] * on H: Since B is a * -algebra, we can use Eq. Let us end this subsection with two remarks: • First note that L a * = L † a and R a * = R † a for zero forms; but L d This will be important below.

Comparison with traditional NCG formalism
Now that we have introduced the algebra B, we would like to see which aspects of the traditional NCG formalism follow from the requirement that B is an associative Eilenberg * -DGA.
To facilitate the comparison, it will be convenient to introduce two important operators on H that we have not discussed yet : D and γ.
(i) Introducing D. From the operator d H (which is not hermitian), we can construct another operator that is Hermitian: D † = D. D is the generalized Dirac operator that appears in the traditional NCG spectral triple {Â, H, D}.
(ii) Introducing γ. Since H is graded, with two parts, we are free to define a corresponding operator on H which detects this grading. To match the usual NCG notation, we do this by defining an operator γ that equals −1 on H L and +1 on H R ; so it is block diagonal in the {H L , H R } basis, and given by Now, let us see which of the traditional NCG axioms and assumptions follow from the fact that B is associative. Associativity of B is equivalent to the requirement that the associator [b, b , b ] vanishes, for all b, b , b ∈ B. Thus, let us list all the different types of associators that arise in this way, and the consequences of requiring them to vanish.
• From [â,â ,â ] we recover the usual assumption that the coordinate algebraÂ (i.e. the algebra that appears in the traditional spectral triple {Â, H, D}) is associative; • From [a, a , a ] we recover the usual assumption that A, the universal algebra of forms overÂ, is associative.
• From [â, h,â ] we recover [Lâ, Râ ] = 0 or, in other words, [π(â), π(â ) 0 ] = 0, which is the traditional "order zero" axiom of NCG. Note that, unlike the previous associators, which all corresponded to traditional NCG axioms and assumptions, these final two associators correspond to new "second-order conditions" which are not normally imposed as axioms in NCG. These new secondorder conditions were first pointed out in [23]; but the fact that they are only nontrivial when = −1 is one of the most important new results in the present paper, and is important for resolving a key puzzle in the NCG construction of the standard model of particle physics. (We will return to this point in the following section.) Next, as explained in Subsection 3.2, from the assumption that B is a * -algebra we recover: • the assumption thatÂ and A are * -algebras; • the assumption that H is equipped with an invertible anti-linear operator J; • the assumption that J 2 = with = ±1; and • the assumption thatÂ is not just left-represented or right-represented on H, but rather left-right bi-represented on H, with the right representation related to the left representation by the familiar NCG formula (4.5).
Finally, let us see what follows from the fact that B is * -DGA (i.e. from including the differential-graded structure of B): • From Eqs. (4.12, 4.13), we recover the assumption that {D, γ} = 0 (which, from our new perspective, follows from the fact that d is an order-one operator on H).
• From the fact that, as explained in point (v) of Section 2 the * -operation must map m-forms to ( m)-forms, we recover the assumption that Jγ = γJ where = ±1 is the * -DGA sign choice explained in Section 2.
Note that, unlike the previous conditions which all corresponded to traditional NCG axioms and assumptions, this final condition corresponds to a new condition (like the new second-order condition described above); and (again like the second order condition) this new condition is only non-trivial when = −1.
So far in this subsection, we have focused on explaining how the requirement B is an associative * -DGA unifies and explains a long list of the traditional axioms and assumptions of NCG. Let us end this Subsection by summarizing a few of the key ways that formalism we have set up differs from traditional NCG: • We obtain another novel constraint d[d[â * ] = 0 which is only non-trivial when = −1.
In the following section, we will see how these differences resolve several key puzzles in the NCG formulation of the standard model of particle physics.

Application to the Standard Model
In the previous section, we defined a particularly simple type of Eilenberg algebra B = A ⊕ H, where A was the universal * -algebra of differential forms over some "algebra of coordinates"Â, and H was the simplest non-trivially graded space (with just two components, H R and H L ); and we explained how the traditional list of NCG axioms and assumptions could, to a remarkable degree, be unified in the requirement that B was an associative * -DGA. In order to specify a particular B of this type, it just remains to choose the following three final inputs: (i) a particular algebra of coordinatesÂ; (ii) a particular left-representation ofÂ on H; and (iii) particular values for the ± signs and .
In this section, we choose these final inputs to be the ones used in the traditional NCG description of the standard model, and then investigate the consequences. In the traditional approach, the geometry of the standard model is, itself, the product of two geometries: (i) one that describes an ordinary, continuous, commutative four-dimensional spacetime; and (ii) another that describes a finite, discrete, non-commutative "internal space". In our language, this corresponds to an Eilenberg algebra B that is, itself, the product of two Eilenberg algebras B c and B f . We discuss these two algebras in turn -for each, we begin by listing the three final inputs mentioned in the previous paragraph.
For a pedagogical introduction to the traditional NCG construction of the standard model, see [20,21]. Note that in this section we discuss the geometry of the standard model. Traditionally one then uses the so-called "spectral action" formula to convert this geometry into an ordinary field theory Lagrangian (which, for this particular choice of geometry, turns out to be the Lagrangian for the standard model of particle physics coupled to Einstein gravity). For an introduction to this conversion process (which we do not discuss here), see [20]. Also note that, in what follows, it will be convenient to discuss the continuous and finite parts of the geometry separately (since one can read off the physical content of the geometry by examining these two parts separately); but it is important to understand that, in reality, the physical fields of the theory (and, in particular, the gauge fields and Higgs fields) arise from the interaction between these two parts (again see [20] for an introduction).

The continuous geometry, B c
The continuous geometry is described by the so-called "canonical spectral triple" corresponding to ordinary (commutative) four-dimensional Riemannian geometry (see Section 2.1 in [20]). In this case, the operator D (4.12) is just the ordinary curved-space Dirac operator where ∇ S µ is the Levi-Civita spin connection and the γ µ are the gamma matrices. In the previous section we showed that our formalism predicts new geometrical constraints, in addition to those of the traditional NCG formalism -but these new constraints are only non-trivial when = −1. As a result, since the continuous geometry described above has = +1, it only needs to satisfy the traditional NCG constraints (which it does), and it does not need to satisfy any additional second-order constraint.
This neatly resolves a key puzzle that was noted at the end of our earlier paper [23] and emphasized in [25]. In our earlier construction [23], the second-order condition gave a nontrivial constraint on both the finite and continuous parts of the geometry; and, although the constraint on the finite part was successful (in that it neatly removed all of the unwanted terms in the finite Dirac operator), the constraint on the continuous part was too strong. By constrast, here we see that the second-order condition only gives a non-trivial constraint in the = −1 case (so it still gives a successful non-trivial constraint on the finite geometry, which has = −1, but it gives no additional constraint on the continuous geometry, which has = +1, and hence is perfectly compatible with it).

The finite geometry, B f
The discrete "internal space" is described by a finite spectral triple (with a finite-dimensional algebra, represented on a finite-dimensional Hilbert space) -see Section 6 in [20].
(i) In this case, the algebra of coordinatesÂ is the direct sum of the complex numbers C, the quaternions H and the 3 × 3 complex matrices M 3 (C): (ii) Next, we describe the left-representation ofÂ on H. For clarity, we will describe a single generation of standard model fermions; the extension to the full set of three generations is straightforward. The Hilbert space H is C 32 (here 32 is the number of fermionic degrees of freedom in a single standard model generation, if we include a right-handed neutrino in each generation to account for the observed neutrino masses). The left-representation of A on H is block-diagonal: let us start by giving physically-appropriate names to the corresponding subspaces of H on which these blocks act. For starters, as explained in Subsection 4.2, the grading splits C 32 into two copies of C 16 : H = H L ⊕ H R . Next, each C 16 splits into two copies of C 8 : H L =F R ⊕ F L and H R = F R ⊕F L (here the subspaces F L and F R contain the left-and right-handed fermions, whileF L andF R contain the corresponding anti-fermions). Finally, each C 8 splits into a lepton part (C 2 ) and a quark part (C 2 ⊗ C 3 ): Now, if we consider an arbitrary elementâ = (λ, q, m) ∈Â, where λ ∈ C is a complex number, q ∈ H is a quaternion, and m ∈ M 3 (C) is a 3 × 3 complex matrix, and we write q = α β −βᾱ and q λ = λ 0 0λ (5.4) where α and β are complex numbers, then (in the {L R ,Q R , L L , Q L , L R , Q R ,L L ,Q L } basis just described) the left-action ofâ on H has the following block diagonal form where I 2 and I 3 denote the 2 × 2 and 3 × 3 identity matrices, respectively. (iii) In this case, the space has KO dimension 6: the corresponding signs are = +1 and = −1 (along with DJ = JD, again see the table in Section 2.2.2 of [20]). In the basis just described, J and γ are then given by • c.c. and γ = −I 0 0 +I (5.6) where I is the 16 × 16 identity matrix, and "c.c." stands for "complex conjugation" -a reminder that J is anti-linear. From the traditional NCG conditions D F = D † F , {D F , γ F } = 0, [D F , J F ] = 0 and the order one condition, one finds that D F is constrained to the form where ∆ is a 16 × 16 symmetric matrix of the form Here Y l and Y q are two arbitrary 2 × 2 matrices, M is a 2 × 2 symmetric matrix given by where a and b are arbitary constants, and N is a 6 × 2 matrix given by where c and d are two arbitrary 3 × 1 column vectors. As reviewed in Ref. [23], these traditional NCG constraints are not strong enoughi.e. they do not constrain D to be of the phenomenologically desired form. Instead, they allow non-zero values for the parameters b, c and d, which give rise to phenomenologically unwanted Yukawa couplings and scalar fields if they are not eliminated. This problem was highlighted in [15] and the concluding section of [16]. In the traditional formalism, one is forced to introduce an extra (empirically-motivated, non-geometric) condition (called the "massless photon" condition [15,16]) in order to eliminate these extra, unwanted terms.
In our formalism, something nice happens instead. As explained in Subection 4.3, when = −1 (as is the case for the finite geometry), we obtain non-trivial "second- These second-order conditions yield exactly the same restrictions on D F that we previously obtained from the second-order conditions in Ref. [23] (or that Brouder et al later obtained from the second-order conditions in [25]). In particular, as shown in [23], these second-order constraints may be satisfied in four different ways by setting (i) b = c = d = 0; (ii) Y q,11 = Y q,21 = b = 0; (iii) Y l,11 = Y l,21 = c = d = 0; or (iv) Y l,11 = Y l,21 = Y q,11 = Y q,21 = c = 0. In particular, solution (i) precisely corresponds to setting the seven unwanted coeffients (b, c, d) to zero, without having to introduce the extra non-geometrical massless photon condition.
Thus, our current formalism neatly resolves the paradox in our earlier paper [23]: that our new second-order constraint seemed to provide such a phenomenologically successful constraint on the finite part of the geometry, while at the same time providing an unwanted over-constraint on the continuous part of the geometry. In our new formalism, the successful constraint on the finite part of the geometry is retained, while the overconstraint on the continuous part is eliminated. This resolution is satisfying, as it directly follows from thinking more clearly about the basic structure of a * -DGA -see Remark (v)' in Section 2.

Some interesting points
In the remainder of this section, we assume that the operator {d H , d † H } (or equivalently the operator D 2 ) is generic -in other words, it is non-singular, with non-vanishing determinant.
Note that this is enough to select the first option (i.e. the phenomenologically desired option) from among the four solutions of the second-order constraint listed at the end of Subsection 5.2; and it is also implies that the quark and lepton Yukawa matrices Y q and Y l are each generic (i.e. have non-vanishing determinant).
Note the strong parallel between the following two pairs of operators: (i) the operators d H and d † H (which generate the up and down transitions between the two levels of a fermionic Hilbert space and satisfy , and (ii) the usual fermionic annihilation and creation operators a and a † (which also generate the up and down transitions between the two levels of a fermionic Hilbert space and satisfy {a, a} = {a † , a † } = 0). The non-degeneracy of {d H , d † H } (which, as we have just seen, selects the phenomenologically correct solution from among the four solutions of the second-order constraint) is then the analogue of the usual condition {a, a † } = 1. This seems to be a contact point between geometry and quantization that deserves further thought and attention.
Note that the standard model geometry as we have described it looks like a real algebra represented on a complex Hilbert space. Really, we should interpret this as short-hand for a real algebra represented on a real Hilbert space of double the dimension (and this is also what we must do if we want to regard A and H as two subspaces of a common vector space B = A ⊕ H). This has an awkward consequence: in Section 4, we argued that the antilinearity of J (over C) could be derived from the fact that B is a * -algebra; but if B is a * -algebra over R, then this is no longer true, and the anti-linearity of J (over C) must be put in by hand. We regard this awkward point as a clue that there is something important that remains to be understood here: we suspect that ironing out this wrinkle is connected, on the one hand, with properly analyzing the geometry as a whole (rather than treating the finite and continuous parts separately); and, on the other hand, with properly handling the connection between lorenzian and euclidean signature. Indeed, in [13], Barrett showed that by thinking in Lorentzian signature, one could argue that the dimension and signature of the full geometry should be 10 and 8, respectively (mod 8), precisely because in this case the spinors could be taken to be simultaneously weyl (as we need for our graded representation) and majorana (i.e. real, which would alleviate the awkwardness mentioned above). It would, of course, be very satisfying if the dimension and signature of spacetime were actually determined by the requirement that A and H both be real, so that they can be naturally combined into B. We regard the sorting out of this seemingly detailed point as an important topic for future work. Now, as explained in Subection 4.3, when = −1 (as is the case for the finite geometry), in addition to the second-order condition, we also obtain another new type of constraint: If we look back at the construction of the algebra A from the coordinate algebraÂ in Subsection 4.1, we see that this constraint has a differerent logical status from the other constraints that we have discussed: it is not actually needed for the definition, construction or consistency of the algebra A; it is merely needed if we want to formally "extend" d fromÂ to A (i.e. from having its argument be any element ofÂ to having its argument be any element of A), so that we can reinterpret the graded * -algebra A as a * -DGA. With this in mind, note that when we apply the constraint L d[d[â] * ] = {d H , L d[â] * } = 0 to the finite geometry, it translates into: (These same constraints apply to both the quark and lepton Yukawa matrices, Y q and Y l .) Eqs. (5.11) are only satisfied non-trivially (i.e. for q and λ non-zero and anti-hermitian) if the Yukawa couplings are of the form (note the sign change compared to the convention we used in [23]): where {y ν , y e , y u , y d } are the Yukawa couplings for the neutrino, electron, up quark and down quark, and Φ = {φ 1 , φ 2 } is the Higgs doublet. 7 We note that the same Higgs doublet appears in both Y l and Y q : this is a consequence of the fact that Eqs. (5.11) must be satisfied for given values of q and λ, regardless of whether Y is Y l or Y q . If we include all three generations, Y l and Y q still have the form shown in Eq. (5.12): y ν , y e , y u and y d become 3 × 3 Yukawa coupling matrices, but there is still just a single Higgs doublet. Then, to see the physical meaning of Eqs. (5.11), first note that they reduce to If we recall that the inner derivations ofÂ generate the gauge transformations of the resulting physical theory, we see that Eq. (5.13) is nothing but the usual equation that determines which linear combination of the electroweak SU (2) × U (1) generators remains unbroken by a given Higgs expectation value (see e.g. Section 2.5 in Ch. 5 of [32])! In other words, imposing the condition L d[d[a] * ] = 0 seems to correctly describe electroweak symmetry breaking in the NCG SM. We find this new geometric encoding of electroweak symmetry breaking to be remarkable.

Discussion
Let us briefly recap. We start by (i) choosing an "algebra of coordinates"Â; (ii) defining A, the corresponding universal * -algebra of differential forms over A; and (iii) taking the simplest non-trivially graded representation of A on H (i.e. the one where H has just two non-zero components, H L and H R ). As we explain in Section 4, nearly all of the axioms and assumptions of the traditional (spectral triple) formulation of NCG are then recovered from the simple requirement that the corresponding Eilenberg algebra B = A ⊕ H (a particularly simple type of super-algebra) is a * -DGA. But this requirement also implies other, novel, geometric constraints. As we explain in Section 5, when we apply this new formalism to the specific NCG data traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct.
Our new formalism improves on our earlier framework [23,24] in a number of important respects. From a mathematical standpoint, our earlier definition of B in [23,24] already unified some of the NCG axioms and assumptions, but now we are able to go much further: roughly speaking, the earlier definition of B unified the axioms and assumptions that did not involve γ, while the new definition also incorporates those that do involve γ. It is encouraging that this improved unification goes hand-in-hand with the improved structure of B which (with its new definition) is now a proper * -DGA. From a physical standpoint, our new formalism resolves a paradox which arose in our earlier work [23,24]: namely, in our earlier formalism, the second-order condition (when applied to the standard model geometry) seemed to give a phenomenologically successful constraint on the finite part of the geometry, but a problematic over-constraint on the continuous part of the geometry. In our current formalism, the unwanted over-constraint on the continuous geometry is automatically eliminated (as explained in Subsection 5.1), while the successful constraint on the finite geometry is automatically retained (as explained in Subsection 5.2). It is satisfying that this resolution follows from thinking more clearly about the basic structure of a * -DGA -see Remark (v)' in Section 2. As explained in Subsection 5.3, our new formalism also, on the one hand, suggests an interesting connection between geometry and quantization and, on the other hand, yields another new constraint that turns out to correspond to a geometric re-encoding of the structure of electroweak symmetry breaking.
Our new formalism also inherits a few other nice features from our earlier work: (i) first, the conceptually nice reinterpretation of the symmetries of the standard model, and the structure of the gauge-Higgs sector, as arising from the requirement that the action should be invariant under automorphisms of B (see [24]); and (ii) second, the corresponding implication that the traditional "standard model geometry" actually yields a slight extension of the standard model which, in addition to including a right-handed partner for each left-handed neutrino, also includes an extra U (1) B−L gauge symmetry and, correspondingly, two new particles: a new U (1) B−L gauge boson, and a new complex scalar field σ that is a singlet under the standard model gauge group SU (3) × SU (2) × U (1), but is charged under the new U (1) B−L , and is responsible for Higgsing this symmetry (so that it is unseen at low energies). As emphasized in [24], this extension of the standard model is phenomenologically viable, and resolves the discrepancy between the traditional NCG Higgs mass prediction (∼ 170 GeV) with that of the observed Higgs mass (∼ 125 GeV), and can account for several cosmological observations that cannot be accounted for by the standard model alone [33].
There are many possible directions for future work. (i) First, as noted in [25], the DGA structure of B suggests a connection to the BRST/BV formalism, and may be an important clue about how to quantize correctly. (ii) It is interesting to consider how our formalism interacts with other recent interesting proposals for how to go beyond the traditional NCG formulation of the standard model (see e.g. [34][35][36][37][38][39][40][41][42][43]). (iii) We believe that there are interesting issues to be sorted out involving the relation between euclidean and lorentzian signature -see the comment in Subsection 5.2. (iv) Finally, although we have mostly restricted our attention in this paper to the case where the algebrasÂ, A and B are associative, we were originally led to Eilenberg's approach by the fact that it was designed to naturally generalize to the non-associative case [23,31]. In other words, one of the advantages of our new formalism is that it is naturally suited to generalizing from non-commutative to nonassociative geometry; and this continues to seem like a very interesting avenue to explore. In particular, it is intriguing to explore physical models based on a coordinate algebraÂ involving the octonions, or the exception Jordan algebra.
operators d L and d R . In this section we wish to highlight how some of these conditions arise, and where matters of convention enter. To do so we consider generalized * -DGAs which satisfy the following generalizations of the conditions given in (2.3b) and Eqs. (2.4): d L [a m a n ] = ρ n d L [a m ]a n + η m a m d L [a n ], (A.1a) (a m a n ) * = χ m,n a * n a * m , (A.1b) for a m , a n ∈ A, and where the coefficients ρ n , η m , and χ m,n are valued in F. We maintain the usual conditions d 2 = 0 and (a * ) * = a. Our goal will be to understand what restrictions are forced on the coefficients in Eqs. (A.1).

A.1.1 The graded Leibniz rule
Let's start by considering the coefficents ρ n , η m in Eq. (A.1a). By applying d L twice to a pair of algebra elements a m a n , and making use of Eq. (A.1a) we find Our first requirement for a DGA is that we want d 2 L = 0 on A, and from (A.2) we see that this implies η m = (−ρ n+1 /ρ n ) m η 0 , and ρ m = (−η n+1 /η n ) m ρ 0 . This implies that ρ, and η are of the form: ρ m = κ m ρ 0 , and η m = (−κ) m η 0 for some κ, η 0 , ρ 0 ∈ F. To further fix η 0 = ρ 0 = 1, we can use one of the following three requirements -any one of them will do the job. First, we could require that d L should obey the ordinary (ungraded) Leibniz rule on the sub-algebra of zero forms A 0 : which directly fixes η 0 = ρ 0 = 1. Second, we could require that the Leibniz rule is the one that makes sense for a unital algebra A (with unit e ∈ A 0 ), in which case which again fixes η 0 = 1. An analogous argument from the right fixes ρ 0 = 1. Third, we could require that the Leibniz rule is the one that makes sense for an associative algebra A, so that it does not matter in which order we use the Leibniz rule to break up the expression d[a m a n a p ] into parts; this implies η m η n = η m+n which, once again, implies η 0 = ρ 0 = 1. All roads lead in the same direction, and so we have: Next notice that given a differential operator d L satisfying the generalized Leibniz rule given in eq (A.1a) and (A.5), we can always define a new differential operator d L [a m ] = (±κ) −m d L [a m ] which will satisfy the graded Leibniz rule d L [a m a n ] = (±1) n d L [a m ]a n + (∓) m a m d L [a n ]. Without any loss of generality we therefore choose the simplest case κ = 1, in eq (A.5) such that we arrive at the 'left' graded Leibniz rule (2.3b). It is called the 'left' graded Leibniz rule because signs are picked up whenever passing over an algebra element from the left. An equally good choice would have been to select κ = −1, which would have resulted in a 'right' acting differential operator.

A.1.2 Properties of graded involutions
Let us next consider what role the function χ m,n plays in our construction. In Eq. (2.5) we introduced d R = * d L * as a 'right-acting' differential. Notice however that this interpretation depended on the form of χ m,n . In particular, for a * -DGA satisfying the generalized condition (A.1), d R satisfies the generalized Leibniz rule: d R [a m a n ] = (−1) n χ m,n χ n, m+1 d R [a m ]a n + χ m,n χ n+1, m a m d R [a n ].
(A. 6) We see that our choice χ m,n = 1 in Section 2 indeed results in d R = * d L * acting as a right differential; but what freedom did we have in this choice? Our first constraint is that we require (a * m ) * = a m , which implies χ m,n = (χ n, m ) −1 . And for the involution to make sense for associative algebras we need χ m,n+p χ n,p = χ m,n χ m+n,p . A choice compatible with both of these restrictions is to select χ m,n = (−1) mn ; but for that choice the operator d R = * • d L • * acts as a 'left' differential, and it is instead the operator d R [a m ] = (−1) m * d L * [a m ] which acts as a right differential. Notice that these two conventional choices (χ m,n = 1 or χ m,n = (−1) mn ) work for both real and complex * -DGAs, and for * -DGAs of both types ( = ±1). Both conventions turn out to be very useful in different situations as we will show below in Subsection A.2. More generally, we can write where χ mn = e iαmn , σ m = ±e −iαm , and we will refer to the special cases α = 0 and α = 1 as "convention I" and "convention II", respectively. 8 For complex DGAs satisfying = 1, we could consider the more general situation in which −π ≤ α ≤ π. Notice however that whichever value of α is chosen, we can always construct a new involution * which is given by a * m = e iτ (m−1)m/2 (a m ) * . For this new involution * one finds a new coefficient χ m,n = χ m,n e iτ mn . Thus we can always pick τ to recover either convention I or II.

A.2 Graded tensor product conventions and definitions
Next let us think about the origin of the (Kozul) tensor product rule given in (2.8). Suppose H and H are graded vector spaces on which graded operators O m : H p → H p+m and O n : H q → H q+n act respectively (where the subscripts denote the order of each operator). Define the product space H = H ⊗ H and product operator O m+n = O m ⊗ O n such that the action of O m+n on elements of H is given by: for h p ∈ H p , h q ∈ H q , and where Ψ n,p are F valued coefficients that we wish to constrain. Note that * -DGAs can be thought of as particular examples of graded vector spaces in which each element a m ∈ A m can be considered as an operator of degree m. Given two * -DGAs (A , d , * ), (A , d , * ), define their tensor A = A ⊗ A following Eq. (A.8) such that the product between algebra elements is given by (a m ⊗ a n )(a p ⊗ a q ) = Ψ n,p (a m a p ⊗ a n a q ) (χ n,p ∈ F) (A.9a) and where the differential and involution are given by where θ is an F valued function of the grading, which again we wish to determine. Let's first determine the form of the function Ψ n,m . We want (A, d) to be a DGA (i.e. we want d to satisfy the appropriate left graded Leibniz rule on A), which implies Ψ n,p = (−1) np Ψ 0,0 9 . This, in turn, is enough to ensure that: • (i) if A and A are both associative, then A is also associative; and • (ii) if A and A are both graded-commutative, then A is also graded-commutative.
(We stress that we do not assume associativity or graded-commutativity in this appendix.) In order to further fix Ψ 0,0 = 1, we can use one of the following two requirementseither one will do the job. First, we could require that, on the sub-algebra of zero-forms A 0 ⊗ A 0 , the multiplication rule should reduce to the standard one for the tensor product of two ungraded algebras A 0 and A 0 : (a 0 ⊗ a 0 )(ã 0 ⊗ã 0 ) = a 0ã 0 ⊗ a 0ã 0 (A.10) which directly fixes Ψ 0,0 = 1. Second, we could require that the product is the one that makes sense for unital algebras A , A and A ⊗A (with units e , e and e ⊗e , respectively), so that (a 0 ⊗ a 0 ) = (e ⊗ e )(a 0 ⊗ a 0 ) = χ 0,0 (e a 0 ⊗ e a 0 ) = χ 0,0 (a 0 ⊗ a 0 ) (A.11) which also fixes Ψ 0,0 = +1. Either path leads back to the (Kozul) product rule (2.8).
Next let us consider the function θ. In Subsection A.1.2 we introduced two good conventions for defining the involution on a differential graded algebra labelled by the signs χ m,n = e iαmn , σ m = ±e −iαm for α = 0 or α = π. Once a convention is chosen, if we want it to remain stable under the tensor product of * -DGAs, then this places restrictions on the function θ. In particular, it should be: θ(a m ⊗ a n ) = e i(π−α)mn (a m ⊗ a n ) (A. 12) with α = 0 (for convention I) or α = π (for convention II). The convention chosen in the body of this paper corresponds to the choice χ m,n = σ m = 1, and θ m,n = (−1) mn .
In future work, we will discuss the implications of this formalism for taking tensor products of spectral triples, since this is an interesting story in its own right [44]. 9 Note that we could have instead asked that an appropriate right-leibniz rule be satisfied in which case we would have arrived at a different convention for the Kozul sign.