A mathematical characterization of the gauge group \(\text{U}(1)\times \text{SU}(2)\times \text{SU}(3)\) of the standard model from the viewpoint of noncommutative geometry is presented. Let \(M\) be the four-dimensional space-time, then a geometric picture of the standard model is given as the noncommutative geoemtry described by the pair \(({\mathcal H},J,{\mathcal A},\gamma,D)\). Here \({\mathcal H}\) is a finite-dimensional Hilbert space whose noncommutative space is denoted \(F\), \(J\) is an antilinear isometry of \({\mathcal H}\) with \(J^2=\varepsilon\), \({\mathcal A}\) is an involutive algebra over \(\mathbb{R}\) acting in \({\mathcal H}\) such that
\[ [a, b^0]= 0,\quad a,b\in{\mathcal A},\quad b^0= Jb^* J^{-1}. \]
\(\gamma\) is a \(\mathbb{Z}/2\)-gading of \({\mathcal H}\) such that \(J\gamma= \varepsilon''\gamma J\), and \(D\) is a self-adjoint operator in \({\mathcal H}\) such that \(JD= \varepsilon'DJ\). \(\varepsilon\), \(\varepsilon'\), \(\varepsilon''\) are either of \(1\) or \(-1\). Then it is shown that if \((\varepsilon,\varepsilon',\varepsilon'')= (1,1,-1)\), then \({\mathcal A}\) is determined to be \(M_2(\mathbb{H})\oplus M_4(\mathbb{C})\), and set \({\mathcal A}_F\) to be the subalgebra of \({\mathcal A}\) on which \(D\) satisfies order 1 condition;
\[ {\mathcal A}_F= \{a,b\in{\mathcal A}\mid[[D, a], b^0]= 0\}, \]
then \({\mathcal A}_F\cong\mathbb{C}\oplus \mathbb{H}\oplus M_3(\mathbb{C})\) (Th.4.3).
To show this, first irreducible triple \(({\mathcal A}, {\mathcal H},j)\) is classified and show there are two cases; \(Z({\mathcal A}_{{\mathbb{C}}})= \mathbb{C}\) or \(Z({\mathcal A}_{{\mathbb{C}}})= \mathbb{C}\oplus \mathbb{C}\) (Lemma 2.2). Then to show the first case does not occur if \((\varepsilon, \varepsilon', \varepsilon'')=(1,1, -1)\) (Lemma 3.1), Theorem 4.3 is proved. The authors show in this case combining the results of A. H. Chamseddine, A. Connes and M. Marcolli [Adv. Theor. Math. Phys. 11, No. 6, 991–1089 (2007; Zbl 1140.81022)] the standard model minimally coupled with Einstein gravity appears naturally as pure gravity on the space \(M\times F\), \(F\) the noncommutative space corresponding to \({\mathcal H}\).
The choice \((\varepsilon, \varepsilon', \varepsilon'')=(1,1, -1)\) specifies the \(K\)-theoretic dimension of \(({\mathcal A},{\mathcal H},J)\) to be 6 modulo 8. Hence the \(K\)-theoretic dimension of \(M\times F\) is 10 modulo 8. Therefore we can use the antisymmetric bilinear form \(\langle J\xi,D\eta\rangle\) to define the fermionic action, so that the functional integral over fermions delivers a Pfaffian rather than a determinant. The authors say this is the “raison d’être” for crossing by \(F\) is to shift the \(K\)-theoretic dimension from 4 to 10 modulo 8.
In conclusion, some further problems such as understanding the role of quaternions, search of mathematical reasons why the generation is 3, and the mathematical meaning of the massless condition, are listed with some discussions.