15d. Small dimensions

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We would like to end this chapter with something refreshing, namely a preliminary study of the free analogue of [math]P^2_\mathbb R[/math]. We recall that the projective space [math]P^{N-1}_\mathbb R[/math] is the space of lines in [math]\mathbb R^N[/math] passing through the origin, the basic examples being as follows:

(1) At [math]N=2[/math] each such a line, in [math]\mathbb R^2[/math] passing through the origin, corresponds to 2 opposite points on the unit circle [math]\mathbb T\subset\mathbb R^2[/math]. Thus, [math]P^1_\mathbb R[/math] corresponds to the upper semicircle of [math]\mathbb T[/math], with the endpoints identified, and so we obtain a circle, [math]P^1_\mathbb R=\mathbb T[/math].

(2) At [math]N=3[/math] the situation is similar, with [math]P^2_\mathbb R[/math] corresponding to the upper hemisphere of the sphere [math]S^2_\mathbb R\subset\mathbb R^3[/math], with the points on the equator identified via [math]x=-x[/math]. Topologically speaking, we can deform if we want the upper hemisphere into a square, with the equator becoming the boundary of this square, and in this picture, the [math]x=-x[/math] identification corresponds to the “identify opposite edges, with opposite orientations” folding method for the square, leading to a space [math]P^2_\mathbb R[/math] which is obviously not embeddable into [math]\mathbb R^3[/math].

In what follows we will be interested in the free analogue [math]P^2_+[/math] of this projective space [math]P^2_\mathbb R[/math]. Our main motivation comes from the fact that, according to the work of Bhowmick-D'Andrea-Dabrowski , later on continued with Das , the quantum isometry group [math]PO_3^+=PU_3^+[/math] of the free projective space [math]P^2_+[/math] acts on the quark part of the Standard Model spectral triple, in Chamseddine-Connes formulation , .

We recall that the free projective space is defined by the following formula:

[[math]] C(P^{N-1}_+)=C^*\left((p_{ij})_{i,j=1,\ldots,N}\Big|p=p^*=p^2,Tr(p)=1\right) [[/math]]

Let us first discuss, as a warm-up, the 2D case. Here the above matrix of projective coordinates is as follows, with [math]a=a^*[/math], [math]b=b^*[/math], [math]a+b=1[/math]:

[[math]] p=\begin{pmatrix}a&c\\ c^*&b\end{pmatrix} [[/math]]

We have the following computation:

[[math]] p^2=\begin{pmatrix}a&c\\ c^*&b\end{pmatrix}\begin{pmatrix}a&c\\ c^*&b\end{pmatrix} =\begin{pmatrix}a^2+cc^*&ac+cb\\ c^*a+bc^*&c^*c+b^2\end{pmatrix} [[/math]]

Thus, the equations to be satisfied are as follows:

[[math]] a^2+cc^*=a [[/math]]

[[math]] b^2+c^*c=b [[/math]]

[[math]] ac+cb=c [[/math]]

[[math]] c^*a+bc^*=c^* [[/math]]

The 4th equation is the conjugate of the 3rd equation, so we remove it. By using [math]a+b=1[/math], the remaining equations can be written as:

[[math]] cc^*=c^*c=ab [[/math]]

[[math]] ac+ca=0 [[/math]]

We have several explicit models for this, using the spheres [math]S^1_{\mathbb R,+}[/math] and [math]S^1_{\mathbb C,+}[/math], as well as the first row spaces of [math]O_2^+[/math] and [math]U_2^+[/math], which ultimately lead us to [math]SU_2[/math] and [math]\bar{SU}_2[/math]. These models are known to be all equivalent under Haar, and the question is whether they are identical. Thus, we must do computations as above in all models, and compare. These are all interesting questions, whose precise answers are not known, so far.

In the 3D case now, that of projective space [math]P^2_+[/math], that we are mainly interested in here, the matrix of coordinates is as follows, with [math]r,s,t[/math] self-adjoint, [math]r+s+t=1[/math]:

[[math]] p=\begin{pmatrix} r&a&b\\ a^*&s&c\\ b^*&c^*&t\end{pmatrix} [[/math]]

The square of this matrix is given by:

[[math]] p^2=\begin{pmatrix} r&a&b\\ a^*&s&c\\ b^*&c^*&t\end{pmatrix} \begin{pmatrix} r&a&b\\ a^*&s&c\\ b^*&c^*&t\end{pmatrix} [[/math]]

We obtain the following formula:

[[math]] p^2=\begin{pmatrix} r^2+aa^*+bb^*&ra+as+bc^*&rb+ac+bt\\ a^*r+sa^*+cb^*&a^*a+s^2+cc^*&a^*b+sc+ct\\ b^*r+c^*a^*+tb^*&b^*a+c^*s+tc^*&b^*b+c^*c+t^2 \end{pmatrix} [[/math]]

On the diagonal, the equations for [math]p^2=p[/math] are as follows:

[[math]] aa^*+bb^*=r-r^2 [[/math]]

[[math]] a^*a+cc^*=s-s^2 [[/math]]

[[math]] b^*b+c^*c=t-t^2 [[/math]]

On the off-diagonal upper part, the equations for [math]p^2=p[/math] are as follows:

[[math]] ra+as+bc^*=a [[/math]]

[[math]] rb+ac+bt=b [[/math]]

[[math]] a^*b+sc+ct=c [[/math]]

On the off-diagonal lower part, the equations for [math]p^2=p[/math] are those above, conjugated. Thus, we have 6 equations. The first problem is that of using [math]r+s+t=1[/math], in order to make these equations look better. Again, many interesting questions here.

Observe the analogy with the basic discussion about hypersurfaces, and about basic affine geometry in general, from the end of chapter 13. In both cases indeed we are led to an interesting mix of basic algebraic geometry and operator theory, and with the operator theory component potentially ranging from very basic to very complicated.

Finally, let us remind again that all this mathematical fun is potentially interesting, in connection with questions in quantum physics, because according to , , the quantum isometry group [math]PO_3^+=PU_3^+[/math] of the free projective space [math]P^2_+[/math] acts on the quark part of the Standard Model triple, in Chamseddine-Connes formulation , .

You might say here, not serious all this, because modern physics means doing complicated QFT, or string theory, ADS/CFT, and so on. But hey, isn't modern physics coming from Pauli discovering the Pauli matrices, then Dirac discovering the Dirac matrices, then Gell-Mann discovering the Gell-Mann matrices. So, there are probably still many things to be discovered, simple and useful, why not in relation with the above.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].