3.5 The Question of Compatibility

We now have two routes to the ${\rm Spin}(10)$ theory. In Section 3.2 we saw how to reach it via the ${\rm SU}(5)$ theory:

$$\xymatrix{ {G_{\mbox{\rm SM}}}\ar[r]^-\phi \ar[d] & {\rm SU}... ...\mathbb C}^5) \ar[r]^-1 & {\rm U}(\Lambda {\mathbb C}^5) \\ }$$

$$\xymatrix{ & \ar@{~>}[r]_{\txt{More Unification}} & }$$

Our work in that section and in Section 3.1 showed that this diagram commutes, which is a way of saying that the ${\rm Spin}(10)$ theory extends the Standard Model.

In Section 3.4 we saw another route to the ${\rm Spin}(10)$ theory, which goes through ${\rm Spin}(4) \times {\rm Spin}(6)$:

$$\xymatrix{ {G_{\mbox{\rm SM}}}\ar[r]^-\theta \ar[d] & {\rm S... ...3) \ar[r]^-{{\rm U}(g)} & {\rm U}(\Lambda {\mathbb C}^5) \\ }$$

$$\xymatrix{ & \ar@{~>}[r]_{\txt{More Unification}} & }$$

Our work in that section and Section 3.3 showed that this diagram commutes as well. So, we have another way to extend the Standard Model and get the ${\rm Spin}(10)$ theory.

Drawing these two routes to ${\rm Spin}(10)$ together gives us a cube:

$$\xymatrix{ & {G_{\mbox{\rm SM}}}\ar[rr]^\phi \ar[dl]_\theta ... ...r[rr]^-{{\rm U}(g)} & & {\rm U}(\Lambda {\mathbb C}^5) & \\ }$$

Are these two routes to ${\rm Spin}(10)$ theory the same? That is, does the cube commute?

Theorem 7   . The cube commutes.

Proof. We have already seen in Sections 3.1-3.4 that the vertical faces commute. So, we are left with two questions involving the horizontal faces. First: does the top face of the cube

$$\xymatrix{ & {G_{\mbox{\rm SM}}}\ar[rr]^\phi \ar[dl]_\theta ... ...) \times {\rm Spin}(6) \ar[rr]^\eta & & {\rm Spin}(10) & \\ }$$

commute? In other words: does a symmetry in ${G_{\mbox{\rm SM}}}$ go to the same place in ${\rm Spin}(10)$ no matter how we take it there? And second: does the bottom face of the cube commute? In other words: does this triangle:

$$\xymatrix{ F \oplus F^* \ar[r]^f \ar[d]_h & \Lambda {\mathbb... ...da {\mathbb C}^2 \otimes \Lambda {\mathbb C}^3 \ar[ru]_g \\ }$$

commute?

In fact they both do, and we can use our affirmative answer to the second question to settle the first. As we remarked in Section 3.4, applying the map $g$ to the Pati-Salam binary code given in Table 6, we get the ${\rm SU}(5)$ binary code given in Table 4. Thus, the linear maps $f$ and $gh$ agree on a basis, so this triangle commutes:

$$\xymatrix{ F \oplus F^* \ar[r]^f \ar[d]_h & \Lambda {\mathbb... ...da {\mathbb C}^2 \otimes \Lambda {\mathbb C}^3 \ar[ru]_g \\ }$$

This in turn implies that the bottom face of the cube commutes, from which we see that the two maps from ${G_{\mbox{\rm SM}}}$ to ${\rm U}(\Lambda {\mathbb{C}}^5)$ going around the bottom face are equal:

$$\xymatrix{ {G_{\mbox{\rm SM}}}\ar[dr] & & \\ & {\rm U}(F \... ...3) \ar[r]^-{{\rm U}(g)} & {\rm U}(\Lambda {\mathbb C}^5) \\ }$$

The work of Section 3.1 through Section 3.4 showed that the vertical faces of the cube commute. We can thus conclude from diagrammatic reasoning that the two maps from ${G_{\mbox{\rm SM}}}$ to ${\rm U}(\Lambda {\mathbb{C}}^5)$ going around the top face are equal:

$$\xymatrix{ {G_{\mbox{\rm SM}}}\ar[r]^\phi \ar[d]_\theta & {\... ...n}(10) \ar[dr] & \\ & & {\rm U}(\Lambda {\mathbb C}^5) \\ }$$

Since the Dirac spinor representation is faithful, the map ${\rm Spin}(10) \to {\rm U}(\Lambda {\mathbb{C}}^5)$ is injective. This means we can drop it from the above diagram, and the remaining square commutes. But this is exactly the top face of the cube. So, the proof is done. $\sqcap$ $\sqcup$