orientation

The congruence link complement $M$ has no orientation-reversing isometry, and its mirror image is the link complement $\bar M$ at the conjugate prime; the interior has the octonion table and its Weil mirror. Let $G$ act on $M$ through reduction modulo $\mathfrak p=(3+\omega)$, $\omega$ a primitive cube root of unity, and on $\R^8$ through the signed permutations of the Weil representation.

(a) Complex conjugation $c$ is an orientation-reversing isometry from $M$ to $\bar M=\Gamma(\bar{\mathfrak p})\backslash\mathbb H^3$. It is a seam over the identity of $G$, $r_{\bar{\mathfrak p}}(\bar\gamma)=r_{\mathfrak p}(\gamma)$ for every $\gamma\in\SL(2,\Z[\omega])$, and it keeps the labels of cusps and of tetrahedra. (b) $D=\mathrm{diag}(-1,1)\in\GL(2,\Z[\omega])$ is an orientation-preserving isometry of $M$ and reduces to the improper element $z\mapsto-z$. So the outer class of $\Aut(G)$ acts on $M$ by rotations. It exchanges the two classes of tetrahedra: the seven quadruples $\{\infty\}\cup(x+\{0,1,3\})$ form one class, and $\{\infty\}\cup(x+\{0,4,6\})$ the other. (c) The signed permutations of the sixteen vectors that normalize the Weil group are the 672 elements of the image of $\GL(2,7)$. The proper ones commute with $\theta=\sqrt{-7}$ and the improper ones anticommute with it. The improper ones exchange the table $A$ and its Weil mirror $A'$, the table $e_xe_{x+1}=e_{x+5}$; the lattices $E_8^a$ and $E_8^b$; the primes $\lambda$ and $\bar\lambda$ over 2 at which the sixteen vectors collapse; and the quartets $\mathbf4$ and $\bar{\mathbf4}$. The improper ones that fix $v_\infty$ and reverse $v_0$ carry $A$ to $A'$. (d) Hence the prime over 7 and the orientation of the interior (the table, the quartet, the lattice, the prime over 2) are independent. Each is changed by an operation that keeps the other: $c$ changes the prime and keeps every datum defined on the projective line, and $D$ with its Weil action changes the interior’s orientation and keeps the prime. No seam over a single automorphism of $G$ changes the one exactly when it changes the other.

In $M$ the table’s ordered quadruple $(\infty,x,x+1,x+3)$ lifts to the ideal tetrahedron $(\infty,a,a+1,a+1+\bar\omega)$ of the tessellation of $\mathbb H^3$ by regular ideal tetrahedra, and it is negatively oriented. The Weil mirror’s quadruple $(\infty,x,x+1,x+5)$ lifts to $(\infty,a,a+1,a+1+\omega)$ and is positively oriented. In $\bar M$ both signs reverse. So the sign is changed by $c$ and, in these sign conventions, by the outer class; the second change is a change of convention.

In $\Q(\zeta_{21})\supset\Q(\sqrt{-3},\sqrt{-7})$ the Frobenius at 2, $\sigma_2$, negates $\sqrt{-3}$ and fixes $\sqrt{-7}$, so it carries $\mathfrak p$ to $\bar{\mathfrak p}$ and fixes $\lambda$ and $\bar\lambda$; $\sigma_{13}$ fixes $\sqrt{-3}$ and negates $\sqrt{-7}$; complex conjugation negates both. The link complement is defined over $\Q(\sqrt{-3})$, and the Weil data over $\Q(\zeta_7)\supset\Q(\sqrt{-7})$. On these fields the change of (a) acts as $\sigma_2$, and that of (c) as $\sigma_{13}$. The two orientations are the two independent generators of $\mathrm{Gal}(\Q(\sqrt{-3},\sqrt{-7})/\Q)\cong C_2\times C_2$.

(a) The boundary scattering matrices and cup products of the local systems $\C$, $V$, $\bar V$ and $\mathrm{Sym}^2V$ on $M$, $V$ the spinor system, have entries in $\Q(\omega)$, and $D$ carries each of them to itself. Where one of them separates $\mathbf4$ from $\bar{\mathbf4}$, it does so through the Paley matrix $\Theta$. (b) Every structure defined on $\Proj^1(\F_7)$ through the reduction modulo $\mathfrak p$ is invariant under the mirror $c$. (c) Hence an invariant that is odd under $c$ and odd under the outer class, separately, must combine the orientation of $M$ with an orientation of data on the finite line. Among the objects of the book none does: the gauge-invariant content of the oriented Cayley data is even under the outer class.

Galois type is not parity. In $\Q(\sqrt{-3},\sqrt{-7})$ the eigenvalue $v_{\mathbf4}=-(24\sqrt{-7}+2\sqrt{21})/49$ of the scattering on the quartet has a component along $\sqrt{21}$, which $\sigma_2$ and $\sigma_{13}$ each negate, but it is not odd under both flips: the outer class negates $\Theta$ and exchanges the quartets, and $v_{\mathbf4}$, the eigenvalue on the quartet so labelled, is fixed. The identification of the outer class with $\sigma_{13}$ holds for data on the finite line, not for constants of the local systems on $M$, which lie in $\Q(\omega)$.

Attach to the cusp $\infty$ the unit 1 and to the cusp $x$ the unit $e_x$. The Cayley 4-form of an octonion algebra, $\Phi(w,x,y,z)=\langle w,\,x\times y\times z\rangle$ with $x\times y\times z=\tfrac12(x(\bar yz)-z(\bar yx))$, is invariant under $\mathrm{Spin}(7)$, and on a basis of units it is $\pm1$ exactly on the fourteen quadruples that span Cayley planes, the blocks of a Steiner system $S(3,4,8)$, and 0 on the other fifty-six.

$\Phi_A$ is $\pm1$ exactly on the 14 tetrahedra of $M$ of the class $\{\infty\}\cup(x+\{0,1,3\})$ together with the complements of the lines $x+\{0,1,3\}$, and $\Phi_{A'}$ exactly on the 14 of the other class. Equivalently, the product of the four units of a quadruple of cusps, in any order and association, is $\pm1$ exactly on the Cayley quadruples of $A$, and is $\pm$ an imaginary unit on the other 56.

Paired with the oriented cells of $M$, the Cayley form gives a sum $S=\sum_T\varepsilon(T)\Phi(T)$ of the shape of Dijkgraaf and Witten’s actions, the pairing of a 3-cochain with the fundamental cycle of the end compactification of $M$; no classical instance of this pairing is known to the book, and it claims no novelty for it. Fix the signs of the units in which $A$ reads $e_xe_{x+1}=+e_{x+3}$ and $A'$ reads $e_xe_{x+1}=+e_{x+5}$, the cyclic gauges. In $M$ every one of $A$’s 14 Cayley cells has $\varepsilon\Phi_A=-1$, so $S(A)=-14$; the 7 cells of $A'$ through $\infty$ have $\varepsilon\Phi_{A'}=+1$ and their complements $-1$, so $S(A')=0$; in $\bar M$ every orientation reverses. The map $W(1)=1$, $W(e_x)=-e_{-x}$ is an isomorphism $A\to A'$, and $D$ carries the one class of cells onto the other.

Changing the signs $s_c$ of the units at the cusps multiplies $\varepsilon(T)\Phi(T)$ by $\prod_{c\in T}s_c$. (a) On the 14 Cayley cells of $A$ these changes realize exactly 16 sign patterns, and the pattern $-1$ on every cell, which is what the mirror $c$ does, is not one of them. (b) Pulled back by $D$, the pattern of $A'$ in its cyclic gauge equals the pattern of $A$ times the pattern of the signs $+1$ at $\infty$ and $-1$ elsewhere, the gauge into which $W$ carries $A$’s. (c) On the gauge orbit of $A$’s pattern, $S$ takes the values $-14$, 0 and 2. (d) For each of the 28 triples $\{T_1,T_2,T_1\triangle T_2\}$ of Cayley cells, which cover every cusp an even number of times, $\prod\varepsilon\Phi_A=-1$ in $M$, and likewise for $A'$; in $\bar M$ every such product is $+1$.

So the gauge-invariant content of the oriented Cayley data is odd under $c$ and invariant under the outer class, and $S$ is not an invariant.

The sign of the table’s cells in the cyclic gauge compares the table in its cyclic gauge with the Weil mirror in its own. The outer class carries the first to the mirror in the opposite gauge, where the sign agrees with the table’s. Every gauge-invariant function of the oriented Cayley data takes the same value for the table and for its Weil mirror, so none of them is odd under each flip separately.