Orientation et charge

Deux orientationsTwo orientations

On a fixed basis $1,e_0,\dots,e_6$ there are 480 octonion multiplication tables with $e_ae_b=\pm e_c$: thirty Fano planes, each with sixteen orientations. A table’s 3-form orients $\R^7$, and the 480 fall into two classes of 240; a table and its opposite lie in different classes. The table $A$ and its Weil mirror $A'$ lie in the same class, while the mirror obtained by relabelling the units by $y\mapsto-y$, and the opposite of $A$, lie in the other: the Weil mirror is not the classical mirror of the 480 tables. The group $2{\cdot}A_7$ of the Hermitian $E_8$ acts on $\Q(\sqrt{-7})^4$ through a character with values $(1\pm\sqrt{-7})/2$ on elements of order 7; its invariant lattice is $E_8$ over the integers of $\Q(\sqrt{-7})$, and the group of order 168 is the stabilizer of one cross.

In $M$ the table’s ordered quadruple $(\infty,x,x+1,x+3)$ lifts to the ideal tetrahedron $(\infty,a,a+1,a+1+\bar\omega)$, negatively oriented, and the Weil mirror’s $(\infty,x,x+1,x+5)$ to $(\infty,a,a+1,a+1+\omega)$, positively oriented, since $1+\bar\omega\equiv3$ and $1+\omega\equiv5$ modulo $\mathfrak p$ while $\operatorname{Im}(1+\bar\omega)<0<\operatorname{Im}(1+\omega)$. In $\bar M$ both signs reverse. The sign is changed by $c$ and, in these conventions, by the outer class; the second change will turn out to be a change of convention.

La diffusion du système spinorielThe scattering of the spinor system

The orientation of $M$ appears in a constant. Let $V$ be the local system on $M$ given by the defining representation of $\SL(2,\Z[\omega])$, restricted to $\Gamma(\mathfrak p)$. At a cusp with translations $t_b$ a cocycle of $V$ restricts to $z(b)=(sb^2/2+Ab+B\bar b,\ sb)$, with $s$ holomorphic in the parameter of the cusp and $B$ antiholomorphic, and the boundary image of $H^1(M;V)$ is the graph $s=L_VB$ of an antisymmetric matrix $L_V$ with zero diagonal, indexed by the eight cusps. The lattices stable under $\SL(2,\Z[\omega])$ are all homothetic to $\Z[\omega]^2$, and it fixes the scattering up to one sign: in the basis $(ue_1,ve_2)$ the ratio $s/B$ is multiplied by $u^2\bar v/(v^2\bar u)$, which for units is $(u/v)^3$, equal to 1 exactly when $\mathrm{diag}(u,v)$ reduces to an inner automorphism of $\SL(2,7)$.

The outer class $D=\mathrm{diag}(-1,1)$ acts by a cochain map, by a signed permutation $R_D$ over $c\mapsto-c$ on the coordinates $B$ and by $-R_D$ on $s$, with $R_D\Theta=-\Theta R_D$. It exchanges the quartets and leaves $L_V$ and $y$ unchanged: the sign that tells the quartets apart belongs to $\Theta$. The mirror $c$ carries $L_V$ to its complex conjugate, $y\mapsto\bar y$, and since $v_{\mathfrak p}(y)-v_{\bar{\mathfrak p}}(y)=-2$ while a common rescaling of the cusps changes this difference by multiples of 3, in no rational normalization is $y$ even or odd under $c$. The scattering matrices and cup products of $\C$, $V$, $\bar V$ and $\mathrm{Sym}^2V$ are carried to themselves by $D$, and every structure defined on $\Proj^1(\F_7)$ through the reduction is invariant under $c$: each orientation is blind to the other’s flip.

Galois type is not parity. The eigenvalue $v_{\mathbf4}=-(24\sqrt{-7}+2\sqrt{21})/49$ has a component along $\sqrt{21}$, which $\sigma_2$ and $\sigma_{13}$ each negate, but it is not odd under both flips, since the outer class negates $\Theta$ and exchanges the quartets. Around a cusp the parabolic $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$, of order 7, acts on $\mathbf4$ with eigenvalues $\zeta^k$, $k\in\{0,1,2,4\}$, and on $\bar{\mathbf4}$ with $k\in\{0,3,5,6\}$: one fixed line together with the quadratic residues, or with the non-residues.

La forme de Cayley sur les cellulesThe Cayley form on the cells

Attach to the cusp $\infty$ the unit 1 and to the cusp $x$ the unit $e_x$. The Cayley 4-form $\Phi(w,x,y,z)=\langle w,\,x\times y\times z\rangle$, 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. For the table $A$ these are 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 for $A'$ 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, and $\pm$ an imaginary unit on the other 56.

Paired with the oriented cells of $M$, $\Phi$ 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 it is known, and no novelty is claimed. In the cyclic gauges, where $A$ reads $e_xe_{x+1}=+e_{x+3}$ and $A'$ reads $e_xe_{x+1}=+e_{x+5}$, every Cayley cell of $A$ has $\varepsilon\Phi_A=-1$ in $M$, so $S(A)=-14$, while $S(A')=0$; in $\bar M$ every orientation reverses. But $S$ is not an invariant, and the outer class carries the table to its mirror in the opposite gauge, where the sign agrees with the table’s: no gauge-invariant function of the oriented Cayley data is odd under each flip separately.

Le système spinoriel aux pointesThe spinor system at the cusps

The cusps of one tetrahedron single out a line of $V$ at each cusp. Let $T_0$ have cusps $\infty,0,1,1+\omega$, with primitive vectors $\xi_\infty=(1,0)$, $\xi_0=(0,1)$, $\xi_1=(1,1)$ and $\xi_{1+\omega}=(1+\omega,1)$, the cusp of $(x,y)$ being $x/y$. They are pairwise unimodular, the null vectors $N_c=\xi_c\xi_c^\dagger$ have $\langle N_a,N_b\rangle=\tfrac12$ and form a $\Z$-basis of $\mathrm{Herm}_2(\Z[\omega])$, and at each cusp $u_c(t)=I+t\,\xi_c\xi_c^{\mathsf T}J$, $J=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$, is parabolic for $t\ne0$ and fixes exactly the line of $\xi_c$. Of the 24 permutations of the cusps exactly the 12 even ones are induced by Möbius maps, all in $\PSL(2,\Z[\omega])$, with lifts forming the binary tetrahedral group $2T$; modulo $\mathfrak p$ the stabilizer of the quadruple and its complement has 48 elements with the element orders of $2O$, whose elements of order 8, of traces 3 and 4, act on $M$ without fixed points.

The six pairs of cusps give six unit timelike vectors $T_{ab}=N_a+N_b$, with $\langle T_{ab},T_{ac}\rangle=\tfrac32$ and $\langle T_{ab},T_{cd}\rangle=2$. Each of the 24 moves $\{c,a\}\to\{c,b\}$ is realized by a parabolic element fixing $\xi_c$, and each of the 6 moves to the opposite pair by an element of order 4 of $2T$; all are holomorphic, so each keeps the class of $V$, while the reflections of $T_0$ are anti-Möbius. Restricted to $2O$ the quartets have no doublet summand: $\SL(2,7)$ has no two-dimensional irreducible representation, $\mathbf4$ and $\bar{\mathbf4}$ both restrict to $\mathbf4_s=E_+\otimes\mathbf2=E_-\otimes\mathbf2$, and the faithful doublets $E_\pm=\rho_\pm$ occur only in $\mathbf6_s$, $\mathbf6_s'$ and $\mathbf8_s$.

L’algèbre de consignationThe commit algebra

In $\C\otimes\Oct$ fix an imaginary unit $e_p$ and put $D=-iL_{e_p}$. For each of the six units $e_a$ orthogonal to $e_p$, $L_{e_a}^2=-I$ and $L_{e_a}$ anticommutes with $L_{e_p}$, hence with $D$, and the product of the six is $\pm L_{e_p}$, whatever the signs of the units: by alternativity $L_xL_y+L_yL_x=-2\langle x,y\rangle I$ for imaginary $x,y$. So along a word of moves of $T_0$, each paired with the left multiplication by a unit orthogonal to $e_p$, the spinor factor keeps the class of $V$ while $D$ changes sign at every move, and after $n$ moves the preserved operator is $(-1)^nD$. The commit algebra asks which operators commute with everything that writes the moves.

For the three lines $\{p,q,r\}$ through $p$, ordered so that $e_qe_r=e_p$, put $a_j=(L_{e_q}+iL_{e_r})/2$ and $N=\sum_ja_j^\dagger a_j$, and on a second factor $\C^2$ write $X$ and $Z$ for the Pauli matrices. On $\Oct$ alone the seven $L_{e_x}$ make a single module of $\mathrm{Cl}_7$, whose volume element is a scalar, and there is one sector. The counter’s $I\otimes Z$ stands in the place of $L_{e_p}$, the volume element becomes $\pm L_{e_p}\otimes Z$, which is not a scalar, and its two eigenspaces are the sectors. With the counter replaced by $\ell^2(\mathbb N)$ the commutant is still spanned by $I$ and $D\otimes(-1)^n$: the parity carries all of the counter’s contribution.

What the centre leaves out: a function of $N$, tensored with $I$ or with $Z$, commutes with the algebra only if it lies in $\mathrm{span}\{I,(-1)^N\otimes Z\}$; the products $L_{e_a}L_{e_b}$ generate $M_4(\C)$ on each eigenspace of $D$; and $J_0=L_u/\sqrt7$, $u=e_0+\dots+e_6$, is not in the algebra, since $L_{e_a}L_uL_{e_a}^{-1}=2L_{e_a}-L_u$ is never $\pm L_u$. The $+i$-eigenspaces of $J_0$ and of $L_{e_p}$ are isoclinic, at the angle with $\cos^2=\tfrac12(1\pm1/\sqrt7)$.

Formes bilinéaires d’une seule mainBilinears of one hand

$V$ and its mirror $\bar V$ are inequivalent, and that fixes the bilinear invariants: for $G=\SL(2,\Z[\omega])$, $\dim(V\otimes V)^G=1$, spanned by $\varepsilon$, $\dim(V\otimes\bar V)^G=0$ and $\dim(\bar V\otimes\bar V)^G=1$, so $\mathrm{Hom}_G(V,\bar V)=0$. At a clock $e_p$, with the quartet $\mathbf4$ the $-1$-eigenspace of $D$, the fifteen products $L_{e_a}L_{e_b}$ span $\mathfrak{su}(4)$, and an element of $(V\otimes V)\otimes(\mathbf4\otimes\bar{\mathbf4})$ invariant under $G$ and $\mathfrak{su}(4)$ is unique up to scale, $\varepsilon\otimes\sum_ie_i\otimes\bar e_i$.

A tensor annihilated by $D_p\otimes I+I\otimes D_p$ for all seven $p$ is a multiple of $\tau_0=1\otimes1+\sum_ke_k\otimes e_k$, the octonion norm’s, which is symmetric; at one clock the $\mathfrak{su}(4)$-invariant ones are spanned by $\tau_0$ and the antisymmetric tensor $\tau_D$ of $D_p$, with $\sum_ie_i\otimes\bar e_i=\tfrac12(\tau_0-\tau_D)$. So the invariant pair has a half that makes a pair only for fermions and a half only for bosons. The symmetric pair tensors invariant under the $\mathrm{SU}(3)$ fixing $e_p$ and annihilated by $D_p\otimes I+I\otimes D_p$ are spanned by $\tau_0$ and $\tau_0\circ(P_\ell-\tfrac13P_q)$, with $P_\ell$ and $P_q$ the projections onto $\langle1,e_p\rangle$ and onto the six other units, and $P_\ell-\tfrac13P_q=-\sigma(B-L)D_p$ for $B-L=\tfrac23N-1$ and $D_p=\sigma(-1)^N$: a second structure, with the factor $-\tfrac13$ between its parts.

Holes in a fermionic Fock space over a finite-dimensional space transform by $(g^{-1})^{\mathsf T}=\varepsilon g\varepsilon^{-1}$, so the holes of $V$ carry $V$ again, whatever positive inner product is used, and none is invariant, since $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$ is a nontrivial Jordan block. The vectors the moves pass between do carry invariant forms: for positive $T$ of determinant one, $\langle\xi,\eta\rangle_T=\xi^\dagger T^{-1}\eta$ makes every $g\in\SL(2,\C)$ an isometry onto $\langle\,,\rangle_{gTg^\dagger}$, and on $\ell^2(G\cdot T_0)\otimes V$ the action is unitary, the discrete form of the representation of $\SL(2,\C)$ induced from $\mathrm{SU}(2)$. On the cusp lines a hole carries the conjugate character, $\bar\chi$ where the cusp’s vector carries $\chi$.