gauge

Let $\mathcal G$ be a connected graph, each edge taken with both orientations, $\bar e$ the reverse of $e$. A seam system over $\mathcal G$ for an object $X$ assigns an incarnation $Y_v$ to each vertex and a seam $s_e\colon Y_v\to Y_w$ to each oriented edge $e$ from $v$ to $w$, with $s_{\bar e}=s_e^{-1}$. Let $A=\Aut_G(X)\cong N_G(H)/H$.

(a) A choice of alignments $\varphi_v\colon X\to Y_v$, one per vertex, a gauge, turns the system into link variables $U_e=\varphi_w^{-1}s_e\varphi_v\in A$, with $U_{\bar e}=U_e^{-1}$. Another gauge $\varphi_v'=\varphi_va_v$ replaces $U_e$ by $a_w^{-1}U_ea_v$, a gauge transformation, and every family $(U_e)$ with $U_{\bar e}=U_e^{-1}$ arises from a seam system.

(b) The monodromy of a cycle $(e_1,\dots,e_m)$ at $v$ is $\varphi_v(U_{e_m}\cdots U_{e_1})\varphi_v^{-1}$, the holonomy read through the alignment at $v$. Its conjugacy class in $A$ does not depend on the gauge, so every class function of the holonomy, such as $\chi(U_{e_m}\cdots U_{e_1})$ for a character $\chi$ of $A$, a Wilson loop, is gauge invariant; if $A$ is abelian the holonomy itself is.

(c) The system is coherent if and only if every holonomy is trivial, if and only if it is gauge equivalent to the system with all $U_e=1$.

(d) If $\mathcal G$ is the 1-skeleton of a 2-complex $\mathcal K$, the holonomy around every 2-cell is trivial (the connection is flat) if and only if holonomy defines a homomorphism $\pi_1(\mathcal K,v)\to A$; the system is then coherent if and only if this homomorphism is trivial.

(e) For fixed incarnations, the gauge classes of seam systems over $\mathcal G$ correspond to the homomorphisms $\pi_1(\mathcal G,v)\to A$ up to conjugation in $A$.

Call two of the 28 pairs of points of $\Proj^1(\F_7)$ adjacent when they share a point: the Johnson graph $J(8,2)$, with 168 edges. Its 280 triangles of pairs through a common point are the 2-cells of a 2-complex $\mathcal K$, the complex of stars. In the Weil representation of $\SL(2,7)$, read through the octonions, there is a family of algebras $\mathfrak{su}(3)$, one over each pair. Along the edge from $\{c,d\}$ to $\{c,e\}$, transport by the odd lift $u$ of the unique element of order 7 fixing $c$ with $d\mapsto e$.

(a) Conjugation by $u$ carries the algebra over $\{c,d\}$ onto the algebra over $\{c,e\}$, so the transports form a seam system for the family. (b) Around every 2-cell the transports compose to the identity: the connection is flat. (c) In a gauge along a spanning tree every link variable lies in the stabilizer $K$, of order 12, of the base pair, and the holonomy group is all of $K$; around a triangle $\{a,b\},\{b,c\},\{c,a\}$, which is not a 2-cell, the holonomy exchanges $a$ and $b$. (d) Every holonomy element acts on the algebra over the base pair by an inner automorphism, and $-I$ acts trivially, so the holonomy acts on $\mathfrak{su}(3)$ through a group of order 6. (e) The Wilson loops of the traces on $\R^8$ and on $\mathfrak{su}(3)$ take the values $(8,8)$, $(-8,8)$, $(2,-1)$, $(0,0)$, $(-2,-1)$ on holonomy elements of orders 1, 2, 3, 4, 6.

The complex of stars has free fundamental group of rank 21, that of the complete graph on the eight points; a loop is a closed walk on the eight points, and its holonomy exchanges the two points of the base pair exactly when the walk has odd length. Around a triangle the holonomy is the involution exchanging $a$ with $b$ and $c$ with its harmonic conjugate with respect to $a$ and $b$; around a 4-cycle its multiplier is the square of the cross-ratio of the two diagonals.

For a closed walk $(a_0,\dots,a_{k-1})$ traced by a loop of $\mathcal K$ at $\{a_0,a_1\}$, choose nonzero vectors $\hat a_i\in\F_7^2$ on the points $a_i$ and put $\kappa_i=[\hat a_i,\hat a_{i-1}]/[\hat a_i,\hat a_{i+1}]$, the corner factors, with indices modulo $k$. Let $h\in\SL(2,7)$ be the holonomy, and $P_{\mathrm{odd}}$, $P_{\mathrm{even}}$ the products of the $\kappa_i$, $1\le i\le k$, over odd and even $i$. If $k$ is even, $h\hat a_0=P_{\mathrm{odd}}\hat a_0$ and $h\hat a_1=P_{\mathrm{even}}\hat a_1$; if $k$ is odd, $h\hat a_0=P_{\mathrm{odd}}\hat a_1$, $h\hat a_1=P_{\mathrm{even}}\hat a_0$ and $h^2=-I$. For $k=4$ the eigenvalue on $\hat a_0$ is the cross-ratio $(a_0,a_2;a_1,a_3)$ itself.

A holonomy element fixing the base pair acts on the algebra over it by an inner automorphism when it commutes with left multiplication by $e_0$ on the complement of $\mathrm{span}(1,e_0)$, and by an outer one, complex conjugation followed by an inner automorphism, when it anticommutes. An element of $\PGL(2,7)$ is proper if it lies in $\PSL(2,7)$ and improper otherwise.

(a) An element of $\PGL(2,7)$ carries the family of algebras to itself if and only if it is proper; an improper one carries the algebra over the base pair to an algebra outside the family. (b) On the stabilizer of the base pair in $\PGL(2,7)$, the six proper elements are inner, and the six improper ones neither commute nor anticommute. (c) So every connection on a graph on the 28 pairs whose transports come from $\PGL(2,7)$ and carry the family has inner holonomy. (d) Of the twelve $\PSL(2,7)$-equivariant connections on $J(8,2)$ with transports in $\PGL(2,7)$, six are proper on every edge and carry the family with inner holonomy, among them the connection on the complex of stars; six are improper and do not carry it, among them the reflection connection, whose holonomy is improper exactly around the loops of odd length. (e) The connection on the Coxeter graph given by the bracket rule is proper and has inner holonomy.

The prediction made before the computation, that improper transports would give outer holonomy around odd loops, failed, and the alternative registered with it holds. The improper elements carry the family to a second family, that of the mirror table, the octonion table relabelled by $y\mapsto-y$; at each pair the unique seam between the two families is conjugation by the harmonic reflection of the pair, one of the 28 polarities of the Fano plane, and an improper connection is a proper one followed by that seam. The table is one point of a circle of products $A_z$, $|z|=1$, each with its own family of algebras, and the improper elements act on the circle by a reflection exchanging the table and its mirror. Its fixed points lie at the phase of the prime over 2, $z=\pm\bar\lambda/|\lambda|$ with $\lambda=(1+\sqrt{-7})/2$, and the two families there are invariant under all of $\PGL(2,7)$, an improper element fixing a pair acting on the algebra over it by an outer automorphism. But the sixteen vectors of the Weil representation multiply among themselves only at the table and its mirror: integrality forces the two families apart, and with them inner holonomy.

The group $\Gamma_S=\SL(2,\Z[\omega][1/\mathfrak p])$ carries the base vertex $t_0$ of the tree $T_{\mathfrak p}$ at 7, the scale tree, to other vertices and back, and the octonion table with its signs is carried along. Fix the edge $e_0=(t_0,t_\infty)$, whose stabilizer is the Iwahori subgroup $\Gamma_I$ of the matrices of $\SL(2,\Z[\omega])$ with lower left entry in $\mathfrak p$; choose one parent for each vertex, rooted at $e_0$, and frames in which every parent sits at $\infty$. The holonomy $h(\gamma,v)\in\PSL(2,7)$ of $\gamma$ at $v$ acts, through the double life read in Klein’s lattice and with the table’s light direction $a^*$ at $\infty$, on the seven points of the Fano plane, and the signed relabellings of the table $e_xe_{x+1}=e_{x+3}$, which form a group $2^3{\cdot}L_3(2)$ of order 1344, realize each collineation in eight ways. The residual $r(h)$ is the least number of units $e_x$ sent to $-e_{g(x)}$ by a relabelling over $h$: it is 0 on the Borel subgroup fixing $a^*$, of order 21, and 2 on the other 147 elements, and when it is 2 exactly three relabellings reverse two units, each the two other points of one of the three lines through a point $c_0$, and none reverses $c_0$.

(1) A loop $\gamma\in\Gamma_S$ at $t_0$ is realized by a relabelling that reverses no unit if and only if $\gamma(t_\infty)$ is the parent of $\gamma t_0$, equivalently $h(\gamma,t_0)(\infty)=\infty$; otherwise every relabelling that realizes it reverses at least two units, and the best reverse exactly two. (2) The condition depends only on the coset $\gamma\Gamma_I$ and is unchanged by $\Gamma_I$ on the left. In the decomposition of $\Gamma_S$ into double cosets $\Gamma_Iw\Gamma_I$ by the infinite dihedral group generated by $s_0$, fixing $t_0$, and $s_1$, fixing $t_\infty$, it holds exactly when the reduced word of $w$ ends in $s_1$. (3) For every loop $\gamma_0$, the 168 loops $\gamma_0\delta$, with $\delta$ running over $\SL(2,\Z[\omega])$ modulo the kernel of its action on the link of $t_0$, have 168 distinct holonomies, 21 of residual 0 and 147 of residual 2. At distance $2k$ the oriented edges reached split into $8\cdot7^{2k-1}$ that point toward $e_0$ and $8\cdot7^{2k}$ that do not. (4) The proportion one in eight is the same for every such choice of parents and every base vertex.

So the edge stabilizer $\Gamma_I$ is the largest subgroup on which the signed table is carried by relabellings without reversals.

$\Gamma_S$ has infinitely many primitive hyperbolic conjugacy classes of each translation length, and with $\pi=3+\omega$ the elements $\mathrm{diag}(\pi^{-1},\pi)$ and $\mathrm{diag}(\pi,\pi^{-1})$ are conjugate in $\SL(2,\Z[\omega])$ while their residuals at $t_0$ are 2 and 0. So the residual is not a function of the conjugacy class, the signed table defines no homomorphism on the loop group, and there is no twist of Ihara’s zeta function by it on any quotient of the scale tree.

The route through Ihara’s zeta function and its Artin–Ihara twists is closed twice: $\Gamma_S$ is not a tree lattice, and the signed table gives no class function. A third obstruction is internal: the relabelling group $2^3{\cdot}L_3(2)$ is a non-split extension, and every relabelling over a collineation of order four has order eight, so $\PSL(2,7)$ does not act on the signs at all.

Let $u=(e_0+\dots+e_6)/\sqrt7$ and $J_0=L_u$, a complex structure on $\Oct$ that splits $\Oct\otimes\C$ into two quartets. $G_2$ is the stabilizer of a unit spinor in $\mathrm{Spin}(7)$, and the stabilizer of $u$ is $\mathrm{SU}(3)_u$ in $G_2$ and $\mathrm{SU}(4)_u$ in $\mathrm{Spin}(7)$, whose chiral spinors are the two quartets. Call a holonomy misaligned when it moves the table’s light direction $a^*$, and let $g$ be its collineation, one of the 147 outside $F_{21}$. Then no relabelling $R$ over $g$ preserves $J_0$ or reverses it: $\langle u,R(u)\rangle$ is $\tfrac37$ for the three relabellings of residual 2, $-\tfrac17$ for four and $-\tfrac57$ for one.

(1) There is an orthogonal map $A_g$ of $\Oct$, unique up to sign, with $A_gL_{e_x}A_g^{-1}=L_{e_{g(x)}}$ for every $x$. It commutes with $J_0$, and it is an automorphism of $\Oct$, up to sign, exactly when $h$ fixes $a^*$; the maps $\pm A_g$ form a copy of $\SL(2,7)$. (2) Every relabelling over $g$ is $\pm A_gL_{e_{x_1}}\cdots L_{e_{x_k}}$, where $x_1<\dots<x_k$ are the units it reverses. (3) Of the 128 realizations of $g$ in $\mathrm{Spin}(7)$, exactly 2 preserve $J_0$, namely $\pm A_g$, exactly 8 preserve the product, the relabellings, and none preserves both unless $h$ fixes $a^*$.

For a misaligned holonomy let $T$ be the table $e_xe_{x+1}=e_{x+3}$ with its light direction and $T'=A_g\cdot T$ the table that $A_g$ carries across the loop. The unit of $T'$ is $A_g(1)=\pm e_{c_1}$, where $c_1=g(c_0)$ is the point of the antiflag whose pair is $\{a^*,h(a^*)\}$. $T$ and $T'$ have the same metric, the same $J_0$ and the same Cayley form of left type, the 4-form preserved by the spin group that the products $L_{e_x}L_{e_y}$ generate; their associative 3-forms differ, and their common derivations form $\mathfrak{su}(3)_{c_1}$.

(1) The following agree for $T$ and $T'$: the edges of the Coxeter graph and their involutions; the Fano orientation on the labels, modulo sign changes of the units; the assignment of points read from each table’s own tour; the face half-turns; for each of the 70 four-sets of $\Proj^1(\F_7)$, whether the product of its four units is real and otherwise which point its imaginary unit names; $J_0$; and the Cayley form of left type. (2) The unit moves from the line of 1 to the line of $e_{c_1}$, and the plane $\mathrm{span}\{1,e_c\}$ is a unital subalgebra of $T'$ only for $c=c_1$, where $T'$ exchanges the roles of the unit and $e_{c_1}$. (3) The planes carried by the maps $A_g$ from the seven pairs through $a^*$ form a family that every $A_g$ carries to itself, and it agrees with the planes $\mathrm{span}\{1,e_c\}$ exactly on those seven.

So at one scale only the table’s unit tells the two transports apart. An expectation registered before the computation, that the two tables share the Cayley form built from the triple cross product, failed: that form is preserved by the spin group of right multiplications, and the form of left type, identified afterwards, is the shared one.

Let each of the 28 antiflags carry its own copy of the table, the copies compared only along the edges of the Coxeter graph by the flat class of comparisons. Write $c(o)$ for the point of an antiflag $o$ and $\mathrm{Col}(c)$ for the 96 relabellings fixing $e_c$, which lie in $\mathrm{SU}(3)_c$. Over a collineation $g$, a per-observer transport chooses a relabelling $R_o$ over $g$ at each antiflag, with signs $s_o$ given by $R_oe_{c(o)}=s_oe_{g\,c(o)}$; it is clock-keeping when every $s_o=+1$, and a single map when every $R_o$ is the same. Along every edge $(o,o')$ the transformed comparison carries $e_{g\,c(o)}$ to $s_os_{o'}e_{g\,c(o')}$, so it keeps the edge’s class exactly when $s_os_{o'}=+1$: the signs form a $\Z/2$-cocycle on the Coxeter graph, and since the graph is connected they are constant.

Let $h$ be a misaligned holonomy and $g$ its collineation. (1) At every antiflag exactly four of the eight relabellings over $g$ keep its point, and any two of them differ by an element of $\mathrm{Col}(g\,c(o))$, so the clock-keeping transport is unique up to $\mathrm{Col}$ at every antiflag. (2) Every clock-keeping choice satisfies all 42 edges; one takes for $R_o$ a relabelling of residual 2 whose reversed pair avoids $c(o)$. (3) The transformed comparisons are gauge equivalent to the flat class by elements of the groups $\mathrm{Col}$. (4) For every antiflag $o$, the transport back along any path composed with $R_o$ lies in $\mathrm{Col}(c(o))$. (5) In every antiflag’s frame, $\mathrm{SU}(3)$ at the target point restores $J_0$.

Every pair of distinct points is joined by exactly two edges of the Coxeter graph, and no edge joins antiflags with the same point. A single map over $g$ that reverses a set $S$ of points fails the edge condition exactly on the $2|S|(7-|S|)$ of the 42 edges that join $S$ to its complement, also after any element of $\mathrm{Col}$ at the targets. Over a misaligned holonomy the eight single maps fail at 20 edges (the three of residual 2), 24 (four) and 12 (one), so none is consistent with the edges; over an aligned holonomy the unsigned map fails nowhere.

Let $X$ be the Coxeter graph with its 24 heptagons as faces. (1) $H_1(X;\Z)=0$ and $\pi_1(X)=1$, so a comparison system that is trivial around every heptagon is trivial around every closed path, whatever its structure group. (2) For one broken loop the flat $\Z/2$ data on the mapping torus form $\Z/2$: a reversal of every antiflag together, never of a pattern of points. (3) $\operatorname{Hom}(\SL(2,\Z[\omega]),\Z/2)=0$ and $\operatorname{Hom}(\Gamma_S,\Z/2)=0$; so if the symmetries of the scale tree act by lifts consistent with the edges, the global sign they assign is trivial.

Over the stabilizer $S_3$ of an antiflag the extension $2^3{\cdot}L_3(2)$ splits, faithfully, by clock-keeping lifts, and the induced lifts form an honest action of the 168 collineations on the 28 copies; so the relabellings, which cannot act on one table without leaving signs, act honestly antiflag by antiflag. A flat comparison system invariant under this action would give a homomorphism of $\PSL(2,7)$ into a group $\mathrm{Col}$ of order 96 restricting to that faithful splitting, which simplicity forbids; with values in $\mathrm{SU}(3)$ it would be one of Klein’s representations $\mathbf3$ or $\bar{\mathbf3}$.

Each vertex of the scale tree carries a Coxeter graph on the 28 pairs of its link. Take the root $t_0$ and its child $v$ in direction 0. A site of the two-frame truncation is a pair of its 14 boundary classes: the 21 root-side pairs, the 21 fine pairs in the link of $v$, and 49 mixed sites, seen at $t_0$ as a pair $\{0,d\}$ and at $v$ as $\{u,\infty_v\}$; two sites meet at a frame when both are its pairs and are joined there by an edge of its Coxeter graph. In $\Proj^1(\Z/49)$ a fine pair has exactly 49 coarse partners, all over one pair of the child frame, and the coarse pairs over a residue disk join the two stars by a complete bipartite graph $K_{7,7}$.

(1) The truncation has 91 sites and 336 meetings and no triangles. Its 882 four-cycles lie inside one frame, through two sites over one pair, and their holonomy is the identity for every frame-dependent link. (2) It has exactly 882 five-cycles, all crossing scales, 441 with three meetings at each frame; with the frames flat, all are trivial exactly when the identifications across scales are constant. (3) With each frame’s lifted heptagons and four-cycles as faces the complex has $H_1\cong\Z^{48}$, without torsion; adding the five-cycles gives $H_1=0$ and a simply connected complex, so across one edge of the tree a flat comparison system is unique up to gauge once the five-cycles are flat. (4) For a hopping with one hop of unit amplitude per meeting, carrying the block of its comparison, in every background with reversible links $\operatorname{tr}H^k$ for $k\le4$ does not depend on the background, and $\operatorname{tr}H^5=60\,(882-W_5)$, with $W_5=\sum\bigl(1-\tfrac13\operatorname{Re}\operatorname{tr}h\bigr)$ over the five-cycles.

Put Wilson’s weight $\beta$ on the heptagons and $w\beta$ on each five-cycle, and integrate out the finest frames. A star, the seven pairs through one point of a frame’s projective line, has members pairwise at distance three with a unique 3-path, and the 21 transports along these paths form its star connection; the 3-path map from the cycle space of $K_7$ to the frame’s is an isomorphism. At leading order in $1/\beta$, integrating out a child frame with any positive semi-definite effective form on its 42 links, together with the 49 cross links, induces on the parent’s star connection the form $7w\,P_{\mathrm{cycle}}$, whatever the child’s form, with $P_{\mathrm{cycle}}$ the projector onto the cycle space of $K_7$. With its seven children integrated out a frame’s effective form is reached after one level and unchanged by deeper levels; on the 15 physical directions its eigenvalues against the heptagon form are $1+w\kappa$, with $\kappa=12\pm\sqrt{15/2}$ (six times each), $21/2$ (twice) and $129/8$ (once).

Every plaquette weight of the tower is ferromagnetic, and for the $\Z/2$ factors of the signs and for $U(1)$ every Wilson loop of a frame is non-decreasing as children are attached, by the inequalities of Griffiths and Ginibre. Refining each meeting to level two copies loops rather than subdividing them: every meeting becomes seven disjoint copies of $K_{7,7}$, every heptagon lifts to exactly $7^7$ heptagons of full length, no loop is subdivided, and at leading order the coarse coupling is $7^7$ times the fine one.