La monodromie des sutures

Systèmes de suturesSeam systems

The object $X_{24}=G/C_7$ has automorphism group $N_G(C_7)/C_7\cong C_3$, so between any two of its incarnations there are three seams. Its incarnations include the classes $7A$ of $g\colon z\mapsto z+1$ and $7B$ of $g^{-1}$, each of 24 elements; the 24 nonzero vectors of $\F_7^2$ up to sign; the 24 flexes and the 24 flex tangents of the Klein quartic; the cyclic labellings of the Fano plane modulo translation, for $\{0,1,3\}$ and for $\{0,4,6\}$; the 24 perfect matchings of the Heawood graph and the 24 heptagons of the Coxeter graph; and, from the literature, the faces of Klein’s map and the cusps of $X(7)$.

A seam system for an object is a family of incarnations with a set $E$ of seams between members of the family, a seam from an incarnation to itself allowed. A cycle is a closed walk $\gamma=(s_1^{\epsilon_1},\dots,s_m^{\epsilon_m})$ at an incarnation $Y$, and its monodromy is $s_m^{\epsilon_m}\circ\cdots\circ s_1^{\epsilon_1}\in\Aut_G(Y)$; the system is coherent if every cycle has trivial monodromy. The word is used as for coverings: going around a loop of identifications returns a permutation of the fibre, here an automorphism of the object.

Monodromy measures exactly how far a family of seams is from being induced by one choice of alignments. For a non-rigid object any automorphism is the monodromy of some cycle, so the notion has content only for seams that the theories give rather than seams that are chosen. These are called natural, without formalizing the word, and each is recorded with the construction that defines it.

La puissanceThe power

For a cyclic stabilizer the automorphisms can be named by numbers. If $H=\langle x\rangle$, of order $n$, is its own centralizer, the automorphisms of the conjugacy class $x^G$ are the power maps $y\mapsto y^k$ for which $x^k$ is conjugate to $x$, and these $k$ form a group $K_H\cong N_G(H)/H$. Carried along any seam to any class $(x^j)^G$, an automorphism of any incarnation becomes one and the same power map: its power.

For the group of order 168 the lemma applies to $C_3$ and $C_4$, with $K=\{\pm1\}$, and to $C_7$, with $K_{C_7}=\{1,2,4\}$, the squares modulo 7; it does not apply to $C_2$, whose involutions have centralizer $D_8$. For $X_{24}$ every automorphism of every incarnation has a power in $\{1,2,4\}$, and two automorphisms of two incarnations correspond under some seam, equivalently every seam, exactly when they have the same power. Hall’s multiplier is an example: 2 is a multiplier of the difference set $\{0,1,3\}$, by Hall’s theorem because 2 is the order of the plane, so $2\ell$ is again a cyclic labelling, and its Singer map is the fourth power of the Singer map of $\ell$: the multiplier has power 4.

Les sutures naturellesNatural seams

Each theory supplies seams by its own conventions. The rotation uses the complex structure, through the choice $\zeta=e^{2\pi i/7}$; the transvection uses the symplectic form $\det$ on $\F_7^2$; the Singer map uses the translation $x\mapsto x+1$ of $\Z/7$. Each is a convention of its theory, and the question is how they fit.

Within the quartic there are two natural seams between the flexes and their tangents: a flex goes to its tangent line, and a tangent goes to its residual point. A flex tangent meets the quartic at its flex with multiplicity 3 and at exactly one other point, again a flex: at $(1:0:0)$ the tangent is $y=0$, which meets the curve where $z^3x=0$, at $(1:0:0)$ three times and at $(0:0:1)$ once.

Un cycle cohérentA coherent cycle

Seams fixed by the conventions of their theories are consistent wherever they meet. From a labelling $\ell$ for $\{0,1,3\}$ the Singer map lands in $7B$; inversion takes it to $7A$; the Singer map read backwards returns the labelling $-\ell$ for $\{0,4,6\}$; and reversal read backwards returns $\ell$. The cycle closes because the Singer map of $-\ell$ is the inverse of the Singer map of $\ell$: part (a) of the theorem.

On the projective line the same happens for the object of size 56. The element of order 3 fixing $a$ and $b$ with multiplier 2 at $a$, the element rotating a three-subset in the cyclic order $(a,b,c)$ for which $[a,b][b,c][c,a]$ is a square in $\F_7$, and the map from ordered pairs to three-subsets that this square class defines form a triangle of natural seams, and it is coherent: the multiplier 2, the square class and the cyclic order fit together without monodromy. The square class is invariant under $\SL(2,7)$ and changes under transpositions, since $-1$ is not a square modulo 7.

La tangente et le point résiduelThe tangent and the residual point

A single theory may supply two natural seams between the same two incarnations, and then a cycle of length two already has nontrivial monodromy. Going from a flex to its tangent and from the tangent to its residual point composes to $\tau$, which sends each flex to the other flex on its tangent. On the coordinate triangle $\tau(1:0:0)=(0:0:1)$, $\tau(0:0:1)=(0:1:0)$ and $\tau(0:1:0)=(1:0:0)$, and $\tau$ turns every one of the eight flex triangles cyclically: the monodromy is the cyclic order the tangents put on each flex triangle, a fact of the projective geometry of the quartic.

Its power is 4: the rotation of $(0:0:1)$ is $g$, the element acting at $(0:1:0)$ by $\zeta$ is $g^4$, so the rotation seam carries $\tau$ to $g\mapsto g^4$. The roles of a point in its line give a second example: the three role seams differ pairwise by automorphisms, with relative powers 1, 2 and 4. Such a monodromy is an invariant of the theory, and the power makes it comparable across theories: under every seam between the flexes and the cyclic labellings, $\tau$ corresponds to Hall’s multiplier 2, both being the fourth-power map on $7A$ and $7B$. Read as a lattice gauge connection, the two seams between the flexes and their tangents form a cycle of length two whose holonomy is $\tau$, of order 3 in $C_3$, and since that group is abelian $\tau$ itself is gauge invariant.

La courbe modulaireThe modular curve

The cusps of the modular curve $X(7)$ join the vectors of $\F_7^2$ to the flexes. Let $\Gamma(7)$ be the principal congruence subgroup of level 7 of $\SL(2,\Z)$ and $X(7)$ the compactified quotient of the upper half-plane. The group $\SL(2,\Z)/\pm\Gamma(7)=\PSL(2,\Z/7)=G$ acts on $X(7)$, the cusps $a/c$, with $a$ and $c$ coprime, correspond equivariantly to the vectors $\pm(a,c)$ modulo 7, and $X(7)$ is isomorphic to the Klein quartic.

At the cusp $\infty$ the local coordinate is $q^{1/7}=e^{2\pi i\tau/7}$, and $g\colon\tau\mapsto\tau+1$ multiplies it by $\zeta$. The cusp $2/7$ is $\gamma\infty$ for $\gamma=\left(\begin{smallmatrix}2&1\\7&4\end{smallmatrix}\right)$, and $\gamma g\gamma^{-1}\equiv g^4$ modulo 7, so $g^4$ acts by $\zeta$ at $2/7$ and $g$ by $\zeta^2$; similarly $g$ acts by $\zeta^4$ at $3/7$. These are the three fixed points of $g$, and the numbers are holomorphic invariants. On the quartic, $\rho(y)$ acts at its fixed flexes by $\zeta,\zeta^2,\zeta^4$ for $y\in7A$ and by $\zeta^3,\zeta^5,\zeta^6$ for $y\in7B$, so any isomorphism carries the modular action to $\rho$ through an inner automorphism, and can be corrected to be equivariant.

Les systèmes de sutures sont des connexionsSeam systems are connections

A seam system over a graph is a lattice gauge connection in disguise, with the stabilizer principle supplying the gauge group. Let a system over a connected graph $\mathcal G$, each edge taken with both orientations, assign an incarnation $Y_v$ to each vertex and a seam $s_e\colon Y_v\to Y_w$ to each oriented edge, with $s_{\bar e}=s_e^{-1}$, and let $A=\Aut_G(X)\cong N_G(H)/H$. A choice of alignments, one per vertex, is a gauge; it turns the seams into link variables in $A$, and holonomy is monodromy read in the gauge.

None of this is new: it is the dictionary between local systems on a graph and representations of its fundamental group, in the language of lattice gauge theory. What seam theory supplies is the gauge group, $N_G(H)/H$, and the examples. The plate shows one from Chapter 11: a connection on the 28 pairs of points of $\Proj^1(\F_7)$, two pairs adjacent when they share a point, whose triangles of pairs through a common point are the 2-cells of a complex. The connection is flat, and around a triangle of pairs that is no 2-cell the holonomy is nontrivial, so the system is not coherent.

La rangée de SingerThe Singer row

The object $G/(7{:}3)$ of size 8 is rigid. Its incarnations include the points of $\Proj^1(\F_7)$, the Sylow 7-subgroups, the flex triangles, the triples of Coxeter heptagons, and the eight cyclic orientations of the Fano plane. The last come from Singer’s description of $\mathrm{PG}(2,2)$ as $\Z/7$ with lines $x+\{1,2,4\}$: a Singer cycle permutes the points cyclically, and a Sylow 7-subgroup of $\GL(3,2)$ is the group it generates.

The cyclic orientation of the difference set $\{0,1,3\}$ orients each line $\{x,x+1,x+3\}$ as $(x,x+1,x+3)$. These 21 oriented triples are those of the octonion multiplication table $e_xe_{x+1}=e_{x+3}$, and they form a triangle presentation, in the sense of Cartwright, Mantero, Steger and Zappa, compatible with $\lambda(x)=x+\{1,2,4\}$; through their theorem it gives a building of type $\tilde A_2$ whose vertex links are Fano incidence graphs, read in Chapter 9 as a completion of the Fano plane. What the seam table adds is the place of $\lambda$ among the objects.

So the antiflag of the object of size 28 and the Singer cycle of the object of size 24 meet in the triangle presentation: $\lambda$ chooses, for each Sylow 7-subgroup, one of its four orbits on antiflags, and the multiplier group $N(P)/P\cong C_3$, which acts on the object of size 24 by the powers, is exactly what makes the choice canonical. At the point 0 the four lines missing it are $\lambda(0)=\{1,2,4\}$, $\{2,3,5\}$, $\{3,4,6\}$ and $\{1,5,6\}$, one in each orbit of the translations; the multiplier $x\mapsto2x$ keeps $\{1,2,4\}$ and carries the other three around a cycle.

Le second blocThe second block

The octonions meet the group through their multiplication table. Number the imaginary units $e_x$, $x\in\Z/7$, so that $e_xe_{x+1}=e_{x+3}$, and identify the plane of the table with the Fano plane by a cyclic labelling for $\{0,1,3\}$: the unit lines are the points, of class $S_4^a$, and the quaternion subalgebras the lines, of class $S_4^b$. What is new is the integral structure. For a binary code $C$ of length 8, on coordinates indexed by $1,e_0,\dots,e_6$, let $L_C=\Z^8+\tfrac12C$; if $C$ is doubly even of dimension 4, $L_C$ is a copy of $E_8$ scaled to minimal norm 1, with the 240 minimal vectors $\pm1$, $\pm e_x$ and $\tfrac12(\pm u_1\pm u_2\pm u_3\pm u_4)$ over the codewords of weight 4.

The closed lattices are the integral octonions of Coxeter; the lattice of the single fixed plane, with the half-units $\tfrac12(1+e_a+e_b+e_c)$ over the table’s own lines, is the one Kirmse proposed, which is not closed. The bridge from an octavian order to a point of the Fano plane is built, and the other orbits are new rows of the table: the 14 lattices whose plane shares one line with the table incarnate $G/A_4^b$, and the 8 sharing none $G/(7{:}3)$.

The two graphs refine as well. In the Heawood graph the cycles of length 6, 8, 10, 12 and 14 form one orbit each, of classes $S_3$, $D_8$, $C_2$, $C_3$ and $C_7$. In the Coxeter graph the cycles of length 7, 8, 9 and 10 form one orbit each, of classes $C_7$, $D_8$, $C_3$ and $C_2$, there are none of length 11, and those of length 12 form two orbits, of classes $C_3$ and $C_2$; pairs of vertices at distance 2 form one orbit, of class $C_2$, at distance 3 a regular orbit of 168, and at distance 4 the pairs of antiflags with a common point (class $V_4^b$) or a common line ($V_4^a$). With these, every one of the fifteen objects has an incarnation in the graphs too.