seam monodromy

Floor 2, Les sutures · introduced in Chapter 4, La monodromie des sutures

What does going around a loop of natural identifications do?

The composite of seams around a closed walk, an automorphism of the incarnation; it measures how far a family of seams is from one choice of alignments.

Plate 2.6The monodromy

\tau

on one flex triangle of the Klein quartic: each flex goes along its tangent to the other flex on it, and three steps return.

Definition(Seam system, monodromy)

The monodromy of a cycle $\gamma=(s_1^{\epsilon_1},\dots,s_m^{\epsilon_m})$ of a seam system, starting and ending at $Y$ , is

\mathrm{mon}(\gamma)=s_m^{\epsilon_m}\circ\cdots\circ s_1^{\epsilon_1}\in\Aut_G(Y).

The word is used as for coverings: going around a loop of identifications returns a permutation of the fibre. Here the fibre is an incarnation and the permutation is an automorphism of the object; through an alignment it is an element of $N_G(H)/H$ , well defined up to conjugation.

Proposition

If $N_G(H)/H$ is abelian, then for each incarnation $Y$ the isomorphism $\Aut_G(Y)\cong\Aut_G(X)$ given by an alignment does not depend on the alignment, and monodromy is a homomorphism from the fundamental group of the graph of the system to $\Aut_G(X)\cong N_G(H)/H$ .

Proof

Two alignments differ by an automorphism $a$ of $X$ , and the two isomorphisms differ by conjugation by $a$ , which is trivial in an abelian group. Concatenating cycles composes monodromies.

Theorem(Monodromy of the object of size 24)

Let $\tau$ be the composite of the tangent and residual-point seams: a flex of the Klein quartic goes to the other flex on its tangent. Then $\tau$ is an automorphism of the flexes of power 4, so the cycle flexes $\to$ flex tangents $\to$ flexes, along the two natural seams, has monodromy of order 3. It permutes each flex triangle cyclically:

\tau\colon\ (1:0:0)\mapsto(0:0:1)\mapsto(0:1:0)\mapsto(1:0:0).

Proof

The rotation of $(0:0:1)$ is $g$ , and the rotation of $(0:1:0)$ is the element acting there by $\zeta$ , which is $g^4$ , since $\rho(g)$ acts there by $\zeta^2$ and so $\rho(g)^4$ by $\zeta^8=\zeta$ . As $\tau(0:0:1)=(0:1:0)$ , the rotation seam carries $\tau$ to a map sending $g$ to $g^4$ , which is the fourth-power map.

Remark

So the answer for non-rigid objects is mixed. Seams fixed by the conventions of their theories are consistent wherever they meet. But a single theory may supply two natural seams between the same two incarnations, and then a cycle of length two already has nontrivial monodromy: the contact point and the residual point of a flex tangent, or the roles of a point in its line, whose three seams have relative powers 1, 2 and 4. The monodromy is then an invariant of the theory; here it is the cyclic order that the tangents put on each flex triangle, a fact of the projective geometry of the quartic.

Theorem(The Coxeter edges and the marking)

Let the Coxeter graph be in its antiflag model, with $G$ acting through a marking $\mu$ . Each edge has the form $\{(p,B),(q,B')\}$ , with $B$ and $B'$ meeting in the third point $c$ of the line $pq$ .

(a) The point rule, which goes from each point of $B$ off $pq$ to the third point of its line with $p$ , and from each point of $B'$ off $pq$ to the third point of its line with $q$ , traces a directed 4-cycle on the quadrangle complementary to $pq$ . The line rule traces, dually, a directed 4-cycle on the four lines missing $c$ . Each rule, followed by the element of order 4 that advances its cycle one step, is a seam from the edges to $4A$ , and the two rules give mutually inverse elements.

(b) The vertex seam sends an antiflag to the pair of points of $\Proj^1(\F_7)$ with the same stabilizer, and an edge to a harmonic pair of disjoint pairs $\{\{a,b\},\{c,d\}\}$ . Of the two directed 4-cycles $a\to c\to b\to d\to a$ and $a\to d\to b\to c\to a$ , exactly one has $[a,c][c,b][b,a]$ a nonzero square, and the bracket rule sends the edge to the element of order 4 advancing that cycle one step.

(c) If $\mu$ differs from $\mu_A$ by an inner automorphism, the bracket rule agrees with the point rule on every edge; if by an outer one, it agrees with the line rule.

Consequently the seam system for $G/C_4$ formed by the Coxeter edges, the harmonic pairs of pairs and the class $4A$ , with the vertex seam, the bracket rule and the point rule, is coherent when the marking is in the class of $\mu_A$ , and its monodromy is the nontrivial automorphism otherwise.

Proof

(a) The rules use only incidence and treat the two antiflags of an edge alike, so they are $G$ -maps; that they give inverse elements was checked by machine. (b) In $[a,c][c,b][b,a]$ each point occurs twice, so its square class does not depend on the coordinate vectors, and it is invariant under $\SL(2,7)$ . With $a=0$ , $b=\infty$ , harmonicity gives $d=-c$ , and the products for the cycle $a\to c\to b\to d$ are all in the square class of $c$ , while the reverse cycle gives that of $-c$ ; as $-1$ is not a square modulo 7, exactly one cycle has a square product. (c) For $\mu_A$ the agreement was checked on all 42 edges. An inner change of marking is induced by a collineation, which commutes with all the constructions. An outer change, by conjugation with a Möbius map of non-square determinant, multiplies every bracket by a non-square, so it reverses the bracket rule.

Proposition(Monodromy as the kernel of holonomy)

Let a seam system over a connected graph $\mathcal G$ have holonomy $\mathrm{hol}\colon\pi_1(\mathcal G,v)\to A$ , read as a description; its kernel $N$ is the group of loops around which the seams close up. The system is coherent if and only if $N=\pi_1(\mathcal G,v)$ , and the group of monodromies is $\pi_1(\mathcal G,v)/N$ . So monodromy is what remains of the loops once the kernel of the holonomy is divided out.

Proof

Holonomy is a homomorphism on the fundamental group, and the system is coherent exactly when every holonomy is trivial.

Examplecomputed

Monodromy can be the spinor sign. In the lattice $E_8$ preserved by $\SL(2,7)$ for one class of tetrahedra of Thurston’s manifold, the 224 half-roots form two copies $O_1$ and $O_2$ of the new object of size 112, whose automorphism group is $C_4$ . The reflection seam, changing the sign of a half-root at the point of its support fixed by its stabilizer, is a seam from $O_1$ to $O_2$ and back, and the cycle it forms has trivial monodromy, since its square is the identity. The sign seam, the sign pattern of $Cr$ on the support of $r$ , with $C$ the conference matrix of the Weil representation, equals the reflection seam on $O_1$ and its negative on $O_2$ : the cycle it forms has monodromy $-I$ . Half of $Cr$ off the support is an automorphism of $O_2$ of order 4 with square $-1$ , the integral shadow of multiplication by $i$ , and it generates the automorphism group.

Open questionopen

On the object $G/V_4^b$ the automorphism group is $S_3$ , not abelian, so monodromy is defined only up to conjugation. A natural seam system with non-abelian monodromy is not known: the natural seams found there, the elation and centre seams, carry swaps to swaps and rotations to rotations, and are coherent.

Built from: seam systemcoherencealignment
Builds: gaugeseam theoryGalois category
Objects: the object of size 24the object of size 42, cyclicthe object of size 42, class b
In the Esquisse: 5Courte marche à travers la théorie de Galois10La table en deux, en sept et à l’infini11Le revêtement double et le miroir16Une loi de réciprocité
In the volume: XIRulial Relativity