A flat domain can still twist a field

Carry a branch
around a hole.

The annulus is orientable and has a global Cartesian frame. Yet a line or polyvector field can return with its local branch permuted. This isolates field monodromy without surface curvature or a Möbius twist.

ANNULUS · ONE GENERATORthe line closes even when its selected arrow reverses

unordered field transported branch chart cut

SHOW

selected loophole 1

branch transportsheet 0 → 1

hole 1 monodromy+1 mod 2

hole 2 monodromynot present

loop around both+1 mod 2

defect index1/2

Topology of the domain

Rotational symmetry N

Charge around hole 1branch shift m₁ modulo N1

Loop to transport around

Carry the selected branchzero to one material circuit0.00 turns

MONODROMY LEDGER

\rho:\pi_1(M)\longrightarrow\mathbb Z_N

For two holes, the two inner loops are independent. A counterclockwise loop enclosing both receives their sum modulo N.

Start with the line field

The arrow is local.
Its square is global.

For $N=2$ , a chosen unit branch $p$ and its negative describe the same line. Instead of storing either arrow, store the symmetric traceless tensor that both produce.

PROJECTIVE STATE

Q=p\otimes p-\tfrac12|p|^2g

Q(-p)=Q(p)

Cross a chart cut and $p$ may jump to $-p$ . The displayed line and $Q$ remain continuous.

01
Pick one arrow locally.
The orange arrow is a lift of the unordered field to one branch sheet.
02
Transport, do not reselect.
Following the same branch around a charged hole can land on another root of the same power field.
03
Forget the branch globally.
The line tensor or polynomial coefficients glue without remembering the local root label.

Why add a second hole?

Two loops.
Two independent choices.

The disk with two holes is a pair of pants. Its fundamental group has two generators, so an $N$ -symmetric field can assign independent cyclic shifts $m_1$ and $m_2$ .

(m_1,m_2)\in\mathbb Z_N\times\mathbb Z_N

m_{\mathrm{both}}=m_1+m_2\pmod N

For a line field, $(m_1,m_2)=(1,1)$ reverses the arrow around either hole, but a loop enclosing both reverses twice and returns to its original branch.

Where are the Clebsch variables?

A Clebsch atlas,
not one global triple.

On every simply connected chart, the selected branch is an ordinary one-form and can be written as $p=d\phi+\alpha\,d\beta$ . Charged loops prevent that signed branch—and therefore one scalar triple—from closing globally.

ONE-HOLE · EXPLICIT LINE LIFT

\phi=\rho\cos\tfrac\theta2,\quad\beta=\theta,\quad\alpha=-\tfrac\rho2\sin\tfrac\theta2

p=d\phi+\alpha\,d\beta=\cos\tfrac\theta2\,d\rho-\rho\sin\tfrac\theta2\,d\theta

After one circuit, $(\phi,\alpha,p)$ change sign. The tensor $Q(p)$ does not.

GENERAL N · POLYNOMIAL STATE

P_x(\xi)=\prod_{k=0}^{N-1}(\xi-p_k)

P_x(\xi)=\xi^N+c_1\xi^{N-1}+\cdots+c_N

The local Clebsch branches may permute; the symmetric coefficients remain global. This is the polyvector analogue of replacing $p$ by $Q$ .

Carry a brancharound a hole.

Two loops.Two independent choices.

Carry a branch
around a hole.

Two loops.
Two independent choices.