Fixed cone and oriented holonomy

Use Minkowski coordinates $z=(T,x,y)$ and

For a prescribed deficit $0<{\mu }_{\mathrm{c}}<2\pi$, set $\alpha =1-{\mu }_{\mathrm{c}}/(2\pi )$. The punctured cone metric is

Its completion has a timelike conical axis. Its spinless point-particle interpretation is standard DeserJackiwHooft1984; here the metric is prescribed rather than derived.

On the universal cover, let

With

we have

Choose angular chart lifts on

Refine if needed to injective developing charts. The two overlap components have different transitions:

The positively oriented particle loop has ordered affine word

The overlap-component index is essential. No flat chart is asserted to contain the axis.

These matrices act on spacetime tangent vectors. More generally, if

a lifted parallel vector has constant developed components. At the endpoint, differentiation of this identity gives

Thus the tangent holonomy in the starting developed frame is $L_{\gamma }$. This is gluing data for the spacetime, not an unrelated internal fibre; compare the polygon-model formulations in tHooft1992,Kadar2005.

For the certificate fix

Its value is independent of the slicing slopes introduced next. The future unit holonomy axis is $e_0=(1,0,0)$. For a marked spacelike face with future unit normal $n$, define

Proper future-preserving chart changes transform both vectors and conjugate the holonomy, preserving this quantity. It is a particle–slice observable, not a particle invariant alone.