Seminarium Teorii Względności i Grawitacji

This talk is about recent results on the Poisson algebra of Wilson lines on null initial surfaces. Our results are markedly different from the spacelike case. On a spacelike hypersurface, Wilson lines commute under the Poisson bracket. On a mull initial surface, this is no longer true because electric and magnetic fields are no longer functionally independent. We compute the Poission brackets and find that Wilson lines commute unless they intersect the same light ray. If they intersect the same light ray, the Poisson commutation relations are determined by the geometry at the intersection and the conformal class of the pull-back of the metric at the null initial surface. Finally, we propose a quantisation of the classical Poisson commutation relations. The non-commutativity of the classical Wilson lines is realised at the quantum level by fermionic anti-commutation relations. Each Wilson lines turns into an anti-commuting Grassmann variable. This effect is known from CFTs in which the n-point functions of exponentials of bosons behave as fermions. The talk is based on arXiv:2511.19756.