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We study contact interactions for long world-lines on a curved surface,focusing on the average number of times two world-lines intersect as a functionof their end-points. The result can be used to extend the concept ofpath-ordering, as employed in the Wilson loop, from a closed curve into theinterior of a surface spanning the curve. Taking this surface as a stringworld-sheet yields a generalisation of the string contact interactionpreviously used to represent the Abelian Wilson loop as a tensionless string.We also describe a supersymmetric generalisation.

Intersection of world-lines on curved surfaces and path-ordering of the Wilson loop