PS on shapes of strings and wormholes

What is the relation of the shape functions of string and wormhole?
Easy in the toy model below, for the string defect

g -> x^2 + y^2 far from the string

for the wormhole

f -> (x^2 + y^2 + z^2)^1/2

therefore

f^2 = z^2 + g

for a "vortex string" on the z-axis for the Vacuum ODLRO field of the fabric of space-time.

whose 3D Hodge * dual is a wormhole (Einstein-Rosen Bridge)