  The tensionless limit of classical string theory may be formulated as a
topological theory on the world-sheet. A vector density carries geometrical
information in place of an internal metric. It is found that path-integral
quantization allows for the definition of several, possibly inequivalent
quantum theories. String amplitudes are constructed from vector densities with
zeroes for each in- or out-going string. It is shown that independence of a
metric in quantum mechanical amplitudes implies that the dependence on such
vector density zeroes is purely topological. For example, there is no need for
integration over their world-sheet positions.
