  We study matrix quantum mechanics at a finite temperature equivalent to one
dimensional compactified string theory with vortex (winding) excitations. It is
explicitly demonstrated that the states transforming under non-trivial U(N)
representations describe various configurations vortices and anti-vortices. For
example, for the adjoint representation the Feynman graphs (representing
discretized world-sheets) contain two faces with the boundaries wrapping around
the compactified target space which is equivalent to a vortex-anti-vortex pair.
A technique is developed to calculate partition functions in a given
representation for the standard matrix oscillator. It enables us to obtain the
partition function in the presence of a vortex-anti-vortex pair in the double
scaling limit using an analytical continuation to the upside-down oscillator.
The Berezinski-Kosterlitz-Thouless phase transition occurs in a similar way and
at the same temperature as in the flat 2D space. A possible generalization of
our technique to any dimension of the embedding space is discussed.
