  The quantum torus algebra plays an important role in a special class of
solutions of the Toda hierarchy. Typical examples are the solutions related to
the melting crystal model of topological strings and 5D SUSY gauge theories.
The quantum torus algebra is realized by a 2D complex free fermion system that
underlies the Toda hierarchy, and exhibits mysterious "shift symmetries". This
article is based on collaboration with Toshio Nakatsu.
