  In this paper a formulation of U(1) gauge theory on a fuzzy torus is
discussed. The theory is regulated in both the infrared and ultraviolet. It can
be thought of as a non-commutative version of lattice gauge theory on a
periodic lattice. The construction of Wilson loops is particularly transparent
in this formulation. Following Ishibashi, Iso, Kawai and Kitazawa, we show that
certain Fourier modes of open Wilson lines are gauge invariant. We also
introduce charged matter fields which can be thought of as fundamentals of the
gauge group. These particles behave like charges in a strong magnetic field and
are frozen into the lowest Landau levels. The resulting system is a simple
matrix quantum mechanics which should reflect much of the physics of charged
particles in strong magnetic fields.
  The present results were first presented as a talk at the Institute for
Mathematical Science, Chennai, India; the author wishes to thank Prof. T. R.
Govindarajan and the IMS for hospitality and financial support, and the
audience for pointing me out the work of Ambjorn, Makeenko, Nishimura and
Szabo. The author is also grateful to the organizers of the Summer School in
Modern Mathematical Physics, Sokobanja, Serbia, Yugoslavia, for financial
support and hospitality during presentation of the present work at
international conference FILOMAT 2001, Nis, Yugoslavia.
