  We construct a matrix representation of compact membranes analytically
embedded in complex tori. Brane configurations give rise, via Bergman
quantization, to U(N) gauge fields on the dual torus, with
almost-anti-self-dual field strength. The corresponding U(N) principal bundles
are shown to be non-trivial, with vanishing instanton number and first Chern
class corresponding to the homology class of the membrane embedded in the
original torus. In the course of the investigation, we show that the proposed
quantization scheme naturally provides an associative star-product over the
space of functions on the surface, for which we give an explicit and
coordinate-invariant expression. This product can, in turn, be used the
quantize, in the sense of deformation quantization, any symplectic manifold of
dimension two.
