Index Theory in the Moyal Algebra

Working within the Moyal algebra, we define an index for Laurent series of smooth complex matrix-valued functions of 2d real variables. For d = 1, we prove that the index of a Laurent series is equal to a multiple of the winding number of its initial term’s determinant. For d > 1, under certain conditions, we prove that the index of a Laurent series depends only upon its initial term’s behaviour on a (2d− 1)-sphere. We show that the index is a homotopy invariant, and finally prove the main result: the index is quantized, with the constant of quantization depending on d.