A Matrix Model for Heterotic $Spin(32)/Z_2$ and Type I String Theory

We consider Heterotic string theories in the DLCQ. We derive that the matrix model of the Spin(32)/$Z_2$ Heterotic theory is the theory living on N D-strings in type I wound on a circle with no Spin(32)/$Z_2$ Wilson line on the circle. This is an O(N) gauge theory. We rederive the matrix model for the $\e8$ Heterotic string theory, explicitly taking care of the Wilson line around the lightlike circle. The result is the same theory as for Spin(32)/$Z_2$ except that now there is a Wilson line on the circle. We also see that the integer N labeling the sector of the O(N) matrix model is not just the momentum around the lightlike circle, but a shifted momentum depending on the Wilson line. We discuss the aspect of level matching, GSO projections and why, from the point of view of matrix theory the $\e8$ theory, and not the Spin(32)/$Z_2$, develops an 11'th dimension for strong coupling. Furthermore a matrix theory for type I is derived. This is again the O(N) theory living on the D-strings of type I. For small type I coupling the system is 0+1 dimensional quantum mechanics.