Anomalies and large N limits in matrix string theory

We study the loop expansion for the low energy effective action for matrix string theory. For long string configurations we find the result depends on the ordering of limits. Taking $g_s\to 0$ before $N\to \mathrm{\infty }$ we find free strings. Reversing the order of limits however we find anomalous contributions coming from the large $N$ limit that invalidate the loop expansion. We then embed the classical instanton solution into a long string configuration. We find the instanton has a loop expansion weighted by fractional powers of $N$. Finally we identify the scaling regime for which interacting long string configurations have a well defined large $N$ limit. The limit corresponds to large “classical” strings and can be identified with the "dual of the 't Hooft limit, $g_{SYM}^2\sim N$.