Black Hole Entropy in M-Theory

Extremal black holes in M-theory compactification on $M\times S^1$ are microscopically represented by fivebranes wrapping $P\times S^1$, where $M$ is a Calabi-Yau threefold and $P$ is a four-cycle in $M$. Additional spacetime charges arise from momentum around the $S^1$ and expectation values for the self-dual three-form field strength in the fivebrane. The microscopic entropy of the fivebrane as a function of all the charges is determined from a two-dimensional $(0,4)$ sigma model whose target space includes the fivebrane moduli space. This entropy is compared to the macroscopic formula. Precise agreement is found for both the tree-level and one-loop expressions.