  We verify a conjecture of Gillet-Soul\'{e}. We prove that the determinant of
the Laplacian on a line bundle over $\mathbb{CP}^{1}$ is always bounded from
above. This can also be viewed as a multi-particle generalization of the
Moser-Trudinger Inequality. Furthermore, we conjecture that this functional
achieves its maximum at the canonical metric. We give some evidence for this
conjecture, as well as links to other fields of analysis.
