On a multi-particle Moser-Trudinger Inequality

We verify a conjecture of Gillet-Soul'{e}. We prove that the determinant of the Laplacian on a line bundle over ${\mathbb{C}\mathbb{P}}^1$ 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.