Nikulin's $K3$ surfaces, adiabatic limit of equivariant analytic torsion, and the Borcherds $\Phi$-function

Abstract

In this note, we prove that the “adiabatic limit” of the equivariant analytic torsion of a Nikulin's $K3$ surface converges to the value of norm of the Borcherds $\Phi$-function at its period point after a certain renormalization.