On $τ$-function of conjugate nets

Abstract

We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a $τ$-function. We investigate the potential using the non-local $\bar\partial$ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally, we give some arguments extending that interpretation to multicomponent Kadomtsev-Petviashvili hierarchy.