Principal hierarchies of infinite-dimensional Frobenius manifolds: the extended 2D Toda lattice

We define a dispersionless tau-symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0$/$\mathrm{\infty }$ respectively, which extends the dispersionless 2D Toda hierarchy of Takasaki and Takebe. Then we construct the deformed flat connection of the infinite-dimen-sional Frobenius manifold $M_0$ introduced by Carlet, Dubrovin and Mertens in Math. Ann. 349 (2011) 75–115 and, by explicitly solving the deformed flatness equations, we prove that the extended 2D Toda hierarchy coincides with principal hierarchy of $M_0$.