Dressing operators in equivariant Gromov-Witten theory of $\mathbb{CP}^1$.

Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. A technical clue of their method is a pair of dressing operators on the Fock space of 2D charged free fermion fields. We reformulate these operators as difference operators in the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to the question of why the equivariant Toda hierarchy emerges in the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. Moreover, the non-equivariant limit of these operators turns out to capture the integrable structure of the non-equivariant Gromov-Witten theory correctly.