| Some integrable hierarchies, such as the Korteweg-de Vries (KdV) hierarchy and the Toda hierarchy, arise by modeling a physical process with a base differential equation then adding other evolutionary equations that commute with it. This allows one to create a flow of solutions of the first equation. In addition to this, Gromov-Witten invariants of certain projective manifolds (or more generally, orbifolds) can be organized into a generating function that is a solution to an integrable hierarchy. The extended Toda hierarchy (ETH) was introduced to find such a hierarchy that governs the Gromov-Witten invariants of CP 1. Similarly, the extended bigraded Toda hierarchy (EBTH) was found to govern the Gromov-Witten theory of certain orbifolds of CP 1 after its introduction.;The evolutionary differential equations that forma given integrable hierarchy produce actions that commute with one another and can be solved simultaneously; in other words, they are symmetries of one another. These equations form flows of solutions. For some hierarchies, such as KdV and Kadomtsev-Petviashvili (KP), these are not the only flows that commute with the flows of the hierarchy. These additional commuting flows are fittingly called additional symmetries. After reviewing the tools necessary to the discussion of additional symmetries in the context of the KdV and KP hierarchies, we discuss the Toda hierarchy, its symmetries, and the ETH. We define additional flows for the extended Toda in Lax form, proving that these flows are indeed symmetries of the ETH and that they commute with one another as the Virasoro algebra.We then show how these symmetries act on the tau function of the hierarchy. Finally, we move to the EBTH, defining additional symmetries for it and investigating these additional flows' actions on the hierarchy's Lax operator, wave functions, and tau function. |
