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

Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of $\mathbb {C}P^1$ orbifolds

“…The classical limit of the periods I (p) a has been computed in [15] using the fact that they satisfy certain differential equations. Let us sketch here how to directly obtain such limit from the definition (6).…”

“…The fact that the logarithmic flows of the EBTH do not originate as restrictions of the 2D Toda flows, points to the existence of a larger hierarchy, which we might call 'extended 2D Toda', defined for a continuous space variable, which should include the usual 2D Toda flows and contain extra flows of logarithmic type. At the dispersionless level, such an extension has been recently found in [7], as the principal hierarchy associated with the infinite-dimensional Frobenius manifold discovered in [5]. The definition of a suitable dispersive version of such hierarchy is still an open problem.…”