Frobenius Structures on Double Hurwitz Spaces

We present a construction Frobenius structures of “dual type” on the moduli space of ramified coverings of $\mathbb{P}^1$ with given ramification type over two points, generalizing Dubrovin's construction of [4, 5]. A complete hierarchy of hydrodynamic type is obtained from the corresponding deformed flat connection. This provides a suitable framework for the theory of weakly deformed soliton lattices of an enlarged class of integrable equations; as applications we consider the q-deformed Gelfand–Dickey hierarchy and the sine-Gordon equation, and explicitly compute the corresponding solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.