It is known, since Witten and Kontsevich’s work regarding the description of the intersection theory of $\psi$-classes on $\overline{\mathcal{M}}_{g, n}$ in terms of the $\tau$-function of a solution to the KdV hierarchy, that the connection between Integrable Systems and 2D Topological Field Theory is a deep one.

We will begin by introducing the fundamentals of Symplectic Field Theory (SFT), a branch of symplectic topology that studies holomorphic curves with boundaries in symplectic manifolds. The potential counting these curves is interpreted as an Hamiltonian corresponding to a Quantum Integrable System. In particular, we will consider the commuting Hamiltonians obtained from the quantisation of the dispersionless KdV hierarchy, that arise naturally in the context of SFT. A complete set of common eigenvectors of these operators is found in terms of Schur polynomials and used to compute the SFT-potential of a disk.

We will finish by providing some remarks on some more recent developments in this area, namely the case of the Quantised Toda Lattice and Double Ramification Hierarchy.