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José Gustavo Elias Rebelo, International School for Advanced Studies

Symplectic Field Theory and Quantum Integrable Systems

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.