On the connection between Hodge theory and integrable systems

Abstract

We investigate the relationship between Hodge theory, a field of mathematics, and integrable systems, a concept in physics. There are several reasons to believe the two notions are related and this suspicion has been strengthened in two recent papers by Grimm and Monnee. There, the authors managed to show that the Weil operator from Hodge theory provides a solution to both the 𝜆-model and the bi- Yang-Baxter model, which are integrable systems. To be precise, they showed that for the bi-Yang- Baxter model the SL(2)-approximation of the Weil operator, coming from the SL(2)-orbit theorem of Hodge theory, provided a solution, whereas the full Weil operator solved the 𝜆-model. In this work we try to build upon their construction to further strengthen the connection between Hodge theory and integrable systems. In particular, we review the relevant mathematical and physical background, as well as the construction of Grimm and Monnee. A concept deeply intertwined with integrable systems, called Poisson-Lie T-duality, seems to play a role in the relation between Hodge theory and integrable systems. We give a self-contained review of the duality and provide one of the necessary steps in the case of SU(2). Moreover, we identify opportunities and difficulties regarding the interplay of Hodge theory, integrable systems and Poisson-Lie T-duality.