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Zhihao Duan, École Normale Supérieure Paris

Instantons in the Hofstadter butterfly: resurgence and quantum mirror curves

Recently an interesting connection between topological string theory and lattice models in condensed matter physics was discussed by several authors. In this talk, we will focus on the Harper-Hofstadter Hamiltonian. For special values of the magnetic flux, its energy spectrum can be exactly solved and its graph has a beautiful shape known as Hofstadter's butterfly. We are interested in the non-perturbative information inside the spectrum. First we consider the weak magnetic field limit and write down a trans-series ansatz for the energies. We then discuss fluctuations around instanton sectors as well as resurgence relations. For the second half of the talk, our goal is to present another powerful way to compute those fluctuations using the topological string formalism, after reviewing all the necessary background. The talk will be based on arXiv: 1806.11092.