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Description
The Gaudin model is a quantum integrable system originally introduced to describe the interaction of multiple charged particles on a line. It consists of $n$ commuting Hamiltonian operators, dependent on $n$ pairwise distinct complex parameters and acting on the tensor product of $n$ irreducible representations of the Lie algebra $\mathfrak{sl}_2$. One of the main tasks of the Gaudin model is to diagonalize these operators and understand how their joint spectrum changes as the parameters vary. In joint work with Natalia Amburg, branched coverings of the parameter space with joint spectra of Hamiltonians were studied in the case of $n = 3$. The base of such coverings is the Riemann sphere, and algebraic curves act as total spaces. The remarkable structure of these curves will be described, along with their connection to isomonodromic deformations.
