Spectral problems for the E6 Minahan-Nemeschansky theory

According to Nekrasov and Shatashvili the Coulomb vacua of four-dimensional N=2 theories of "class S'', subjected to the Omega background in two of the four dimensions, correspond to the eigenstates of a quantisation of the Hitchin integrable system. The vacua may be found as the intersection between two Lagrangian branes in the Hitchin moduli space, one of which is the space of opers (or quantum Hamiltonians) and one is defined in terms of a system of Darboux coordinates on the corresponding moduli space of flat connections. I will introduce such a system of Darboux coordinates on the moduli space of SL(3) flat connections on the three-punctured sphere through a procedure called abelianization and describe the spectral problem characterising the corresponding quantum Hitchin system. This talk is based on work to appear with Andrew Neitzke.