I'll explain how the quantization of Hitchin integrable system can be formulated in the N=2 supersymmetric gauge theory with the help of half-BPS surface defects. I'll first review the universal oper for the Gaudin model constructed from a current algebra, and relate it to the constraints for the coinvariants of the affine Kac-Moody algebra with the twisted vacuum module. In the N=2 gauge theory side, we consider two types of surface defects, the "canonical" surface defect and the "regular monodromy" surface defect, inserted on top of each other. The correlation function of the surface defects is shown to give a basis of coinvariants with the twisted vacuum module. The insertion of twisted vacuum module is known to give the action of Hecke modification on the coinvariants. I'll define the Hecke operator as an integral of the image of Hecke modifications, which is shown to factorize due to the cluster decomposition of the two surface defects. The factorization explains why the action of the Hecke operator is diagonal. Using this factorization property and the relation with the universal oper, I show the sections of the Hecke eigensheaf give common eigenfunctions of the quantum Hitchin Hamiltonians (with the eigenvalues parametrizing the space of opers), explaining the statement of Beilinson and Drinfeld in the N=2 gauge theory framework.
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