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The affine Gaudin model, associated with an untwisted affine Kac-Moodyalgebra, is known to arise from a certain gauge fixing of 4-dimensional mixedtopological-holomorphic Chern-Simons theory in the Hamiltonian framework. Weshow that the finite Gaudin model, associated with a finite-dimensionalsemisimple Lie algebra, or more generally the tamely ramified Hitchin system onan arbitrary Riemann surface, can likewise be obtained from a similar gaugefixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

3-dimensional mixed BF theory and Hitchin's integrable system