  We study the Nekrasov partition function of the five dimensional U(N) gauge
theory with maximal supersymmetry on R^4 x S^1 in the presence of codimension
two defects. The codimension two defects can be described either as monodromy
defects, or by coupling to a certain class of three dimensional quiver gauge
theories on R^2 x S^1. We explain how these computations are connected with
both classical and quantum integrable systems. We check, as an expansion in the
instanton number, that the aforementioned partition functions are
eigenfunctions of an elliptic integrable many-body system, which quantizes the
Seiberg-Witten geometry of the five-dimensional gauge theory.
