  We prove a duality, recently conjectured in arXiv:1103.5726, which relates
the F-terms of supersymmetric gauge theories defined in two and four dimensions
respectively. The proof proceeds by a saddle point analysis of the
four-dimensional partition function in the Nekrasov-Shatashvili limit. At
special quantized values of the Coulomb branch moduli, the saddle point
condition becomes the Bethe Ansatz Equation of the SL(2) Heisenberg spin chain
which coincides with the F-term equation of the dual two-dimensional theory.
The on-shell values of the superpotential in the two theories are shown to
coincide in corresponding vacua. We also identify two-dimensional duals for a
large set of quiver gauge theories in four dimensions and generalize our proof
to these cases.
