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Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed as identical. In particular, we are able to give an alternative and much simpler geometric proof of a result of E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado on the construction of bases of affine Lie algebra representations. At the same time, we give a simple parametrization of the irreducible components of Nakajima quiver varieties associated to infinite and cyclic quivers. We also define new varieties whose irreducible components are in one-to-one correspondence with bases of the highest weight representations of affine gl_{n+1}.