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Permutation-equivariant quantum K-theory IV. $D_q$-modules

In Part II, we saw how genus-0 permutation-equivariant quantum K-theory of a manifold with isolated fixed points of a torus action can be reduced via fixed point localization to permutation-equivariant quantum K-theory of the point. In Part III, we gave a complete description of genus-0 permutation-equivariant quantum K-theory of the point by means of adelic characterization. Here we apply the adelic characterization to introduce the action on this theory of a certain group of $q$-difference operators. This action will enable us to prove that toric $q$-hypergeometric functions represent K-theoretic GW-invariants of toric manifolds.