Reconstruction and Convergence in Quantum K-Theory via Difference Equations

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We give a new reconstruction method of big quantum K-ring based on the q-difference module structure in quantum K-theory. The q-difference structure yields commuting linear operators Ai, com on the K-group as many as the Picard number of the target manifold. The genus-zero quantum K-theory can be reconstructed from the q-difference structure at the origin t=0 if the K-group is generated by a single element under the actions of Ai, com. This method allows us to prove the convergence of the big quantum K-rings of certain manifolds, including the projective spaces and the complete flag manifold Fl₃.