Reconstruction and Convergence in Quantum K-Theory via Difference Equations

We give a new reconstruction method of big quantum K-ring based on the q-difference module structure in quantum K-theory [12, 13]. The q-difference structure yields com-muting 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.