Reconstruction and Convergence in Quantum $K$-Theory via Difference Equations

Understand

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 $A_{i,\rm com}$ 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 $A_{i,\rm com}$.
• 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 $\operatorname{Fl}_3$.