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

Abstract

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 $A_{i,\mathrm{com}}$ 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 $A_{i,\mathrm{com}}$. 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 ${\mathrm{Fl}}_3$.

36 pages

References in corpus (5)

• The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory
• Semi-infinite Schubert varieties and quantum K-theory of flag manifolds
• Twisted orbifold Gromov-Witten invariants
• Quantum Grothendieck Polynomials
• Quantum cohomology via D-modules

Cited by in corpus (13)

• Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
• Quantum K-Theory of Calabi-Yau Manifolds
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• Permutation-equivariant quantum K-theory VIII. Explicit reconstruction
• On the quantum K-ring of the flag manifold
• Non-archimedean quantum K-invariants
• Quantum K theory rings of partial flag manifolds
• Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles
• Darboux coordinates on the BFM spaces
• Quantum K-theory of flag varieties via non-abelian localization
• Positivity determines the quantum cohomology of Grassmannians
• Quantum $K$-theory and $q$-Difference equations