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We introduce the most general to date version of the permutation-equivariantquantum K-theory, and express its total descendant potential in terms ofcohomological Gromov-Witten invariants. This is the higher-genus analogue ofadelic characterization from the paper [7] by Givental-Tonita, and is based onthe application of Kawasaki's Riemann-Roch formula to moduli spaces of stablemaps.

Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera