Permutation-equivariant quantum K-theory VII. General theory

TL;DR

This paper develops a general theory for permutation-equivariant quantum K-theory, introducing new invariants and proving a key correspondence, with implications for the structure of the big J-function in genus 0.

Contribution

It introduces K-theoretic GW-invariants of mixed nature and proves the ancestor-descendant correspondence formula, advancing the theoretical framework of permutation-equivariant quantum K-theory.

Findings

Proves the ancestor-descendant correspondence formula.

Shows the range of the big J-function is overruled in genus 0.

Establishes the structure of permutation-equivariant quantum K-theory.

Abstract

We introduce K-theoretic GW-invariants of mixed nature: permutation-equivariant in some of the inputs and ordinary in the others, and prove the ancestor-descendant correspondence formula. In genus 0, combining this with adelic characterization, we derive that the range ${\mathcal L}_X$ of the big J-function in permutation-equivariant theory of a target space $X$ is overruled.