Two reconstruction theorems in permutation equivariant quantum K-theory

TL;DR

This paper extends the permutation equivariant quantum K-theory framework by generalizing the AD correspondence, enabling reconstruction of genus 0 and 1 invariants from simpler invariants, and explores the geometric structure of the big J-function.

Contribution

It introduces a generalized AD correspondence for permutation equivariant quantum K-theory and demonstrates how to reconstruct higher genus invariants from genus one data.

Findings

Reconstructed genus 0 and 1 invariants from 1-point invariants.

Established the Lagrangian cone structure of the big J-function.

Generalized the AD correspondence to arbitrary permutative inputs.

Abstract

In this paper, we first generalize the K-theoretic Ancestor-Descendant (AD) correspondence in \cite{perm7} to allow arbitrary permutative inputs. With this version of AD correspondence, we reconstruct K-theoretical descendant $g=0$ invariants, and $g=1$ invariants with point target space, from $1$-point invariants of the corresponding genus. In the appendix, we show that the graph of big $\mathcal{J}$ function also forms a Lagrangian cone in the permutation equivariant setting.