Two reconstruction theorems in permutation equivariant quantum K-theory

TL;DR

The paper develops a completely permutable framework for permutation-equivariant K-theoretic Gromov-Witten invariants, extending the Ancestor-Descendant correspondence to reconstruct higher-genus data from 1-point correlators. Genus-0 invariants are shown to be recoverable via a finite algorithm from 1-point data, while genus-1 invariants for a point target are reconstructed through a structured combination of genus-1 terms and permuted contributions with a fixed-point choice ensuring $ar{\mathbf{t}}(1)=0$. A detailed graph-sum and vanishing analysis underpins the reconstruction, with a contraction mapping guaranteeing convergence for the auxiliary tau-parameters. The appendix establishes Givental’s Lagrangian cone formalism in this setting, showing the big J-function's range forms an overruled Lagrangian cone, and provides the adelic characterization of this cone. Overall, the work yields a coherent computational scheme for permutation-equivariant K-theoretic Gromov-Witten invariants and clarifies their symplectic-geometric structure.

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.

Theorems & Definitions (61)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Lemma 1.1
• proof