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K theoretic virtual fundamental cycle of global Kuranishi chart

Dun Tang

TL;DR

The paper constructs a $K$-theoretic virtual fundamental cycle for almost complex global orbifold Kuranishi charts as an element of $K_0(|I\mathcal{M}|)$ and develops a compatible framework of orbifold $K$-theory, fake $K$-homology, and the choose-an-operator map. It defines a sector-wise, index-theoretic pairing with the inertia orbifold and shows independence from chart choices, recovering the invariants of Abouzaid23. The central mechanism uses a $K$-theoretic cap product, wrong-way maps, and a Kawasaki-index-guided push-forward to a point, ensuring that the resulting invariants agree with the established targets in symplectic Gromov–Witten theory and quantum K-theory. The approach extends naturally to virtual orbifolds and provides a robust, chart-independent description of the virtual fundamental cycle in $K$-theory, enabling applications to permutation-equivariant quantum K-theory and beyond.

Abstract

In this paper, we define the virtual fundamental cycle of a global Kuranishi chart as an element in the (analytic) orbispace K-homology of the virtual orbifold and verify that it defines the same invariants as those in \cite{Abouzaid23}.

K theoretic virtual fundamental cycle of global Kuranishi chart

TL;DR

The paper constructs a -theoretic virtual fundamental cycle for almost complex global orbifold Kuranishi charts as an element of and develops a compatible framework of orbifold -theory, fake -homology, and the choose-an-operator map. It defines a sector-wise, index-theoretic pairing with the inertia orbifold and shows independence from chart choices, recovering the invariants of Abouzaid23. The central mechanism uses a -theoretic cap product, wrong-way maps, and a Kawasaki-index-guided push-forward to a point, ensuring that the resulting invariants agree with the established targets in symplectic Gromov–Witten theory and quantum K-theory. The approach extends naturally to virtual orbifolds and provides a robust, chart-independent description of the virtual fundamental cycle in -theory, enabling applications to permutation-equivariant quantum K-theory and beyond.

Abstract

In this paper, we define the virtual fundamental cycle of a global Kuranishi chart as an element in the (analytic) orbispace K-homology of the virtual orbifold and verify that it defines the same invariants as those in \cite{Abouzaid23}.

Paper Structure

This paper contains 22 sections, 33 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. The case of manifolds
  3. Results
  4. Orbifold K-theory
  5. (Compactly supported) K-group
  6. Twisted sectors, inertia orbispace, and fake K-group
  7. K-homology
  8. Embedding relatively compact sets into compact orbifolds
  9. The 'choose-an-operator' map
  10. The pairing
  11. The Cap product
  12. $K^{0}(\mathcal{Y}, \partial \mathcal{Y})$-module structures
  13. K-homological maps $i^!$ and $i_\ast$ and $K$-theoretical map $i_!$
  14. Compatibility
  15. Statement of results
  16. ...and 7 more sections

Key Result

Theorem 1

$[M] = \mathcal{O}^{vir}$.

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • proof
  • Corollary 2.1
  • ...and 48 more