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Permutation-Equivariant Quantum K-Theory of the Quintic Singularity

Maxime Cazaux

Abstract

We compute the genus-0 permutation-equivariant quantum K-theory of the singularity of the quintic polynomial, parallel to Givental--Lee's quantum K-theory of the quintic threefold. We obtain a generating function which recovers the entire 25-dimensional space of solutions to the $q$-difference equation satisfied by the permutation-equivariant K-theoretic $I$-function of the quintic.

Permutation-Equivariant Quantum K-Theory of the Quintic Singularity

Abstract

We compute the genus-0 permutation-equivariant quantum K-theory of the singularity of the quintic polynomial, parallel to Givental--Lee's quantum K-theory of the quintic threefold. We obtain a generating function which recovers the entire 25-dimensional space of solutions to the -difference equation satisfied by the permutation-equivariant K-theoretic -function of the quintic.

Paper Structure

This paper contains 30 sections, 46 theorems, 174 equations, 1 figure.

Table of Contents

  1. Overview of the main results.
  2. Outline of the paper.
  3. Defining the invariants
  4. Spin curves and the fundamental class
  5. Multiplicities
  6. Fundamental class
  7. A symplectic space
  8. The invariants
  9. The fake theories
  10. The fake invariants
  11. Fake $J$-functions
  12. Lagrangian cones
  13. An extension of the fake theory
  14. The spine CohFT
  15. Stable maps to $B\mathbb{Z}_m$ and roots of the trivial bundle
  16. ...and 15 more sections

Key Result

Theorem A

Let $f$ be a $\boldsymbol{\mu}_r$-invariant element of $\mathcal{K}$. Then $f$ lies in the image of the $J$ function if and only if

Figures (1)

  • Figure 1: Decomposition into head and arms

Theorems & Definitions (107)

  • Theorem A
  • Corollary
  • Theorem B
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Remark 1.5
  • Proposition 1.6
  • ...and 97 more