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The $p$-curvature of quantum connections for Calabi-Yau threefolds

Shaoyun Bai, Jae Hee Lee

Abstract

We show that for closed Calabi-Yau manifolds of real dimension six, the quantum Steenrod operations on divisor classes agree with the $p$-curvature of the quantum connection.

The $p$-curvature of quantum connections for Calabi-Yau threefolds

Abstract

We show that for closed Calabi-Yau manifolds of real dimension six, the quantum Steenrod operations on divisor classes agree with the -curvature of the quantum connection.
Paper Structure (35 sections, 27 theorems, 128 equations)

This paper contains 35 sections, 27 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Statement of main result
  3. Contexts for the result
  4. Arithmetic aspects of mirror symmetry
  5. Computation of quantum Steenrod operation
  6. Proof method
  7. Further discussions
  8. $Q\Sigma_b = \psi_b$ for general semi-positive targets
  9. Quantum Steenrod operations from virtual counts
  10. Arithmetic properties of quantum connection
  11. Organization of the paper
  12. Quantum operations
  13. $p$-pseudocycles
  14. Quantum product using $p$-pseudocycles
  15. Quantum connection and $p$-curvature
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $(X,\omega)$ be a symplectic Calabi--Yau manifold with $\dim_{\mathbb{R}} X = 6$, and $b \in \mathrm{im}(H^2(X;\mathbb{Z}) \to H^2(X;\mathbb{F}_p))$ a degree $2$ cohomology class. Then that is the quantum Steenrod operation for $b$ and the $p$-curvature endomorphism for $b$ are equal.

Theorems & Definitions (73)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4: bai-pomerleano-xu
  • ...and 63 more