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P-adic splittings of the quantum connection

Paul Seidel

TL;DR

The article develops a p-adic framework for splitting the quantum connection of a monotone symplectic manifold by lifting idempotent-induced decompositions from quantum cohomology to a completed, $p$-adic setting. It constructs a tower of equivariant operations $Q\Sigma_m$ parametrized by $\Gamma_m=\mathbb{Z}/p^m$, analyzes their covariance under the quantum connection, and takes an inverse limit to obtain a canonical splitting on $H^*(M)[q^{\pm 1}]\langle\langle t\rangle\rangle$ whose $t=0$ reduction recovers the idempotent splitting. A key technical achievement is the p-adic divisibility property of the projection matrices, ensuring integrality in the $p$-adic sense and robustness under the limit. The approach blends equivariant cohomology, quantum Steenrod-type operations, and degeneration arguments to address splitting conjectures in contexts where completion and p-adic methods are natural, with potential implications for Fukaya-categorical perspectives and semisimplicity questions in quantum cohomology.

Abstract

We introduce operations with p-adic integer coefficients, associated to idempotents in the quantum cohomology of a monotone symplectic manifold, and apply them to the structure of the quantum connection.

P-adic splittings of the quantum connection

TL;DR

The article develops a p-adic framework for splitting the quantum connection of a monotone symplectic manifold by lifting idempotent-induced decompositions from quantum cohomology to a completed, -adic setting. It constructs a tower of equivariant operations parametrized by , analyzes their covariance under the quantum connection, and takes an inverse limit to obtain a canonical splitting on whose reduction recovers the idempotent splitting. A key technical achievement is the p-adic divisibility property of the projection matrices, ensuring integrality in the -adic sense and robustness under the limit. The approach blends equivariant cohomology, quantum Steenrod-type operations, and degeneration arguments to address splitting conjectures in contexts where completion and p-adic methods are natural, with potential implications for Fukaya-categorical perspectives and semisimplicity questions in quantum cohomology.

Abstract

We introduce operations with p-adic integer coefficients, associated to idempotents in the quantum cohomology of a monotone symplectic manifold, and apply them to the structure of the quantum connection.

Paper Structure

This paper contains 14 sections, 19 theorems, 85 equations, 2 figures.

Table of Contents

  1. Overview
  2. The splitting problem
  3. State of the art
  4. $p$-adic coefficients
  5. Idea of the construction
  6. Equivariant cohomology
  7. Group cohomology
  8. Equivariant cohomology with coefficients
  9. The $p$-adic operation
  10. Construction for fixed $m$
  11. Covariant constancy
  12. Increasing $m$
  13. Proof of Theorem \ref{['th:p-adic']}
  14. Elementary considerations

Key Result

Theorem 1.8

Take $R$ as in eq:p-adic-number-ring. Let $(e_i)$ be a collection of idempotents eq:idempotents. Then there is a canonical splitting of $H^*(M)[q^{\pm 1}]\langle\space\langle t \rangle\space\rangle$, which is invariant under $\nabla_{tq\partial_q}$, and whose $t = 0$ reduction agrees with the splitt

Figures (2)

  • Figure 1: Picture of \ref{['eq:quantum-steenrod-1']}.
  • Figure 2: Schematic picture of the constraints on marked points from \ref{['eq:parametrized-maps']}.

Theorems & Definitions (37)

  • Conjecture 1.1
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Example 1.12
  • Example 1.13
  • Remark 2.1
  • Lemma 2.2
  • ...and 27 more