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Isomorphism in the augmentation category

Honghao Gao, Hanming Liu

TL;DR

The paper proves that two augmentations are isomorphic in the positive augmentation category for a Legendrian submanifold if and only if they differ by a composition of a $dga$-homotopy and a dilation, extending known results from knots/links to arbitrary dimensions using quantum flow trees instead of an explicit $n$-copy dga. It develops a chain-level model of $H^*\\mathcal{A}ug_+(V,\\Lambda)$ with a two-term $A_\\infty$ structure $(m_1,m_2)$ and analyzes cocycles and products in $\\mathrm{Hom}(\\epsilon_1,\\epsilon_2)$ to identify dilated augmentation homotopies as the relevant obstructions. The work derives corollaries describing when isomorphisms reduce to $dga$-homotopies or to dilations, particularly in characteristic $2$, and discusses implications for the relation between augmentation categories and microlocal rank-1 sheaf categories. Overall, it provides a dimension-free understanding of augmentation isomorphisms and highlights limitations in matching augmentation data with microlocal sheaves in higher dimensions.

Abstract

Given a Legendrian submanifold in any dimension, we prove that two augmentations are isomorphic within the positive augmentation category exactly when they differ by a combination of a dga homotopy and a dilation. This extends the corresponding statement for Legendrian knots and links, but instead of relying on the dga for consistent copies, we make use of quantum flow tree techniques. Consequently, we can strengthen and clarify a result of the first author as follows: for knot contact homology, the augmentation category is not in general equivalent to the microlocal rank 1 sheaf category.

Isomorphism in the augmentation category

TL;DR

The paper proves that two augmentations are isomorphic in the positive augmentation category for a Legendrian submanifold if and only if they differ by a composition of a -homotopy and a dilation, extending known results from knots/links to arbitrary dimensions using quantum flow trees instead of an explicit -copy dga. It develops a chain-level model of with a two-term structure and analyzes cocycles and products in to identify dilated augmentation homotopies as the relevant obstructions. The work derives corollaries describing when isomorphisms reduce to -homotopies or to dilations, particularly in characteristic , and discusses implications for the relation between augmentation categories and microlocal rank-1 sheaf categories. Overall, it provides a dimension-free understanding of augmentation isomorphisms and highlights limitations in matching augmentation data with microlocal sheaves in higher dimensions.

Abstract

Given a Legendrian submanifold in any dimension, we prove that two augmentations are isomorphic within the positive augmentation category exactly when they differ by a combination of a dga homotopy and a dilation. This extends the corresponding statement for Legendrian knots and links, but instead of relying on the dga for consistent copies, we make use of quantum flow tree techniques. Consequently, we can strengthen and clarify a result of the first author as follows: for knot contact homology, the augmentation category is not in general equivalent to the microlocal rank 1 sheaf category.
Paper Structure (2 sections, 7 theorems, 20 equations)

This paper contains 2 sections, 7 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Proof of the main theorem

Key Result

Theorem 1.2

Let $\epsilon_1,\epsilon_2:\mathcal{A}\to k$ be two augmentations. Then, $\epsilon_1$ is quasi-isomorphic to $\epsilon_2$ in $\mathcal{A}ug_+(V,\Lambda)$ if and only if there is a dilated augmentation homotopy $K$ between $\epsilon_1$ and $\epsilon_2$.

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Lemma 2.2
  • proof
  • ...and 4 more