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Deformations of Gentle $A_\infty$-Algebras

Raf Bocklandt, Jasper van de Kreeke

TL;DR

The paper develops deformation theory for gentle $A_ ablafty$-algebras associated to arc collections on marked surfaces by introducing orbigons to encode curved higher products, and then computes the Hochschild cohomology and Gerstenhaber structure. It shows that when the arc collection has no loops or two-cycles, the Hochschild dgla is formal and all curved deformations are gauge-equivalent to explicit ${}^rbmu$-type structures, with a two-parameter extension ${}^{r,s}bmu$ available via weights. The work connects to mirror symmetry and dimer model perspectives, aligns with Koszul duality, and provides explicit bases and formulas for Hochschild generators, cup products, and brackets. Overall, it clarifies how curved deformations of gentle algebras are controlled by zeroth and first Hochschild components and how formality emerges in the NL2 regime, offering tools for studying wrapped Fukaya categories and related B-model structures.

Abstract

In this paper we calculate the Hochschild cohomology of gentle $A_\infty$-algebras of arc collections on marked surfaces without boundary components. When the underlying arc collection has no loops or two-cycles, we show that the dgla structure of the Hochschild complex is formal and give an explicit realization of all deformations up to gauge equivalence.

Deformations of Gentle $A_\infty$-Algebras

TL;DR

The paper develops deformation theory for gentle -algebras associated to arc collections on marked surfaces by introducing orbigons to encode curved higher products, and then computes the Hochschild cohomology and Gerstenhaber structure. It shows that when the arc collection has no loops or two-cycles, the Hochschild dgla is formal and all curved deformations are gauge-equivalent to explicit -type structures, with a two-parameter extension available via weights. The work connects to mirror symmetry and dimer model perspectives, aligns with Koszul duality, and provides explicit bases and formulas for Hochschild generators, cup products, and brackets. Overall, it clarifies how curved deformations of gentle algebras are controlled by zeroth and first Hochschild components and how formality emerges in the NL2 regime, offering tools for studying wrapped Fukaya categories and related B-model structures.

Abstract

In this paper we calculate the Hochschild cohomology of gentle -algebras of arc collections on marked surfaces without boundary components. When the underlying arc collection has no loops or two-cycles, we show that the dgla structure of the Hochschild complex is formal and give an explicit realization of all deformations up to gauge equivalence.
Paper Structure (11 sections, 26 theorems, 59 equations, 2 figures)

This paper contains 11 sections, 26 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Curved gentle algebras
  3. Arc collections and gentle algebras.
  4. Orbigons
  5. $A_\infty$-structures on the gentle algebra.
  6. Hochschild Cohomology of Gentle $A_\infty$-algebras
  7. Definitions
  8. Reduction to the zeroth and first component
  9. A set of generators
  10. Gerstenhaber structure product on Hochschild cohomology
  11. Classifying curved deformations

Key Result

Lemma 2.1

Figures (2)

  • Figure 2.1: The possible diagrams that can occur in as double products
  • Figure 3.1:

Theorems & Definitions (79)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Example 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.2
  • ...and 69 more