$ A_\infty $-Deformations and their Derived Categories

TL;DR

This paper develops methods to define and analyze derived categories of curved $A_ abla$-deformations, overcoming traditional obstacles posed by curvature through explicit constructions and adaptations of classical theorems.

Contribution

It introduces a framework for handling curved $A_ abla$-deformations, including a deformed Kadeishvili theorem and techniques for constructing their derived categories.

Findings

Curved $A_ abla$-deformations can be incorporated into derived categories.

Infinitesimally curved deformations are manageable with proper definitions.

The $A_ abla$-structure on minimal models is explicitly constructed using trees.

Abstract

Handling curved $ A_\infty $-deformations is challenging and defining their derived categories seems impossible. In this paper, we show how to welcome the curvature and build derived categories despite the apparent difficulties. We explicitly describe their $ A_\infty $-structure by a deformed Kadeishvili theorem. This paper starts with a review of $ A_\infty $-deformations for non-specialists. We explain why $ A_\infty $-deformations require curvature and why curvature supposedly causes problems. We prove that infinitesimally curved $ A_\infty $-deformations are in fact non-problematic and provide the correct definitions for their twisted completion and minimal model. The classical Kadeishvili theorem collapses for curved deformations. We show how to adapt it by iteratively optimizing the curvature and changing the homological splitting infinitesimally. The final $ A_\infty $-structure on the minimal model is constructed in terms of trees.