# The Bogomolov-Tian-Todorov Theorem Of Cyclic $A_\infty$-Algebras

## TL;DR

This paper proves a non-commutative analogue of the Bogomolov-Tian-Todorov theorem for cyclic $A__$-algebras, showing smoothness of certain deformation functors under specific conditions.

## Contribution

It establishes the smoothness of deformation functors for cyclic $A__$-algebras satisfying the Hodge-to-de Rham degeneration, extending classical deformation theory results.

## Key findings

- Deformation functor of Hochschild cochains is smooth.
- Deformation functor of cyclic Hochschild cochains is smooth.
- Results apply to finite-dimensional smooth unital cyclic $A__$-algebras.

## Abstract

Let $A$ be a finite-dimensional smooth unital cyclic $A_\infty$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Bogomolov-Tian-Todorov theorem: the deformation functor associated with the differential graded Lie algebra of Hochschild cochains of $A$ is smooth. Furthermore, the deformation functor associated with the DGLA of cyclic Hochschild cochains of $A$ is also smooth.