# On the formality of the little disks operad in positive characteristic

## TL;DR

This paper constructs a Grothendieck-Teichmüller group action on the completed little n-disks operad using a variant of the Boardman-Vogt tensor product, establishing partial formality in positive characteristic and providing a new proof in characteristic zero.

## Contribution

It introduces a novel approach to relate the Grothendieck-Teichmüller group with the little disks operad, advancing understanding of operad formality in different characteristics.

## Key findings

- Partial formality established for $E_n$ with mod $p$ coefficients
- New proof of formality theorem in characteristic zero
- Construction of Grothendieck-Teichmüller group action on operad completion

## Abstract

Using a variant of the Boardman-Vogt tensor product, we construct an action of the Grothendieck-Teichm\"uller group on the completion of the little n-disks operad $E_n$. This action is used to establish a partial formality theorem for $E_n$ with mod $p$ coefficients and to give a new proof of the formality theorem in characteristic zero.