On induced L-infinity action of diffeomorphisms on Cochains

TL;DR

This paper develops an $L_{ abla}$-infinity action of diffeomorphisms on cochains within a De Rham algebra framework, providing explicit computations for simple space-time manifolds, advancing quantum gravity formulations.

Contribution

It introduces a method to induce diffeomorphism actions on cochains via homotopy transfer, resulting in an $L_{ abla}$-infinity structure, with explicit examples for basic manifolds.

Findings

Explicit $L_{ abla}$-infinity actions computed for interval, circle, and square.

Demonstrates how to transfer diffeomorphism actions to cochains using homotopy transfer.

Provides a finite-dimensional algebraic approach to quantum gravity models.

Abstract

One of the approaches to quantum gravity is to formulate it in terms of De Rham algebra, choose a triangulation of space-time, and replace differential forms by cochains (that form a finite dimensional vector space). The key issue of general relativity is the action of diffeomorphisms of space-time on fields. In this paper, we induce the action of diffeomorphisms on cochains by homotopy transfer (or, equivalently, BV integral) that leads to a $L_{\infty}$ action. We explicitly compute this action for the space-time being an interval, a circle, and a square.