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On induced L-infinity action of diffeomorphisms on Cochains

Andrey Losev, Dmitrii Sheptunov, Xin Geng

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.

On induced L-infinity action of diffeomorphisms on Cochains

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 action. We explicitly compute this action for the space-time being an interval, a circle, and a square.

Paper Structure

This paper contains 10 sections, 73 equations.

Table of Contents

  1. Introduction
  2. BV formalism, $L_\infty$ module structure, BV integral and homotopic transfer
  3. How to understand $L_{\infty}$ module
  4. BV formalism
  5. $L_{\infty}$ algebra and $L_{\infty}$ module from the BV point of view
  6. Induction of the $L_{\infty}$ module by contraction of the acyclic subcomplex in the BV formalism
  7. Application to the action of diffeomorphisms on cochains
  8. Interval
  9. Circle
  10. Square