# Levy Laplacian on manifold and Yang-Mills heat flow

## TL;DR

This paper introduces a covariant Levy Laplacian on infinite dimensional manifolds and establishes its connection to solutions of the Yang-Mills heat equations via flow of parallel transports.

## Contribution

It provides a covariant definition of the Levy Laplacian and links it to Yang-Mills heat flow solutions through the flow of parallel transports.

## Key findings

- Covariant Levy Laplacian defined on infinite dimensional manifolds.
- Yang-Mills heat equations characterized by the flow of parallel transports.
- Equivalence between solutions of Yang-Mills heat equations and covariant Levy Laplacian heat flow.

## Abstract

A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang-Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Laplacian on the infinite dimensional manifold.