Integration of differential graded manifolds

TL;DR

This paper develops a method to integrate L_-algebroids, represented as differential graded manifolds, into L_-groupoids using a combination of integral transformations, gauge conditions, and homotopy coherence, with applications to symplectic structures.

Contribution

It introduces a novel approach to integrate differential graded manifolds into local Lie k-groupoids, including a gauge fixing technique and an A_-functor framework for symplectic cases.

Findings

Constructed a big Kan simplicial manifold for solutions of Maurer-Cartan equations.

Established a gauge condition that produces finite-dimensional local Lie k-groupoids.

Showed that m-symplectic differential graded manifolds integrate into local m-symplectic Lie k-groupoids.

Abstract

We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fr\'echet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer-Cartan equation to closed differential forms. Following ideas of Ezra Getzler we then impose a gauge condition which cuts out a finite-dimensional simplicial submanifold. This "smaller" simplicial manifold is (the nerve of) a local Lie k-groupoid. The gauge condition can be imposed only locally in the base of the L_\infty-algebroid; the resulting local k-groupoids glue up to a coherent homotopy, i.e. we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local Lie k-groupoids. Finally we show that a m-symplectic differential graded manifold integrates to a local m-symplectic Lie k-groupoid; globally these assemble to form an A_\infty-functor. As a particular case for m=2 we obtain integration of Courant algebroids.