Integration of Differential Graded Manifolds

Source abstract

Abstract We consider the problem of integration of $L_\infty$-algebroids (differential non-negatively graded manifolds) to $L_\infty$-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet 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 the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold is (the nerve of) a local Lie $\ell$-groupoid. The gauge condition can be imposed only locally in the base of the $L_\infty$-algebroid; the resulting local $\ell$-groupoids glue up to a coherent homotopy, that is, we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local $\ell$-groupoids. Finally, we show that a $k$-symplectic differential non-negatively graded manifold integrates to a local $k$-symplectic Lie $\ell$-groupoid; globally, these assemble to form an $A_\infty$-functor. As a particular case for $k=2$, we obtain integration of Courant algebroids.