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Towards an adjunction between the homotopy theories of dg manifolds and Lie ∞-groupoids

Banff International Research Station for Mathematical Innovation and Discovery

Lie ∞-groupoids are simplicial manifolds which satisfy conditions similar to the horn filling conditions for Kan simplicial sets. Lie ∞-groupoids are to non–negatively graded dg manifolds, or L∞-algebroids, as Lie groups are to Lie algebras. In particular, there is an integration procedure based on a smooth analog of Sullivan’s realization functor from rational homotopy theory that pro- duces a Lie ∞-groupoid from dg– manifold. There is also a differentiation functor due to Sˇevera, which uses supergeometry to construct the 1-jet of a simplicial manifold. In this talk, I will present joint work (arXiv:1609.01394) in progress with Chenchang Zhu in which we study the relationship between these integration and differentiation procedures, in analogy with Lie’s Second Theorem. A crucial first step involves constructing a user–friendly homotopy theory for Lie ∞-groupoids. This is a subtle problem, due to the fact that the category of manifolds lacks limits. I will describe how results of Behrend and Getzler can be generalized to develop a homotopy theory for Lie ∞-groups/groupoids that is compatible with the well–known homotopy theory of L∞- algebras/algebroids. If time permits, I will mention some possible applications to AKSZ σ-models via Kotov–Strobl’s theory of characteristic classes for (non–trivial) Q-bundles.

Mathematics; Category theory; homological algebra; Quantum theory; Mathematical physics

video/mp4

Author affiliation: University of Nevada

2017-12-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63831

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/