Homotopy Lie theory: Lie's Second Theorem for Lie $n$-groupoids

• Applicant: Professorin Dr. Chenchang Zhu, Ph.D.
• Subject Area: Mathematics
• Term: from 2015 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 273215788

Integration and differentiation form a pair of adjoint functorsbetween the category of Lie algebras and that of Lie groups. Thiscorrespondence between infinitesimal and global symmetries is acornerstone of modern Lie theory. Our project aims at a higherhomotopical correspondence between Lie n-algebras and Lie n-groups. The motivation for this stems from the rolehigher Lie theory plays in string geometry, gaugetheory, and the theory of higher stacks.There already exist notions of integration and differentiation inhigher Lie theory, and there are also several models forhigher categories of Lie n-algebras and Lie n-groupoids,some more explicit than others. Both of these aspects are crucial for applications.However, currently missing is a precise understanding of thecompatibility between the two. Namely, it is not known howintegration and differentiation interact with the higher categorical/homotopical structure.The main goal of thisproject is to fill this gap.In analogy with Lie's Second Theorem, we propose that this compatibility bepresented by integration and differentiation functors which induce anequivalence between mapping spaces (rather thanhom-sets) of Lie n-algebras and Lie n-groups.To achieve this, we will first construct a simplicial category of Lie n-groupoidsusing a minimal amount of abstract theory, so that itis maximally accessible to differential geometers and mathematicalphysicists. We will perform a detailed comparison of this model withthose of Schreiber and Pridham. We will upgrade the integration and differentiationprocedures of Getzler-Henriques and Severa, respectively, to simplicialfunctors. The simplicial category we will use to represent the highercategory of Lie n-algebras is the one recently constructed byDolgushev and the first applicant. Finally, using these functors, we will establish a naturalhomotopy equivalence of mapping spaces which generalizes theclassical adjunction between Lie algebras and Lie groups.

• DFG Programme: Research Grants
• Co-Investigator: Professor Christopher Rogers, Ph.D.