Cotangent bundles for higher Lie groupoids

• Applicant: Professorin Dr. Chenchang Zhu, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 543037407

This project is to figure out the explicit description of a cotangent bundle for a Lie $n$-groupoid. As we know, one of the key ingredients in symplectic dynamical system and classical mechanics is the natural symplectic structure $\omega_c$ on the cotangent bundle $T^*M$ of a manifold $M$. Many natural symplectic structures come from the symplectic reduction of $(T^*M, \omega_c)$ as reduced phase spaces. When we go higher, Pantov-Toen-Vaquie-Vezzosi and Calaque show that the shifted cotangent bundle of a derived higher stack carries a shifted symplectic structure. In contrast to algebraic geometry, where a cotangent bundle can be simply defined as spectrum of symmetric algebra of tangent sheaves, the concept of cotangent bundle of a higher Lie groupoid as higher vector bundle ($\VB$) groupoid can not be directly obtained. There are many Lie $n$-groupoids presenting the same $n$-stack. But we believe there is a canonical $\VB$ Lie $n$-groupoid presenting the cotangent bundle as soon as we choose a fixed Lie $n$-groupoid $X$ presenting the $n$-stack, just as the tangent bundle is canonically given by the tangent $\VB$ $n$-groupoid $TX$. Notice that the level-wise naive dual $T^*X_i$ does not make sense at all: even for $n=1$, the cotangent groupoid of a Lie groupoid $G:=G_1\rightrightarrows G_0$ is $T^*G_1 \rightrightarrows A^*$, where $A$ is the Lie algebroid of $G$. The lack of an explicit expression of the higher cotangent bundle blocks us from further developing higher and derived structures in dynamical system and mechanics. This project is designed to figure out all these and especially the canonical shifted symplectic structure on the shifted cotangent bundle.

• DFG Programme: Research Grants