A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

TL;DR

This paper establishes a local model for Lagrangians in shifted symplectic derived schemes, extending Darboux-type theorems and enabling advances in Poisson geometry, Fukaya categories, and Donaldson-Thomas theory.

Contribution

It proves a Lagrangian Neighbourhood Theorem providing explicit local models for Lagrangians in shifted symplectic derived schemes for negative shifts, building on Darboux form models.

Findings

Local models for Lagrangians as twisted shifted conormal bundles

Extension of Darboux Theorem to Lagrangian neighborhoods

Partial results for the case when k=0

Abstract

Pantev, Toen, Vaqui\'e and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks ${\bf X}$ for $k\in\mathbb Z$, and Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or \'etale local models for $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$ presenting them as twisted shifted cotangent bundles. We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for ${\bf X}$. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when $k=0$. We expect our results will have future applications to $k$-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.