A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Abstract

Pantev, Toen, Vaquié and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks $\mathbf{X}$ for $k\in \mathbb{Z}$, and Lagrangians $\mathbf{f}:\mathbf{L}\to \mathbf{X}$ 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 étale local models for $k$-shifted symplectic derived schemes $\mathbf{X}$ 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 $\mathbf{f}:\mathbf{L}\to \mathbf{X}$ in $k$-shifted symplectic derived schemes $\mathbf{X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for $\mathbf{X}$. 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'.

68 pages

References in corpus (6)

• Shifted Poisson Structures and Deformation Quantization
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Derived coisotropic structures II: stacks and quantization
• Shifted Poisson and symplectic structures on derived N-stacks
• Braces and Poisson additivity
• Perversely categorified Lagrangian correspondences

Cited by in corpus (2)

• Notes on Derived Geometric Formulations in Physics
• Shifted Contact Structures and Their Local Theory