A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

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'.Comment: 68 page

Braces and Poisson additivity

We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad $O$ to associative algebras in the category of homotopy $O$-algebras. As an example, we identify the category of $P_{n+1}$-algebras with the category of associative algebras in $P_n$-algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.Comment: 49 page

Quantum moment maps

We introduce quantum versions of Manin pairs and Manin triples and define quantum moment maps in this context. This provides a framework that incorporates quantum moment maps for actions of Lie algebras and quantum groups for any quantum parameter. We also show how our quantum moment maps degenerate to known classical versions of moment maps and describe their fusion.Comment: 38 page

Symplectic implosion and the Grothendieck-Springer resolution

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to reduce a Hamiltonian $G$-space to a Hamiltonian $H$-space where $H$ is the maximal torus of $G$. We show that this procedure coincides with an algebraic version of symplectic implosion of Guillemin, Jeffrey and Sjamaar. We explain how to obtain generalizations of this picture to quasi-Hamiltonian spaces and their elliptic version.Comment: 24 page

Quantum moment maps

We introduce quantum versions of Manin pairs and Manin triples and define quantum moment maps in this context. This provides a framework that incorporates quantum moment maps for actions of Lie algebras and quantum groups for any quantum parameter. We also show how our quantum moment maps degenerate to known classical versions of moment maps and describe their fusion.Comment: 38 page