Related papers: Shifted symplectic reduction of derived critical l…

We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a…

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any $B\mathbb{G}_m$-action on a non-positively-shifted symplectic…

We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of…

The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived…

In this paper, we formally define the concept of shifted contact structures on derived (Artin) stacks and study their local properties in the context of derived algebraic geometry. In this regard, for negatively shifted contact derived…

Let $(X,\omega_X)$ be a derived scheme with a 0-symplectic form and suppose there is a Hamiltonian $G$-action with a moment map for $G$ a reductive group. We prove, under no further assumptions, that symplectic reduction along any coadjoint…

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion…

Given a regular function $\phi$ on a smooth stack, and a $(-1)$-shifted Lagrangian $M$ on the derived critical locus of $\phi$, under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix…

Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration…

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the…

The classical transversality lemma of contact geometry constructs a contact structure on a hypersurface transverse to a Liouville vector field using point-set topology and local flows. This paper translates the classical transversality…

We will quickly explore the derived geometry of zero loci of sections of vector bundles, with particular emphasis on derived critical loci. In particular we will single out many of the derived geometric structures carried by derived…

In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type…

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks.…

Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $\sigma$ compatible with the $T$-action. The real locus of…

Let $G$ be a linear connected complex reductive Lie group. The purpose of this paper is to give explicit symplectic isomorphisms from twisted cotangent bundles of the complex generalized flag varieties, whose transition functions are given…

For a $(-1)$-shifted Lagrangian in a critical locus, we construct a homomorphism from the $K$-group of matrix factorisations of the critical locus to the $K$-group of the Lagrangian, partially answering the Joyce-Safronov conjecture. The…

We construct a quadratic Morse-Bott function on the real Grassmannian of a symplectic vector space from a compatible linear complex structure. We show that its critical loci consist of linear subspaces that split into isotropic and complex…

In this article we prove a derived version of the Marsden-Weinstein-Meyer symplectic reduction theorem. We model the symplectic quotient as a dg-groupoid. We then construct the reduced symplectic form inside the Bott-Shulman complex of the…

The Marsden-Weinstein-Meyer symplectic reduction has an analogous version for cosymplectic manifolds. In this paper we extend this cosymplectic reduction to the context of groupoids. Moreover, we prove how in the case of an algebroid…