Related papers: Derived Symplectic Reduction in Differential Geome…
Gromov-Witten invariants of a symplectic manifold are a count of holomorphic curves. We describe a formula expressing the GW invariants of a symplectic sum $X# Y$ in terms of the relative GW invariants of $X$ and $Y$. This formula has…
In this paper we study quotients of Lie algebroids and groupoids endowed with compatible differential forms. We identify Lie theoretic conditions under which such forms become basic and characterize the induced forms on the quotients. We…
A log symplectic manifold is a Poisson manifold which is generically nondegenerate. We develop two methods for constructing the symplectic groupoids of log symplectic manifolds. The first is a blow-up construction, corresponding to the…
The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the…
Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the…
This paper establishes semiorthogonal decompositions for derived Grassmannians of perfect complexes with Tor-amplitude in $[0,1]$. This result verifies the author's Quot formula conjecture [J21a] and generalizes and strengthens Toda's…
This is a survey paper on derived symplectic geometry, that will appear as a chapter contribution to the book "New Spaces for Mathematics and Physics", edited by Mathieu Anel and Gabriel Catren. Our goal is to explain how derived stacks can…
We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as…
The Adler-Gelfand-Dikii Poisson structure arises naturally in the study of $n$-th order differential operators on the circle and plays a central role in Poisson geometry and integrable systems. Let $G$ be one of the Lie groups…
We describe the symplectic reduction construction for the physical phase space in gauge theory and apply it for the BF theory. Symplectic reduction theorem allows us to rewrite the same phase space as a quotient by the gauge group action,…
Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new…
On symplectic manifolds, we introduce a Morse-type complex with elements generated by pairs of critical points of a Morse function. The differential of the complex consists of gradient flows and an integration of the symplectic structure…
There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for any two functions f and g. We notice that this non-commutative differential algebra has a geometrical…
In this paper we determine conditions of existence of an induced Riemannian structure on the symplectic quotient of a symplectic and Riemannian manifold following the action of a Lie group acting upon it in a hamiltonian way with…
By using the theory of analytic vectors and manifolds modelled on normed spaces, we provide a rigorous symplectic differential geometric approach to $t$-dependent Schr\"odinger equations on separable (possibly infinite-dimensional) Hilbert…
In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin…
Unfortunately, some proofs in the first version of this paper were incorrect. In this revised version, some minor gaps are fixed, one serious mistake found. The main theorem is now claimed only under a restrictive technical assumption. This…
The imploded cross-section of a symplectic manifold is a stratified space allowing for an abelianization of its symplectic reduction. After recalling symplectic and Poisson reduction and reviewing the basics of symplectic implosion, we…
In this work we study representations of the Poincare group defined over symplectic manifolds, deriving the Klein-Gordon and the Dirac equation in phase space. The formalism is associated with relativistic Wigner functions; the Noether…
We show (Theorem 3) that the symplectic reduction of the spatial $n$-body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions.…