Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

McKay equivalence for symplectic resolutions of singularities

Abstract: Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived category $D^b_G(Coh(V))$ of $G$-equivariant coherent sheaves on $V$.

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Abstract: We show that the strong form of Heisenberg’s inequalities due to Robertson and Schrodinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the “minimum volume ellipsoid” together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what “quantum scale” is. We take the opportunity to discuss the principle of the symplectic camel, which is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics.

Lefschetz fibrations and symplectic homology

Abstract: We show that for each k>3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space ℝ2k which are pairwise distinct as symplectic manifolds.

Symplectic reflection algebras

Abstract: We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on category O.

Symplectic reduction and Riemann-Roch formulas for multiplicities

Abstract: It is well-known that the presence of conserved quantities in a Hamiltonian dynamical system enables one to reduce the number of degrees of freedom of the system. This technique, which goes back to Lagrange and was treated in a modern spirit in papers of Marsden and Weinstein [17] and Meyer [21], is nowadays known as symplectic reduction. In their paper [8] Guillemin and Sternberg considered the problem: what is the quantum analogue of symplectic reduction? In other words, when one quantizes both a mechanical system with symmetries and its reduced system, what is the relationship between the two quantum-mechanical systems that one obtains? Recently a number of authors have made substantial progress in solving this problem, on which I shall report in this note. This development was brought about by work of Witten [27] and subsequent work of Jeffrey and Kirwan [11], Kalkman [13] and Wu [29] on cohomology rings of symplectic quotients. Another important idea turned out to be Lerman’s technique of symplectic cutting or equivariant symplectic surgery [16], a generalization of the notions of blowing up and symplectic reduction.