This is a glossary of properties and concepts in symplectic geometry in mathematics. The terms listed here cover the occurrences of symplectic geometry both in topology as well as in algebraic geometry (over the complex numbers for definiteness). The glossary also includes notions from Hamiltonian geometry, Poisson geometry and geometric quantization.
In addition, this glossary also includes some concepts (e.g., virtual fundamental class) in intersection theory that appear in symplectic geometry as they do not naturally fit into other lists such as the glossary of algebraic geometry.
A
AKSZ
C
coisotropic
completely integrable system
D
Darboux chart
dilating
derived symplectic geometry
Derived algebraic geometry with symplectic structures.
F
Fukaya
1. Kenji Fukaya.
2. Fukaya category.
H
Hamiltonian
K
Kontsevich formality theorem
L
Lagrangian
3. Lagrangian fibration
4. Lagrangian intersection
Liouville form
The volume form \( {\displaystyle \omega ^{n}/n!} \) on a symplectic manifold \( (M, \omega) \) of dimension 2n.
M
Maslov index
(sort of an intersection number defined on Lagrangian Grassmannian.)
moment
Moser's trick
P
Poisson
1.
2. Poisson manifold.
5. The Poisson sigma-model, a particular two-dimensional Chern–Simons theory.[1]
S
shifted symplectic structure
A generalization of symplectic structure, defined on derived Artin stacks and characterized by an integer degree; the concept of symplectic structure on smooth algebraic varieties is recovered when the degree is zero.[2]
Springer resolution
symplectic action
A Lie group action (or an action of an algebraic group) that preserves the symplectic form that is present.
symplectic reduction
symplectic variety
An algebraic variety with a symplectic form on the smooth locus.[3] The basic example is the contangent bundle of a smooth algebraic variety.
V
virtual fundamental class
A generalization of the fundamental class concept from manifolds to a wider notion of space in higher geometry, in particular to orbifolds.
Notes
https://ncatlab.org/nlab/show/Poisson+sigma-model
Pantev, T.; Toen, B.; Vaquie, M.; Vezzosi, G. (2013). "Shifted Symplectic Structures". Publications mathématiques de l'IHÉS. 117: 271–328. arXiv:1111.3209. doi:10.1007/s10240-013-0054-1.
Is the generic deformation of a symplectic variety affine?
References
Kaledin, D. (2006-08-06). "Geometry and topology of symplectic resolutions". arXiv:math/0608143.
Kontsevich, M. Enumeration of rational curves via torus actions. Progr. Math. 129, Birkhauser, Boston, 1995.
Meinrenken's lecture notes on symplectic geometry
Guillemin, V.; Sternberg, S. (1984). Symplectic Techniques in Physics. New York: Cambridge Univ. Press. ISBN 0-521-24866-3.
Woodward, Christopher T. (2011), Moment maps and geometric invariant theory, arXiv:0912.1132, Bibcode:2009arXiv0912.1132W
External links
http://arxiv.org/pdf/1409.0837.pdf (tangentially related)
Hellenica World - Scientific Library
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