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glossary of symplectic geometry

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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.

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A

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{{term|Arnold}}

{{defn|Arnold conjecture.}}

{{term|AKSZ}}

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C

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{{term|coisotropic}}

{{term|completely integrable system}}

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D

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{{term|Darboux chart}}

{{term|deformation quantization}}

{{defn|1=deformation quantization.}}

{{term|dilating}}

{{term|derived symplectic geometry}}

{{defn|1=Derived algebraic geometry with symplectic structures.}}

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E

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{{term|Noether}}

{{defn|Emmy Noether's theorem}}

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F

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{{term|Floer}}

{{defn|Floer homology}}

{{term|Fukaya}}

{{defn|no=1|1=Kenji Fukaya.}}

{{defn|no=2|1=Fukaya category.}}

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H

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{{term|Hamiltonian}}

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I

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{{term|integrable system}}

{{defn|integrable system}}

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K

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{{term|Kontsevich formality theorem}}

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L

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{{term|Lagrangian}}

{{defn|no=3|Lagrangian fibration}}

{{defn|no=4|Lagrangian intersection}}

{{term|Liouville form}}

{{defn|1=The volume form \omega^n / n! on a symplectic manifold (M, \omega) of dimension 2n.}}

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M

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{{term|Maslov index}}

{{defn|(sort of an intersection number defined on Lagrangian Grassmannian.)}}

{{term|moment}}

{{term|Moser's trick}}

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N

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{{term|Novikov}}

{{defn|Novikov ring}}

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P

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{{term|Poisson}}

{{defn|no=1|1=}}

{{defn|no=2|1=Poisson algebra.}}

{{defn|no=3|1=A Poisson manifold generalizes a symplectic manifold.}}

{{defn|no=4|1=A Poisson–Lie group, a Poisson manifold that also has a structure of a Lie group.}}

{{defn|no=5|1=The Poisson sigma-model, a particular two-dimensional Chern–Simons theory.{{Cite journal|title=Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries|author1=Martin Bojowald|author2=Alexei Kotov|author3=Thomas Strobl|journal=Journal of Geometry and Physics|volume=54|issue=4|date=August 2005|pages=400–426|doi=10.1016/j.geomphys.2004.11.002|arxiv=math/0406445|bibcode=2005JGP....54..400B |s2cid=15085408 }}}}

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Q

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{{term|quantized}}

{{defn|no=1|1=quantized algebra}}

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S

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{{term|shifted symplectic structure}}

{{defn|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.{{cite journal|title=Shifted Symplectic Structures|last1=Pantev|first1=T.|last2=Toen|first2=B.|last3=Vaquie|first3=M.|last4=Vezzosi|first4=G.|arxiv=1111.3209|journal=Publications mathématiques de l'IHÉS|volume=117|year=2013|pages=271–328|doi=10.1007/s10240-013-0054-1|s2cid=11246087}}}}

{{term|Spectral invariant}}

{{defn|Spectral invariants.}}

{{term|Springer resolution}}

{{term|symplectic action}}

{{defn|A Lie group action (or an action of an algebraic group) that preserves the symplectic form that is present.}}

{{term|symplectic reduction}}

{{term|symplectic variety}}

{{defn|An algebraic variety with a symplectic form on the smooth locus.[https://mathoverflow.net/q/49111 Is the generic deformation of a symplectic variety affine?] The basic example is the cotangent bundle of a smooth algebraic variety.}}

{{term|symplectomorphism}}

{{defn|A symplectomorphism is a diffeomorphism preserving the symplectic forms.}}

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T

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{{term|Thomas–Yau conjecture}}

{{defn|see Thomas–Yau conjecture}}

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V

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{{term|virtual fundamental class}}

{{defn|A generalization of the fundamental class concept from manifolds to a wider notion of space in higher geometry, in particular to orbifolds.}}

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Notes

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References

  • {{Cite arXiv|title = Geometry and topology of symplectic resolutions|eprint = math/0608143 |date = 2006-08-06|first = D.|last = Kaledin}}
  • Kontsevich, M. Enumeration of rational curves via torus actions. Progr. Math. 129, Birkhauser, Boston, 1995.
  • Meinrenken's [http://www.math.toronto.edu/mein/teaching/lectures.html lecture notes on symplectic geometry]
  • {{cite book|first1 = V.|last1 = Guillemin|first2 = S.|last2 = Sternberg|title = Symplectic Techniques in Physics|location = New York|publisher = Cambridge Univ. Press|year = 1984|isbn = 0-521-24866-3}}
  • {{citation |last=Woodward |first=Christopher T. |arxiv=0912.1132 |title=Moment maps and geometric invariant theory |year=2011|bibcode=2009arXiv0912.1132W }}

External links

  • http://arxiv.org/pdf/1409.0837.pdf (tangentially related)

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Category:Symplectic geometry

Symplectic

Category:Wikipedia glossaries using description lists