nLab symplectic geometry

Idea

Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations; it originated as a formalization of the mathematical apparatus of classical mechanics and geometric optics (and the related WKB-method in quantum mechanics and, more generally, the method of stationary phase in harmonic analysis). A wider branch including symplectic geometry is Poisson geometry and a sister branch in odd dimensions is contact geometry. A special and central role in the subject belongs to certain real-like half-dimensional submanifolds, called lagrangian (or Lagrangean) submanifolds, which are in some sense classical points. Symplectic geometry radically changed after the 1985 article of Mikhail Gromov on pseudoholomorphic curves and the subsequent work of Andreas Floer giving birth to symplectic topology or “hard methods” of symplectic geometry.

Applications

• In its application to physics, symplectic geometry is the fundamental mathematical language for Hamiltonian mechanics, geometric quantization, geometrical optics.

• A tremendous amount of insight into higher Lie theory (Lie groupoids, Lie ∞-groupoids, Lie ∞-algebroids) has derived from Alan Weinstein‘s long-term project of understanding the role of symplectic geometry in geometric quantization. See there for more details.

Zoran Skoda: it is true that large influence of Weinstein’s program can not be overestimated, but it is not the origin of these considerations; it rather builds up on earlier fundamental works of Kirillov, Kostant, Souriau who invented geometric quantization, all of them originally in symplectic context; and the florishing of the subject from mid 1960s till mid 1980-s is related to their work; and other related tracks of Guillemin, Sternberg, Kashiwara, Karasev, Arnold and so on; and vast developments in harmonic analysis and representation theory (Kostant, Auslander, Vogan, Wallach, Stein…), microlocal analysis (Kashiwara, Saito, Hormander, Maslov, Karasev, Duistermaat…), integrable systems/quantum groups (this is more into more general Poisson geometry: Lie-Poisson groups, classical r-matrices, bihamiltonian systems…), and related approaches to quantization (Berezin method, coherent states…).

• T-duality and (related) mirror symmetry interchange the symplectic data and complex algebraic geometry data. Some cases of both the symplectic and complex algebraic picture can be unified and studied in a broader concept of generalized complex geometry.

Related concepts

• There is a vertical categorification of symplectic geometry to higher symplectic geometry. This involves multisymplectic geometry and the geometry of symplectic Lie n-algebroids. And their combination.

  See higher symplectic geometry

• holomorphic symplectic geometry

• Arnold conjecture

• phase space, Hamiltonian flow, prequantum line bundle

• symplectic formulation of quantum mechanics

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

Textbook accounts:

• Rolf Berndt, An introduction to symplectic geometry, Graduate Studies in Mathematics 26 (2001) [ams:gsm-26, pdf]

• Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer (2008) [doi:10.1007/978-3-540-45330-7, pdf]

Lecture notes:

• Konstantin Athanassopoulos, Notes on Symplectic Geometry, lecture notes (2015) [pdf, pdf]

Survey of the field:

• Ivan Contreras, Diego Martinez, Nicolas Martinez, Diego Rodriguez: Symplectic Geometry, Poisson Geometry, and Beyond [arXiv:2411.12551]

Discussion from the point of view of homological algebra of abelian sheaves is in

• C. Viterbo, An introduction to symplectic topology through sheaf theory (2010) (pdf)

The notion of symplectic geometry may be understood as the mathematical structure that underlies the physics of Hamiltonian mechanics. A classical monograph that emphasizes this point of view:

• Vladimir Arnol'd, Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, Springer (1978) [doi:10.1007/978-1-4757-1693-1]

For more on this see Hamiltonian mechanics.

Symplectic geometry is also involved in geometric optics, geometric quantization and the study of oscillatory integrals and microlocal analysis. Books concentrating on these topics include

• Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys and Monographs 14, AMS (1977) [ams:surv-14]

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

• J.J. Duistermaat, Fourier integral operators, Progress in Mathematics, Birkhäuser 1995 (and many other references at microlocal analysis).

• Alan Weinstein, Symplectic geometry, (survey) Bull. Amer. Math. Soc. 5 (1981), 1-13, doi

• N. R. Wallach, Symplectic geometry and Fourier analysis, Math. Sci. Press, Brookline, Mass., 1977.

• wikipedia symplectic geometry Application: “Jacobi’s elimination of nodes” is moved to equivariant localization and elimination of nodes.

A proof of the Arnold conjecture in symplectic geometry is in

• Mohammed Abouzaid, Andrew J. Blumberg, Arnold Conjecture and Morava K-theory, (arXiv:2103.01507)