Geometric quantization

Symplectic geometry



This is polarization of phase space (or any symplectic manifold) into canonical coordinates and canonical momenta. Distinguish the polarization identity (such as in an inner product space or a Jordan algebra) or wave polarization? (such as polarized light).


For a symplectic manifold (X,ω)(X, \omega) regarded as the phase space of a physical system, a choice of polarization is, locally, a choice of decomposition of the coordinates on XX into “canonical coordinates” and “canonical momenta”.

The archtypical example is that where X=T *ΣX = T^* \Sigma is the cotangent bundle of a manifold Σ\Sigma. In this case the canonical canonical coordinates are those parameterizing Σ\Sigma itself, while the canonical canonical momenta are coordinates on each fiber of the cotangent bundle.

But for general symplectic manifolds there is no such canonical choice of coordinates and momenta. Moreover, in general there is not even a global notion of canonical momenta. Instead, a choice of (real) polarization is a foliation of phase space by Lagrangian submanifolds and then

  • the “canonical coordinates” are coordinates on the corresponding leaf space (parameterizing the leaves);

  • the “canonical momenta” are coordinates along each leaf. If there is no typical leaf then this is not a globally defined notion, only the polarization itself is.

Locally this is a choice of coordinate patch ϕ: 2nX\phi : \mathbb{R}^{2n} \to X such that the symplectic form takes the form

ϕ *ω= i=1 ndq idp i \phi^* \omega = \sum_{i = 1}^n \mathbf{d} q^i \wedge \mathbf{d}p_i

where the {q i: 2n n× n}\{q^i : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\} are the canonical coordinates on the first n\mathbb{R}^n-factor of the Cartesian space 2n\mathbb{R}^{2n}, and where {p o: 2n n× n}\{p_o : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\} are canonical coordinates on the second n\mathbb{R}^n-factor.


The traditional notion of polarization applies to a symplectic manifold.

Symplectic manifold are the lowest step in a tower of notions in higher symplectic geometry which proceeds with n-plectic geometry for all nn and manifolds refined to smooth infinity-groupoids. The next simplest cases in this tower are symplectic Lie n-algebroids, which for n=1n=1 are Poisson Lie algebroids and for n=2n = 2 are Courant Lie 2-algebroids:

Of a symplectic manifold

Let (X,ω)(X, \omega) be a symplectic manifold.


A real polarization of (X,ω)(X, \omega) is a foliation by Lagrangian submanifolds.

For instance (Weinstein, p. 9).

More generally


A polarization of (X,ω)(X,\omega) is a choice of involutive Lagrangian subbundle 𝒫T X\mathcal{P} \hookrightarrow T_{\mathbb{C}} X of of the complexified tangent bundle of XX.

For instance (Bates-Weinstein, def. 7.4)

Of a Poisson Lie algebroid

A Poisson Lie algebroid 𝔓\mathfrak{P} is a symplectic Lie n-algebroid for n=1n = 1. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form ω\omega. One can then say


A dg-Lagrangian submanifold of (𝔓,ω)(\mathfrak{P}, \omega) is also called a Λ\Lambda-structure. (Ševera, section 4).

Hence we might say real polarization of (𝔓,ω)(\mathfrak{P}, \omega) is a foliation by dg-Lagrangian submanifolds.


For (X,π)(X, \pi) the Poisson manifold underlying a Poisson Lie algebroid (𝔓,ω)(\mathfrak{P}, \omega), a dg-Lagrangian submanifold of (𝔓,ω)(\mathfrak{P}, \omega) corresponds to a coisotropic submanifold of (X,π)(X, \pi).

(Ševera, section 4)


The dg-Lagrangian submanifolds also correspond to branes in the Poisson sigma-model (see there) on (𝔓,ω)(\mathfrak{P}, \omega).

Of a Courant Lie 2-algebroid

A Courant Lie algebroid \mathfrak{C} is a symplectic Lie n-algebroid for n=2n = 2. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form ω\omega. One can then say


A dg-Lagrangian submanifold of (,ω)(\mathfrak{C}, \omega) is also called a Λ\Lambda-structure. (Ševera, section 4).

Hence we might say real polarization of (,ω)(\mathfrak{C}, \omega) is a foliation by dg-Lagrangian submanifolds.


The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid (,ω)(\mathfrak{C}, \omega) correspond to Dirac structures on (,ω)(\mathfrak{C}, \omega).

(Ševera, section 4)

Of a symplectic Lie nn-algebroid


Of an nn-plectic smooth \infty-groupoid

A simple notion of a real polarization for 2-plectic manifolds is considered within the context of higher geometric quantization in Rogers, Chap. 7.


Real polarizations


Kähler polarizations

If the symplectic manifold (X,ω)(X,\omega) lifts to the structure of a Kähler manifold (X,J,g)(X, J, g), hence with Riemannian metric g(,)=ω(,I())g(-,-) = \omega(-,I(-)), then the holomorphic/antiholomorphic decomposition induced by the complex manifold structure is a polarization of (X,ω)(X,\omega). Polarizations of this form are therefore called Kähler polarizations.

Quantum states and wave functions

Upon (geometric) quantization of the physical system described by the symplectic manifold (X,ω)(X, \omega) a quantum state is supposed to be a function on XX – or rather a section of a prequantum line bundle which is a “wave-function that only depends on the canonical coordinates”, not on the canonical momenta. In terms of polarizations this is formalized by saying that a quantum state is a section which is covariant constant along the leaves of the polarization (along the “momentum direction”).

Bohr-Sommerfeld leaves

After a choice of prequantum line bundle \nabla lifting ω\omega, a Bohr-Sommerfeld leaf of a (real) polarization is a leaf on which the prequantum line bundle is not just flat, but also trivializable as a circle bundle with connection.

Liouville integrability

If a polarization on an 2n2n-dimensional symplectic manifold is generated from nn Hamiltonian vector fields whose Hamiltonians commute with each other under the Poisson bracket (and one of them is regarded as that generating time evolution of a mechanical system) then one speaks of a Liouville integrable system.

Refinement to higher geometric quantization

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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)

Refinement to higher quantization by push-forward

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory KUKUthird integral SW class W 3W_3spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class w 4w_4
2integral Morava K-theory K˜(2)\tilde K(2)seventh integral SW class W 7W_7Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation



Lecture notes include

  • Alan Weinstein, Lectures on Symplectic manifolds Lecture 2 Lagrangian splittings, real and complex polarizations, Kähler manifolds, CBMS Regional Conference Series in Mathematics, AMS (1977)
  • Kristin Shaw, An introduction to polarizations (pdf)

and section 4 and 5 of

or section 5 of

  • A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, Mathematical Foundations of Geometric Quantization Extracta Math. 13 (1998) 135-238 (arXiv:math-ph/9904008)


  • Yuichi Nohara, Independence of Polarization in Geometric Quantization (pdf)

Lagrangian submanifolds of L-infinity algebroids are considered in

A candidate for polarizations for higher geometric quantization in nn-plectic geometry is discussed in Chapter 7 of

Revised on October 11, 2013 20:10:34 by Toby Bartels (