nLab
higher symplectic geometry

Context

Symplectic geometry

Higher geometry

Contents

Idea

Higher symplectic geometry is the generalization of symplectic geometry to the context of higher geometry.

It involves two kinds of generalizations:

  1. the symplectic form generalizes from a 2-form to a form of arbitrary arity. This aspect is called multisymplectic geometry.

  2. the base manifold is generalized to a smooth ∞-groupoid or ∞-Lie algebroid. For binary symplectic forms this is called a symplectic Lie n-algebroid.

In the full higher symplectic geometry both of these aspects are unified: a multisymplectic \infty-groupoid is a smooth ∞-groupoid equipped with a differential n-form on smooth ∞-groupoids satisfying some condition.

Examples of higher symplectic geometries arise naturally as the covariant phase spaces “over the point” or “in top codimension” (in the sense of extended topological quantum field theory) in systems of ∞-Chern-Simons theory: their \infty-multisymplectic form is the invariant polynomial that defines the theory.

Definition

Definition

Let 𝔞\mathfrak{a} be an L-∞ algebroid. For nn \in \mathbb{N}, an n-plectic form or multisymplectic form of nn arguments on 𝔞\mathfrak{a} is

  • an invariant polynomial ω\omega on 𝔞\mathfrak{a} which is nn-linear (takes nn arguments):

    ωW n(𝔞)=Ω n(𝔞); \omega \in W^n(\mathfrak{a}) = \Omega^n(\mathfrak{a}) \,;
  • such that the contraction morphism

    ι ()ω:T𝔞W n1(𝔞)=Ω n1(𝔞) \iota_{(-)}\omega : T \mathfrak{a} \to W^{n-1}(\mathfrak{a}) = \Omega^{n-1}(\mathfrak{a})

in injective.

Example

If 𝔞\mathfrak{a} is a Lie 0-algebroid (over a smooth manifold) then it is simply that smooth manifold, 𝔞=X\mathfrak{a} = X. In this case W(𝔞)=Ω (X)W(\mathfrak{a}) = \Omega^{\bullet}(X) is the ordinary de Rham complex and an invariant polynomial is a closed differential form of positive degree.

In this case an nn-plectic form on 𝔞\mathfrak{a} is a closed nn-form ω(,,.,)\omega(-,-,.\dots, -) on XX such that for every vector field vΓ(TX)v \in \Gamma(T X) we have

(ω(v,,,)=0)(v=0). (\omega(v,-,\cdots,-) = 0) \;\; \Rightarrow \;\; (v = 0).

Applications

Higher Chern-Simons theory

Every invariant polynomial ω\omega induces an ∞-Chern-Simons theory action functional

S CS:Ω(Σ,𝔞). S_{CS} : \Omega(\Sigma,\mathfrak{a}) \to \mathbb{R} \,.

The variation of that functional is

δS CS:A Σω(δA,F A,,F A). \delta S_{CS} : A \mapsto \int_\Sigma \omega(\delta A, F_A, \cdots, F_A) \,.

Therefore the condition that the invariant polynomial is nn-plectic amounts to saying that S CSS_{CS} has no spurious global symmetries.

(…)

Isbell duality between algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}geometryA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A}superalgebra
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}algebraA\phantom{A}A\phantom{A}geometryA\phantom{A}
A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}AQFTA\phantom{A}A\phantom{A}FQFTA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

∞-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)

References

Discussion of what here we call “higher symplectic geometry over Lie 0-algebroids” (multisymplectic geometry) is in

For more references see multisymplectic geometry.

Last revised on February 8, 2013 at 02:02:25. See the history of this page for a list of all contributions to it.