nLab
Hamiltonian symplectomorphism infinity-group

Contents

Idea

The generalization of the traditional notion to symplectomorphism group of a symplectic manifold to n-plectic spaces in higher geometry

Definition

For H\mathbf{H} a cohesive (∞,1)-topos equipped with differential cohesion, let 𝔾Grp(H)\mathbb{G} \in Grp(\mathbf{H}) by an ∞-group equipped with braided ∞-group structure. Write B𝔾 conn\mathbf{B}\mathbb{G}_{conn} for the corresponding moduli ∞-stack of 𝔾\mathbb{G}-principal ∞-connections (see there). For :XB𝔾 conn\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn} a 𝔾\mathbb{G}-principal ∞-connection on some object XHX \in \mathbf{H}, which here we call a prequantum ∞-bundle. the corresponding quantomorphism ∞-group by definition comes with a canonical homomorphism

QuantMorph()Aut(X) \mathbf{QuantMorph}(\nabla) \to \mathbf{Aut}(X)

to the automorphism ∞-group of XX. The Hamiltonian symplectomorphism \infty-group HamSym()\mathbf{HamSym}()\nabla is the 1-image of this map

QuantMorph()HamSymp()Aut(X). \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSymp}(\nabla) \hookrightarrow \mathbf{Aut}(X) \,.

higher and integrated Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

References

Last revised on August 17, 2018 at 10:19:25. See the history of this page for a list of all contributions to it.