geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
The notion of Heisenberg Lie $n$-algebra is the generalization to n-plectic geometry of the notion of Heisenberg Lie algebra in symplectic geometry.
The Heisenberg Lie $n$-algebra integrates to the Heisenberg n-group.
We discuss a generalization of the notion of Heisenberg Lie algebra from ordinary symplectic geometry to a notion of Heisenberg Lie n-algebra in higher geometric quantization of n-plectic geometry.
The following definition is naturally motivated from the fact that:
Poisson bracket Lie algebra, the one underlying the corresponding Poisson algebra (see below) on the constant and linear functions.
In view of this, the following definition takes the Heisenberg Lie $n$-algebra to be the sub-Lie $n$-algebra of the Poisson Lie n-algebra on the linear and constant differential forms.
First we need the following definition, which is elementary, but nevertheless worth making explicit once.
Let $n \in \mathbb{N}$, let $(V, \omega)$ be an n-plectic vector space.
The corresponding $n$-plectic manifold is the n-plectic manifold $(V, \mathbf{\omega})$, with $V$ now the canonical smooth manifold structure on the given vector space, and with
the differential form obtained by left (right) translating $\omega$ along $V$.
Explicitly, for all vector fields $\{v_i \in \Gamma(T V)\}_{i = 1}^n$ and all points $x \in V$ we set
Here on the right – and in all of the following – we are using that every tangent space $T_x V$ of $V$ is naturally identified with $V$ itself
Let $n \in \mathbb{N}$, let $(V, \omega)$ be an n-plectic vector space and let $(V, \mathbf{\omega})$ be the corresponding n-plectic manifold.
The Heisenberg Lie $n$-algebra $Heis(V,\omega)$ is the sub-Lie n-algebra of the Poisson Lie n-algebra $\mathcal{P}(V, \omega)$ on those differential forms which are either linear or constant (with respect to left/right translation on $V$).
All one has to observe is:
This is indeed a sub-Lie $n$-algebra.
We need to check that the linear and constant forms are closed under the L-infinity algebra brackets of $\mathcal{P}(V, \omega)$.
The only non-trivial such brackets are the unary one, and the ones on elements all of degree 0.
The unary bracket is given by the de Rham differential. Since this sends a linear form to a constant form and a constant form to 0, our sub-complex is closed under this.
Similarly, the brackets on elements all in degree 0 is given by contraction of $\mathbf{\omega}$ with the Hamiltonian vector fields of linear or constant forms. Since $\mathbf{\omega}$ is a constant form, and since the de Rham differential of a linear or constant form is constant (or even 0), these Hamiltonian vector fields are necessarily constant. Hence their contraction with $\mathbf{\omega}$ gives a constant form.
slice-automorphism ∞-groups in higher prequantum geometry
cohesive ∞-groups: | Heisenberg ∞-group | $\hookrightarrow$ | quantomorphism ∞-group | $\hookrightarrow$ | ∞-bisections of higher Courant groupoid | $\hookrightarrow$ | ∞-bisections of higher Atiyah groupoid |
---|---|---|---|---|---|---|---|
L-∞ algebras: | Heisenberg L-∞ algebra | $\hookrightarrow$ | Poisson L-∞ algebra | $\hookrightarrow$ | Courant L-∞ algebra | $\hookrightarrow$ | twisted vector fields |
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
The topological part of the Heisenberg Lie 2-algebra of the string sigma-model called the WZW model has been discussed (not under this name, though) in
and shown to be the string Lie 2-algebra.
General discussion in the broader context of higher differential geometry and higher prequantum geometry is in
Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)
Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)
Last revised on July 20, 2015 at 16:56:07. See the history of this page for a list of all contributions to it.