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
A Heisenberg $n$-group is a sub-∞-group of a quantomorphism n-group of a n-plectic space $(G, \omega)$ equipped with ∞-group structure, on those whose underlying n-plectomorphisms act by lect ∞-action of $G$ on itself. This is the higher refinement of the traditional notion of Heisenberg group.
For $(G, \omega)$ an n-plectic geometry with higher geometric prequantization $(G, \nabla)$ and for $G$ equipped with ∞-group structure, the corresponding Heisenberg $\infty$-group is the sub-∞-group of the quantomorphism n-group on those elements whose corresponding n-plectomorphism is given by such a left action.
The corresponding Lie n-algebra is the Heisenberg Lie n-algebra.
For $G$ a compact simply connected simple Lie group, there is the “WZW gerbe”, hence the circle 2-bundle with connection on $G$ whose curvature 3-form is the left invariant extension $\langle \theta \wedge [\theta \wedge \theta]\rangle$ of the canonical Lie algebra 3-cocycle to the group
The string 2-group is the smooth 2-groupo of automorphism of $\mathcal{L}_{WZW}$ which cover the left action of $G$ on itself (hence the “Heisenberg 2-group” of $\mathcal{L}_{WZW}$ regarded as a prequantum 2-bundle)
This is due to (Fiorenza-Rogers-Schreiber 13, section 6.2.1).
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 $X$)
Last revised on October 17, 2013 at 22:35:46. See the history of this page for a list of all contributions to it.