An article that we have written:
Domenico Fiorenza, $\;$ Chris Rogers $\;$and$\;$ Urs Schreiber:
Higher geometric prequantum theory
published version renamed and more expository:
Higher $U(1)$-gerbe connections in geometric prequantization
Reviews in Mathematical Physics 28 06 (2016)
v3: pdf
lecture notes:
Higher geometric prequantum theory and The brane bouquet
talk at Bayrischzell workshop 2013
on the higher geometric refinement of prequantum field theory and geometric prequantization formulated generally in cohesive higher geometry.
In the companion article L-∞ algebras of local observables from higher prequantum bundles we discuss the L-∞ algebraic aspects in more detail. See also the discussion at higher Atiyah groupoid.
For more background see
and
Applications include
We develop the refinement of geometric prequantum theory to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show how there is canonically a tower of higher “gauge groupoids” assigned to a higher prequantization and establish the higher and integrated Kostant-Souriau extension of higher Hamiltonian symplectomorphisms fairly generally. In the special case of higher differential geometry over smooth manifolds we find the L-∞ algebra extension of Hamiltonian vector fields which is the higher-Poisson bracket of local observables and show that it is equivalent to the construction in n-plectic geometry proposed by Rogers. Finally we indicate examples of applications of higher prequantum theory in the extended geometric quantization of local quantum field theories and specifically in string geometry.
The differential concretification formula given in Remark 2.10.2 of the article is wrong as stated: it works in degree 1 but produces spurious contributions in degree 0.
This error had kindly been pointed out by Marco Benini and Alexander Schenkel.
(The error results from assuming that the evident model of the map $\sharp_1 [X, \mathbf{B}G_{conn}] \to \sharp_1 [X, \mathbf{B}G]$ by simplicial presheaves over the site of Cartesian manifolds $U$ — namely in degree 0 by sending horizontal smooth $\mathfrak{g}$-valued forms on $X \times U$ to the point, and in degree 1 by sending their gauge transformations with potentially discontinuous families of component functions to these families of component functions — is a $U$-wise Kan fibration. If it were, then the homotopy fiber product proposed in 2.10.2 would be computed by the 1-categorical fiber product with this map, which does give the claimed result. Alas, the map fails to satisfy the Kan lifting property in lowest degree. When computing the proposed homotopy fiber product properly it comes out right in degree 1, but it picks up spurious contributions in degree 0.)
A correct formula for the differential concretification process of $[X, \mathbf{B}G_{conn}]$ was published in
Section 3.2 (pp. 17) of
Marco Benini,$\;$ Alexander Schenkel,$\;$ Urs Schreiber:
The stack of Yang-Mills fields on Lorentzian manifolds
CMP 359 (2018) 765-820
differential cohomology in a cohesive topos
Higher geometric prequantum theory
Last revised on February 21, 2023 at 06:22:32. See the history of this page for a list of all contributions to it.