Higher U(1)-gerbe connections in geometric prequantization

Abstract

We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields -- which is the higher Poisson bracket of local observables -- and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry.Comment: Title changed. Exposition revised. 55 page

The Rational Higher Structure of M‐theory

“…This is reflected in the fact that non‐homotopy theory has no answer to the evident followup question; if 2‐cocycles classify central extensions, then: For example, the Lie algebra itself (for ) carries no non‐trivial 2‐cocycle, but it carries, a non‐trivial 3‐cocycle (in fact precisely one, up to rescaling) given on elements by where is the Lie bracket, and the Killing form. This cocycle controls both the SU( n ) WZW‐model as well as SU( n )p‐Chern‐Simons theory (see [] for the higher‐structure perspective on this phenomenon) and hence is crucial both in field theory (gauge instantons) as well as in string theory (rational CFT compactifications).…”

Parametrised Homotopy Theory and Gauge Enhancement

“…We prove that the topological charge structure of fundamental super M-branes produces twisted K-theory charges of Type IIA D-branes under double dimensional reduction when all torsion effects are ignored 1 . Our result is based upon Sati's conjecture [2] that the topological M-brane charge is classified by degree-4 cohomotopy, and our result in turn contributes to a growing body of evidence for this conjecture (see [3][4][5][6][7][8][9][10]).…”

Lie n-algebras of BPS charges

“…But in [21,22,39] we explained that the WZW-type sigma models for super p 2 -branes with (higher) gauge fields on the their worldvolume and on which super p 1 -branes may end, are globally defined not on target superspacetime X itself, but on the total space X of a super p 1 -stack extension X → X of superspacetime, which itself is a differential refinement of the p 1 -gerbeX that underlies the WZW term of the p 1 -branes. Moreover, in [16] we showed that the higher Heisenberg-Kostant-Souriau extensions of remark 2.11 generalizes to such higher stacky base spaces. Schematically this follows now by the following picture, the full details are in [38,39]: .…”

“…We indicate how the higher stacky Heisenberg-Kostant-Souriau extension of [16] generalizes this to higher gauged WZW models (such as the Green-Schwarz-type sigmamodel for the M5-brane) and produces cohomological corrections to the naive brane charges.…”

“…By the results of [16] the group of currents induced by this, as discussed above, is the "quantomorphism group" of this bundle regarded as a prequantum line bundle in the sense of Souriau. It is a famous fact [30] (see [37, around proposition 42]), that this is the Kac-Moody central extension of the loop group of G, whose Lie algebra is the corresponding affine Lie algebra of G. This is "the" current algebra as seen in most of the mathematics literature.…”