Abelian gerbes as a gauge theory of quantum mechanics on phase space

Abstract

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A, B, H : a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H = dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constanth. U(1) gauge transformations acting on the triple A, B, H are also defined, parametrized either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness versus classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H ]/2π i is an integral class in de Rham cohomology is related to the discretization of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction to generalized complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.

On the Noncommutative Eikonal

“…The term 2i1 can be interpreted as a constant potential, and can therefore be dropped. As it stands, (49) is strictly equivalent to the Hamilton-Jacobi equation (36) when U = 0, and we can declare…”

“…Now (36) and (39) cannot be balanced dimensionally in terms of just one dimensionful parameter; at least one more dimensionful parameter is needed for homogeneity. Planck's constant does precisely that job.…”

“…(42) and (43) above re-express the Hamilton-Jacobi equations (36) and (39) in terms of the noncommutative variables X, Y, P X , P Y . However, in general one should stop short of calling (42) and (43) Hamilton-Jacobi equations in the strict sense of the word.…”

“…Thus Eqs. (36) and (39) are bona fide HamiltonJacobi equations. However, neither the Bopp shift (4) nor its inverse is a canonical transformation, because the algebra satisfied by X, Y, P X , P Y differs from that satisfied by Q A , Q B , P A , P B .…”

A supermembrane with central charges on a G2 manifold

“…It has a natural interpretation in terms of the existence of fluxes on the compact sector of the target space. In fact, the existence of fluxes is equivalent to the existence of a bundle gerbe or higher order bundle on the target space [54][55][56][57][58][59][60] In the case of MIM2 on a T 7 target, we should then consider all possible immersions and impose for each of them the topological or central charge condition. This is a geometrical argument emphasizing that we should consider the summation of all possible immersions from to the target; see also [61].…”