Higher prequantum geometry by Urs Schreiber in 2017

1. Prequantum field theory

The Faraday tensor \[F\] encodes the classical Lorentz force that the electromagnetic field on a spacetime \[X\] exerts on an electron (propagating in that field).

Within a particular region \[U\] the quantum electron couples to the vector potential which is a differential 1-form \[A_U\], such that \[dA_U = F|_U\]. This is to say that the differential of the vector potential within the region \[U\] is equivalent to the restriction of the global Faraday tensor \[F\] to the given region.

To define the quantized electron globally requires lifting of the locally defined vector potentials to an \[\mathbb{R}/\mathbb{Z}\]-principal connection on a \[\mathbb{R}/\mathbb{Z}\]-principal bundle over spacetime \[X\]. The first Chern class of the principle bundle is quantized as the magnetic charge whose induced force the electron feels. This is a prerequisite to the quantization of the electron dynamics and is an example of pre-quantization.

The coupling of quarks to the week and strong nuclear forces is described locally by a \[\mathfrak{su}(n)\] Lie algebra valued 1-form \[A_U\] from which the strength of the nuclear force field is encoded by the 2-form \[F|_U = dA_U + 1/2 [ A_U \wedge A_U ]\]. For the consistent quantization of quarks as required for the global definition of Wilson loop observables, this local data must be lifted to an \[SU(n)\]-principal connection on a \[SU(n)\]-principal bundle over spacetime \[X\]. The second Chern class of this bundle is quantized and is physically interpreted as the number of instantons. Instantons are expressed in physics via Chern-Simons 3-forms which are mathematically the pre-quantization of the 4-form \[ \mathrm{tr}(F \wedge F)\] to a 2-gerbe with 2-connection.