Abstract
In the early years of the 19th Century, after having studied the trajectories of some celestial objects as Asteroid Ceres or Comet Biela, Carl Freidrich Gauss set forth his famous formula for the linking number, thus providing one of the first mathematical tools allowing a characterization of celestial orbits configurations.
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Notes
- 1.
The \(U(1)\) gauge fields used by physicists are related to the top component of the corresponding DB 1-cocycles according to \(\mathcal{A}_\alpha = 2 \pi (\hbar c / e) A_\alpha \), where \(e\) is the charge of the electron, \(\hbar \) is the Planck constant and \(c\) is the speed of light; the field strength tensor is then \(\mathcal{F} = 2 \pi (\hbar c / e) \mathbf{F}\).
- 2.
which in \(\check{\text{ C }}\)ech cohomology is the equivalent of the exterior product.
- 3.
The Chern-Simons action can be generalized to \((4l+3)\)-dimensional manifolds as it is the only dimension where the DB square \(\bar{\mathbf{A}} \star \bar{\mathbf{A}}\) is not zero, \(\bar{\mathbf{A}}\) being a \((2l+1)\)-connection.
- 4.
Strictly speaking there is also a factor \((e/2 \pi \hbar c)^2\) in front of the abelian and non-abelian lagrangians for the reason explained in the first footnote.
- 5.
The first component of this DB cocycle defines a closed \(1\)-current which doesn’t have integral periods, these periods being defined as intersections with \(\varSigma / 2k\).
- 6.
This class rather belongs to the translation group \(\varOmega ^2_{\mathbb {Z}}(M)^* \).
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Thuillier, F. (2015). Deligne-Beilinson Cohomology in U(1) Chern-Simons Theories. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_8
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