| Gauge Field Terminology | Fiber Bundle Terminology |
|---|---|
| spacetime space of phase factors symmetry or gauge group gauge transformations gauge principle classical fields gauge potential gauge field strength | base space bundle space structure group inner automorphisms of the bundle G-equivalence principle cross section of a vector bundle connection 1-form curvature 2-form |
| electromagnetism electromagnetism with monopoles Dirac's monopole quantization | dynamical theory with a connection in a U(1)-bundle connection in a nontrivial U(1)-bundle over \(\mathbb{R}^{2} \times \mathrm{S}^{2}\) classification of U(1)-bundles according to the first Chern class |
| Yang-Mills theory instanton number | dynamical theory with a connection in a SU(2)-bundle classification of SU(2)-bundles over \(\mathrm{S}^{4}\) according to the second Chern class |
columns: Gauge Field Terminology | Fiber Bundle Terminology
| potential | field strength | Bianchi identity | Noether identity |
|---|---|---|---|
| \(\vartheta^{\alpha}\) | \(T^{\alpha}=D \vartheta^{\alpha}\) | \(D T^{\alpha} \equiv R_{\gamma}{ }^{\alpha} \wedge \vartheta^{\gamma}\) | \[ \begin{aligned} \widehat{D} \Sigma_{\alpha} & \left.\cong\left(e_{\alpha}\right\rfloor R_{\beta}^{\gamma}\right) \wedge \Delta^{\beta} \gamma \\ & \left.-\frac{1}{2}\left(e_{\alpha}\right\rfloor Q_{\beta \gamma}\right) \wedge \sigma^{\beta \gamma} \end{aligned} \] |
| \(\Gamma_{\alpha}{ }^{\beta}\) | \[ \begin{aligned} & R_{\alpha}{ }^{\beta}=d \Gamma_{\alpha}{ }^{\beta} \\ & -\Gamma_{\alpha}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\beta} \end{aligned} \] | \(D R_{\alpha}{ }^{\beta} \equiv 0\) | \(D \Delta^{\alpha}{ }_{\beta}+\vartheta^{\alpha} \wedge \Sigma_{\beta} \cong \sigma^{\alpha}{ }_{\beta}\) |
columns: potential | field strength | Bianchi identity | Noether identity
| Objects | \(p\)-form | Components | \(\mathrm{n}=4\) | 3 | 2 | |
|---|---|---|---|---|---|---|
| \(\vartheta^{\alpha}\) | Vector | 1 | \(n^{2}\) | 16 | 9 | 1 |
| \(\Gamma_{\alpha}^{\star}\) | Vector | 1 | \(n^{2}\) | 16 | 9 | 1 |
| \(T^{\alpha}\) | Vector | 2 | \(n^{2}(n-1) / 2\) | 24 | 9 | 2 |
| \(R^{\alpha \beta}\) | Bivector | 2 | \(n^{2}(n-1)^{2} / 4\) | 36 | 9 | 1 |
| \(\Sigma_{\alpha}\) | Vector | \(n-1\) | \(n^{2}\) | 16 | 9 | 4 |
| \(\tau_{\alpha \beta}\) | Bivector | \(n-1\) | \(n^{2}(n-1) / 2\) | 24 | 9 | 2 |
| \(\eta_{\alpha}\) | Vector | \(n-1\) | \(n^{2}\) | 16 | 9 | 4 |
columns: | Objects | \(p\)-form | Components | \(\mathrm{n}=4\) | 3 | 2
| Spin | Gravitational | YM anomaly |
|---|---|---|
| 1/2 | 1 | 1 |
| 3/2 | -21 | 3 |
columns: Spin | Gravitational | YM anomaly
| := | : Equal by definition |
|---|---|
| \(\equiv\) | : Identically equal |
| \(\approx\) | : Isomorphic |
| \(\simeq\) | : Approximately equal |
| ~ | : Asymptotic expansion |
| × | : Topological product |
| ⊗ | : Direct product |
| ⊕ | : Direct sum |
| ⊗ | : Semidirect product |
| \(\mathrm{G}_{\circ}\) | : Connected component of the group |
| G | : Simply connected covering group |
| \([\mathrm{A}, \mathrm{B}]:=\mathrm{AB}-\mathrm{BA}\) | : Commutator |
| \(\{\mathrm{A}, \mathrm{B}\}:=\mathrm{AB}+\mathrm{BA}\) | : Anticommutator |
| A*= C A | : Complex-conjugate matrix |
| \(\mathrm{A}^{+}=\mathrm{A}^{* \mathrm{~T}}\) | : Hermitian adjoint matrix |
| \(\pi^{-1}(\mathrm{~N})\) | : Original domain of N with respect to the mapping \(\pi: \mathrm{M} \rightarrow \mathrm{N}\) |
| \(\mathrm{T}_{(\mu \nu)}:=\frac{1}{2}\left(\mathrm{~T}_{\mu \nu}+\mathrm{T}_{\nu \mu}\right)\) | : Symmetrized second-rank tensor |
| \(\mathrm{T}_{[\mu \nu]}:=\frac{1}{2}\left(\mathrm{~T}_{\mu \nu}-\mathrm{T}_{\nu \mu}\right)\) | : Antisymmetrized second-rank tensor |
columns: := | : Equal by definition