| \(d\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| \(\epsilon\) | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 |
| \(\epsilon^{\prime}\) | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| \(\epsilon^{\prime \prime}\) | 1 | -1 | 1 | -1 |
columns: \(d\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
| \(D=3\) CS Lagrangians | Characteristic classes | Groups |
|---|---|---|
| \[ L_{3}^{(A) d S}=\epsilon_{a b c}\left(R^{a b} \pm \frac{e^{a} e^{b}}{3 l^{2}}\right) e^{c} \] | \(\mathbf{E}_{4}=\epsilon_{a b c}\left(R^{a b} \pm \frac{e^{a} e^{b}}{l^{2}}\right) T^{c}\) | \begin{tabular}[t]{l} \(S O(4)\), \\ \(S O(3,1)\) \\ or \(S O(2,2)\) \end{tabular} |
| \(L_{3}^{\text {Lor }}=\omega^{a}{ }_{b} d \omega^{b}{ }_{a}+\frac{2}{3} \omega^{a}{ }_{b} \omega^{b}{ }_{c} \omega^{c}{ }_{a}\) | \(\mathbf{P}_{4}^{\text {Lor }}=R^{a}{ }_{b} R^{b}{ }_{a}\) | \(S O(2,1)\) |
| \(L_{3}^{T o r}=e^{a} T_{a}\) | \(\mathbf{N}_{4}=T^{a} T_{a}-e^{a} e^{b} R_{a b}\) | \(S O(2,1)\) |
| \(L_{3}^{U(1)}=A d A\) | \(\mathbf{P}_{4}^{U(1)}=F F\) | \(U(1)\) |
| \(L_{3}^{S U(N)}=\operatorname{Tr}\left[\mathbf{A} d \mathbf{A}+\frac{2}{3} \mathbf{A} \mathbf{A} \mathbf{A}\right]\) | \(\mathbf{P}_{4}^{S U(4)}=\operatorname{Tr}[\mathbf{F F}]\) | \(S U(N)\) |
columns: \(D=3\) CS Lagrangians | Characteristic classes | Groups
| \(D=7 \mathrm{CS}\) Lagrangians | Characteristic classes |
|---|---|
| \(L_{7}^{\text {Lor }}=\omega(d \omega)^{3}+\frac{8}{5} \omega^{3}(d \omega)^{2}+\cdots+\frac{4}{7} \omega^{7}\) | \(\mathbf{P}_{8}^{\text {Lor }}=R^{a}{ }_{b} R^{b}{ }_{c} R^{c}{ }_{d} R^{d}{ }_{a}\) |
| \(L_{7}^{I}=\left(\omega^{a}{ }_{b} d \omega^{b}{ }_{a}+\frac{2}{3} \omega^{a}{ }_{b} \omega^{b}{ }_{c} \omega^{c}{ }_{a}\right) R^{a}{ }_{b} R^{b}{ }_{a}\) | \(\left(\mathbf{P}_{4}^{\text {Lor }}\right)^{2}=\left(R^{a}{ }_{b} R^{b}{ }_{a}\right)^{2}\) |
| \(L_{7}^{I I}=\left(e^{a} T_{a}\right)\left(R^{a}{ }_{b} R^{b}{ }_{a}\right)\) | \[ \begin{aligned} & \mathbf{N}_{4} \mathbf{P}_{4}^{\text {Lor }}= \\ & \left(T^{a} T_{a}-e^{a} e^{b} R_{a b}\right) R^{c}{ }_{d} R^{d}{ }_{c} \end{aligned} \] |
columns: \(D=7 \mathrm{CS}\) Lagrangians | Characteristic classes
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| \(\varepsilon\) | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| \(\varepsilon^{\prime}\) | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 |
| \(\varepsilon^{\prime \prime}\) | 1 | -1 | 1 | -1 |
columns: n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
| \(L\) | \(p p p p\) |
|---|---|
| k \(k k\) k | \(\bar{L}\) |
columns: \(L\) | \(p p p p\)
| \(L\) | \(P\) | \(Q\) |
|---|---|---|
| \(K\) | \(L\) | \(S\) |
| \(S\) | \(\bar{Q}\) | \(R\) |
columns: \(L\) | \(P\) | \(Q\)
| \(L\) | \(p p p p\) |
|---|---|
| k \(k k k\) | \(\bar{L}\) |
columns: \(L\) | \(p p p p\)
| \(\mathfrak{s} \mathfrak{u}(2 \mid 2)\) | create \(8_{B}+8_{F}\) magnons |
|---|---|
| destroy 8 + 8 magnons | \(\overline{\mathfrak{s} \mathfrak{u}(2 \mid 2)}\) |
columns: \(\mathfrak{s} \mathfrak{u}(2 \mid 2)\) | create \(8_{B}+8_{F}\) magnons