【ゲージ理論】いくらかの公式（１）

(1)
$\mathfrak{g}$の基底を$\{E_a\}$とし、$K=K^a{E}_a,K^a\in {\mathrm{\Omega }}^k(M)$などと書く。
$$\begin{array}{rl}[K\wedge L]& =K^a\wedge L^b[E_a,E_b]=-(-1{)}^{kl}{L}^b\wedge K^a[E_b,E_a]\\ & =(-1{)}^{kl+1}[L\wedge K]\end{array}$$

(2)
$$\begin{array}{rl}(-1{)}^{mk}[[K\wedge L]\wedge M]& =(-1{)}^{mk}{K}^a\wedge L^b\wedge M^c[[E_a,E_b],E_c]\\ (-1{)}^{kl}[[L\wedge M]\wedge K]& =(-1{)}^{kl}{L}^b\wedge M^c\wedge K^a[[E_b,E_c],E_a]=(-1{)}^{kl+k(l+m)}{K}^a\wedge L^b\wedge M^c[[E_b,E_c],E_a]\\ (-1{)}^{lm}[[M\wedge K]\wedge L]& =(-1{)}^{lm}{M}^c\wedge K^a\wedge L^b[[E_c,E_a],E_b]=(-1{)}^{lm+m(k+l)}{K}^a\wedge L^b\wedge M^c[[E_c,E_a],E_b]\end{array}$$
より
$$\begin{array}{rl}& (-1{)}^{mk}[[K\wedge L]\wedge M]+(-1{)}^{kl}[[L\wedge M]\wedge K]+(-1{)}^{lm}[[M\wedge K]\wedge L]\\ & =(-1{)}^{mk}{K}^a\wedge L^b\wedge M^c([[E_a,E_b],E_c]+[[E_b,E_c],E_a]+[[E_c,E_a],E_b])=0\end{array}$$

(3)
$$\begin{array}{rl}d[K\wedge L]& =d(K^a\wedge L^b)[E_a,E_b]=(d{K}^a)\wedge L^b+(-1{)}^k{K}^a\wedge d(L^b)[E_a,E_b]\\ & =[dK\wedge L]+(-1{)}^k[K\wedge dL]\end{array}$$

(1)
$d_A\psi =d\psi +[A\wedge \psi ]$より
$$\begin{array}{rl}{d}_A{d}_A\psi & =d(d\psi +[A\wedge \psi ])+[A\wedge (d\psi +[A\wedge \psi ])]\\ & =d(A\wedge \psi -(-1{)}^k\psi \wedge A)+A\wedge d\psi -(-1{)}^{k+1}d\psi \wedge A+[A\wedge [A\wedge \psi ]]\\ & =dA\wedge \psi -\psi \wedge dA+A\wedge [A\wedge \psi ]-(-1{)}^{k+1}[A\wedge \psi ]\wedge A\\ & =dA\wedge \psi -\psi \wedge dA+A\wedge (A\wedge \psi -(-1{)}^k\psi \wedge A)-(-1{)}^{k+1}(A\wedge \psi -(-1{)}^k\psi \wedge A)\wedge A\\ & =dA\wedge \psi -\psi \wedge dA+A\wedge A\wedge \psi -\psi \wedge A\wedge A\\ & =F\wedge \psi -\psi \wedge F=[F\wedge \psi ]\end{array}$$

(2)
$$\begin{array}{rl}{d}_{A}F& =dF+[A\wedge F]=d(dA+A\wedge A)+A\wedge F-F\wedge A\\ & =dA\wedge A-A\wedge dA+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0\end{array}$$

(1)
$$\begin{array}{rl}{d}_A\psi & =d\psi +[A\wedge \psi ]\\ & =\frac{1}{k!}({\partial }_{i_1}{\psi }_{i_2\cdots i_{k+1}}+[A_{i_1},{\psi }_{i_2\cdots i_{k+1}}])d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_{k+1}}\\ & =\frac{1}{k!}({\partial }_{[i_1}{\psi }_{i_2\cdots i_{k+1}]}+[A_{i_1},{\psi }_{i_2\cdots i_{k+1}}])d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_{k+1}}\\ & =\frac{1}{k!}({\mathrm{\nabla }}_{[i_1}{\psi }_{i_2\cdots i_{k+1}]}+[A_{i_1},{\psi }_{i_2\cdots i_{k+1}}])d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_{k+1}}\\ & =\frac{1}{k!}({\mathrm{\nabla }}_{i_1}{\psi }_{i_2\cdots i_{k+1}}+[A_{i_1},{\psi }_{i_2\cdots i_{k+1}}])d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_{k+1}}\\ & =\frac{1}{k!}(D_A^g{)}_{[i_1}{\psi }_{i_2\cdots i_{k+1}]}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_{k+1}}\end{array}$$

(2)
$$(D_A{)}_j{\psi }_{{\ell }_1\cdots {\ell }_k}={\partial }_j{\psi }_{{\ell }_1\cdots {\ell }_k}+[A_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]$$
より
$$(D_A{)}_i(D_A{)}_j{\psi }_{{\ell }_1\cdots {\ell }_k}={\partial }_i{\partial }_j{\psi }_{{\ell }_1\cdots {\ell }_k}+[{\partial }_i{A}_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_j,{\partial }_i{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_i,{\partial }_j{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_i,[A_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]]$$
であるから
$$[(D_A{)}_i,(D_A{)}_j]{\psi }_{{\ell }_1\cdots {\ell }_k}=[{\partial }_i{A}_j-{\partial }_j{A}_i,{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_i,[A_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]]-[A_j,[A_i,{\psi }_{{\ell }_1\cdots {\ell }_k}]]$$
である。ここでLie環のJacobi恒等式
$$[A_i,[A_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]]+[A_j,[{\psi }_{{\ell }_1\cdots {\ell }_k},A_i]]+[{\psi }_{{\ell }_1\cdots {\ell }_k},[A_i,A_j]]=0$$
を使うと
$$\begin{array}{rl}[(D_A{)}_i,(D_A{)}_j]{\psi }_{{\ell }_1\cdots {\ell }_k}& =[{\partial }_i{A}_j-{\partial }_j{A}_i,{\psi }_{{\ell }_1\cdots {\ell }_k}]-[A_j,[{\psi }_{{\ell }_1\cdots {\ell }_k},A_i]]-[{\psi }_{{\ell }_1\cdots {\ell }_k},[A_i,A_j]]-[A_j,[A_i,{\psi }_{{\ell }_1\cdots {\ell }_k}]]\\ & =[{\partial }_i{A}_j-{\partial }_j{A}_i,{\psi }_{{\ell }_1\cdots {\ell }_k}]+[[A_i,A_j],{\psi }_{{\ell }_1\cdots {\ell }_k}]\\ & =[F_{ij},{\psi }_{{\ell }_1\cdots {\ell }_k}]\end{array}$$
を得る。

(3)
ほぼ(2)と同様に
$$(D_A^g{)}_i(D_A^g{)}_j{\psi }_{{\ell }_1\cdots {\ell }_k}={\mathrm{\nabla }}_i{\mathrm{\nabla }}_j{\psi }_{{\ell }_1\cdots {\ell }_k}+[{\mathrm{\nabla }}_i{A}_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_j,{\mathrm{\nabla }}_i{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_i,{\mathrm{\nabla }}_j{\psi }_{{\ell }_1\cdots {\ell }_k}]+[A_i,[A_j,{\psi }_{{\ell }_1\cdots {\ell }_k}]]$$
となるから、
$$\begin{array}{r}[(D_A^g{)}_i,(D_A^g{)}_j]{\psi }_{{\ell }_1\cdots {\ell }_k}=({\mathrm{\nabla }}_i{\mathrm{\nabla }}_j-{\mathrm{\nabla }}_j{\mathrm{\nabla }}_i){\psi }_{{\ell }_1\cdots {\ell }_k}+[F_{ij},{\psi }_{{\ell }_1\cdots {\ell }_k}]\end{array}$$
を得る。