Author: Eiko

Tags: connection, differential algebra, flat connection

Time: 2024-10-26 10:05:41 - 2024-10-26 19:05:41 (UTC)

Coordinate Form

Following connections and curvature, we know the algebraic connections defined on a vector bundle goes as

$${\mathrm{\nabla }}^{(p)}:{\mathrm{\Omega }}^p\otimes V\to {\mathrm{\Omega }}^{p+1}\otimes V$$

with

$$\mathrm{\nabla }(\omega \otimes v)=d\omega \otimes v+(-1{)}^p\omega \wedge \mathrm{\nabla }v$$

where $\omega \in {\mathrm{\Omega }}^p$ and $v\in V$. The connection is flat if the curvature $R$ is zero, i.e. $R={\mathrm{\nabla }}^2={\mathrm{\nabla }}^{(1)}\circ {\mathrm{\nabla }}^{(0)}=0$.

Locally when we choose a coordinate isomorphism ${\mathcal{O}}^{\oplus n}\cong V$, we can write

$${\mathrm{\nabla }}^{(0)}:{\mathcal{O}}^{\oplus n}\to ({\mathrm{\Omega }}^1{)}^{\oplus n}$$

as ${\mathrm{\nabla }}^{(0)}=d\cdot I_n+\mathrm{\Lambda }\in \mathrm{Hom}({\mathcal{O}}^{\oplus n},({\mathrm{\Omega }}^1{)}^{\oplus n})$.

Coordinate Form of Higher Derivatives

What are the coordinate form for ${\mathrm{\nabla }}^{(p)}$ where $p\ge 1$? By definition it is ${\mathrm{\nabla }}^{(p)}:({\mathrm{\Omega }}^p{)}^n\to ({\mathrm{\Omega }}^{p+1}{)}^n$, and computation of $\mathrm{\nabla }(\sum {\omega }_i\otimes e_i)$ shows

$$\underset{―}{\omega }\mapsto d\underset{―}{\omega }+(-1{)}^p\mathrm{\Lambda }\cdot (\underset{―}{\omega }\wedge )$$

here $\underset{―}{\omega }=({\omega }_i{)}_i$ is the column vector form and $\mathrm{\Lambda }\cdot (\underset{―}{\omega }\wedge ):=(\sum _j{\omega }_j\wedge {\mathrm{\Lambda }}_{ij}{)}_i$. We can also simply write the above as

$${\mathrm{\nabla }}^{(p)}=d\cdot I_n+(-1{)}^{p+1}(\mathrm{\Lambda }\wedge )\in \mathrm{Hom}(({\mathrm{\Omega }}^p{)}^n,({\mathrm{\Omega }}^{p+1}{)}^n).$$

Coordinate Criterion of Flat Connection

The composition ${\mathrm{\nabla }}^{(1)}\circ {\mathrm{\nabla }}^{(0)}$ will map $\underset{―}{f}\in {\mathcal{O}}^n$ to

$$\begin{array}{rl}{\mathrm{\nabla }}^{(1)}(d\underset{―}{f}+\mathrm{\Lambda }\underset{―}{f})& =d(\mathrm{\Lambda }\underset{―}{f})-\mathrm{\Lambda }\cdot (d\underset{―}{f}+\mathrm{\Lambda }\underset{―}{f})\wedge \\ & =d\mathrm{\Lambda }\cdot \underset{―}{f}+\mathrm{\Lambda }\cdot d\underset{―}{f}-\mathrm{\Lambda }\cdot d\underset{―}{f}-\mathrm{\Lambda }\cdot (\mathrm{\Lambda }\underset{―}{f}\wedge )\\ & =d\mathrm{\Lambda }\cdot \underset{―}{f}-\mathrm{\Lambda }\cdot (\mathrm{\Lambda }\underset{―}{f}\wedge )\\ & =d\mathrm{\Lambda }\cdot \underset{―}{f}+\mathrm{\Lambda }\cdot (\wedge \mathrm{\Lambda }\underset{―}{f}).\end{array}$$

i.e. 

$${\mathrm{\nabla }}^2=d\mathrm{\Lambda }+\mathrm{\Lambda }\wedge \mathrm{\Lambda }$$

here $A\wedge B:=(\sum _k{a}_{ik}\wedge b_{kj}{)}_{ij}$. This means the coordinate criterion for a connection being flat is

$$d\mathrm{\Lambda }+\mathrm{\Lambda }\wedge \mathrm{\Lambda }=0.$$

Multi-dimensional Differentials

(Locally) when we have ${\mathrm{\Omega }}^1$ is generated by multiple independent differentials $d{x}_i$, we can write

$$\mathrm{\Lambda }=\sum _i{\mathrm{\Lambda }}^{(i)}d{x}_i,{\mathrm{\Lambda }}^{(i)}\in M_n(\mathcal{O})$$

Then the integrable condition evaluates to

$$\begin{array}{rl}d\mathrm{\Lambda }+\mathrm{\Lambda }\wedge \mathrm{\Lambda }& =\sum _{j,i}({\mathrm{\Lambda }}_i^{(j)}+{\mathrm{\Lambda }}^{(i)}{\mathrm{\Lambda }}^{(j)})d{x}_i\wedge d{x}_j\\ & =\sum _{i<j}({\mathrm{\Lambda }}_i^{(j)}-{\mathrm{\Lambda }}_j^{(i)}+[{\mathrm{\Lambda }}^{(i)},{\mathrm{\Lambda }}^{(j)}])d{x}_i\wedge d{x}_j.\end{array}$$

i.e. there are $(\genfrac{}{}{0ex}{}{d}{2})$ equations

$${\mathrm{\Lambda }}_j^{(i)}-{\mathrm{\Lambda }}_i^{(j)}=[{\mathrm{\Lambda }}^{(i)},{\mathrm{\Lambda }}^{(j)}],1\le i<j\le d.$$