Connection Cochain of Abelian Extensions and Connection $1$-Forms

TL;DR

This paper explores the relationship between connection cochains of abelian extensions and connection 1-forms on principal bundles, clarifying their interplay in the context of abelian groups.

Contribution

It extends the concept of connection cochains to abelian extensions and relates them explicitly to connection 1-forms in principal bundles.

Findings

Connection cochains of abelian extensions are shown to correspond to connection 1-forms.

The paper clarifies the relationship between algebraic and geometric notions of connections.

Provides a framework for understanding connections in abelian principal bundles.

Abstract

In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure group is abelian.

Equations

$$\tau (ga)=\tau (g)+a$$

$${\tau }_{\ast ,1_{\text{Aut}(P)}}(X)(p)=0,$$

$$\tau (\tilde{g}a)=\tau (\tilde{g})+a,$$

$$\mu (g)a={\tilde{g}}^{-1}a\tilde{g},$$

$$\pi ({\tilde{g}}^{\prime }{\tilde{g}}^{-1})=g{g}^{-1}=1_G.$$

$${\tilde{g}}^{\prime }{\tilde{g}}^{-1}=a^{\prime }.$$

$${\tilde{g}}^{\prime -1}a{\tilde{g}}^{\prime }={\tilde{g}}^{-1}{a}^{\prime -1}a{a}^{\prime }\tilde{g}={\tilde{g}}^{-1}a\tilde{g}.$$

$$\nu (\tilde{g})a={\tilde{g}}^{-1}a\tilde{g}.$$

$$\displaystyle\mu_{*,1_{G}}(V)a=\frac{d}{dt}\bigg{|}_{t=0}\mu(c_{V}(t))a$$

$$(g_1,g_2)\mapsto s(g_1{g}_2{)}^{-1}s(g_1)s(g_2),$$

$$\chi (g_2,g_3)-\chi (g_1{g}_2,g_3)+\chi (g_1,g_2{g}_3)-\mu (g_3)\chi (g_1,g_2)=0.$$

$$(g_1,a_1)(g_2,a_2)=(g_1{g}_2,\chi (g_1,g_2)+\mu (g_2)a_1+a_2),$$

$$(g,a)\mapsto s(g)a$$

$$\Phi ((g_1,a_1)(g_2,a_2))$$

$$=s(g_1{g}_2)s(g_1{g}_2{)}^{-1}s(g_1)s(g_2)s(g_2{)}^{-1}{a}_{1}s(g_2)a_2$$

$$=s(g_1)a_{1}s(g_2)a_2$$

$$=\Phi (g_1,a_1)\Phi (g_2,a_2),$$

$$\tilde{g}\mapsto (g,s(g{)}^{-1}\tilde{g}).$$

$$\Phi \circ \Psi (\tilde{g})=\Phi (g,s(g{)}^{-1}\tilde{g})=s(g)s(g{)}^{-1}\tilde{g}=\tilde{g},$$

$$\Psi \circ \Phi (g,a)=\Psi (s(g)a)=(g,a),$$

$$C^{\infty }(P,G{)}^G=\{\gamma \in C^{\infty }(P,G)\mid \gamma (pg)=\gamma (p)\text{ for all p∈P and g∈G}\},$$

$$\text{Aut}(P)=\{\phi \in \text{Diff}(P)\mid \phi (pg)=\phi (p)g\text{ for all p∈P and g∈G}\}.$$

$$\iota (\gamma )(p)=p\gamma (p)$$

$$\rho (\phi )(x)=\pi (\phi (p)),$$

$$\mathfrak{X}(P{)}^r=\{X\in \mathfrak{X}(P)\mid {r_g}_{\ast }{X}_p=X_{pg}\text{ for all p∈P and g∈G}\},$$

$$\displaystyle{r_{g}}_{*}\frac{d}{dt}\bigg{|}_{t=0}\varphi(p,t)=\frac{d}{dt}\bigg{|}_{t=0}\varphi(p,t)g=\frac{d}{dt}\bigg{|}_{t=0}\varphi(pg,t),$$

$$C^{\infty }(P,\mathfrak{g}{)}^G=\{\gamma \in C^{\infty }(P,\mathfrak{g})\mid \gamma (pg)=\gamma (p)\text{ for all p∈P and g∈G}\},$$

$$\begin{array}{rl}(\mu (f)\gamma )(p)& =s(f{)}^{-1}\circ \iota (\gamma )\circ s(f)(p)=s(f{)}^{-1}(s(f)(p)\gamma (s(f)(p)))=p\gamma (s(f)(p))\\ & =\gamma \circ \sigma \circ f\circ \pi (p),\end{array}$$

$$\displaystyle\begin{split}(\mu_{*,1_{\text{Aut}(P)}}(X)\gamma)(p)&=\frac{d}{dt}\bigg{|}_{t=0}(\mu(\varphi_{X}(-,t))\gamma)(p)=\frac{d}{dt}\bigg{|}_{t=0}\gamma\circ\sigma\circ\varphi_{X}(-,t)\circ\pi(p)\\ &=\gamma_{*,\sigma(\pi(p))}\sigma_{*,\pi(p)}(X_{\pi(p)})\\ &=\gamma_{*,p}(\tilde{X}_{p}),\end{split}$$

$$\displaystyle\gamma_{*,p}(\tilde{X}_{p}-\tilde{X}_{p}^{\prime})=\gamma_{*,p}{j_{p}}_{*,1_{G}}V=\frac{d}{dt}\bigg{|}_{t=0}\gamma(pc_{V}(t))=\frac{d}{dt}\bigg{|}_{t=0}\gamma(p)=0,$$

$$g\mapsto pg.$$

$$\displaystyle\gamma_{*,pg}(\tilde{X}^{\prime\prime}_{pg})=\gamma_{*,pg}({r_{g}}_{*}\tilde{X}_{p})=\frac{d}{dt}\bigg{|}_{t=0}\gamma(c_{\tilde{X}_{p}}(t)g)=\frac{d}{dt}\bigg{|}_{t=0}\gamma(c_{\tilde{X}_{p}}(t))=\gamma_{*,p}(\tilde{X}_{p}),$$

Full text

1 Introduction

In [1], Moriyoshi has introduced the concept of the connection cochain with respect to central extensions.

Definition 1**.**

[1] Suppose that there is a central extension \textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1} . A cochain $\tau:G\to A$ that satisfies the condition

[TABLE]

for $g\in G$ and $a\in A$ is called a connection cochain.

In Definition 3, we apply this idea to the abelian extensions. From Remark 4 to Remark $7$, we review some basic concepts of abelian extensions. In Example 8, we study the abelian extension

[TABLE]

corresponding to a principal $G$-bundle, where $G$ is an abelian group. We give an explicit expression of the action of $\text{Diff}(M)_{P}$ on $C^{\infty}(P,G)^{G}$ in Equation (4), and the corresponding infinitesimal action in Equation (5). With the constructions above, we show

Theorem 2** (Theorem 9).**

Let $\tau:\text{Aut}(P)\to C^{\infty}(P,G)^{G}$ be a conenction cochain of abelian extension with respect to (3) and $\tau_{*,1_{\text{Aut}(P)}}:\mathfrak{X}(P)^{r}\to C^{\infty}(P,\mathfrak{g})^{G}$ its differential map over $1_{\text{Aut}(P)}$. If for all $X\in\mathfrak{X}(P)^{r}$ with $X_{p}=0$,

[TABLE]

then ${l_{\tau(1_{\text{Aut}(P)})^{-1}}}_{*}\tau_{*,1_{\text{Aut}(P)}}$ is a connection $1$-form over $P$.

2 Preliminary

Definition 3**.**

For an abelian group extension

[TABLE]

a connection cochain of abelian extensions is a map $\tau:\tilde{G}\to A$ such that

[TABLE]

for all $\tilde{g}\in\tilde{G}$ and $a\in A$.

Remark 4**.**

Define $\mu:G\to\text{Diff}(A)$ by

[TABLE]

where $a\in A$, $g\in G$ and $\tilde{g}$ is a lift of $g$ in $\tilde{G}$. Note that (3) is independent of the choice of the lifts. Indeed, let $\tilde{g}^{\prime}$ be another lift of $g$. Then, since

[TABLE]

we see that there exists $a^{\prime}\in A$ such that

[TABLE]

It follows that

[TABLE]

It’s easy to see that $\mu$ induces an action of $G$ on $A$. When (2) is a Lie group extension, then $\mu$ is a smooth action. Similarly, we can define an action of $\tilde{G}$ on A given by

[TABLE]

These action (3) and (4) induces the coboundary map of the Lie group cochains $C^{*}(G,A)$ and $C^{*}(\tilde{G},A)$, respectively.

Remark 5**.**

If (2) is a Lie group extension, then there exists the following corresponding Lie algebra extension

[TABLE]

The infinitesimal action of (3) is the differential $\mu_{*,1_{G}}:\mathfrak{g}\to\mathfrak{X}(A)$ given by

[TABLE]

where $V\in\mathfrak{g}$, $a\in A$ and $c_{V}(t)$ is a curve in $G$ with initial vector $V$. Note that the curve induced by the exponential map $\exp(tV)$ may be a choice of $c_{V}(t)$. Similarly, we have the action $\nu_{*,1_{\tilde{G}}}$ corresponding to $\nu$. These actions may be used to give the boundary map of the associated Lie algebra cohomology.

Remark 6**.**

The abelian extension (2) induces a $2$-cocycle $\chi:G\times G\to A$ associated with a section $s:G\to\tilde{G}$, given by

[TABLE]

called the Euler cocycle. The terminology $2$-cocycle means that

[TABLE]

Remark 7**.**

Let $G\times_{\chi}A$ be the group with multiplication

[TABLE]

the group structure of which is ensured by the fact that $\chi$ is a $2$-cocycle. It’s easy to see that $G\times_{\chi}A$ is isomorphic to $\tilde{G}$. Indeed, define $\Phi:G\times_{\chi}A\to\tilde{G}$ by

[TABLE]

where $s:G\to\tilde{G}$ is the section inducing the Euler cocycle. Since

[TABLE]

we see that $\Phi$ is a homomorphism. On the other hand, define $\Psi:\tilde{G}\to G\times_{\chi}A$ by

[TABLE]

Since

[TABLE]

and

[TABLE]

we deduce that $\Phi$ is an isomorphism.

Example 8**.**

Let $G$ be an abelian Lie group and $\pi:P\to M$ a principal $G$-bundle. Then, we have the following abelian Lie group extension

[TABLE]

where

[TABLE]

and

[TABLE]

Moreover, $\iota:C^{\infty}(P,G)^{G}\to\text{Aut}(P)$ is given by

[TABLE]

for $p\in P$, and $\rho:\text{Aut}(P)\to\text{Diff}(M)_{P}$ is given by

[TABLE]

where $x\in M$ and $p$ is a lift of $x$ in $P$. Note that $\text{Diff}(M)_{P}=\text{Im }\rho$. The corresponding Lie algebra extension is

[TABLE]

where

[TABLE]

called the right-invariant vector field over $P$. Indeed, for any curve $\varphi(t)$ in $\text{Aut}(P)$ through $1_{\text{Aut}(P)}$, we have

[TABLE]

which implies that $\text{Lie}(\text{Aut}(P))=\mathfrak{X}(P)^{r}$. Moreover,

[TABLE]

which is the Lie algebra of $C^{\infty}(P,G)^{G}$. The maps $\iota_{*}$ and $\rho_{*}$ are the differentials of $\iota$ and $\rho$ over the identities of the corresponding groups, respectively. The action of $\text{Diff}(M)_{P}$ on $C^{\infty}(P,G)^{G}$ is given by

[TABLE]

for $f\in\text{Diff}(M)_{P}$, $\gamma\in C^{\infty}(P,G)^{G}$ and $p\in P$, where $s:\text{Diff}(M)_{P}\to\text{Aut}(P)$ is a section of $\rho:\text{Aut}(P)\to\text{Diff}(M)_{P}$ and $\sigma:M\to P$ a section of $P$ which is smooth around $f(\pi(p))$. Note that (6) is independent of the choice of $\sigma$ and we have identified $C^{\infty}(P,G)^{G}$ with its image in $\text{Aut}(P)$. The infinitesimal action of $\mu$ for $X\in\mathfrak{X}(M)$, $\gamma\in C^{\infty}(P,G)^{G}$, and $p\in P$ is given by

[TABLE]

where $\varphi_{X}(-,t)$ is the flow of $X$ and $\tilde{X}_{p}$ is a lift of $X_{\pi(p)}$ in $T_{p}P$. We need to show that (7) is independent of the choice of the lift. Indeed, let $\tilde{X}^{\prime}_{p}$ be another lift in $T_{p}P$. Then, since

[TABLE]

where $c_{V}(t)$ is a curve in $G$ with initial vector $V$, we see that (7) is independent of the choice of the lift with respect to a fixed $p$. Note that since $\tilde{X}_{p}-\tilde{X}^{\prime}_{p}$ is vertical, there exists $V\in\mathfrak{g}$ such that ${j_{p}}_{*,1_{G}}V=\tilde{X}_{p}-\tilde{X}_{p}^{\prime}$, where $j_{p}:G\to P$ is given by

[TABLE]

If $\tilde{X}^{\prime\prime}_{pg}$ is another lift in $T_{pg}P$, then since ${r_{g}}_{*}\tilde{X}_{p}$ is also a lift of $X$ in $T_{pg}P$, we see that

[TABLE]

where $c_{\tilde{X}_{p}}(t)$ is a curve in $P$ with initial vector $\tilde{X}_{p}$. Therefore, (5) is independent of the choice of the lift.

3 Theory

Theorem 9**.**

Let $\tau:\text{Aut}(P)\to C^{\infty}(P,G)^{G}$ be a conenction cochain of abelian extension with respect to (3) and $\tau_{*,1_{\text{Aut}(P)}}$ its differential map over $1_{\text{Aut}(P)}$. If for all $X\in\mathfrak{X}(P)^{r}$ with $X_{p}=0$,

[TABLE]

then ${l_{\tau(1_{\text{Aut}(P)})^{-1}}}_{*}\tau_{*,1_{\text{Aut}(P)}}$ is a connection $1$-form over $P$.

Proof.

Recall that any tangent vector $X_{p}\in T_{p}P$ could be extended to a right-invariant vector field $X$ over $P$. Then, ${l_{\tau(1_{\text{Aut}(P)})^{-1}}}_{*}\tau_{*,1_{\text{Aut}(P)}}$ is regarded as a $\mathfrak{g}$-valued in the sense that

[TABLE]

The condition (8) ensures that it is well-defined. Now, we check that ${l_{\tau(1_{\text{Aut}(P)})^{-1}}}_{*}\tau_{*,1_{\text{Aut}(P)}}$ is a connection $1$-form. Let $V\in\mathfrak{g}$ and $\underline{V}$ its fundamental vector field over $P$. Since $G$ is abelian, we see that $\underline{V}$ is rignt-invariant. It follows that

[TABLE]

where $\varphi_{\underline{V}}(-,t)$ is the flow of $\underline{V}$. It’s easy to see that

[TABLE]

since

[TABLE]

It follows that

[TABLE]

where we have regarded $\exp(tV)$ as a constant function in $C^{\infty}(P,G)^{G}$ and $\iota(\exp(tV))(-)=(-)\exp(tV)$. Thus, by the definition of connection cochain, we have

[TABLE]

Moreover, we have

[TABLE]

where $X$ is the right-invariant vector field extending $X_{p}$, which implies that $X_{pg}={r_{g}}_{*}X_{p}$. Therefore, ${l_{\tau(1_{\text{Aut}(P)})^{-1}}}_{*}\tau_{*,1_{\text{Aut}(P)}}$ is a connection $1$-form. ∎