Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be an a loop of diffeomorphisms, $F_0=F_1=$ identity.

For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\pi_1(V)$ is independent of $x\in V$.

I call a Hermitian connection $\nabla$ on $L$ semi-flat (with respect to $\{F_t\}$) if $Hol^\nabla_{\gamma_x}=Id$ for every $x\in V$.

Can you describe a purely topological condition (i.e. it only depends on the class of $\{F_t\}\in \pi_1(Diff_0(V)$) which insures the existence of such connection.

One trivial such condition is the triviality of $L$.

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.

For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\pi_1(V)$ is independent of $x\in V$.

I call a Hermitian connection $\nabla$ on $L$ semi-flat (with respect to $\{F_t\}$) if $Hol^\nabla_{\gamma_x}=Id$ for every $x\in V$.

Can you describe a purely topological condition (i.e. it only depends on the class of $\{F_t\}\in \pi_1(Diff_0(V)$) which insures the existence of such connection.

One trivial such condition is the triviality of $L$.