Vector fields and group actions

In section 4 page 12 of the article Cohomologie équivariante et théorème de Stokes, there is a statement which says that

if we consider the action of the $S^1$ (parametrized by the angle $\phi$) on itself is by rotations. Then this action is generated by the vector field $\frac{\partial}{\partial \phi}$.

what does it mean in for a group action to be generated by a vector field ?

Solution 1:

In general, a vector field gives us an action by $\Bbb R$ by letting the result of $t\in\Bbb R$ acting on the point $x\in X$ be "follow the flow of the vector field for $t$ seconds, starting at $x$". In the concrete example, $2\pi$ "seconds" of flow tkae every point back to the origin, thus allowing us to view the action by $\Bbb R$ as an action by $\Bbb R/2\pi\Bbb Z\cong S^1$.