In string theory a conformal transformation changes the metric, $g_{\mu\nu} \rightarrow \Omega(\tau, \sigma)g_{\mu\nu}$ with $g_{\mu\nu}$ the two-dimensional metric on the (Polyakov) string worldsheet. It otherwise leaves the bosonic fields alone, $X^{\mu}(\tau, \sigma) \rightarrow X^{\mu}(\tau, \sigma)$.
Q1: Is this because the bosonic fields have conformal weight $\mathbb{0}$?
Now to follow up how does this work in point particle theory. Can we form a conformally invariant theory using an action of the form $$S = -\int e^{-1}(\tau) \dot{X}^{2} - m^{2}e(\tau) d\tau$$ by setting $m^{2} = 0$, having the einbein $e$ transform as the square root of a metric determinant and choosing $X^\mu(\tau)$ to have the right conformal weight?
