I've been revisiting my lecture notes on string theory. There's one question about conformal invariance suddenly popped up in my head which made me very confused.
Starting from Polyakov action $$S[X,g]=\frac{1}{4\pi\alpha^{\prime}}\int d^{2}\sigma\sqrt{|g|}g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}\eta_{\mu\nu},$$
it suffers from two gauge redundancies:
worldsheet diffeomorphism invariance $$\sigma^{a}\rightarrow\tilde{\sigma}^{a}(\sigma^{0},\sigma^{1})$$
worldsheet metric Weyl invariance $$g_{ab}\rightarrow e^{2\omega(\sigma)}g_{ab}$$
In conformal gauge, we set the worldsheet metric to be a flat metric $\eta_{ab}$, but there is still a residual gauge freedom $$\sigma^{\pm}\rightarrow \tilde{\sigma}^{\pm}(\sigma)$$
in lightcone coordinates. In complex coordinates, this residual invariance means invariance under holomorphic and anti-holomorphic transformations on a Riemann surface.
After the gauge fixing procedure, we end up with $$Z=\int\frac{\mathcal{D}X\mathcal{D}g}{|\mathrm{Diff}||\mathrm{Weyl}|}e^{-S[X,g]}=\int\mathcal{D}X\mathcal{D}b\mathcal{D}c\exp-\left\{\frac{1}{2\pi\alpha^{\prime}}\int d^{2}z\partial X^{\mu}\bar{\partial}X_{\mu}+\frac{1}{2\pi}\int d^{2}z(b\bar{\partial}c+\bar{b}\partial)\bar{c}\right\}$$
This is where the conformal invariance in string theory comes from. At the moment, it seems that conformal invariance in string theory should be deemed as an unphysical gauge redundancy, because it's a residual freedom in conformal gauge. However, later on we are talking about Noether currents and Ward identities associated with conformal transformations. The "conformal charge" is defined as $$Q_{\epsilon}=\frac{1}{2\pi i}\oint dz T(z)\epsilon(z),$$
where $T(z)$ can be expanded using Virasoro generators. Then, we are treating this conformal symmetry on the worldsheet as a physical symmetry.
So what is going on? Is conformal invariance in string theory a physical symmetry or a gauge redundancy? I never asked this question to my professor when I was studying string theory at school.
