Energy of string in bosonic string theory

Question

When we start from the Polyakov action, we can choose to work in the conformal gauge $h^{\alpha\beta}=\eta^{\alpha\beta}$ where $h^{\alpha\beta}$ is the metric on the world-sheet and $\eta^{\alpha\beta}$ is the 2D Minkowiski metric. The action becomes $$S_p= -\dfrac{T}{2} \int d\sigma d\tau\ \ \partial_\alpha X^m\ \partial^\alpha X^n \ g_{mn} $$ where $g_{mn}$ is the metric of the background spacetime and English indices refer to this background spacetime metric and Greek indices refer to the world-sheet metric (Minkowski).

For choosing a gauge, we have to also impose the constraint $$T_{\alpha\beta}=-\dfrac{2}{T}\dfrac{1}{\sqrt{-h}}\dfrac{\delta S}{\delta h^{\alpha\beta}}=0$$ Most of the resources that I use to study string theory, say that this condition is the vanishing of the stress-energy tensor. But, we can also have a stress-energy tensor given by $$T^{'}_{\mu\nu}=-\dfrac{2}{T}\dfrac{1}{\sqrt{-g}}\dfrac{\delta S}{\delta g^{\mu\nu}}$$ What I am confused about is what is the metric that, say, if we take $\alpha=\beta=0$ or $\mu=\nu=0$ will give the energy density of the string? And, on a deeper level, what is the conceptual, physical difference between the two?

Answer 1

1. On one hand, the world-sheet (WS) stress-energy-momentum (SEM) tensor $T^{\alpha\beta}$ is essentially the functional derivative of the sigma-model (Polyakov) action $S[X,h]$ wrt. the WS metric $h_{\alpha\beta}$. WS reparametrization invariance and Weyl symmetry imply that $T^{\alpha\beta}$ should vanish, aka. the Virasoro-constraints. (This is a hallmark of reparametrization invariant theories, cf. e.g. this Phys.SE post. Note that there is a trace anomaly, which needs to vanish.) This in turn leads to a mass-shell condition, which encodes the energy-spectrum of the string-excitations.

2. On the other hand, the target-space (TS) metric $G_{\mu\nu}$ plays the role of coupling constants in the sigma model. The TS SEM tensor $T^{\mu\nu}$ is essentially the functional derivative of the effective TS action $S_{\rm eff}[G]$ wrt. the TS metric $G_{\mu\nu}$. The functional derivative is essentially a beta function. Weyl invariance implies that it should vanish. This leads to a string-generalization of EFE.