# Energy of string in bosonic string theory

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?

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.
• But, the constraints give that $T_{00}=0$, implying zero energy. Am I misunderstanding something here? – LORENTZo_lamas Feb 5 at 16:31