# Stress-Energy Tensor and Conformal Invariance in String Theory

Since the Euler-Lagrange Equations corresponding to the Polyakov Action implies no dependance on the auxillary metric we arrive at the constraint $$T_{ab}=0$$. We then change to lightcone coordinates $$++$$ and $$--$$ and write $$T_{++}$$, $$T_{+-}$$, $$T_{-+}$$, and $$T_{--}$$ in terms of the $$T_{ab}$$ which all vanish due to the vanishing of the $$T_{ab}$$. One way to see that the trace vanishes is via Weyl Symmetry, but since all of the $$T_{++}$$ etc vanish isn't it obvious that the trace vanishes? And then isn't the equation

$$\partial_{-}T_{++}=0$$

true trivially? Given the importance of these results towards establishing conformal field theory in String Theory I would appreciate any help understanding this reasoning.

The stress-energy-momentum (SEM) tensor $$T_{ab}$$ doesn't vanish as an operator identity/off-shell. The Virasoro constraints $$T_{ab}\approx 0$$ are on-shell equations that hold in quantum average $$\langle T_{ab}\rangle=0.$$
1. Dilation symmetry $$\Rightarrow$$ tracelessness of SEM tensor $$T_{\pm\mp}=0$$.
2. World-sheet (WS) translation symmetry $$\Rightarrow$$ continuity eq. for SEM tensor $$\partial_{\mp}T_{\pm\pm}=0$$.
• $^1$ Concerning the Weyl anomaly, see e.g. this Phys.SE post. – Qmechanic Apr 17 at 8:05