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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.

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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 $\langle T_{ab}\rangle=0.$

If there is no Weyl-anomaly, we may consistently impose off-shell

  1. Dilation symmetry $\Rightarrow$ tracelessness of SEM tensor $T_{\pm\mp}=0$.

  2. World-sheet (WS) translation symmetry $\Rightarrow$ continuity eq. for SEM tensor $\partial_{\mp}T_{\pm\pm}=0$.

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  • $\begingroup$ $^1$ Concerning the Weyl anomaly, see e.g. this Phys.SE post. $\endgroup$ – Qmechanic Apr 17 at 8:05
  • $\begingroup$ Thank you for your comments. My question primarily refers to the classical theory however. I am just a little confused in the literature why after we show all the components of the stress tensor vanish, we then derive a whole host of statements, including the vanishing of the trace of the stress energy tensors and constraints involving the Virasoro generators that hold trivially becuase the stress energy vanishes, but are presented in a way that makes them seem less trivial. I was just wondering if I was overlooking something $\endgroup$ – Canonical Momenta Apr 23 at 3:47

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