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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.

Qmechanic
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Canonical Momenta
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1 Answers1

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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 $\partial_{\mp}T_{\pm\pm}=0$.

Qmechanic
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