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In Becker, Becker and Schwarz' book about string theory, the following symmetries are listed for the $\sigma$-model of the string:

  1. Poincaré transformations
  2. Reparametrizations $\sigma^{\alpha}\rightarrow f^{\alpha}(\sigma)$ and corresponding pullback of the worldsheet metric $h_{\alpha\beta}$
  3. Weyl transformations $h_{\alpha\beta}\rightarrow e^{\phi(\sigma,\tau)}h_{\alpha\beta}$

To get to $h_{\alpha\beta}=\mathrm{diag}(-1,1)$ gauge we use the following: As $h_{\alpha\beta}$ is symmetric it only has $3$ independent parameters. By reparametrization invariance, we get two equations which reduces the number of independent parameters to 1 and the last one is removed by using Weyl transformation symmetry.

But what bugs me is that even after this there is a symmetry left that BBS term residual symmetry, which is related to reparameterization symmetry: A reparameterization satisfying $$\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha}=\Lambda \eta_{\alpha\beta}$$ is still a symmetry after gauge fixing. Since we already "used up" Weyl and reparametrization symmetries, why is this residual symmetry left?

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The whole reason behind gauge fixing is to make the path integrals easier: choosing a slice that cuts each gauge equivalence class once allows us to get rid of the gauge volume via the Fadeev-Popov procedure. While we're at it, we might as well gauge fix the metric to the 2D unit metric (at least locally), since it's nice and simple. However, we could have made any other choice of gauge fixing, provided that it is related to the unit metric via diffeomorphisms and Weyl scalings. It turns out that in 2D, all metrics are related via these symmetries (proof).

Now once we've fixed the metric to the unit metric, we would expect any infinitesimal transformation of $\mathrm{Diff}\times\mathrm{Weyl}$ to push the metric away from our gauge choice. However, there are transformations that don't do that: these are precisely the diffeomorphisms that can be undone by Weyl scalings, i.e. the conformal transformations. This is the "residual symmetry", but it's not "residual" in the sense that it's not covered by $\mathrm{Diff}$ or $\mathrm{Weyl}$. It's "residual" in that it remains even after gauge fixing. All of this stems from the fact that string theory is formulated with a dynamical background metric, which means that conformal transformations are honest diffeomorphisms.

However, we have seen that we kill off 3 degrees of freedom via our gauge transformations, corresponding to the 3 free components of a symmetric metric (or antisymmetric, if we were in Minkowski space). So doesn't the presence of residual conformal symmetry contradict the original statement? No, it's consistent because $\mathrm{Conf}\subset\mathrm{Diff}$ is a measure-zero subset and so sneaks past the counting procedure. As a side point, the presence of the residual symmetry (manifesting as "conformal Killing vectors") is not as annoying as it seems, much of it is handled automatically by boundary conditions.

This is probably easier to illustrate like this: imagine our theory has a symmetry under a group $G$ with an action on the space of metrics $G\times H\to H$. We could gauge fix our metric to some $h\in H$ of our choice using the action of $G$. However, we might notice that there are elements $g\in G$ which act as $gh=h$ on our choice of metric. So just because we have "used up" the symmetry $G$ in gauge fixing does not mean that there can't be residual symmetry. This is precisely what happens for the string worldsheet, except for the fact that the symmetry group is roughly a product of $G=\mathrm{Diff}\times\mathrm{Weyl}$.

Finally, if you want to derive the infinitesimal form of these residual transformations: $$ g'_{\mu\nu}(x')=\Omega^2(x)g_{\mu\nu}(x) \\g'_{\rho\sigma}(x')\frac{\partial x'^\rho}{\partial x^\mu}\frac{\partial x'^\sigma}{\partial x^\nu}=\Omega^2(x)g_{\mu\nu}(x) \\x'_\mu=x_\mu+\epsilon_\mu+\mathcal O(\epsilon^2) \\g'_{\rho\sigma}(x')(\delta^\rho_\mu+\partial_\mu\epsilon^\rho)(\delta^\sigma_\nu+\partial_\nu\epsilon^\sigma)=\Omega^2(x)g_{\mu\nu}(x) \\g_{\mu\nu}+(\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu)+\mathcal O(\epsilon^2)=\Omega^2(x) g_{\mu\nu} \\\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=\kappa(x)g_{\mu\nu}\qquad\square $$

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  • $\begingroup$ Excellent insightful answer. If possible could you expand about "conformal transforms are honest diffeomorphism". Its something I get very confused about. What would you consider a dishonest diffeomorphism. I'm thinking it is like in special relativity with fixed metric, it is dishonest to consider diffeomorphisms which aren't just Lorentz transformations, but in GR we can consider a wider class of diffeomorphisms. Is that the same sort of idea? $\endgroup$
    – isometry
    Commented Apr 20, 2021 at 13:10
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    $\begingroup$ @duality Thanks. Well, that was me being a bit cavalier with words, but for a dynamical metric like in string theory, conformal transformations are diffeomorphisms and hence gauge/unphysical symmetries. However, if you consider the other case, with a fixed background metric, then one might naïvely think that conformal transformations are still diffeomorphisms, but they are actually physical, global symmetries, with corresponding conserved current, etc. $\endgroup$ Commented Apr 20, 2021 at 13:50
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    $\begingroup$ Yes that helps my understanding. $\endgroup$
    – isometry
    Commented Apr 20, 2021 at 14:26
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    $\begingroup$ @duality Check the discussion below eq $(4.1)$ in Tong's string lecture notes $\endgroup$
    – aitfel
    Commented Apr 20, 2021 at 14:29
  • $\begingroup$ @NiharKarve Thanks! the group action explanation really helped though I think there are 2 typos in the answer one is antisymmetric metric tensor in para 3 and the other one is the last 2nd line in the derivation. $\endgroup$
    – aitfel
    Commented Apr 20, 2021 at 14:33

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