# How can you show diffeomorphism invariance of closed string field theory?

String Field theory if it predicts General Relativity should have 26D space-time diffeomorphism invariance (presumably). How can one show that Closed String Field Theory has this symmetry?

(Besides just saying it contains a spin-2 field as one of the components).

The simplest way to start seeing the diffeomorphism of the closed SFT is to look at the free action $$S = \frac{1}{2} \, (\Psi, Q_B \Psi)$$ (the bracket is the standard BPZ product with a $c_0^-$ insertion) and the gauge transformation $$\delta \Psi = Q_B \Lambda.$$ If you expand $\Psi$ and $\Lambda$ on the oscillator basis, you can analyse the gauge transformation of each spacetime field, and in particular of the metric. You will find that the transformation reads $$\delta g_{\mu\nu} = \partial_\mu \xi_\nu + \partial_\nu \xi_\mu$$ where $g_{\mu\nu}$ is the metric field in $\Psi$ and $\xi_\mu$ is one of the gauge parameter in $\Lambda$. Since this is the correct linearized diffeomorphism transformation, this gives a first hint. Then you can also expand the action $S$ itself and you will recognize that the kinetic term for $g_{\mu\nu}$ is in fact the linearisation of the Ricci scalar. You can find a detailed explanation of these facts in general gauges in arxiv:1206.3901.
• Yes, well that shows that possibly one field $g$ might have the correct transformation properties in a certain linear approximation. But other than that not very convincing I'm afraid! I'll check out your link. – zooby Jul 12 '18 at 18:19
• Another possibility is to use the knowledge from the worldsheet. There you can compute scattering amplitudes of this field $g$ with other fields, and you find that this has the correct form of a scattering of a graviton. SFT interactions are defined as pieces from these amplitudes, so you would also get the appropriate low energy Lagrangian. If the interactions have the right form, the invariance follows. – Harold Jul 14 '18 at 12:52
• You may want to check hep-th/0005085 and hep-th/0306041 to see how the gauge invariance of $A_\mu$ is deduced from the open SFT gauge symmetry. Similar computations should be doable for the closed string. – Harold Jul 17 '18 at 5:56