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).
 A: 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.
A concrete analysis leading to the identifications of the CFT states to the low-energy fields (metric…) and how the SFT gauge symmetry descends to the low-energy gauge symmetries (diffeomorphisms…) can be found in arxiv:hep-th/9110038. I don't have time to read the paper in details and make a summary, but you can certainly find the answer to your question there. The analysis proceeds through a perturbative expansion and study the correspondence to the first non-linear order. In principle the analysis can be pushed further. You may also want to look at the more recent paper arxiv:hep-th/0005085; it focuses only on the open string but the general method should also hold for the closed string.
