# 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.

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.

• 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
• Interestingly I can't find any papers that prove diffeomorphism invariance of string field theory so it leads me to doubt whether it is. – zooby Jul 15 '18 at 0:47
• As I said in my previous message: SFT is equivalent to worldsheet string theory. In the latter it is clear beyond doubt (at least in my case) that it is diffeo invariant. Thus SFT must enjoy also this property. The main question is thus how to prove it only in the SFT formalism, not whether it exists. – Harold Jul 16 '18 at 8:54
• 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