# How is spacetime curvature generated in string field theory?

In String Theory, the graviton originates from a mode of vibration of the fundamental strings and spacetime curvature itself is somehow generated by the strings themselves. In String Field Theory, the theory is formulated in the language of Quantum Field Theory and strings are excitations of the quantum fields. In the String Field Theory case, how is space-time curvature generated? In essence, how can the strings themselves (and the vibration) can cause the curvature of spacetime?

PS: I am not an expert in String Theory or String Field Theory, so apologies if any of the facts mentioned above is inaccurate.

• Space-time is not generated by any mechanism in (perturbative) string theory or string field theory. Spacetime always exists as the background in which dynamics occurs. It is, of course, itself dynamical so it can curve and expand and do all sorts of interesting things. Jul 9 '21 at 10:21
• @PraharMitra I see. Is what I heard, that the vibration of strings affects spacetime or generates the curvature of spacetime, incorrect? Jul 9 '21 at 11:55
• It is correct! Vibrations of strings does in fact generate curvature in spacetime -- it does not generate spacetime itself. Jul 9 '21 at 12:16
• @Prahar I started the bounty to increase the detail to the answer, on how string (field) theory can generate curvature of spacetime through vibration of strings. I think I accidentally clicked the reason of starting the bounty as to "draw more attention", but I meant to "increase details". Jul 11 '21 at 8:19
• @Prahar I want to know how spacetime curvature is generated in String (Field) Theory; i.e. how the vibration of strings can possibly lead to curvature. Jul 11 '21 at 8:23

The question of how strings build up spacetime is not limited to the string field theory formalism. I won't reproduce the entire resolution, which is substantively answered here, here, and links therein, but the crux is that the strings themselves do not "build up" the spacetime background. Rather, deformations of this background can be generated by inserting on-shell graviton vertex operators on the worldsheet, which when integrated over the worldsheet (to preserve diffeomorphism invariance), is equivalent to adding a coherent superposition of graviton states as in-states during the scattering. All of this happens of course in a perturbative framework, and so we cannot answer the question as to what the background metric itself is made up of.

The story is similar in its second-quantised cousin string field theory, which is of course perturbatively equivalent to it. Let us stick to closed string field theory, in which the graviton has long been known to exist: here, the string field can be decomposed into an infinite number of quantum fields corresponding to the string modes. Then, the creation operator for the graviton would yield the same result as above in perturbation theory -- choose a background, add gravitonic in-states to "simulate" a deformed background.

However one major premise of string field theory was to provide a background-independent, non-perturbative formulation of the quantum theory. While the latter has not had much success, the former has been proven conclusively for the bosonic closed string field theory. That said, I don't know much string field theory, so I am not entirely certain what this implies for the "structure" of spacetime - perhaps an expert can weigh in.

• Background independence has also been proven for superstring field theory: arxiv:1711.08468. Jul 12 '21 at 15:03
• @Harold thanks! Incidentally, you were the expert I was referring to in my final line :) Jul 12 '21 at 16:17
• Thanks! Unfortunately, I don't have much to add to your answer, which is very good. I will just provide a slightly alternative point of view. Jul 13 '21 at 23:11

The answer from Nihar Karve provides useful information on how to describe spacetime in string theory, and I would like to add another perspective.

Closed string field theory describes spacetime curvature in (almost – see below) the same way as general relativity.

In general relativity, the dynamics of the metric $$g_{\mu\nu}$$ and the curvature of spacetime is described by the Einstein-Hilbert action coupled to some matter. And as you know, Einstein equation shows that the matter affects the curvature through its energy-momentum tensor. At this stage, there is no fixed background to describe GR, the fields do not propagate on something. In general, one looks to classical solutions to the equations of motion and start to study the properties of this specific spacetime and study how fields propagate on it.

Since it can be confusing to not have a background, a solution is to expand the metric around a fixed metric as $$g_{\mu\nu} = \hat g_{\mu\nu} + h_{\mu\nu}$$, where $$\hat g_{\mu\nu}$$ is fixed (and may or may not be a solution to the eom) and $$h_{\mu\nu}$$ is the new dynamical variable. Since the action is non-polynomial in $$h_{\mu\nu}$$, the simpler is to expand it in $$h_{\mu\nu}$$: you will get an infinite number of monomial interactions. As long as you keep all powers of $$h_{\mu\nu}$$, the action is completely equivalent to the original one (although it is now perturbative). The background is just a convenient computational tool, but it does not determine what is the physical spacetime. A nice example can be found in arxiv:2010.08882 where the Schwarzschild metric is recovered from amplitudes.

Closed strings unify both as matter and gravitons, so it is not possible to split them as we do in GR. However, it is possible to introduce an energy-momentum pseudotensor for gravity such that Einstein equations can be reformulated as a conservation law for the total matter+gravity energy-momentum tensor. This shows that there is no strong dichotomy between

Closed string field theory (SFT) provides an action for the string field, which is more conveniently studied in momentum space where you can see all the modes, including the graviton. In the latter form, the classical action just looks like GR expanded around a background (as described above). For the free action of massless fields with general gauge fixing conditions, see arxiv:1206.3901. Everything you can do in GR, you can theoretically do it in SFT. The metric is split as $$g_{\mu\nu} = \hat g_{\mu\nu} + h_{\mu\nu}$$, so it is perturbative, but SFT itself tells you how to compute the action up to any power of $$h_{\mu\nu}$$ and you can also resum expressions (like in GR). The only (technical) limitation is that you are basically forced to start with Minkowski as a background, $$\hat g_{\mu\nu} = \eta_{\mu\nu}$$. (And obviously I am not saying that it is easy to do it in practice, just that it is possible in theory.)

Let me note stress that you don't need a background-independent nor non-perturbative formulation to already extract a lot of information.

To summarize, if the question is how to describe spacetime curvature in SFT, you just do what you do in GR expanded around a background. More generally, SFT is very useful and intuitive because you can see it as a standard QFT and use traditional methods (just being careful with the non-locality of interactions). The drawback is that it's not really teaching anything philosophical on spacetime.

(Let me note that it is my personal point of view: I don't any reference describing this, so maybe it's naive and other researchers may not agree.)

One way to look at strings is that everything happens in a fixed spacetime background. No spacetime dynamics at all.

However, there is a type of excitation (called graviton) which interacts with all other types of excitations (and itself) in such a way as if the spacetime is actually different from the said fixed background and that it is dynamic. And this interaction is so universal and consistent that it is impossible to determine the background - different background can lead to the same results if everything else is changed accordingly. The results have no dependence on the background. Unsurprisingly, this is called background independence.

Physics deals with measurable things, so the background itself is irrelevant because there is no way to measure it. The "effective" spacetime that results from interactions with gravitons (which behaves dynamically) is then considered as the spacetime we all know and love - it is the only spacetime we can see.

One might loosely state that strings "simulate" the dynamic spacetime, but they do it perfectly, completely hiding the background.