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From what I understand the Polyakov action in string theory is essentially something like

$$S(\xi, g, G)=\kappa \int_{\Sigma} d \mu_{g} \operatorname{Tr}_{g} \xi^{*} G$$

where $\Sigma$ is a given topological surface, $g$ is a Riemannian metric on it, $\xi:\Sigma\to M$ is an embedding of $\Sigma$ into a fixed target manifold $M$ with a Riemannian metric $G$. Yang-Mills fields etc are omitted. The action is essentially the energy functional for embedded surfaces. The string length enters through $\kappa$, which is the string tension.

My intuitive understanding of the action is that it essentially computes some kind of energy associated to the embedding $\xi$ of $(\Sigma,g)$ into $(M,G)$, where $(\Sigma,g)$ is the world sheet of a collection of strings. Intersecting this with a spacelike hypersurface would seem to give us the lengths of the strings existing in this timeslice. Embeddings that are stationary for this action are minimal in a precise sense.

In any case, even when restricting to stationary embeddings it isn't clear that this gives us a fixed, unique and well-defined string length, although it could possibly give an order of magnitude (depending on the tension $\kappa$). More importantly, this action is invariant under conformal transformations of $g$. This would seem to make it impossible to assign any kind of length to a string.

Since the string length is considered a fundamental constant of nature in string theory, there must be something wrong in my understanding.

My first guess is that I take the interpretation of the world sheet of a string much too literally, and that in reality this has little to do with an actual set of strings moving through spacetime, something more similar to a Feynman diagram than something that could be seen in a bubble chamber.

It could also be that string length should be taken as a metaphor, and the actual fundamental constant would be e.g. the string tension, which we convert to a length just to have some kind of a mental image.

Finally, maybe the conformal invariance doesn't have any real physical meaning, and it is just an artifact of our description, like a gauge symmetry or a choice of units.

Could anyone shed some light on this?

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  • $\begingroup$ Related, and also. $\endgroup$
    – Cosmas Zachos
    Aug 8, 2019 at 21:42
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    $\begingroup$ @CosmasZachos My actual question is how a scale invariant system can have a characteristic length? One way to show how is to actually derive it, as is done in Becker Becker Schwarz, but I'm a bit lost in the details there, e.g. does it depend on a specific gauge fixing (that breaks the confirmation invariance)? I would be particularly interested in understanding if my reasoning above is essentially correct, or how it goes wrong. $\endgroup$
    – doetoe
    Aug 8, 2019 at 23:56
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    $\begingroup$ I think you are confusing the absence of a notion of length in $\Sigma$ with the presence of the notion of a length in $M$. The theory is not conformally-invariant in $M$, so the notion of length is a perfectly well-defined classical notion there. The theory is conformally-invariant in $\Sigma$, so people usually do not speak of a characteristic length scale in $\Sigma$ unless they are introducing a scale by hand as a means of regularisation (eg, close to the boundary of the moduli space of $\Sigma$, or in sigma-model calculations of the beta functions associated to target space fields, etc.). $\endgroup$
    – Wakabaloola
    Aug 9, 2019 at 12:17
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    $\begingroup$ (I think you mean to say `Polyakov' action) $\endgroup$
    – Wakabaloola
    Aug 9, 2019 at 12:19
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    $\begingroup$ @Wakabaloola I think that's basically it. Could you make your comment into an answer? $\endgroup$
    – MannyC
    Aug 9, 2019 at 23:16

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