Action dimensions in string theory

Question

Disclaimer: I'm tagging this as "string theory" because the doubts came to me studying the actions in string theory, but the underlying problem could be about the concept of the action itself. Also, the discussion will consider the bosonic case only for simplicity.

From my understanding, in string theory (at least type II) the dynamical fundamental objects are D-branes, open strings, and closed strings. D-branes move in the $26$ dimensional spacetime, open strings move on the $p+1$ dimensional D$p$-brane, and closed strings again move in spacetime.
The classical string action (both open and closed), the Polyakov action, is integrated in $2$ dimensions, therefore on the string worldsheet, in the same way, a point particle action is integrated in $1$ dimension, on the worldline. But when studying the low energy effective action for the closed string, we found $$S_{LEE}=-\frac{1}{2k^2}\int d^{26}x \mathfrak{g}e^{-2\Phi}\left(\mathcal{R}-4(\partial\Phi)^2+\frac{1}{12}H^2\right)$$ so an action in $26$ dimensions. Why is this? Shouldn't this be the string action, therefore propagating on the string?
Now for the branes and open strings. We studied the Dirac-Born-Infeld action $$S_{DBI}=-T_p\int d^{p+1}\xi\left(1-\frac{1}{4}F^2+\frac{1}{2}(\partial\phi^I)^2\right)$$ that is in $p+1$ dimensions. Is this an action for the open strings or for the branes? If it's for the open strings, then the same question as above: why isn't it two-dimensional? Also, what is the action of the brane? If instead it is an action for the brane itself (because the fields on the brane give it a dynamical identity) then why isn't it in $26$ dimensions, if branes are propagating in spacetime?

Answer 1

First let's recall how QFT works. We have fields (metric, gauge fields, scalars, whatever) which have an action in spacetime. If these fields are not interacting, the Fourier modes of these fields are just a bunch of independent harmonic oscillators. Quantum mechanically, the occupation numbers of the Fourier modes are "particles." In a weakly interacting theory, these oscillators interact with each other, and scatter. We compute scattering amplitudes with Feynman diagrams.

Now, you can think about string theory as one very special quantum field theory$^*$ in 26 dimensional spacetime, with many special properties. This very special quantum field theory has an infinite number of fields. It has a massless scalar field, a massless 2-form field, a massless metric field, an infinite tower of massive fields whose masses increase with spin, and a tachyonic scalar field we don't talk about. You can scatter the particles of these fields off of each other, etc.

Where the strings come in is that, even though you could compute the scattering amplitudes using the 26 dimensional spacetime action, you could also compute them using string diagrams.

You see, a string has many different states. The different ways in which it can wiggle are considered to be single particles for any of these fields. If you compute a scattering amplitudes using the string worldsheet path integral you will get the same exact result as if you used the target space effective action and regular QFT Feynman diagrams.

So that 26 dimensional action you wrote down comes from reverse engineering the target space action that gives you the same scattering amplitudes as the string worldsheet path integrals.

However, there's a totally separate way you can get that very action, by considering a single string propagating in a fixed classical background, and demanding Weyl invariance on the worldsheet so that the quantization of the string is well defined. The fact that you get the same action doing this radically different procedure is confirmation that you should really think of the strings themselves as, when put into collective coherent states, spacetime, B fields, dilaton fields, etc. (It's also guaranteed by the state operator correspondence.)

Similarly, the DBI action literally is the action of an honest-to-god brane moving through spacetime. However, just like in the closed string case, the quanta of the fields on the brane worldvolume have the interpretation of strings oscillating in some particular mode. The particles/quanta of the gauge field on the brane are strings in the vector mode, for instance.

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Footnote$^*$ there are many ways in which string theory isn't really "just" a particular quantum field theory, but for the purposes of this answer it is very helpful to think of it in that way.

Answer 2

I'll answer your first question on the string action, and if I have time later edit this to address your second too. (Really we want to be studying the Beta functions to answer this better, but the technical details really aren't straight forward enough to answer here - I can go into it if you'd like though).

The spacetime action $S_{LEE}$ comes directly from the string action (i.e. the two-dimensional interacting field theory), but we're now interpreting the fields $X^{\mu}$ as target space coordinates and looking at the low energy effective action in spacetime. This is the background spacetime in which the string itself propagates. This obviously doesn't explain how we arrive at the specific action though, just what we're looking at in the effective action.

Remember that what's important is relating the perturbative expansion parameter at low energies with the $e^{\phi}$ term in the effective action. For the bosonic string action you may find these answers also useful: How are low energy effective actions derived in string theory? "low-energy effective action" but in what sense?