There is obviously a gap in my knowledge of the origin of effective actions in string theory.

As far as I understand it, the strategy is straightforward (at least in principle):

  1. Write down the worldsheet action for the superstring.
  2. Take compactification and (or) localization into account, obtain a low-energy action for the brane.
  3. Re-interpret this action as a QFT on the brane with a natural cut-off of order $l_s$ (string length).

OK, so that explains why we have a field-theoretic action of some form. However there is no reason (besides our intuition) for it to generate a QFT.

In path integral formulation, the fundamental measure comes from the (supersymmetric version of) Polyakov path integral over the fields on the superstring worldsheet.

It seems to me that it could (in general) generate another low-energy "effective measure", spanning another QFT.

How does one show that this is not the case?

  • $\begingroup$ I am a little bit confused about the wording in this question. So you understand that you will get a field-theoretic (FT) action of some form in the low energy limit, but you do not understand why a QFT? Is the Q (quantum) part the problem? $\endgroup$
    – Heterotic
    Apr 2, 2015 at 8:57
  • $\begingroup$ Or do you think you should be getting a particular QFT and you are inquiring about the possibility of getting alternative QFTs from this prescripion? $\endgroup$
    – Heterotic
    Apr 2, 2015 at 8:59
  • $\begingroup$ @Heterotic I am confused, too, The Q is obviously the problem here. However I can imagine getting a different QFT (with another action) because of the measure. $\endgroup$ Apr 3, 2015 at 7:58
  • $\begingroup$ I think the remarks on LEEA in GSW chapter 13 may be helpful. see relevant question link an link $\endgroup$ Apr 6, 2015 at 11:20

1 Answer 1


In general, once you fix the high energy theory, then there will be a unique EFT following from running down that theory to some lower energy $E_L$ (which will depend of course on $E_L$). Note however that deriving this EFT can be highly complicated sometimes. If for any reason the EFT you get at $E_L$ is not what you wanted, then you have to modify the high energy theory and start again. There is no other option.

The EFT you get might be quantum or not, sometimes depending on how far down you go. For example, you can take QFT down to QM or even all the way down to Newtonian physics (at least in principle). Mathematically, you can see if the resulting theory is quantum or not by looking at the commutators of the operators in the EFT and checking if they are small enough to be ignored or not.

  • $\begingroup$ but in the string case, the effective field theory is got by making the amplitudes (string side and qft side) match. I don't think the uniqueness of EFT for string theory is trivial. $\endgroup$ Apr 6, 2015 at 11:18

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