In the second chapter of string theory book by Green-Schwarz-Witten, the advantages of string theory over others is discussed. It is stated that the higher-dimensional analogs, with the action proportional to the world-volume of a "brane" object, are not perturbatively renormalizable QFTs. Their action is as follows: $$ S \sim -T_p \int d^{n+1}\sigma \sqrt{h}\, h^{ij} (\sigma) \partial_i X^\mu \partial_j X^\nu G_{\mu\nu} (X) $$ From their discussion it follows that they are referring to the "superficial" renormalizability, with the power-counting method. I am unable to understand the details of this argument.

The question Renormalizability of the Polyakov Action seems to provide some answer for this. However, it looks like this argument is substantially based on the fact that metric in the target space (which is actually not a dynamic variable) is not constant, which is not the case, say, when it is a usual Minkowski space. In this case, the considered terms that ruin renormalizability do not appear in the expansion of the action. Does it mean that in case of Minkowski target space the problem is absent? Or can one see the similar problem from the terms that come from the expansion of world-sheet metric?


1 Answer 1


The curvature has dimensions (length)^−2

, and therefore the D-dimensional

Newton constant G_D must have dimension (length)^D−2

. This is proportional to the square of the gravitational coupling constant, which therefore has negative mass dimension for D > 2. Ordinarily, barring some miracle, this is an indication of nonrenormalizability. 1 It has been shown by explicit calculation that no such miracle occurs in the case of pure gravity in D = 4. There is no good reason to expect miraculous cancellations in other cases with D > 3, either, though it would be nice to prove that they don’t occur.



D11 supergravity required infinite corrections to reach M theory.

M theory is the union of the open ball d11 and the closed ball defined as it's bound. Or at least that's how I like to view it.

Edit:it is an open question for 11>D>4

  • 2
    $\begingroup$ This answer has been flagged as low-quality and voted for deletion. $\endgroup$
    – joseph h
    Commented Oct 26, 2020 at 2:14
  • 1
    $\begingroup$ Use Latex it makes it much more accessible $\endgroup$
    – Physiker
    Commented Oct 26, 2020 at 5:34
  • $\begingroup$ Flagged as low-quality answer. $\endgroup$ Commented Oct 26, 2020 at 20:16

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