String theory in arbitrary number of dimensions

[Demystifier]

A long time ago, physicists thought that only a small class of quantum field theories (QFT's) makes physical sense - those which are renormalizable. But then gradually it became accepted (Weinberg was the most influential figure in that regard) that QFT does not really need to be renormalizable, as long as it is viewed as an effective theory, not a fundamental one.

Analogously, string theorists are used to think that only a small class of string theories make physical sense - those that live in the right number of dimensions (which unfortunately exceeds the observed number of dimensions). Even though they do not longer think that strings are fundamental (because they believe in existence of a more fundamental M-theory, which nobody really understands), they still seem to think that the number of dimensions is fixed. However, a recent paper
http://lanl.arxiv.org/abs/1204.6263
argues that string theory, viewed as an effective theory, makes sense in any number of dimensions.

I am not sure about the technical details, but conceptually it makes a lot of sense to me. What do you think?

 

[atyy]

Tong's notes http://arxiv.org/abs/0908.0333 (section 5.3.2) have some comments on the non-critical string: "Although it's a slight departure from the our main narrative, it's worth pausing to mention what Polyakov actually did in his four page paper. His main focus was not critical strings, with D = 26, but rather non-critical strings with D ≠ 26. From the discussion above, we know that these suffer from a Weyl anomaly. But it turns out that there is a way to make sense of the situation"

 

[Demystifier]

The Polyakov's non-critical strings is indeed an old and well-known idea, but the idea in this new paper seems to be different.

 

[fzero]

The major gap that I can identify in the paper has to do with the gauge condition used to fix worldsheet reparameterization invariance, eq (17):

$$\nabla_\mu \left( \tilde{g}^{\mu\nu} \sqrt{-\tilde{g}} \right) = 0.$$

This equation is nonlinear in the fields and under any quantization, including deformation quantization, we have to specify an ordering of operators. See the remarks in the 3rd paragraph on page 24 about deformation quantization in the sense of a formal power series.

Now, this issue of the reparameterization constraint is completely ignored in the section on quantization. In the ordinary approaches (canonical, light-cone, etc. quantization) to the string, it is precisely the ordering ambiguity in the reparameterization constraint that leads to the central extension of the Virasoro algebra. Here, while the authors talk about renormalizing the BV operator and insuring nilpotency of the BRST charge, it is still not clear how the authors have really dealt with the constraint.

Incidentally, cancelling the anomaly in the quantum string just amounts to coupling the theory to an "internal" CFT with the right central charge. The noncritical string is one way to cancel the anomaly in 4 free bosons, but there are obviously many other ways. In the original formulation of the string, additional free bosons are "added," which have an interpretation as additional dimensions, but most of the internal CFTs would have no such geometrical interpretation. It's not obvious that thinking of the extra degrees of freedom as extra dimensions is necessarily correct, though it's often useful in practice to have the additional geometric framework.

 

[Demystifier]

Fzero, good comment!