Today begins discussion of what many consider some of the flaws or weaknesses of string theory (or, as it is often called, superstring theory, due to its "merger" with supersymmetry, which I will discuss later).
The first concerns whether there are dimensions of space beyond the three we're used to thinking about: west-east (or left-right), north-south (or forward-backward), and up-down (or higher-lower). In The Fabric of the Cosmos, Brian Greene summarizes the situation as he sees it:
...at the top of the skeptics' list of superstring theory's shortcomings was a feature I've yet to introduce. Superstring theory does indeed provide a successful merger of gravity and quantum mechanics, one that is free of the mathematical inconsistencies that plagued all previous attempts. However, strange as it may sound, in the early years after its discovery, physicists found that the equations of superstring theory do not have these enviable properties if the universe has three spatial dimensions. Instead, the equations of superstring theory are mathematically consistent only if the universe has nine spatial dimensions, or, including the time dimension, they work only in a universe with ten spacetime dimensions! (p. 359)
Further refinements, as Greene also notes, now require 11 spacetime dimensions, seven more than have ever been seen (here's another article that discusses this issue).
Here's how Greene explains the need for the extra dimensions:
If a string can vibrate in a fourth spatial dimension, it can execute more vibrational patterns than it could in only three; if a string can vibrate in a fifth spatial dimension, it can execute more vibrational patterns than it could in only four; and so on... but with nine space dimensions [later found to be ten], the constraint on the number of vibrational patterns is satisfied perfectly (pp. 370-371).
Further, with the work of some creative physicists and mathematicians -- some of it going back roughly 85 years -- ideas for accommodating extra dimensions are readily available.
Under what is known as the Kaluza-Klein model, one can envision how a garden hose, fully elongated, would appear to represent one dimension. However, if one considers the circumference of the hose's width, that could be an extra dimension (I think it's fitting to use the example of a hose to present this "watered-down" explanation). This hose illustration can be seen at the following website, in the first diagram under Kaluza's picture.
A more complex figure, known as a Calabi-Yau manifold, has been proposed as being able to account possibly for six additional dimensions.
Within the past year or so, two major books by physicists have come out, specifically on the topic of extra dimensions. One, by Lawrence Krauss, is entitled Hiding in the Mirror, whereas the other, by Lisa Randall, is entitled Warped Passages. My interest level would warrant reading one of the two books and, based on this recommendation, I chose Krauss's. I found it reasonably interesting.