Greg Moore has written an article for the latest Notices of the AMS entitled “WHAT IS… a Brane?”. He begins by noting that “The term ‘brane’ has come to mean many things to many people” and one of the difficulties of the subject is that one has to figure out from context what sort of brane someone is talking about.
For the case of branes of dimension greater than one, in general no one knows how to consistently quantize such objects. See Warren Siegel’s research summary for comments about this. He notes that now that M-theory shows that non-perturbatively strings are membranes in 11d, one doesn’t have a finite quantum theory of gravity, since membranes are infinite even in perturbation theory. All one has is an effective low-energy supergravity theory, which is what one had before one got involved with string theory. While at Siegel’s web-site, check out his latest parody paper called “The Everything of Theory”, which includes the following lines:
“The real problem with string theory is that there is no alternative. However, the reason there is no alternative is that no one ever bothers to look for one; in fact, there is a strong resistance to even considering looking for one. Consequently, practically all theoretical high energy physics (and even most of phenomenology) is now string theory. Thus, string theory is not so much the Theory of Everything (since it explains nothing), but rather the “Everything of Theory”, since it now encompasses all of theory. This era in string research is strongly reminiscent of the Dutch tulip trade just before the Tulip Crash of 1637. ”
I, for one, am missing the joke here…
Update: Lubos Motl has a long posting about branes and M-theory, which explains many things. As usual though, he insists that the full dynamics of the branes in M-theory is completely determined and unique even though he doesn’t know what it is in any phenomenolgically realistic background. To get a finite quantum theory of gravity that has anything to do with the real world out of M-theory, you need to show that you can get well-defined, finite results for the dynamics of these branes in the case of four large dimensions, the rest small. As far as I can see any claim to have this now is purely wishful thinking.