Yesterday Joe Polchinski gave a lunch-time talk at the KITP on the topic of What is String Theory? No answer to the question, but he provided an outline of three topics being discussed at the current KITP workshop program that have something to do with it.
At the end, someone asked about the LHC and supersymmetry, Polchinski responded that string theory didn’t require LHC-scale supersymmetry, but if supersymmetry was discovered at the LHC then there would be a “sociological” effect encouraging to string theorists. I also noticed recently that Polchinski has a web-page On some criticisms of string theory.
In his discussion of the pure spinor formalism, he noted that supersymmetry doesn’t seem to “resonate” with mathematicians, but that pure spinors are more something they recognize. This is certainly true, with supersymmetry something frustratingly close to some standard mathematical constructions, but quite different in other ways. Pure spinors occur naturally when one tries to construct spinors geometrically. Projectively, the space of pure spinors is SO(2n)/U(n), a space which has some quite beautiful properties. In the Borel-Weil geometric construction of representations, spinors are holomorphic sections of a line bundle over this space (for details of this, see the chapter on spinors in Loop Groups, the book by Pressley and Segal).
For the superstring, one is interested in the case of n=5, and a certain sigma model with target space the space of pure spinors. There’s a more general class of sigma models of which this is a special case, and for more about some of the interesting connections of this to other subjects, see the recent KITP talks by Nekrasov and Frenkel. The Frenkel talk is especially interesting, since it involves several other quite beautiful related ideas. He describes one motivation for studying some of these sigma models that comes from geometric Langlands. While he was at Santa Barbara, Frenkel also gave two nice survey talks about geometric Langlands, see here.
Update: Clifford Johnson explains here that not only do we not know what string theory is, but we can’t even say anything useful about what it isn’t, other than “it is not a theory of strings”. The problem with this situation, according to him is:
people who don’t know what they’re talking about, and sometimes with an axe to grind, shouting loudly (and sometimes deliberately misleadingly) about it.
Update: More thoughts from Clifford on the question of how to deal with string theory critics.