Nikita Nekrasov is giving a very interesting series of talks at the Jerusalem Winter School on the topic of “Introduction to modern covariant superstring theory.” The first of his talks was yesterday and is now on-line. In it he outlined the two main formalisms for superstring theory and discussed their advantages and drawbacks, while also giving a beautiful discussion of the quantization of the superparticle, and the use of twistor and pure-spinor methods in 10d super-Yang-Mills.
One of these two formalisms, the NSR formalism, uses supersymmetry on the world-sheet, with target space a usual (bosonic) space (i.e. 10d space-time). The advantage of this is that amplitudes are computed using a linear theory, supergravity on the worldsheet. One disadvantage of this is that spacetime supersymmetry is not manifest, only recovered after GSO projection. A very serious technical problem is that, while one ultimately wants to construct amplitudes by summing over spin structures and integrating over the moduli space, the formalism gives one for each spin structure an amplitude on the super-moduli space, not the moduli space (and these super-moduli spaces are different for different spin structures). In recent years D’Hoker and Phong have been able to deal with this problem for genus 2 (and they have some results for genus 3), but for higher genus how to consistently get amplitudes on moduli space remains an open problem. Note that the problem with these multi-loop amplitudes is not only that you aren’t sure they are finite, but you aren’t sure that they are even well-defined. Presumably this is purely a technical problem, not evidence of an inherent inconsistency problem with such amplitudes, but one can’t be sure of this until someone finds a way of resolving the problem.
The other formalism, the so-called Green-Schwarz formalism, uses a bosonic worldsheet, but takes the target space to be a supermanifold. This has the advantage of making space-time supersymmetry manifest, and avoiding the problem of integrating over super-moduli space, but it carries its own disadvantages. The world-sheet theory is now a highly non-linear, constrained theory, with both first-class and second-class constraints, constraints that Nekrasov describes as “hard to separate in a covariant way”. No one knows how to quantize this theory preserving super-Poincare invariance, so one typically uses a non-covariant gauge-fixing like light-cone gauge, something that runs into trouble at genus 2 or higher.
In recent years, Berkovits has been developing an improved version of the Green-Schwarz formalism, sometimes called the Berkovits formalism, and this is the main topic of Nekrasov’s lectures. Presumably Nekrasov will be discussing in his next two lectures how this works and some of the interesting problems with it, problems that he wrote a paper about a couple months ago, one which was discussed here. In his talk, Nekrasov seemed rather nervous that he would get into trouble because people might think he was raising the possibility of superstring perturbation theory being inconsistent. At one point he said that his “policy statement” was that he hoped that things could be made to work at any genus. He also seemed concerned in his talk yesterday about how his remarks might be reported today, saying:
There are really conc… well… I don’t want to call them conceptual problems because these days everything is recorded. If I say something is a conceptual problem, tomorrow there will be a blog on that, or a paper. So, there are some technical difficulties….