4 - Strings with world-sheet supersymmetry

Summary

The bosonic string theory that was discussed in the previous chapters is unsatisfactory in two respects. First, the closed-string spectrum contains a tachyon. If one chooses to include open strings, then additional open-string tachyons appear. Tachyons are unphysical because they imply an instability of the vacuum. The elimination of open-string tachyons from the physical spectrum has been understood in terms of the decay of D-branes into closed-string radiation. However, the fate of the closed-string tachyon has not been determined yet.

The second unsatisfactory feature of the bosonic string theory is that the spectrum (of both open and closed strings) does not contain fermions. Fermions play a crucial role in nature, of course. They include the quarks and leptons in the standard model. As a result, if we would like to use string theory to describe nature, fermions have to be incorporated. In string theory the inclusion of fermions turns out to require supersymmetry, a symmetry that relates bosons and fermions, and the resulting string theories are called superstring theories. In order to incorporate supersymmetry into string theory two basic approaches have been developed

• The Ramond–Neveu–Schwarz (RNS) formalism is supersymmetric on the string world sheet.

• The Green–Schwarz (GS) formalism is supersymmetric in ten-dimensional Minkowski space-time. It can be generalized to other background space-time geometries.

These two approaches are actually equivalent, at least for ten-dimensional Minkowski space-time.