Fermions on the worldsheet of effective strings via coset construction

In this paper the detailed Coleman-Callan-Wess-Zumino procedure for introducing fermions on the world sheet of a string propagating in flat space-time is presented. The theory of nonlinear realizations is used to derive the transformation as well as the interactions of fermionic matter fields under arbitrary spinorial representations of the unbroken subgroup. This demonstrates that even for nonsupersymmetric spinors, the interactions are still severely restricted by the nonlinearly realized symmetry. We also explain how supersymmetric models provide an example for this construction with Goldstinos as matter fields, and how one can use the $\kappa$-symmetry of the Green Schwarz action in particular, to verify this nonlinear transformation for a specific matter field representation. We finally restrict the target space dimension without reference to supersymmetry, but rather by imposing one-loop integrability on a fermionic string that nonlinearly realizes Poincare symmetry. This singles out the critical dimension $D=10$ for heterotic, Green-Schwarz and Ramond-Neveu-Schwarz supersymmetric strings.

Figure 1

The bosonic one loop contribution to the 4 boson annihilation. ($s$, $t$ and $u$ channels).

Figure 2

The fermionic one loop contribution to the 4 boson annihilation. ($s$, $t$ and $u$ channels).

Figure 3

Two loop s channel contribution to annihilations.

Figure 4

Higher loop diagrams that will give rise to $C_2(R)$.