Spinning string

The spinning string adds worldsheet Majorana fermions and local worldsheet supersymmetry to the embedding coordinates. Its gauge-fixed quantum description is the Ramond–Neveu–Schwarz formulation of the superstring.

The Gliozzi–Scherk–Olive projection selects consistent worldsheet fermion-number and chirality sectors in an RNS string theory. In suitable supersymmetric choices it removes the tachyonic Neveu–Schwarz sector ground state and keeps a chosen Ramond sector chirality. Together with the consistent sector combinations it produces spacetime fermions and, for theories such as Type II superstring theory, a tachyon-free spectrum. The mere presence of classical worldsheet supersymmetry does not specify this projection or guarantee tachyon removal.

Antiperiodic worldsheet Majorana fermions have half-integer modes in this sector.

Periodic worldsheet Majorana fermions have integer modes and fermionic zero modes in this sector.

Local supersymmetry on a string worldsheet relates embedding coordinates to their worldsheet Majorana fermion partners. Fixing this fermionic gauge symmetry introduces commuting superconformal ghosts.

The worldsheet gravitino is the fermionic vector-spinor gauge field for local worldsheet supersymmetry, paired with the worldsheet metric in two-dimensional supergravity. In a locally supersymmetric string action it couples to the worldsheet supercurrent. Fixing its gauge and the bosonic conformal gauge leaves the usual flat rigid transformations on embeddings and worldsheet Majorana fermions; it should not be silently omitted when claiming an arbitrary curved-worldsheet locally supersymmetric action.

In flat conformal gauge, take $\mathcal{L}=\partial X\cdot \partial X+i{\alpha }^{\prime }\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$ and rigid transformations $\delta X=iA\overline{ϵ}\psi$, $\delta \psi =B{\gamma }^{\nu }{\partial }_{\nu }Xϵ$, with $A={\alpha }^{\prime }B$. The Majorana Grassmann bilinear interchange and Clifford algebra combine the variations intoThe identity is off shell. The integrated action is invariant when its boundary flux vanishes, for example on a closed worldsheet or with compatible supersymmetric endpoint conditions. Constant spinors and ordinary derivatives here refer to flat gauge; arbitrary curved worldsheets require covariant spinor derivatives and the locally supersymmetric completion.

The commuting beta-gamma system produced by worldsheet supersymmetry gauge fixing has conformal weights $(3/2,-1/2)$ and central charge $11$ per chiral sector.

Each real chiral fermion has conformal weight $1/2$ and contributes $1/2$ to the central charge. The Ramond sector and Neveu–Schwarz sector differ by periodic versus antiperiodic boundary conditions.