Critical dimension of string theory

The critical dimension is the target-spacetime dimension in which quantum anomalies obstructing Lorentz or Weyl symmetry cancel. It is 26 for the bosonic string.

Here critical means cancellation of the string’s quantum anomalies; it does not mean the upper or lower critical dimension of a phase transition.

For the FP ghost vacuum annihilated by $b_{n\ge 0}$ and $c_{n>0}$, the FP ghost level-one and level-two descendant coefficients are $-2$ and $-17$. Matter contributes ${\alpha }_0^2$ and $2{\alpha }_0^2+D/2$. Virasoro closure from BRST nilpotence at level one fixes the total highest weight to ${\alpha }_0^2/2-1$; level two then requires $D=26$. Imposing ground-state BRST closure gives ${\alpha }_0^2=2$ and the open bosonic tachyon. Interpret the coefficients with the stated ghost zero-mode pairing.

Each of $D$ embedding bosons contributes one to the chiral central charge, and each real worldsheet Majorana fermion contributes one half. The reparameterization bc system contributes $-26$, and the superconformal ghosts contribute $+11$. Vanishing total worldsheet Weyl anomaly therefore requires $3D/2-26+11=0$, or $D=10$. This differs from the critical dimension 26 of the bosonic string, which has no worldsheet fermions or superconformal ghosts.

In one critical chiral RNS string sector, light-cone gauge in string theory leaves eight transverse directions. The Neveu–Schwarz sector massless states $b_{-1/2}^i|0;k⟩$ give eight spacetime bosonic polarizations. The Ramond zero-mode Clifford algebra gives sixteen ground-state polarizations, split into two eight-dimensional spinor chiralities. The supersymmetric GSO projection keeps one chirality, giving eight spacetime fermionic polarizations. Without that projection the massless counts do not agree. Counting a chiral sector is distinct from tensoring left and right sectors into closed-string states.