T = 1/ (2 π α^{'})

and take the closed-string spatial coordinate periodic with period

2 π

. One convenient Lorentzian convention for the conformal gauge action is

I = \frac{1}{4 π α ^{'}} \int d τ d σ (\dot{X}^{m} \dot{X}_{m} - X^{' m} X_{m}^{'}) + \frac{i}{2 π} \int d τ d σ [b (\partial_{τ} - \partial_{σ}) c + b (\partial_{τ} + \partial_{σ}) c] .

(1)

The two anticommuting bc ghost systems are the Faddeev-Popov ghosts of the worldsheet metric gauge fixing. Their fields have conformal weights

2

and

- 1

. Ghost normalizations can be rescaled without changing the algebra. After a Wick rotation the two terms become the familiar holomorphic and antiholomorphic

b \overset{ˉ}{\partial} c

and

b \partial c

actions.

Let

L_{n}

and

L_{n}

include both matter and ghost contributions. With the conventional plane normalization of the zero mode, their Virasoro algebra is

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{D - 26}{12} (m^{3} - m) δ_{m + n, 0},

(2)

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{D - 26}{12} (m^{3} - m) δ_{m + n, 0}, [L_{m}, L_{n}] = 0.

(3)

Each free coordinate boson contributes central charge

1

, including the timelike coordinate after the usual analytic continuation. The weight-

(2, - 1)

bc ghost system contributes

- 26

in each chiral sector. Thus the total central charge is

D - 26

The Virasoro constraints originate in a gauge symmetry. A nonzero central term is a quantum obstruction to realizing that gauge symmetry without an anomaly; equivalently, the BRST charge fails to be nilpotent. Cancellation of the worldsheet Weyl anomaly therefore requires

D = 26.

(4)

This is the critical dimension of the bosonic string. The accompanying physical-state normal-ordering condition gives the usual matter intercept

1

in each sector. A shift of the zero-mode convention changes the linear term in the central polynomial, but cannot change the required cancellation of its cubic coefficient.