\partial X^{μ} (z) \partial X^{ν} (w) \sim - \frac{α ^{'}}{2} \frac{η ^{μν}}{( z - w ) ^{2}} .

T_{X} (z) T_{X} (w)

, the double contraction of the two derivatives in each normal-ordered stress tensor has two pairings. It gives

T_{X} (z) T_{X} (w) \sim \frac{D /2}{( z - w ) ^{4}} + \dots .

c = D

δ_{v} T_{X} = v \partial T_{X} + 2 (\partial v) T_{X} + \frac{D}{12} \partial^{3} v,

so in the notation of the question

c_{X} = \frac{D}{12} .

2

Solution

The worldsheet ghosts form an anticommuting bc system of weights

(2, - 1)

. Its double contractions give central charge

c_{b c} = - 2 (6 \cdot 2^{2} - 6 \cdot 2 + 1) = - 26.

Therefore

δ_{v} T_{gh} = v \partial T_{gh} + 2 (\partial v) T_{gh} - \frac{26}{12} \partial^{3} v,

and hence

c_{gh} = - \frac{13}{6} .

Adding the matter and ghost transformations gives

δ_{v} T_{tot} = v \partial T_{tot} + 2 (\partial v) T_{tot} + \frac{D - 26}{12} \partial^{3} v .

For the Laurent modes

L_{n} = \oint d z z^{n + 1} T_{tot} (z) / (2 πi)

, the corresponding Virasoro algebra is

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{D - 26}{12} m (m^{2} - 1) δ_{m + n, 0} .

D = 26

, the total central charge and central term vanish. The total stress tensor then transforms as a genuine conformal tensor of weight two, and the quantum worldsheet gauge anomaly cancels.

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