Note, that here, the $\hat L_n$ are operators on the state given by the sums of the dots of the mode operators, i.e. $\hat L_0=\sum_{k=-\infty}^\infty\hat\alpha_{-n}\cdot\hat\alpha_n$.
Also note that The Virasoro Algebra is the central extension of the Witt/Conformal Algebra so that explains why we have a $D$, it is equivalent to the central charge.
I'll expand on Chris Gerig's answer.
Not only do we need $D=26$, we also need the normal ordering constant $a=1$. The normal ordering constant is the eigenvalue of $\hat L_0$ with the eigenvector the state.
We want to promote the time-like states to spurious, zero-norm states, right? So, we impose the (level 1) spurious state conditions on the state as ffollows ($|\chi\rangle$ are the basis vectors to build the spurious state $\Phi\rangle$ on.)$:
$$ \begin{gathered} 0 = {{\hat L}_1}\left| \Phi \right\rangle \\ {\text{ }} = {{\hat L}_1}{{\hat L}_{ - 1}}\left| {{\chi _1}} \right\rangle \\ {\text{ }} = \left[ {{{\hat L}_{ - 1}},{{\hat L}_1}} \right]\left| {{\chi _1}} \right\rangle + {{\hat L}_{ - 1}}{{\hat L}_1}\left| {{\chi _1}} \right\rangle \\ {\text{ }} = \left[ {{{\hat L}_{ - 1}},{{\hat L}_1}} \right]\left| {{\chi _1}} \right\rangle \\ {\text{ }} = 2{{\hat L}_0}\left| {{\chi _1}} \right\rangle \\ {\text{ }} = 2{c_0}\left( {a - 1} \right)\left| {{\chi _1}} \right\rangle \\ \end{gathered} $$
That means that $a=1$.
Now, for a level 2 spurious state,
$$\begin{gathered} \left[ {{{\hat L}_1},{{\hat L}_{ - 2}} + k{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}} \right]\left| \psi \right\rangle = \left( {3{{\hat L}_{ - 1}} + 2k{{\hat L}_0}{{\hat L}_{ - 1}} + 2k{{\hat L}_{ - 1}}{{\hat L}_0}} \right)\left| \psi \right\rangle {\text{ }} \\ {\text{ }} = \left( {3 - 2k} \right){{\hat L}_{ - 1}} + 4k{{\hat L}_0}{{\hat L}_{ - 1}}{\text{ }}\left( {3 - 2k} \right){{\hat L}_{ - 1}} + 4k{{\hat L}_0}{{\hat L}_{ - 1}}{\text{ }} \\ 0 = {{\hat L}_1}\left| \psi \right\rangle = {{\hat L}_1}\left( {{{\hat L}_{ - 2}} + k{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}} \right)\left| {{\chi _1}} \right\rangle = \left( {\left( {3 - 2k} \right){{\hat L}_{ - 1}} + 4k{{\hat L}_0}{{\hat L}_{ - 1}}} \right)\left| {{\chi _1}} \right\rangle \\ {\text{ }} = \left( {\left( {3 - 2k} \right){{\hat L}_{ - 1}} + 4k{{\hat L}_{ - 1}}\left( {{{\hat L}_0} + 1} \right)} \right)\left| {{\chi _1}} \right\rangle \\ {\text{ }} = \left( {3 - 2k} \right){{\hat L}_{ - 1}}\left| {{\chi _1}} \right\rangle \\ 2k = 3 \\ k = \frac{3}{2} \\ \end{gathered} $$
Since this level 2 spurious state can be written as:
$$ {\left| \Phi \right\rangle = {{\hat L}_{ - 2}}\left| {{\chi _1}} \right\rangle + k{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}\left| {{\chi _2}} \right\rangle }$$
So, then,
$$ \begin{gathered} {{\hat L}_2}\left| \Phi \right\rangle = 0 \\ {{\hat L}_2}\left( {{{\hat L}_{ - 2}} + \frac{3}{2}{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}} \right)\left| {{\chi _2}} \right\rangle = 0 \\ \left[ {{{\hat L}_2},{{\hat L}_{ - 2}} + \frac{3}{2}{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}} \right]\left| {{\chi _2}} \right\rangle + \left( {{{\hat L}_{ - 2}} + \frac{3}{2}{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}} \right){{\hat L}_2}\left| {{\chi _2}} \right\rangle = 0 \\ \left[ {{{\hat L}_2},{{\hat L}_{ - 2}} + \frac{3}{2}{{\hat L}_{ - 1}}{{\hat L}_{ - 1}}} \right]\left| {{\chi _2}} \right\rangle = 0 \\ \left( {13{{\hat L}_0} + 9{{\hat L}_{ - 1}}{{\hat L}_{ - 1}} + \frac{D}{2}} \right)\left| {{\chi _2}} \right\rangle = 0 \\ \frac{D}{2} = 13 \\ D = 26 \\ \end{gathered} $$
Q.E.D.
So, this was done essentially to remove the imaginary norm ghost states and using the Canonical / Gupta - Bleuer formalism.
It's also possible to use , say, e.g. Light Cone Gauge (LCG) quantisation. However, in other quantisation methods, the conformal anomaly is manifest in other forms. E.g., in LCG quantisationn, it is manifest as a failure of lorentz symmetry. A good overview of this method can be found in Kaku Strings, Conformal fields, and M-theory.
Q&A (4870)
