# String Theory-Virasoro Generators -- show commutator relation

1. Apr 17, 2017

### binbagsss

1. The problem statement, all variables and given/known data

(I have dropped the hats on the $\alpha_{n}^{u}$ operators and $L_{m}$)

$[\alpha_{n}^u, \alpha_m^v]=n\delta_{n+m}\eta^{uv}$
$L_m=\frac{1}{2}\sum\limits_{n=-\infty}^{\infty} : \alpha_{m-n}^u\alpha_{n}^v: \eta_{uv}-\delta_{m,0}$

where : denotes normal-ordered.

Show that : $[\alpha_{m}^u,L_n]=m\alpha_{m+n}^u$

2. Relevant equations

see above
3. The attempt at a solution

For a given $n$ we are looking at the following commutator: $[\alpha_m,\alpha_{n-m}\alpha_m]$

to use commutator relation:

$[a,bc]=-a[b,c]-[a,c]b$

$a= \alpha_m$
$b= \alpha_{n-m}$
$c= \alpha_m$

$[a,c]=0$
$[b,c]=(n-m)\delta_{n=0}\eta^{uv}$ using (1)

$\implies [\alpha_{m}^u,L_n]=\alpha_m^u(m)\eta^{uv}$ which is wrong...

thanks in advance

2. Apr 19, 2017

### binbagsss

bump please. many thanks in advance, very grateful.

3. Apr 20, 2017

### king vitamin

The commutator relation you've written down is wrong.

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