String level operator ($N$)

The string level operator is the weighted oscillator number $N={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$. A closed string has independent operators $N$ and $\tilde{N}$ for its two chiral sectors.

The canonical commutation relations imply $[N,{\alpha }_{-k}^i]=k{\alpha }_{-k}^i$ for $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. With zero level assigned to the oscillator vacuum, a Fock space basis state containing $r_{ki}$ creation operators of each mode has level ${\sum }_{k,i}k{r}_{ki}$. The level is therefore a nonnegative integer; it counts oscillator excitation weighted by frequency, not simply the number of creation operators.