Bosonic string mass spectrum

For a closed bosonic string,

A matter vertex with left and right oscillator levels $N,\tilde{N}$ and momentum $k$ has weights $N+{\alpha }^{\prime }{k}^2/4,\tilde{N}+{\alpha }^{\prime }{k}^2/4$. An integrated string vertex operator must have weights $(1,1)$, giving the displayed mass shell and level matching with $M^2=-k^2$. Physical states are further identified modulo null string states, equivalently through BRST cohomology; merely counting all weight-one primaries would include gauge redundancies.

Each level-two chiral sector of the critical closed string is the symmetric traceless square $S_0^{2}V$ of the massive little group vector space. The tensor product of its two sectors contains the exterior square ${\Lambda }^{2}V$: the equivariant map $(S,T)\mapsto ST-TS$ is onto the antisymmetric matrices. Compact-group complete reducibility realizes this quotient as a subrepresentation. It describes a massive two-form field with $M^2=8\pi T=4/{\alpha }^{\prime }$.

In the critical bosonic string theory, the states ${\alpha }_{-1}^i{\tilde{\alpha }}_{-1}^j|0;p⟩$ have $M^2=0$. Their coefficient matrix decomposes into a symmetric trace-free tensor, an antisymmetric tensor, and a scalar trace, representing the graviton, Kalb–Ramond field, and dilaton. Their transverse particle polarization counts add to $(D-2{)}^2$. The normal-ordering constant of a string is one.

The string level operator for a free-ended open string is $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. The quantum condition is ${\alpha }^{\prime }{M}^2=N-a$, where $a$ is the normal-ordering constant of a string. Lorentz-consistent bosonic string theory has $a=1$.

The first levels are the oscillator vacuum, ${\alpha }_{-1}^I|0⟩$, and ${\alpha }_{-2}^I|0⟩,{\alpha }_{-1}^I{\alpha }_{-1}^J|0⟩$. At level two the transverse vector representation and symmetric square assemble into the massive symmetric traceless square of one extra rotation dimension.