The bosonic string spectrum and the explicit states up to level 10 from the lightcone and the chaotic behavior of certain string amplitudes

Abstract We compute the irreps and their multiplicities of bosonic string spectrum up to level 10 and we give explicitly the on shell top level lightcone states which make the irreps. For the irreps up to three indexes and all the totally antisymmetric ones we give the general recipe and the full irreps. It turns out that lightcone is quite efficient in building these low indexes irreps once the top level states are known. For scalars and vectors we compute the multiplicity up to level 22 and 19 respectively. The first scalar at odd level appears at level 11. For the bosonic string in non critical dimensions we argue that at level N there are always states transforming as tensors with $$s\ge \frac{1}{2} N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> indices. Only in critical dimensions there are states with $$s\le \frac{1}{2} N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . Looking at the explicit coefficients of the combinations needed to make the irreps from the lightcone states we trace the origin of the chaotic behavior of certain cubic amplitudes considered in literature to the extremely precise and sensitive mixtures of states. For example the vectors at level $$N=19$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>19</mml:mn> </mml:mrow> </mml:math> are a linear combinations of states and when the coefficients are normalized to be integer some of them have more than 1200 figures.