# Question on the Hagedorn tower in Type I string theory

In a previous question (Mass spectrum of Type I string theory), I had asked about the mass spectrum of Type I string theory. I got a response saying that it is a Hagedorn tower. However, my source does not discuss much about Type I string theory (it only discusses Type IIA and Type IIB) so I searched for more sources. Yet, all the sources (all some lecture notes) I got, for some reason stopped at Type IIA and Type IIB string theories.

So my question is, what is really the mass spectrum defining the Hagedorn tower? Is it something analogous to the Type II Mass Spectrums? $m= \sqrt{\frac{2\pi T}{c_0}\left(N+\tilde N -a-\tilde a\right)}$

Thanks!

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Please don't self-close, such posts may be useful to others. If you managed to answer it yourself I suggest you post your findings as an answer here. – Manishearth May 16 '13 at 16:20

## 2 Answers

Hagedorn spectrum just means that the density of states varies exponentially with the energy/mass. $m^2$ (asymptotically) given by the "level" (N) of the state (upto a sqrt). The number of states at level $N$ corresponds to the possible partitions of $N$ into different oscillator modes. That means that the number of states at level $N$ will increase exponentially (for large $N$). Taking a square-root (sicne we want how #states scales with $m$) still leaves an exponential, giving us a Hagedorn spectrum.

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Here is my solution.

For the Ramond Ramond Sector, $${m_\rm{I}} = \frac{{\left[ {\hat a_ + ^\mu ,\hat{\tilde a}_ + ^\nu } \right]}}{2}{m_{{\rm{IIB}}}}$$

For the Neveu-Schwarz Neveu-Schwarz Sector, $${m_\rm{I}} = \frac{{\left[ {\hat d_ {-1/2} ^\mu ,\hat{\tilde d}_{-1/2} ^\nu } \right]}}{2}{m_{{\rm{IIB}}}}$$

For the Ramond Neveu-Schwarz Sector or the Neveu-Schwarz Ramond Sector, $${m_\rm{I}} = \frac{{{{\hat a}_ + }\hat d_{ - 1/2}^\mu - {{\hat{\tilde a}}_ + }\hat{ \tilde d}_{ - 1/2}^\mu }}{2}{m_{{\rm{IIB}}}}$$

My reasoning is that the mass spectrum is linear in the state vectors.

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