In many sources, the Hagedorn temperature of different string theories is given using the high temperature (or high-level limit) $N\gg1$ in the calculation of the string entropy such that: $$T_H=\left(\frac{\partial S}{\partial E}\right)^{-1}$$ and the entropy is calculated using the Boltzmann formula for counting partitions $$S(E)=k\log(\Omega(E))$$

For Bosonic string theory for example there are $B=D-2=24$ transverse directions, where $D$ is the critical dimension, so for level $N$ the number of partitions is equivalent to the partition obtained by a Boson with spin $B$ in energy level $E$: $$\Omega_B(E)=\exp\left(\pi\sqrt{\frac{B\alpha'}{6}}E\right)=_{D=26}\exp\left(4\pi\sqrt{\alpha'}E\right)$$ Where $\alpha'$ is the string tension coming from the relation of mass squared (Energy squared) to the level $N$.

For Type II superstring theory The number of partitions constitute the partition on each sector on each side (Left and Right moving) as half of states, times degeneracy of $16$ coming from the ground state of the $R$ sector So we get: $$\Omega_{Type\space II}(E)=64(\Omega_B(E)\Omega_F(E))^2$$ When the bosonic part $\Omega_B(E)$ is now taken with spin $B=D-2=8$ for the fermionic field the number of partitions is equivalent to the partition obtained by a fermion with spin $F$ in energy level $E$: $$\Omega_F(E)=\exp\left(\pi\sqrt{\frac{F\alpha'}{12}}E\right)=_{D=10}\exp\left(\pi\sqrt{\frac{2\alpha'}{3}}E\right)$$ So overall we have $$\Omega_{Type\space II}(E)\approx\exp\left(2\pi\sqrt{\frac{(F+2B)\alpha'}{12}}E\right)=_{D=10}\exp\left(2\sqrt{2}\pi\sqrt{\alpha'}E\right)$$

For Heterotic superstring theory The left moving sector is given by a Bosonic string in $\tilde{D}=26$ dimensions while the right moving sector is given by a supersymmetric string in $D=10$ dimensions, taking this into account the number of partitions would be: $$\Omega_{Heterotic}(E)=16(\Omega_\tilde{B}(E))^{\tilde{D}=26}(\Omega_B(E)\Omega_F(E))^{D=10}$$ That is: $$\Omega_{Heterotic}(E)\approx\exp\left(\pi\sqrt{\frac{(F+2B+2\tilde{B})\alpha'}{12}}E\right)=_{\tilde{D}=26,D=10}\exp\left(2\pi\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\alpha'}E\right)$$

Unfortunately, I didn't find in any source explicitly how to derive the number of the partition in order to estimate the entropy thus Hagedorn temperature of Type I superstring theory. According to Wikipedia: "Type I string theory can be obtained as an orientifold of type IIB string theory, with 32 half-D9-branes added in the vacuum to cancel various anomalies giving it a gauge group of SO(32) via Chan-Paton factors". Is this means in particular that Type I superstrings have the same number of partitions so the same entropy and Hagedorn temperature as Type II superstrings?



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