# Quick Calculation 14.7

A First Course in String Theory

Count the number of graviton, Kalb-Ramond, and dilation states in ten dimensions. Add these numbers up and confirm you get 64.

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Equation (13.69):

$\sum_{I,J} \hat S_{IJ} {a_1^I}^\dagger {\bar a}_1^{J \dagger} | p^+, \vec p_T \rangle$

p.292 “… the states (13.69) represent one-particle graviton states … ”

Equation (13.64):

In the closed string state space, the general state of fixed momentum at the massless level is

$\sum_{I,J} R_{IJ} {a_1^I}^\dagger {\bar a}_1^{J \dagger} | p^+, \vec p_T \rangle$

Equation (13.68):

$R_{IJ} = \hat S_{IJ} + A_{IJ} + S' \delta_{IJ}$

$S' = \frac{S}{D-2}$

Equation (10.108):

For the symmetric traceless part, $\hat S_{IJ}$ (with the size $(D-2) \times (D-2)$), the number of independent components is

$n(D) = \frac{1}{2} (D - 2) (D - 1) - 1$

The $-1$ at the end reflects the fact that if the trace is zero, the last component of the trace is not independent anymore.

In ten dimensions ($D = 10$),

$n(D) = \frac{1}{2} (8) (9) -1 = 35$

p.292 For $A_{IJ}$ (a skew-symmetric matrix), the number of independent components is

$n(D) = \frac{1}{2} (D - 3) (D - 2) = 28$

p.293 “The oscillator part of (13.71) has no free indices ($I$ is summed over), so it represents one state. It corresponds to [an] one-particle state of a massless scalar field. This field is called the dilation.”

— Me@2015-06-29