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.


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
2015.06.29 Monday (c) All rights reserved by ACHK