Problem 14.4a3

Closed string degeneracies | A First Course in String Theory

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(a) State the values of \alpha' M^2 and give the degeneracies for the first five mass levels of the closed bosonic string theory.

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But for non-massless states, this probably is not true anymore:

| ~\text{Number of states}
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle~| ~\left[ \frac{(D-2)(D-1)}{2} \right]^2
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_2^{K\dagger} | p^+, \vec p_T \rangle~| ~\left[ \frac{(D-2)(D-1)}{2} \right](D-2)
{a_2^I}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle~| ~(D-2)\left[ \frac{(D-2)(D-1)}{2} \right]
{a_2^I}^\dagger \bar a_2^{K\dagger} | p^+, \vec p_T \rangle~| (D-2)^2

So the total number of states for \frac{1}{2} \alpha' M^2 = 2 (N = \bar N = 2) is

\left[ \frac{(D-2)(D-1)}{2} \right]^2 + \left[ \frac{(D-2)(D-1)}{2} \right](D-2)
+ (D-2)\left[ \frac{(D-2)(D-1)}{2} \right] + (D-2)^2

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Should D be 10 or 26?

p.324 “Out of 26 left-moving bosonic coordinates of the bosonic factor only ten of them are matched by the right-moving bosonic coordinates of the superstring factor.”

D should be 26 for bosonic strings. So the total number of states is

\frac{1}{2} \alpha' M^2~| ~\text{Number of states}
2~| ~104976

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What does the difference of this part and Section 14.6 come from?

This part is for bosonic closed string, while Section 14.6 is for bosonic open string. There is no \bar N to consider in Section 14.6.

p.290 “A basis vector | \lambda, \bar \lambda \rangle belongs to the state space if and only if it satisfies the level-matching constraint”

N^\perp = \bar N^\perp.

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Can we create a formula for the number of states?

— Me@2018-04-23 11:31:16 AM

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2018.04.23 Monday (c) All rights reserved by ACHK