String Theory and M-Theory

Homework Problems 57
PROBLEM 2.11
The open-string states at the N = 2 level were shown in Section 2.5 to form
a certain representation of SO(25). What does this result imply for the
spectrum of the closed bosonic string at the NL = NR = 2 level?
PROBLEM 2.12
Construct the spectrum of open and closed strings in light-cone gauge for
level N = 3. How many states are there in each case? Without actually
doing it (unless you want to), describe a strategy for assembling these states
into irreducible SO(25) multiplets.
PROBLEM 2.13
We expect the central extension of the Virasoro algebra to be of the form
[Lm, Ln] = (m − n)Lm+n + A(m)δm+n,0,
because normal-ordering ambiguities only arise for m + n = 0.
(i) Show that if A(1) = 0 it is possible to change the definition of L0, by
adding a constant, so that A(1) = 0.
(ii) For A(1) = 0 show that the generators L0 and L±1 form a closed
subalgebra.
PROBLEM 2.14
Derive an equation for the coefficients A(m) defined in the previous problem
that follows from the Jacobi identity
[[Lm, Ln], Lp] + [[Lp, Lm], Ln] + [[Ln, Lp], Lm] = 0.
Assuming A(1) = 0, prove that A(m) = (m 3 − m)A(2)/6 is the unique
solution, and hence that the central charge is c = 2A(2).
PROBLEM 2.15
Verify that the Virasoro generators in Eq. (2.91) satisfy the Virasoro algebra.
It is difficult to verify the central-charge term directly from the commutator.
Therefore, a good strategy is to verify that A(1) and A(2) have the correct
values. These can be determined by computing the ground-state matrix
element of Eq. (2.93) for the cases m = −n = 1 and m = −n = 2.