Lectures on String Theory

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In the closed string case we have in addition the condition of level-matching
which requires that for physical states
M 2 L = M 2 R .
Finally, we give expressions for the light-cone action
S l.c = 1 ∫ ((Ẋi d 2 σ ) 2 − (X ′i ) 2 − 2i
8π
¯ψ
)
i ρ a ∂ α ψ i
and the Hamiltonian
(7.1)
H = (p i ) 2 + ∑ n>0
(α i −nα i n + ᾱ i −nᾱ i n) + ∑ r>0
r(b i −rb i r + ¯b i −r¯b i r) − 2a .
Note that every sector, R and NS, has its own Hamiltonian.
Let us now analyze the closed string spectrum. We first discuss the right-moving
part which (up to a mass rescaling by 2) is equivalent to the spectrum of open
fermionic string.
• NS-sector. The ground state is the oscillator vacuum |0〉 with α ′ M 2 = −a. The
first excited state is b i −1/2 |0〉 with α′ M 2 = 1 − a. This is a vector of SO(d − 2),
2
where the critical dimension d = 10. Since the little Lorentz group for massless
states in d-dimensions is SO(d − 2) this state must be massless which gives the
normal ordering constant to be a = 1 . At the next level one has the states
2
α−1|0〉 i and b i −1/2 bj −1/2 |0〉 with α′ M 2 = 1 . The number of these bosonic states
2
is 8 + 28 = 36 = 9×8 , they are comprise an antisymmetric representation of
2
SO(9), the little Lorentz group for massive states.
• R-sector. The Ramond ground state is a spinor of SO(9, 1). The dimension
of the Dirac spinor in d = 10 is 2 d 2 = 2 5 = 32, i.e. it has 32 complex or 64
real components. In ten dimensions it is possible to impose both Majorana
and Weyl conditions 20 which reduce the number of independent components
to 64 = 16. On shell the number of components is further reduced by two
2×2
because the Dirac equation Γ µ ∂ µ ψ relates half of the components to the other
half (which satisfies the Klein-Gordon equation). The 8 remaining components
can be viewed as the components of the Majorana-Weyl spinor of SO(8), the
latter being the little Lorentz group for massless states in d = 10. Indeed, the
spinor of SO(8) should have
(2 8 2 complex components)/(Majorana × Weyl) = 32/4 = 8
20 In general, for the groups SO(p, q) the Majorana and Weyl conditions can be simultaneously
imposed if and only if p − q = 0 mod 8. For Minkowski space, p = d − 1, q = −1, this gives
d = 2 + 2n and for Euclidean space, p = d, q = 0, this gives d = 2n.