344
17 Superstring
The sign difference allows to avoid sign differences between type II and heterotic
string theories in most formulas [2].
The Hilbert space of GSO even states satisfying the b
−
0 = 0 and L
−
0 = 0
conditions is denoted by H T (ghost and picture numbers are arbitrary). This Hilbert
space is the direct sum of the NS and R Hilbert spaces:
H T = H NS ⊕ H R .
(17.25)
The subspace of states with picture number N pic = n is written as H n . The picture
numbers of the NS and R states are, respectively, integer and half-integer. Two
special subspaces of H T play a distinguished role:
H T = H −1 ⊕ H −1/2 ,
H T = H −1 ⊕ H −3/2 .
(17.26)
To understand this, consider the vacuum |p of the φ field with picture number p:
|p = e
pφ (0) |0 .
(17.27)
Then, acting on the vacuum with the β n and γ n modes implies
∀n ≥ −p −
1
2
:
β n |p = 0,
∀n ≥ p +
3
2
:
γ n |p = 0.
(17.28)
For p = −1, all positive modes (starting with n = 1/2) annihilate the vacuum in the
NS sector. This is a positive asset because positive modes that do not annihilate the
vacuum can create states with arbitrary negative energy (since it is bosonic). 3 For
p = −1/2 or p = −3/2, the vacuum is annihilated by all positive modes, but not
by one of the zero-modes γ 0 or β 0 . Nonetheless, one can show that the propagator
in the R sector allows to propagate only a finite number of states if one chooses
H −1/2 ; the role of H −3/2 will become apparent when discussing how to build the
superstring field theory.
Basis states are introduced as in the bosonic case:
H T = Span{|φ r },
H T = Span{|φ
c
r },
(17.29)
such that
φ
c
r |φ s = δ rs .
(17.30)
3 This is not a problem on-shell since the BRST cohomology is independent of the picture number.
However, this matters off-shell since such states would propagate in loops and make the theory
inconsistent.
17 Superstring
The sign difference allows to avoid sign differences between type II and heterotic
string theories in most formulas [2].
The Hilbert space of GSO even states satisfying the b
−
0 = 0 and L
−
0 = 0
conditions is denoted by H T (ghost and picture numbers are arbitrary). This Hilbert
space is the direct sum of the NS and R Hilbert spaces:
H T = H NS ⊕ H R .
(17.25)
The subspace of states with picture number N pic = n is written as H n . The picture
numbers of the NS and R states are, respectively, integer and half-integer. Two
special subspaces of H T play a distinguished role:
H T = H −1 ⊕ H −1/2 ,
H T = H −1 ⊕ H −3/2 .
(17.26)
To understand this, consider the vacuum |p of the φ field with picture number p:
|p = e
pφ (0) |0 .
(17.27)
Then, acting on the vacuum with the β n and γ n modes implies
∀n ≥ −p −
1
2
:
β n |p = 0,
∀n ≥ p +
3
2
:
γ n |p = 0.
(17.28)
For p = −1, all positive modes (starting with n = 1/2) annihilate the vacuum in the
NS sector. This is a positive asset because positive modes that do not annihilate the
vacuum can create states with arbitrary negative energy (since it is bosonic). 3 For
p = −1/2 or p = −3/2, the vacuum is annihilated by all positive modes, but not
by one of the zero-modes γ 0 or β 0 . Nonetheless, one can show that the propagator
in the R sector allows to propagate only a finite number of states if one chooses
H −1/2 ; the role of H −3/2 will become apparent when discussing how to build the
superstring field theory.
Basis states are introduced as in the bosonic case:
H T = Span{|φ r },
H T = Span{|φ
c
r },
(17.29)
such that
φ
c
r |φ s = δ rs .
(17.30)
3 This is not a problem on-shell since the BRST cohomology is independent of the picture number.
However, this matters off-shell since such states would propagate in loops and make the theory
inconsistent.