17.1 Worldsheet Superstring Theory
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number condition: the naive insertion of e φ ∼ δ(β) breaks the BRST invariance.
The PCO zero-mode is obtained from the contour integral:
X 0 =
1
2π i
dz
z
X (z).
(17.20)
It can be interpreted as delocalizing a PCO insertion from a point to a circle, which
decreases the risk of divergence.
17.1.2 Hilbert Spaces
The description in terms of the (η, ξ, φ) fields leads to a subtlety: the bosonization
involves only the derivative ∂ξ and not the field ξ itself, meaning that the zero-mode
ξ 0 is absent from the original Hilbert space defined from (β, γ ). In the bosonized
language, the Hilbert space without the ξ zero-mode is called the small Hilbert
space and is made of state annihilated by η 0 (the η zero-mode)
H small =
|ψ | η 0 |ψ = 0
.
(17.21)
Removing this condition leads to the large Hilbert space: 2
H small = H large ∩ ker η 0 .
(17.22)
A state in H small contains ξ with at least one derivative acting on it.
A correlation function defined in terms of the (η, ξ, φ) system is in the large
Hilbert space and will vanish since there is no ξ factor to absorb the zero-mode of
the path integral. As a consequence, correlation functions (and the inner product)
are defined with a ξ 0 insertion (by convention at the extreme left) or, equivalently,
ξ(z). The position does not matter since only the zero-mode contribution survives,
and the correlation function is independent of z. Sometimes it is more convenient to
work in the large Hilbert space and to restrict later to the small Hilbert space.
The SL(2, C) invariant vacuum is normalized as
k| c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 e
−2φ(z)
|k
= (2π)
D δ
(D) (k + k
).
(17.23)
Remark 17.1 (Normalization in Type II) In type II theory, the SL(2, C) is normalized as
k| c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 e
−2φ(z) e
− ¯
φ( ¯
w)
|k
= −(2π)
D δ
(D) (k + k
).
(17.24)
2 The relation between the small and large Hilbert spaces is similar to the one between the H and
H 0 = b 0 H Hilbert space from the open string since the (b, c) and (η, ξ ) are both fermionic firstorder systems.
343
number condition: the naive insertion of e φ ∼ δ(β) breaks the BRST invariance.
The PCO zero-mode is obtained from the contour integral:
X 0 =
1
2π i
dz
z
X (z).
(17.20)
It can be interpreted as delocalizing a PCO insertion from a point to a circle, which
decreases the risk of divergence.
17.1.2 Hilbert Spaces
The description in terms of the (η, ξ, φ) fields leads to a subtlety: the bosonization
involves only the derivative ∂ξ and not the field ξ itself, meaning that the zero-mode
ξ 0 is absent from the original Hilbert space defined from (β, γ ). In the bosonized
language, the Hilbert space without the ξ zero-mode is called the small Hilbert
space and is made of state annihilated by η 0 (the η zero-mode)
H small =
|ψ | η 0 |ψ = 0
.
(17.21)
Removing this condition leads to the large Hilbert space: 2
H small = H large ∩ ker η 0 .
(17.22)
A state in H small contains ξ with at least one derivative acting on it.
A correlation function defined in terms of the (η, ξ, φ) system is in the large
Hilbert space and will vanish since there is no ξ factor to absorb the zero-mode of
the path integral. As a consequence, correlation functions (and the inner product)
are defined with a ξ 0 insertion (by convention at the extreme left) or, equivalently,
ξ(z). The position does not matter since only the zero-mode contribution survives,
and the correlation function is independent of z. Sometimes it is more convenient to
work in the large Hilbert space and to restrict later to the small Hilbert space.
The SL(2, C) invariant vacuum is normalized as
k| c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 e
−2φ(z)
|k
= (2π)
D δ
(D) (k + k
).
(17.23)
Remark 17.1 (Normalization in Type II) In type II theory, the SL(2, C) is normalized as
k| c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 e
−2φ(z) e
− ¯
φ( ¯
w)
|k
= −(2π)
D δ
(D) (k + k
).
(17.24)
2 The relation between the small and large Hilbert spaces is similar to the one between the H and
H 0 = b 0 H Hilbert space from the open string since the (b, c) and (η, ξ ) are both fermionic firstorder systems.