Nakahara M. Geometry, Topology and Physics
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To carry out the orthogonal decomposition of {δγ },weneedtodefinethe
inner products in various spaces. The most natural choices are
δγ
zz
2
=
d
2
z
√
γ δγ
zz
δγ
zz
(14.109a)
δγ
z ¯z
2
=
d
2
z
√
γ δγ
z ¯z
δγ
z ¯z
(14.109b)
and
δv
z
2
=
d
2
z
√
γγ
z ¯z
δv
z
δv
z
.(14.109c)
Note that δγ
zz
dz ⊗ dz and δγ
z ¯z
dz ⊗ d¯z are different tensors; we have to specify
the inner product separately.
Following the argument in the previous subsection, we introduce the
orthogonal decomposition,
δγ
zz
=∇
(−1)
z
δ ˜v
z
+ δτ
i
µ
izz
=∇
(−1)
z
δ ¯v
z
+ δτ
i
φ
izz
(14.110)
where δ ¯v = δ ˜v + (projection of δτ
i
µ
izz
into {im ∇
(−1)
z
}). The orthogonality of
∇
(−1)
z
δ ¯v
z
and φ
izz
implies
0 = (∇
(−1)
z
δv
z
,φ
izz
) =
d
2
z
√
γ δv
z
(−∇
z
(−2)
φ
izz
)
where we have noted that ∇
(−1)†
z
=−∇
z
(−2)
. Thus, we find that (figure 14.6)
φ
izz
∈ ker ∇
z
(−2)
. (14.111)
The explicit form of ∇
z
(−2)
shows that ∂
¯z
φ
izz
= 0, that is ker ∇
z
(−2)
is the set
of holomorphic tensors of helicity −2. The tensor φ
i
= φ
izz
dz ⊗ dz is called the
quadratic differential while µ
i
= µ
izz
dz ⊗ dz is the Beltrami differential,see
figure 14.7. In practical computations, it is often convenient to specify the gauge
slice by the Beltrami differential, see later. Now we have established that
{ker P
†
1
}={Quadratic differential}={ker ∇
z
(−2)
}. (14.112)
The Riemann–Roch theorem (14.27) takes the form
dim
ker ∇
(−1)
z
− dim ker ∇
z
(−2)
= 3 − 3g. (14.113)
Now we have separated {δγ} into mutually orthogonal pieces
{δγ }={conf}⊕{im ∇
(−1)
z
}⊕{ker ∇
z
(−2)
}+cc (14.114)
which should be compared with (14.80b). The measure becomes
δγ δ X → J d
n
δτ δ ¯v δφ δ
˜
X d
k
δa (14.115)
Figure 14.6. The map ∇
(−)
z
and its adjoint ∇
z
(−2)
.
Figure 14.7. The Beltrami differential {µ
i
} spans the deformation tangent to the gauge
slice while {φ
i
} spans ker ∇
z
(−2)
.
where n and k are the complex dimensions of the Teichm¨uller space and the CKV,
respectively. The Jacobian is obtained by repeating the argument in the previous
subsection and we find that
Z
g
=
γ X
1
V (Diff*Weyl)
e
−S
=
Mod
d
n
τ X
det
+
(1)
V (CKV)
|det(µ, φ)|
2
det(φ, φ) det(, )
e
−S
. (14.116)
Since we are integrating over complex variables, the power of a half in (14.96)
does not appear in (14.116). The X-integration yields
Z
g
=
Mod
d
n
τ
V (CKV)
|det(µ, φ)|
2
det(φ, φ) det(, )
× det
+
(1)
det
d
2
z
√
γ
−13
. (14.117)
14.2.4 Moduli spaces of Riemann surfaces
The spaces Mod(
g
) and Teich(
g
) have been defined as
Mod(
g
) ≡
g
/Diff(
g
) Teich(
g
) ≡
g
/Diff
0
(
g
).
They are related through MCG ≡ Diff(
g
)/Diff
0
(
g
) as Mod(
g
) =
Teich(
g
)/MCG. We look at these objects more closely here. We first note:
g dim CKV CKV dim Teich(
g
) MCG
03 SL(2, C) 0SL(2, R)
11U(1) × U(1) 1SL(2, Z)
≥20 empty 3g − 3?
(14.118)
[Remark:MCGforg≥2 can be expressed by 3g − 1 Dehn twists which are,
however, not minimal.] From (14.118), we immediately conclude that Z
0
= 0
since the Teichm¨uller space is a single point and the volume of SL(2,
) is infinite.
Of course, this does not imply that the three amplitudes with vertex operators
vanish. In general, Mod(
g
) is topologically non-trivial although Teich(
g
) is.
Teich(
g
) is a universal covering space of Mod(
g
) and the topological non-
triviality comes from MCG.
In actual computations, the uniformization theorem is very useful. In the
previous subsection, we first chose the Beltrami differential µ
i
, then changed the
basis to φ
i
∈ ker P
†
1
. Our initial choice µ
i
is motivated by the uniformization
theorem.
Theorem 14.2. (Uniformization theorem)Let
g
be a torus with genus g.Then
it is conformally related to the constant-curvature Riemann surface, which is given
by the following:
g Riemann surface Metric sign
0 ∪{∞} ds
2
= dz ⊗ d¯z/(1 + z ¯z)
2
+
1
/Lds
2
= dz ⊗ d¯z 0
≥2 H/G ds
2
= dz ⊗ d¯z/(Im z)
2
−
(14.119)
where L is a lattice in
(see example 8.2), H the upper half-plane and G ⊂
SL(2,
) is called the Fuchsian group. The metric for g ≥ 2isthePoincar
´
e
metric, see example 7.6.
The proof of this theorem is found in Farkas and Kra (1980), for example.
Thanks to this theorem, we may always take constant-curvature metrics to form
the gauge slice in
g
. This corresponds to a special choice of the Beltrami
differential µ
i
. This slice defines the Weil–Petersson measure:
d
n
τ
|det(µ, φ)|
2
det(φ, φ)
=
d(Weil–Petersson) (14.120)
see D’Hoker and Phong (1986).
Exercise 14.6. Compute the scalar curvature of the metrics given in (14.119).
Verify that they are independent of z and ¯z.
14.3 One-loop amplitudes
As an illustration of the formalism developed in the previous section, we compute
the one-loop vacuum-to-vacuum amplitude of the closed orientable bosonic string
theory. Since dim
Teich(
1
) = 1 and dim ker ∇
(−1)
z
= 1, we have
Z
1
=
Mod
dτ
V (CKV)
|(µ, φ)|
2
(φ, φ) ·(, )
det
+
(1)
det
d
2
ξ
√
γ
−13
. (14.121)
To evaluate (14.121) we need to take several steps.
14.3.1 Moduli spaces, CKV, Beltrami and quadratic differentials
In example 8.2, we have shown that the complex structure, namely the conformal
structure, of the torus is specified by a complex parameter τ(Im τ>0). Figure
8.3 shows the moduli space
Mod(
g
) =
1
/G = Teich(
g
)/SL(2, ) = H /SL(2, )
where H is the upper half-plane.
Take the torus T
τ
specifiedbytheTeichm¨uller parameter τ = τ
1
+iτ
2
(τ
2
>
0). As a representative, we take a torus in figure 14.8. The metric in
naturally
induces a flat metric (as guaranteed by the uniformization theorem)
γ =
1
2
[dz ⊗ d¯z + d¯z ⊗ dz]. (14.122)
The CKV are globally defined holomorphic vectors. We take = α∂/∂z as
the normalized basis of the CKV. The condition (, ) = 1 yields
d
2
z|α|
2
=
τ
2
|α|
2
= 1, that is α = τ
−1/2
2
(we have dropped the phase). The vector
generates translations in the complex plane,
z → z
= z + τ
−1/2
2
(v
1
+ iv
2
). (14.123)
Figure 14.8. The parallelogram whose complex structure is parametrized by τ .
We must note, however, that the translation is defined modulo the lattice;
τ
−1/2
2
(v
1
+ iv
2
) and τ
−1/2
2
(v
1
+ iv
2
) + (m + τ n) yield the identical translation.
This forces τ
−1/2
2
(v
1
+ iv
2
) to lie within the parallelogram of figure 14.8. Since
τ
2
=
d
2
z = τ
−1
2
d
2
v
V (CKV) is found to be
V (CKV) =
d
2
v = τ
2
2
. (14.124)
Our next task is to evaluate the Weil–Petersson measure. On the torus there
is one quadratic differential φ.Sinceφ ∈
−2
is a globally defined holomorphic
differential, it must be of the form,
φ = a dz ⊗ dza∈
. (14.125)
To find the Beltrami differential, we evaluate the change of the metric under
a small variation of τ . For this purpose, it is convenient to introduce the ξ
α
-
coordinate system in figure 14.8. The point A corresponds to (1, 0) and B to
(0, 1). Accordingly, we have z = ξ
1
+ τξ
2
. Under a small change δτ of the
Teichm ¨uller parameter, we have, up to a conformal factor,
|dz|
2
→|dξ
1
+ (τ + δτ)dξ
2
|
2
=|dz + δτdξ
2
|
2
=
dz + dτ
dz − d¯z
2iτ
2
2
=
dz + δτ
id ¯z
2τ
2
.
Comparing this with (14.110), we find that
µ
zz
= i/2τ
2
. (14.126)
Here (δτ )µ is the complex conjugate of (δτ )µ in (14.110). Of course, this is a
reparametrization of the Teichm¨uller space and does not affect the results. If the
reader feels awkward with this, s/he may choose ¯τ as the Teichm¨uller parameter.
From (14.125) and (14.126), we have, up to irrelevant constants,
(µ, φ) =
d
2
z µ
zz
φ
zz
=
i
2τ
2
aτ
2
∝ a
(φ, φ) =
d
2
z φ
zz
φ
zz
= a
2
τ
2
.
Finally, we have obtained
|(µ, φ)|
2
(φ, φ)
= τ
−1
2
. (14.127)
14.3.2 The evaluation of determinants
We first consider det
P
†
1
P
1
= det
+
(1)
. Since we take a flat metric, the Laplacian
takes quite a simple form,
+
(1)
=−2∂
z
∂
¯z
= (14.128)
where is the Laplacian defined by (14.87b). Since
d
2
ξ
√
γ =
d
2
z = τ
2
the amplitude (14.121) reduces to
Z
1
=
Mod
dτ
τ
2
2
det
τ
2
det
τ
2
−13
(14.129)
↑↑↑
V (CKV) W-P
d
2
z
where we have used (14.124) and (14.127). We have factorized the integrand so
that the modular invariance is manifest, see exercise 14.7.
Let us compute the spectrum of . It is convenient to express the Laplacian
in ξ
α
-coordinates. From
ξ
1
= i( ¯τ z − τ ¯z)/2τ
2
ξ
2
= (z −¯z)/2iτ
2
(14.130)
we readily find that
=−
1
2τ
2
2
[|τ |
2
(∂
1
)
2
− 2τ
1
∂
1
∂
2
+ (∂
2
)
2
] (14.131)
where ∂
1
= ∂/∂ξ
1
etc. The eigenfunction satisfying the periodic boundary
condition on the torus is
ψ
m,n
(ξ) = exp[2π i(nξ
1
+ mξ
2
)] (m, n) ∈
2
. (14.132)
Substituting this into (14.131), we find the eigenvalue
λ
m,n
=
2π
2
τ
2
2
(m − τ n)(m −¯τ n). (14.133)
The determinant is expressed as an infinite product:
det
=
m,n
2π
2
τ
2
2
|m + τn|
2
(14.134)
the product being taken for all integers (m, n) = (0, 0).
Clearly det
is ill defined and needs to be regularized. Let us introduce the
Eisenstein series (Siegel 1980, Lang 1987) defined by
E(τ, s) ≡
m,n
τ
s
2
|m + τn|
2s
(14.135)
the summation being taken for all integers (m, n) = (0, 0). This series converges
for Re s > 1 and can be analytically continued to the complex s-plane. The series
E(τ, s) has a simple pole at s = 1 where we have a Laurent expansion,
E(τ, s) =
π
s − 1
+ 2π[γ − ln 2 −ln(
√
τ
2
|η(τ)|
2
)]+ (s − 1). (14.136)
This expression is known as the Kronecker first limit formula and is essential
for our purposes. In (14.136), γ = 0.57721 ...is Euler’s constant and η(τ) is the
Dedekind η-function
η(τ) ≡ e
iπτ/12
n>1
(1 −e
2iπnτ
). (14.137)
Neglecting constant factors, we have
det
τ
2
= exp
−ln τ
2
+
ln
|m + τn|
2
τ
2
2
= exp
−ln τ
2
−
∂
∂s
[τ
s
2
E(τ, s)]
s=0
= exp{−ln τ
2
[1 + E(τ, 0)]−E
(τ, 0)}. (14.138)
To evaluate the exponent, we note the functional equation,
π
−s
(s)E(τ, s) = π
−(1−s)
(1 − s)E(τ, 1 − s). (14.139)
Taking the limit s → 0 in (14.139), we have
sE(τ, 1 − s) = π
1−2s
(1 + s)
(1 − s)
E(τ, s)
= π(1 −2s ln π +···)
(1 −γ s +···)
(1 +γ s +···)
[E(τ, 0) + E
(τ, 0)s +···]
= π E(τ, 0) +[−2(ln π + γ)E(τ, 0) + E
(τ, 0)]πs +···.
From (14.136), we also have
sE(τ, 1 − s) =−π + 2π s[γ − ln 2 − ln(
√
τ
2
|η(τ)|
2
)]+···.
Equating the coefficients of s
0
and s
1
,wefindthat
E(τ, 0) =−1 (14.140a)
E
(τ, 0) =−2[ln 2π + ln(
√
τ
2
|η(τ)|
2
)]. (14.140b)
Substituting (14.140) into (14.138), we obtain
det
τ
2
= exp[−E
(τ, 0)]=τ
2
|η(τ)|
4
. (14.141)
Finally, it follows from (14.129) and (14.141) that
Z
1
=
Mod
dτ
τ
2
2
τ
−12
2
|η(τ)|
−48
. (14.142)
A neat form of Z
1
is obtained if we define the discriminant
(τ ) ≡ (2π)
12
η(τ)
24
. (14.143)
Up to an irrelevant constant, the one-loop amplitude is
Z
1
=
Mod
dτ
τ
2
2
τ
−12
2
|(τ )|
−2
. (14.144)
(τ ) is known as the cusp form of weight 12, implying
aτ + b
cτ + d
= (cτ + d)
12
(τ ) (14.145)
and c(0) = 0, where the c(n) are the Fourier coefficients,
(τ ) =
n≥0
c(n)e
2πniτ
. (14.146)
Higher genus amplitudes are given by the cusp forms of other weights, see Belavin
and Knizhnik (1986), Moore (1986), Gilbert (1986) and Morozov (1987).
Exercise 14.7. Show that
η(τ + 1) = e
πi/12
η(τ) η(−1/τ) = (−iτ)
1/2
η(τ) (14.147)
where the branch is chosen so that
√
z > 0ifz > 0. Use this result to show
that dτ/τ
2
2
and τ
−12
2
|η(τ)|
−48
are independently invariant under τ → τ + 1and
τ →−1/τ .
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