42
2 Worldsheet Path Integral: Vacuum Amplitudes
for any h ∈ G. Given the Lie algebra g of the group, a general element of the algebra
is a linear combination of the generators T i with coefficients α i
α = α
i T i .
(2.54)
Group elements can be parametrized in terms of α through the exponential map.
Moreover, since a Lie group is a manifold, it is locally isomorphic to R n : this
motivates the use of a flat metric for the Lie algebra, such that
G =
dα :=
i
dα
i .
(2.55)
Finally, it is possible to perform a change of coordinates from the Lie parameters
to coordinates x on the group: the resulting Jacobian is the Haar measure for the
coordinates x.
Remark 2.8 While T g is a manifold, this is not the case of M g for g ≥ 2, which is
an orbifold: the quotient by the modular group introduces singularities [27].
Remark 2.9 (Moduli Space and Fundamental Domain) Given a group acting on a
space, a fundamental domain for a group is a subspace such that the full space is
generated by acting with the group on the fundamental domain. Hence, one can
view the moduli space M g as a fundamental domain (sometimes denoted by F g )
for the group g and the space T g .
In the conformal gauge (2.16), the metric g ab can be parametrized by
g ab = ˆ
g
(f,φ)
ab (t) := e
2f ∗ φ f
∗
ˆ
g ab (t) = f
∗
e
2φ
ˆ
g ab (t)
,
(2.56)
where φ := ω and t denotes the dependence in the moduli parameters. To avoid
surcharging the notations, we will continue to write g when there is no ambiguity.
In coordinates, this is equivalent to
g ab (σ ) = ˆ
g
(f,φ)
ab (σ ; t) := e
2φ(σ )
ˆ
g
ab (σ ; t),
ˆ
g
ab (σ ; t) =
∂σ c
∂σ a
∂σ d
∂σ b ˆ
g cd (σ
; t).
(2.57)
Remark 2.10 Strictly speaking, the matter fields also transform and one should
write = (f ) := f ∗ ˆ
and include them in the change of integration measures
of the following sections. But, this does not bring any particular benefits since these
changes are trivial because the matter is decoupled from the metric.
Remark 2.11 Although the metric cannot be completely gauge fixed, having just
a finite-dimensional integral is much simpler than a functional integral. In higher
dimensions, the gauge fixing does not reduce that much the degrees of freedom and
a functional integral over ˆ
g remains (in similarity with Yang–Mills theories).
2 Worldsheet Path Integral: Vacuum Amplitudes
for any h ∈ G. Given the Lie algebra g of the group, a general element of the algebra
is a linear combination of the generators T i with coefficients α i
α = α
i T i .
(2.54)
Group elements can be parametrized in terms of α through the exponential map.
Moreover, since a Lie group is a manifold, it is locally isomorphic to R n : this
motivates the use of a flat metric for the Lie algebra, such that
G =
dα :=
i
dα
i .
(2.55)
Finally, it is possible to perform a change of coordinates from the Lie parameters
to coordinates x on the group: the resulting Jacobian is the Haar measure for the
coordinates x.
Remark 2.8 While T g is a manifold, this is not the case of M g for g ≥ 2, which is
an orbifold: the quotient by the modular group introduces singularities [27].
Remark 2.9 (Moduli Space and Fundamental Domain) Given a group acting on a
space, a fundamental domain for a group is a subspace such that the full space is
generated by acting with the group on the fundamental domain. Hence, one can
view the moduli space M g as a fundamental domain (sometimes denoted by F g )
for the group g and the space T g .
In the conformal gauge (2.16), the metric g ab can be parametrized by
g ab = ˆ
g
(f,φ)
ab (t) := e
2f ∗ φ f
∗
ˆ
g ab (t) = f
∗
e
2φ
ˆ
g ab (t)
,
(2.56)
where φ := ω and t denotes the dependence in the moduli parameters. To avoid
surcharging the notations, we will continue to write g when there is no ambiguity.
In coordinates, this is equivalent to
g ab (σ ) = ˆ
g
(f,φ)
ab (σ ; t) := e
2φ(σ )
ˆ
g
ab (σ ; t),
ˆ
g
ab (σ ; t) =
∂σ c
∂σ a
∂σ d
∂σ b ˆ
g cd (σ
; t).
(2.57)
Remark 2.10 Strictly speaking, the matter fields also transform and one should
write = (f ) := f ∗ ˆ
and include them in the change of integration measures
of the following sections. But, this does not bring any particular benefits since these
changes are trivial because the matter is decoupled from the metric.
Remark 2.11 Although the metric cannot be completely gauge fixed, having just
a finite-dimensional integral is much simpler than a functional integral. In higher
dimensions, the gauge fixing does not reduce that much the degrees of freedom and
a functional integral over ˆ
g remains (in similarity with Yang–Mills theories).