46
2 Worldsheet Path Integral: Vacuum Amplitudes
2.3.2 Reparametrizations and Analysis of P 1
The properties of the operator P 1 are responsible for both problems preventing a
direct factorization of the measure; for this reason, it is useful to study it in more
detail.
The operator P 1 is an object which takes a vector v to a symmetric traceless
2-tensor T , see (2.65a). Conversely, its adjoint P
†
1 can be defined from the scalar
product (2.34c)
(T , P 1 v) g = (P
†
1 T , v) g ,
(2.70)
and takes symmetric traceless tensors to vectors. In components, one finds
(P
†
1 T ) a = −2∇
b T ab .
(2.71)
The Riemann–Roch theorem relates the dimension of the kernels of both
operators [26]:
dim ker P
†
1 − dim ker P 1 = −3χ g = 6g − 6.
(2.72)
Teichmüller Deformations
We first need to characterize Teichmüller deformations, the variations of moduli
parameters which lead to transformations of the metric independent from diffeomorphisms and Weyl rescalings. This means that the different variations must be
orthogonal for the inner product (2.34).
First, the deformations must be traceless, otherwise they can be compensated by
a Weyl transformation. The traceless metric variations δg which cannot be generated
by a vector field ξ are perpendicular to P 1 ξ (otherwise, the former would be a linear
combination of the latter):
(δg, P 1 ξ) g = 0 ⇒ (P
†
1 δg, ξ ) g = 0.
(2.73)
Since ξ is arbitrary, this means that the first argument vanishes
P
†
1 δg = 0.
(2.74)
Metric variations induced by a change in the moduli t i are in the kernel of P
†
1
δg ∈ ker P
†
1 .
(2.75)
Elements of ker P
†
1 are called quadratic differentials and a basis (not necessarily
orthonormal) of ker P
†
1 is denoted as
ker P
†
1 = Span{φ i },
i = 1, . . . , dim ker P
†
1
(2.76)
2 Worldsheet Path Integral: Vacuum Amplitudes
2.3.2 Reparametrizations and Analysis of P 1
The properties of the operator P 1 are responsible for both problems preventing a
direct factorization of the measure; for this reason, it is useful to study it in more
detail.
The operator P 1 is an object which takes a vector v to a symmetric traceless
2-tensor T , see (2.65a). Conversely, its adjoint P
†
1 can be defined from the scalar
product (2.34c)
(T , P 1 v) g = (P
†
1 T , v) g ,
(2.70)
and takes symmetric traceless tensors to vectors. In components, one finds
(P
†
1 T ) a = −2∇
b T ab .
(2.71)
The Riemann–Roch theorem relates the dimension of the kernels of both
operators [26]:
dim ker P
†
1 − dim ker P 1 = −3χ g = 6g − 6.
(2.72)
Teichmüller Deformations
We first need to characterize Teichmüller deformations, the variations of moduli
parameters which lead to transformations of the metric independent from diffeomorphisms and Weyl rescalings. This means that the different variations must be
orthogonal for the inner product (2.34).
First, the deformations must be traceless, otherwise they can be compensated by
a Weyl transformation. The traceless metric variations δg which cannot be generated
by a vector field ξ are perpendicular to P 1 ξ (otherwise, the former would be a linear
combination of the latter):
(δg, P 1 ξ) g = 0 ⇒ (P
†
1 δg, ξ ) g = 0.
(2.73)
Since ξ is arbitrary, this means that the first argument vanishes
P
†
1 δg = 0.
(2.74)
Metric variations induced by a change in the moduli t i are in the kernel of P
†
1
δg ∈ ker P
†
1 .
(2.75)
Elements of ker P
†
1 are called quadratic differentials and a basis (not necessarily
orthonormal) of ker P
†
1 is denoted as
ker P
†
1 = Span{φ i },
i = 1, . . . , dim ker P
†
1
(2.76)