34
2 Worldsheet Path Integral: Vacuum Amplitudes
The latter two conditions are summarized by
S m [f
∗ g, f
∗ ] = S m [g, ,],
S m [e
2ω g, ,] = S m [g, ,].
(2.19)
The invariance under diffeomorphisms is straightforward to enforce by using only
covariant objects. Since the scalar fields represent embedding of the string in
spacetime, the non-linear sigma model condition means that spacetime is identified
with the target space of the sigma model, of which D dimensions are non-compact,
and the spacetime metric appears in the matter action as in (2.3). The isometries
of the target manifold metric become global symmetries of S m : while they are not
needed in this chapter, they will have their importances in other chapters. Finally,
to make the action consistent with the topology of the worldsheet, the fields must
satisfy appropriate boundary conditions. For example, the scalar fields X μ must be
periodic for the closed string:
X
μ (τ, σ ) ∼ X
μ (τ, σ + 2π).
(2.20)
Remark 2.3 (2d Gravity) The setup in two-dimensional gravity is exactly similar,
except that the system is, in general, not invariant under Weyl transformations. As a
consequence, one component of the metric (usually taken to be the Liouville mode)
remains unconstrained: in the conformal gauge, (2.16) only ˆ
g is fixed.
The symmetries (2.19) of the action have an important consequence: they imply
that the matter action is conformally invariant on flat space g ab = δ ab . A twodimensional conformal field theory (CFT) is characterized by a central charge c m :
roughly, it is a measure of the quantum degrees of freedom. The central charge is
additive for decoupled sectors. In particular, the scalar fields X μ contribute as D,
and it is useful to define the perpendicular CFT with central charge c ⊥ as the matter
which does not describe the non-compact dimensions:
c m = D + c ⊥ .
(2.21)
This will be discussed in length in Part I. For this chapter and most of the book, it
is sufficient to know that the matter is a CFT of central charge c m and includes D
scalar fields X μ :
matter CFT parameters: D, c m .
(2.22)
The energy–momentum is defined by
T m,ab := −
4π
√ g
δS m
δg ab .
(2.23)
2 Worldsheet Path Integral: Vacuum Amplitudes
The latter two conditions are summarized by
S m [f
∗ g, f
∗ ] = S m [g, ,],
S m [e
2ω g, ,] = S m [g, ,].
(2.19)
The invariance under diffeomorphisms is straightforward to enforce by using only
covariant objects. Since the scalar fields represent embedding of the string in
spacetime, the non-linear sigma model condition means that spacetime is identified
with the target space of the sigma model, of which D dimensions are non-compact,
and the spacetime metric appears in the matter action as in (2.3). The isometries
of the target manifold metric become global symmetries of S m : while they are not
needed in this chapter, they will have their importances in other chapters. Finally,
to make the action consistent with the topology of the worldsheet, the fields must
satisfy appropriate boundary conditions. For example, the scalar fields X μ must be
periodic for the closed string:
X
μ (τ, σ ) ∼ X
μ (τ, σ + 2π).
(2.20)
Remark 2.3 (2d Gravity) The setup in two-dimensional gravity is exactly similar,
except that the system is, in general, not invariant under Weyl transformations. As a
consequence, one component of the metric (usually taken to be the Liouville mode)
remains unconstrained: in the conformal gauge, (2.16) only ˆ
g is fixed.
The symmetries (2.19) of the action have an important consequence: they imply
that the matter action is conformally invariant on flat space g ab = δ ab . A twodimensional conformal field theory (CFT) is characterized by a central charge c m :
roughly, it is a measure of the quantum degrees of freedom. The central charge is
additive for decoupled sectors. In particular, the scalar fields X μ contribute as D,
and it is useful to define the perpendicular CFT with central charge c ⊥ as the matter
which does not describe the non-compact dimensions:
c m = D + c ⊥ .
(2.21)
This will be discussed in length in Part I. For this chapter and most of the book, it
is sufficient to know that the matter is a CFT of central charge c m and includes D
scalar fields X μ :
matter CFT parameters: D, c m .
(2.22)
The energy–momentum is defined by
T m,ab := −
4π
√ g
δS m
δg ab .
(2.23)