2.1 Worldsheet Action and Symmetries
31
In order to describe a proper string theory, the worldsheet metric g ab (σ ) should
not be dynamical. This means that the worldsheet has no intrinsic dynamics and
that no supplementary degrees of freedom are introduced when parametrizing
the worldsheet with a metric. A solution to remove these degrees of freedom is
to introduce gauge symmetries with as many gauge parameters as there are of
degrees of freedom. The simplest symmetry is invariance under diffeomorphisms:
indeed, the worldsheet theory is effectively a QFT coupled to gravity and it makes
sense to require this invariance. Physically, this corresponds to the fact that the
worldsheet spatial coordinate σ used along the string and worldsheet time are
arbitrary. However, diffeomorphisms alone are not sufficient to completely fix the
metric. Another natural candidate is Weyl invariance (local rescalings of the metric).
A diffeomorphism f ∈ Diff(( g ) acts on the fields as
σ
a
= f
a (σ
b ),
g
(σ
) = f
∗ g(σ ),
,
(σ
) = f
∗ (σ ),
(2.5)
where the star denotes the pullback by f : this corresponds simply to the standard
coordinate transformation where each tensor index of the field receives a factor
∂σ a /∂σ b . In particular, the metric and scalar fields transform explicitly as
g
ab (σ
) =
∂σ c
∂σ a
∂σ d
∂σ b g cd (σ ),
X
μ (σ
) = X
μ (σ ).
(2.6)
The index μ is inert since it is a target spacetime index: from the worldsheet point
of view, it just labels a collection of worldsheet scalar fields. Infinitesimal variations
are generated by vector fields on g :
δ ξ σ
a
= ξ
a ,
δ ξ = L ξ
δ ξ g ab = L ξ g ab ,
(2.7)
where L ξ is the Lie derivative 2 with respect to the vector field ξ ∈ diff(( g ) T T g .
The Lie derivative of the metric is
L ξ g ab = ξ
c ∂ c g ab + g ac ∂ b ξ
c
+ g bc ∂ a ξ
c
= ∇ a ξ b + ∇ b ξ a .
(2.8)
The Lie algebra generates only transformations in the connected component
Diff 0 (( g ) of the diffeomorphism group which contains the identity.
Transformations not contained in Diff 0 (( g ) are called large diffeomorphisms:
this includes reflections, for example. The quotient of the two groups is called the
modular group g (also mapping class group or MCG):
g := π 0
Diff(( g )
=
Diff(( g )
Diff 0 (( g )
.
(2.9)
2 For our purpose here, it is sufficient to accept the definition of the Lie derivative as corresponding
to the infinitesimal variation.
31
In order to describe a proper string theory, the worldsheet metric g ab (σ ) should
not be dynamical. This means that the worldsheet has no intrinsic dynamics and
that no supplementary degrees of freedom are introduced when parametrizing
the worldsheet with a metric. A solution to remove these degrees of freedom is
to introduce gauge symmetries with as many gauge parameters as there are of
degrees of freedom. The simplest symmetry is invariance under diffeomorphisms:
indeed, the worldsheet theory is effectively a QFT coupled to gravity and it makes
sense to require this invariance. Physically, this corresponds to the fact that the
worldsheet spatial coordinate σ used along the string and worldsheet time are
arbitrary. However, diffeomorphisms alone are not sufficient to completely fix the
metric. Another natural candidate is Weyl invariance (local rescalings of the metric).
A diffeomorphism f ∈ Diff(( g ) acts on the fields as
σ
a
= f
a (σ
b ),
g
(σ
) = f
∗ g(σ ),
,
(σ
) = f
∗ (σ ),
(2.5)
where the star denotes the pullback by f : this corresponds simply to the standard
coordinate transformation where each tensor index of the field receives a factor
∂σ a /∂σ b . In particular, the metric and scalar fields transform explicitly as
g
ab (σ
) =
∂σ c
∂σ a
∂σ d
∂σ b g cd (σ ),
X
μ (σ
) = X
μ (σ ).
(2.6)
The index μ is inert since it is a target spacetime index: from the worldsheet point
of view, it just labels a collection of worldsheet scalar fields. Infinitesimal variations
are generated by vector fields on g :
δ ξ σ
a
= ξ
a ,
δ ξ = L ξ
δ ξ g ab = L ξ g ab ,
(2.7)
where L ξ is the Lie derivative 2 with respect to the vector field ξ ∈ diff(( g ) T T g .
The Lie derivative of the metric is
L ξ g ab = ξ
c ∂ c g ab + g ac ∂ b ξ
c
+ g bc ∂ a ξ
c
= ∇ a ξ b + ∇ b ξ a .
(2.8)
The Lie algebra generates only transformations in the connected component
Diff 0 (( g ) of the diffeomorphism group which contains the identity.
Transformations not contained in Diff 0 (( g ) are called large diffeomorphisms:
this includes reflections, for example. The quotient of the two groups is called the
modular group g (also mapping class group or MCG):
g := π 0
Diff(( g )
=
Diff(( g )
Diff 0 (( g )
.
(2.9)
2 For our purpose here, it is sufficient to accept the definition of the Lie derivative as corresponding
to the infinitesimal variation.