String Theory and M-Theory

30 The bosonic string
p = 0, this reproduces the result in Eq. (2.5) if one makes the identifications
T0 = m and h00 = −m 2 e 2 . ✷
2.3 String sigma-model action: the classical theory
In this section we discuss the symmetries of the string sigma-model action in
Eq. (2.14). This is helpful for writing the string action in a gauge in which
quantization is particularly simple.
Symmetries
The string sigma-model action for the bosonic string in Minkowski spacetime
has a number of symmetries:
• Poincaré transformations. These are global symmetries under which the
world-sheet fields transform as
δX µ = a µ νX ν + b µ
and δh αβ = 0. (2.19)
Here the constants a µ ν (with aµν = −aνµ) describe infinitesimal Lorentz
transformations and b µ describe space-time translations.
• Reparametrizations. The string world sheet is parametrized by two coordinates
τ and σ, but a change in the parametrization does not change the
action. Indeed, the transformations
σ α → f α (σ) = σ ′α
and hαβ(σ) =
∂f γ
∂σα ∂f δ
∂σβ hγδ(σ ′ ) (2.20)
leave the action invariant. These local symmetries are also called diffeomorphisms.
Strictly speaking, this implies that the transformations and
their inverses are infinitely differentiable.
• Weyl transformations. The action is invariant under the rescaling
hαβ → e φ(σ,τ) hαβ and δX µ = 0, (2.21)
since √ −h → e φ√ −h and h αβ → e −φ h αβ give cancelling factors. This
local symmetry is the reason that the energy–momentum tensor is traceless.
Poincaré transformations are global symmetries, whereas reparametrizations
and Weyl transformations are local symmetries. The local symmetries
can be used to choose a gauge, such as the static gauge discussed earlier, or
else one in which some of the components of the world-sheet metric hαβ are
of a particular form.