Introduction to String Theory and D–Branes - School of Natural ...

Therefore we have
from which we can derive
and so substituting into S, we recover the Nambu–Goto action, So.
Let us note some of the symmetries of the action:
• Spacetime Lorentz/Poincaré:
hab − 1
2 γabγ cd hcd = 0 , (14)
γ ab hab = 2(−h) 1/2 (−γ) −1/2 , (15)
X µ → X ′µ = Λ µ νX ν + A µ ,
where Λ is an SO(1, D − 1) Lorentz matrix and A µ is an arbitrary constant D–vector. Just as before
this is a trivial global symmetry of S (and also So), following from the fact that we wrote them in
covariant form.
• Worldsheet Reparametrisations:
δX µ = ζ a ∂aX µ
δγ ab = ζ c ∂cγ ab − ∂cζ a γ cb − ∂cζ b γ ac , (16)
for two parameters ζ a (τ, σ). This is a non–trivial local or “gauge” symmetry of S. This is a large extra
symmetry on the world–sheet of which we will make great use.
• Weyl invariance:
γab → γ ′ ab = e2ω γab , (17)
specified by a function ω(τ, σ). This ability to do local rescalings of the metric results from the fact
that we did not have to choose an overall scale when we chose γ ab to rewrite So in terms of S. This
can be seen especially if we rewrite the relation (15) as (−h) −1/2 hab = (−γ) −1/2 γab.
We note here for future use that there are just as many parameters needed to specify the local symmetries
(three) as there are independent components of the world-sheet metric. This is very useful, as we shall see.
2.2 String Equations of Motion
We can get equations of motion for the string by varying our action (11) with respect to the X µ :
1
δS =
2πα ′
d 2
σ ∂a (−γ) 1/2 γ ab
∂bXµ δX µ
− 1
2πα ′
dτ (−γ) 1/2 ∂σXµδX µ
σ=π
, (18)
σ=0
which results in the equations of motion:
(−γ) 1/2 γ ab ∂bX µ
≡ (−γ) 1/2 ∇ 2 X µ = 0 , (19)
with either:
or:
∂a
X ′µ (τ, 0) = 0
X ′µ
(τ, π) = 0
X ′µ (τ, 0) = X ′µ (τ, π)
X µ (τ, 0) = X µ
(τ, π)
γab(τ, 0) = γab(τ, π)
Open String
(Neumann b.c.’s)
Closed String
(periodic b.c.’s)
We shall study the equation of motion (19) and the accompanying boundary conditions a lot later. We are
going to look at the standard Neumann boundary conditions mostly, and then consider the case of Dirichlet
conditions later, when we uncover D–branes, using T–duality. Notice that we have taken the liberty of
introducing closed strings by imposing periodicity.
8
(20)
(21)