Lectures on String Theory

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2. Varying w.r.t. γ τσ leads to determination of X ′− :
X ′− = − 1
P −
P i X ′i = 2π
p + P iX ′i . (3.61)
If we integrate the last equation over σ we obtain
X − (2π) − X − (0) = 2π
p + ∫ 2π
0
dσP i X ′i . (3.62)
The closed string periodicity condition requires the fulfillment of the following
constraint
V =
∫ 2π
0
dσP i X ′i = 0 . (3.63)
This is the only constraint which remains unsolved and it is known as the level
matching condition. We will impose it on physical states of the theory.
3. Now we can determine the world-sheet metric γ αβ . Equation of motion for P +
is
0 = δL =
δP Ẋ+ −
P − γτσ
+
+ T γττ γ ττ X′+ ,
which with our gauge choice gives
γ ττ = −1 .
4. Equation of motion for Ẋ− gives
which gives
0 = d δL δL
−
dt δẊ− δX = − d p +
− dt 2π − δL δL
=⇒
δX− δX = 0 , −
( ) γ
τσ
∂ σ
γ P ττ − = 0 =⇒ ∂ σ γ τσ = 0 .
For the closed string case this implies that γ τσ = γ τσ (τ) is an arbitrary function
of τ. The presence of this function signals a residual symmetry. Indeed, on the
solutions of the level-matching constraint V = 0 the ratio γτσ
can be shifted by
γ ττ
an arbitrary function f(τ) of τ without affecting the Lagrangian.
5. Varying w.r.t P − we find an evolution equation for X − :
0 = δL
δP −
= Ẋ− − P +
T γ ττ = 0 =⇒ Ẋ− = π
T p + (P iP i + T 2 X ′ iX ′i ) .