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

δχ = −2
δχ = −1
Figure 3: World–sheet topology change due to emission and reabsorption of open and closed strings
g s
Figure 4: The basic three–string interaction for closed strings, and its analogue for open strings. Its strength,
gs, along with the string tension, determines Newton’s gravitational constant GN.
2.3.1 The Stress Tensor
Let us also note that we can define a two–dimensional energy–momentum tensor:
Notice that
T ab (τ, σ) ≡ − 2π
√ −γ
δS
δγab
= − 1
α ′
1
g2 s
∂ a Xµ∂ b X µ − 1
2 γab γcd∂ c Xµ∂ d X µ
. (27)
T a a ≡ γabT ab = 0 . (28)
This is a consequence of Weyl symmetry. Reparametrization invariance, δγS ′ = 0, translates here into (see
discussion after equation (24))
T ab = 0 . (29)
These are the classical properties of the theory we have uncovered so far. Later on, we shall attempt to
ensure that they are true in the quantum theory also, with interesting results.
2.4 Gauge Fixing
Now recall that we have three local or “gauge” symmetries of the action:
2d reparametrizations : σ, τ → ˜σ(σ, τ), ˜τ(σ, τ)
Weyl : γab → exp(2ω(σ, τ))γab . (30)
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