Lectures on String Theory

– 82 –
The last formula can be understood in the sense of calculus of exterior differential
forms ∂f = ∂f and ¯∂f = ∂f
∂
. In particular, the de Rahm operator d = dx + dy ∂ ∂z ∂¯z ∂x ∂y
can be written as
d = ∂ + ¯∂ ≡ dz ∂ ∂z + d¯z ∂ ∂¯z .
Also one has ∂∂ = ¯∂ ¯∂ = 0. Therefore, we can rewrite our formula as
√
hR = 2id(∂φ) .
This shows that √ hR is locally a total derivative and therefore, we see again that
eq.(5.2) cannot change under smooth variations of the metric, i.e. it is a topological
invariant.
On a compact Riemann surface M we consider an abelian differential Ω, which is
a meromorphic differential form. It means that in a given coordinate patch (U α , z α )
it can be written in the form Ω = f α dz α , where f α is a meromorphic function 15 .
We can also suppose that the coordinate patches are chosen in such a way that
every U α contains at most one pole or one zero of Ω. In a patch U α the metric is
ds 2 = 2e φ α
|dz α | 2 . Thus, on the intersections of the patches U α ∩ U β we have
e φ α
e φ β
= ∣ ∣∣ f α
f β
∣ ∣∣
2
.
Thus, there exists a globally defined function
ϕ = eφ α
|f α | 2 on U α for all α .
This function is smooth except for singularities at zeros and poles of Ω. Since log |f α | 2
is harmonic outside zeros and poles of Ω we have
√
hR = 2id(∂ log ϕ) .
Let M ɛ = M − ∪D k,ɛ , where D k,ɛ are small disks around the singularities of Ω.
Then, by Stokes’s theorem we have
∫
M
√
∫
hR = 2i lim d(∂ log ϕ) = −2i ∑
ɛ→0
M ɛ k
∫
lim ∂ log ϕ .
ɛ→0
∂D k,ɛ
To evaluate the integrals over the circles we note that at a zero or pole of Ω the
function ϕ is of the form ϕ = ψ/|z| 2m with a smooth function ψ without zeros and
m is the order of zero or pole (m < 0 in the latter case). Therefore,
∫
lim ∂ log ϕ = lim ∂ log |z|
ɛ→0
∂D ɛ→0 k,ɛ
∫|z|=ɛ
−2m dz
= −m lim
ɛ→0
∫|z|=ɛ z = −2πim .
15 A function f(z) is called meromorphic if it does not have any other singularities except poles.