Lectures on String Theory

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i.e the conformal factor φ and the action of Laplacian on it are
Therefore,
φ = ln 2 − 2 ln(1 + |z| 2 ) =⇒ − △φ =
∫
1
4π C
dxdy √ hR = 1
4π
∫ ∞
0
8
(1 + |z| 2 ) 2 .
8
2πrdr
(1 + r 2 ) = 2 , 2
which is indeed the Euler characteristic for sphere, a compact orientable manifold
with genus g = 0. It is also known that on a torus, a manifold of genus g = 1, there
exists the globally defined flat metric. Therefore, the Euler characteristic of torus
is χ = 0. In fact, the Euler characteristic χ(g) for a Riemann surface of arbitrary
genus g can be found by using the recurrent formula
χ(g 1 + g 2 ) = χ(g 1 ) + χ(g 2 ) − χ(0) = χ(g 1 ) + χ(g 2 ) − 2
and the fact that χ(1) = 0. This again leads to the formula χ(g) = 2 − 2g.
5.3 Moduli space
The moduli space of all the metrics is the same as the moduli space of Riemann
surfaces and it is defined as the space of all metrics devided by diffeomorphisms and
Weyl rescalings
M g =
all metrics
diffeomorphisms × Weyl rescalings .
The moduli space is finite-dimensional and it is parametrized by a finite number of
complex parameters τ i called moduli. The dimension of the moduli space is another
topological invariant and it depends on the genus g only.
Complex geometry
Since we have a system of well-defined complex coordinates on a Riemann surface
we can consider general tensors
V z...z¯z...¯z z...z¯z...¯z(z, ¯z) ,
in particular, V z ∂ z and V ¯z ∂¯z are vector fields and V z dz and V¯z d¯z are one-forms. All
these tensors are one component objects. The metric h z¯z and h z¯z can be used to
convert all ¯z indices into z-indices. Tensors with one type of indices (for example,
z indices) are called holomorphic. Holomorphic tensors which depend on z variable