Lectures on String Theory

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The last term here reflects the dependence of the metric on moduli which cannot be
compensated by diffeomorphisms and Weyl rescalings. We can split this variation
into trace and traceless parts
(
δh trace
αβ = Λ + ∇ γ V γ + 1 ∑ 2 hγδ i
δh traceless
αβ
δτ i
∂
∂τ i
h γδ
)
h αβ
= ∇ α V β + ∇ β V α − h αβ ∇ γ V
} {{ } γ + ∑ i
Operator P
( ∂
δτ i h αβ − 1 ∂τ i 2 h αβh γδ ∂ )
h γδ .
∂τ i
We see that we can always shift the Weyl rescaling parameter Λ as
Λ → Λ − ∇ γ V γ − 1 2 hγδ ∑ i
δτ i
∂
∂τ i
h γδ
so that the trace part of the variation will transform as
δh trace
αβ = Λh αβ .
In the complex coordinates we have h zz = h¯z¯z = 0 and therefore rewriting the
variation formulae in these coordinates we find
δh z¯z = Λh z¯z ,
δh zz = 2∇ (1)
z V
} {{ } z + ∑
Operator P
i
δτ i µ i zz ,
where µ i zz = ∂ τi h zz = h z¯z µ i¯z
z . We see, in particular, that an infinitesimal change of
h z¯z can always be written as a Weyl rescaling. Also the covariant derivative ∇ (1)
z
introduced above should be naturally identified with the operator P . Finally, we
note that decomposition into sum of two terms
δh zz = 2∇ (1)
z V z + ∑ i
δτ i µ i zz
is not orthogonal w.r.t. to the scalar product we introduced above. Denote by φ i zz a
basis of the orthogonal complement of ∇ (1)
z :
(φ i zz|∇ (1)
z V z ) = −(∇ z (2)φ i zz|V z ) for any V z ∈ V (1) .
The last equation is equivalent to
∇ z (2)φ i zz = 0 =⇒ ¯∂φ
i
zz = 0 .
Thus, the kernel (or, in other words, the space of zero modes) of the operator P † =
∇ z (2)
consists of global analytic tensors of the second rank. Such tensors are of special
importance and they are called quadratic differentials. Thus, the dimension of the