String Theory and M-Theory

394 String geometry
Since the prepotential is defined only up to an overall scaling, strictly speaking
it is not a function but rather a section of a line bundle over the moduli
space.
The prepotential determines the metric on moduli space. Using the general
rule for closed three-forms α and β
α ∧ β = −
M
AI
α β −
AI
β α , (9.116)
BI
I
the Kähler potential (9.104) can be rewritten in the form
e −K2,1
I=0
BI
h
= −i
2,1
X I FI
¯ − X I
FI , (9.117)
as you are asked to verify in a homework problem. As a result, the Kähler
potential is completely determined by the prepotential F , which is a holomorphic
homogeneous function of degree two. This type of geometry is
called special geometry.
An important consequence of the product structure (9.98) of the moduli
space is that the complex-structure prepotential F is exact in α ′ . Indeed,
the α ′ expansion is an expansion in terms of the Calabi–Yau volume V ,
which belongs to M 1,1 (M), and it is independent of position in M 2,1 (M),
that is, the complex structure. 20 When combined with mirror symmetry,
this important fact provides insight into an infinite series of stringy α ′ corrections
involving the Kähler-structure moduli using a classical geometric
computation involving the complex-structure moduli space only.
The Kähler transformations
The holomorphic three-form Ω is only determined up to a function f, which
can depend on the moduli space coordinates X I but not on the Calabi–Yau
coordinates, that is, the transformation
Ω → e f(X) Ω (9.118)
should not lead to new physics. This transformation does not change the
Kähler metric, since under Eq. (9.118)
K 2,1 → K 2,1 − f(X) − ¯ f(X), (9.119)
which is a Kähler transformation that leaves the Kähler metric invariant.
20 Since V and α ′ are the only scales in the problem, the only dimensionless quantity containing
α ′ is (α ′ ) 3 /V . So if one knows the full V dependence, one also knows the full α ′ dependence.