String Theory and M-Theory

390 String geometry
where Mr is any complex r-dimensional submanifold of the Calabi–Yau
three-fold. The subset of metric deformations that lead to a Kähler form
satisfying Eq. (9.94) is called the Kähler cone. This space is a cone since
if J satisfies (9.94), so does rJ for any positive number r, as illustrated in
Fig. 9.7.
Fig. 9.7. The deformations of the Kähler form that satisfy Eq. (9.94) build the
Kähler cone.
The five ten-dimensional superstring theories each contain an NS–NS twoform
B. After compactification on a Calabi–Yau three-fold, the internal
(1, 1)-form B a ¯ b has h 1,1 zero modes, so that this many additional massless
scalar fields arise in four dimensions. The real closed two-form B combines
with the Kähler form J to give the complexified Kähler form
J = B + iJ. (9.95)
The variations of this form give rise to h 1,1 massless complex scalar fields
in four dimensions. Thus, while the Kähler form is real from a geometric
viewpoint, it is effectively complex in the string theory setting, generalizing
the complexification of the ρ parameter of the torus. This procedure is called
the complexification of the Kähler cone. For M-theory compactifications,
discussed later, there is no two-form B, and so the Kähler form, as well as
the corresponding moduli space, is not complexified.
The purely holomorphic and antiholomorphic metric components gab and
g ā ¯ b are zero. However, one can consider varying to nonzero values, thereby
changing the complex structure. With each such variation one can associate
the complex (2, 1)-form
Ωabcg c ¯ d δg ¯dē dz a ∧ dz b ∧ d¯z ē . (9.96)
This is harmonic if (9.90) is satisfied. The precise relation to complexstructure
variations is explained in Section 9.6.