String Theory and M-Theory

368 String geometry
Hodge numbers of the Eguchi–Hanson space are h 0,0 = h 1,1 = h 2,2 = 1.
Moreover, the (1, 1)-form is anti-self-dual and is given by
J = 1
1
rdr ∧ (dψ + cos θdφ) −
2 4 r2 sin θdθ ∧ dφ, (9.25)
as you are asked to verify in a homework problem. In terms of the complex
coordinates
i
z1 = r cos (θ/2) exp (ψ + φ)
2
and
i
z2 = r sin (θ/2) exp (ψ − φ) ,
2
(9.26)
the metric is Kähler with Kähler potential
K = log
r2 exp(r4 + a4 ) 1/2
a2 + (r4 + a4 ) 1/2
. (9.27)
Hodge numbers of K3
The cohomology of K3 can be computed by combining the contributions
of the T 4 and the Eguchi–Hanson spaces. The result obtained in this way
remains correct after the metric has been smoothed out.
The Eguchi–Hanson spaces contribute a total of 16 generators to H 1,1 ,
one for each of the 16 spaces used to blow up the singularities. Moreover, on
the T 4 the following four representatives of H 1,1 cohomology classes survive
the 2 identifications:
dz 1 ∧ d¯z 1 , dz 2 ∧ d¯z 2 , dz 1 ∧ d¯z 2 , dz 2 ∧ d¯z 1 . (9.28)
This gives in total h 1,1 = 20. In addition, there is one H 2,0 class represented
by dz 1 ∧ dz 2 and one H 0,2 class represented by d¯z 1 ∧ d¯z 2 . As a result, the
Hodge numbers of K3 are given by the Hodge diamond
1
0 0
1 20 1
0 0
1
(9.29)
Thus, the nonzero Betti numbers of K3 are b0 = b4 = 1, b2 = 22, and the
Euler characteristic is χ = 24. The 22 nontrivial harmonic two-forms consist
of three self-dual forms (b + 2 = 3) and 19 anti-self-dual forms (b−2 = 19).