JHEP07(2017)027
where we split the three-dimensional radial direction into |~x|
2
= r
2
+ (ξ
1
)
2
. The sum
in (2.3) does not converge but can be regularized. After Poisson resummation we obtain
the Ooguri-Vafa metric [18] described by
h(r, ξ
1
) =
˜
R
2
2πR
1
log(µ/r) +
X
n6=0
e
i n ξ
1
/R
1
K
0
|n|r
R
1
, (2.4)
with µ a constant that controls the regulator and absorbs also all other possible constants,
for instance the first term in (2.3). K
0
is the zeroth-order modified Bessel function of the
second kind, whose series expansion for large r reads
K
0
|n|r
R
1
= e
−
|n|r
R
1
∞
X
k=0
(−1)
k
Γ(
1
2
+ k)
2
√
πk!
R
1
2|n|r
k+
1
2
. (2.5)
Hence, the leading semi-flat term in (2.4) (i.e. the logarithm) is a good approximation
of the exact metric far away from the degeneration point up to exponentially suppressed
terms. In fact, the semi-flat approximation of a smooth K3 — repaired with the Ooguri-
Vafa metric at the 24 I
1
points — gives a metric that is a good approximation of the exact
Calabi-Yau metric [19]. The expression (2.4) can also be derived (in a simple way) field-
theoretically [20, 21] and (in a complicated way) by solving explicitly the Riemann-Hilbert
problem with wall-crossing technology [22].
The semi-flat approximation of the above background is a flat two-torus fibration pa-
rameterized by the coordinates (ξ
1
, ξ
2
) over a two-dimensional base R
2
. For the latter, we
introduce polar coordinates (r, θ), and the one-form ω mentioned above can be brought into
the form ω
sf
= f dξ
1
with df = ?
2
dh. Note that in later calculations we take a gauge where
ω
sf
=
˜
R
2
2πR
1
θdξ
1
. (2.6)
Furthermore, we observe that after encircling the defect in the base as θ → θ + 2π, the
shift ω
sf
→ ω
sf
+
˜
R
2
R
1
dξ
1
should be compensated by the shift ξ
2
→ ξ
2
−
˜
R
2
R
1
ξ
1
which, as
expected, corresponds to the action of a Dehn twist on the torus cycles.
4
The corrections
to the semi-flat term in (2.4) explicitly break one of the U(1) isometries of the torus fiber.
This affects also the one-form (2.6), which is corrected (up to gauge transformations) by
modified Bessel functions of the second kind as
ω = ω
sf
−
˜
R
2
πR
1
r
X
k>0
K
1
k r
R
1
sin
k ξ
1
R
1
dθ . (2.7)
The above analysis can be easily extended to a I
n
degeneration. The solution is given
by coalescing n Taub-NUT centers, and the Ooguri-Vafa corrections (2.4), which completely
smooth out the semi-flat metric for n = 1, now replace the semi-flat singularity with an
A
n−1
singularity, as expected.
4
We neglect a constant shift which is not captured by the action on the homology [22].
– 4 –
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