Let
X
\mathcal {X}
be a translation surface of genus
g
>
1
g>1
with
2
g
−
2
2g-2
conical points of angle
4
π
4\pi
and let
γ
\gamma
,
γ
′
\gamma ’
be two homologous saddle connections of length
s
s
joining two conical points of
X
\mathcal {X}
and bounding two surfaces
S
+
S^+
and
S
−
S^-
with boundaries
∂
S
+
=
γ
−
γ
′
\partial S^+=\gamma -\gamma ’
and
∂
S
−
=
γ
′
−
γ
\partial S^-=\gamma ’-\gamma
. Gluing the opposite sides of the boundary of each surface
S
+
S^+
,
S
−
S^-
one gets two (closed) translation surfaces
X
+
\mathcal {X}^+
,
X
−
\mathcal {X}^-
of genera
g
+
g^+
,
g
−
g^-
;
g
+
+
g
−
=
g
g^++g^-=g
. Let
Δ
\Delta
,
Δ
+
\Delta ^+
and
Δ
−
\Delta ^-
be the Friedrichs extensions of the Laplacians corresponding to the (flat conical) metrics on
X
\mathcal {X}
,
X
+
\mathcal {X}^+
and
X
−
\mathcal {X}^-
respectively. We study the asymptotical behavior of the (modified, i.e. with zero modes excluded) zeta-regularized determinant
det
∗
Δ
\textrm {det}^*\, \Delta
as
γ
\gamma
and
γ
′
\gamma ’
shrink. We find the asymptotics
\[
det
∗
Δ
∼
κ
s
1
/
2
Area
(
X
)
Area
(
X
+
)
Area
(
X
−
)
det
∗
Δ
+
det
∗
Δ
−
\textrm {det}^*\,\Delta \sim \kappa s^{1/2}\frac {\textrm {Area}\,(\mathcal {X})}{\textrm {Area}\,(\mathcal {X}^+)\textrm {Area}\,(\mathcal {X}^-)}\,\textrm {det}^*\,\Delta ^+\textrm {det}^*\,\Delta ^-
\]
as
s
→
0
s\to 0
; here
κ
\kappa
is a certain absolute constant admitting an explicit expression through spectral characteristics of some model operators. We use the obtained result to fix an undetermined constant in the explicit formula for
det
∗
Δ
\textrm {det}^*\, \Delta
found in an earlier work by the author and D. Korotkin.