70
5 Open-Closed BV Equation
In the following, we abbreviate L q =
g ¯
h g L g . We get
F ◦ L c =
∞
n=0
1
n!
i+j =n−1
(f
∧i
∧ f ◦ (L q + ¯
hD(ω
−1
c )) ∧ f
∧j ) ◦ Δ n
+
∞
n=0
1
n!
i+j +k=n−2
¯
h
f
∧i
∧ f ◦ D(e i ) ∧ f
∧j
∧ f ◦ D(e
i ) ∧ f
∧k
◦ Δ n
=
f ◦ L c +
1
2
¯
h(f ◦ D(e i ) ∧ f ◦ D(e
i )) ◦ Δ
∧ F
◦ Δ .
Let us turn to the right-hand side of Eq. (5.5). There we have the maps δ and
[−, −]. The defining map δ = π 2 ◦ δ of δ has two outputs and one input. Recall that
the order of a coderivation is the number of outputs of the underlying defining map
(see Sect. B.1). Similarly, we can define higher order derivations by the number of
inputs of the underlying defining map. So we can interpret δ either as a second order
coderivation or as a first order derivation, and
[−, −] as a first order coderivation or
as a second order derivation. For our purpose, the second point of view will be more
useful. Having these properties in mind, one can show that
δ ◦ F =
δ ◦ f ∧ F
◦ Δ
and
[−, −] ◦ F =
[−, −] ◦ f +
1
2
[−, −] ◦
f ∧ f
◦ Δ −
(
[−, −] ◦ f) ∧ f
◦ Δ
∧ F
◦ Δ .
Besides the properties of δ and
[−, −], we also used cocommutativity and coassociativity of Δ. Thus we can equivalently define the QOCHA by
f ◦ L c +
¯
h
2
f ◦ D(e i ) ∧ f ◦ D(e
i )
◦ Δ
(5.6)
= L o ◦ f +
1
2
[−, −] ◦
f ∧ f
◦ Δ −
(
[−, −] ◦ f) ∧ f
◦ Δ .
The individual terms in Eq. (5.6) can be identified with the five distinct sewing
operations of bordered Riemann surfaces with closed string insertions (punctures
in the bulk) and open string insertions (punctures on the boundaries). The sewing
joins either two open string insertions or two closed string insertions. In addition,
the sewing may involve a single surface or two surfaces.
(1) Take an open string insertion of one surface and sew it with another open string
insertion on a second surface. The genus of the resulting surface is the sum of the