5.2 Quantum Open-Closed Homotopy Algebra
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genera of the individual surfaces, whereas the number of boundaries decreases
by one. This operation is identified with
1
2
[−, −] ◦
f ∧ f
◦ Δ −
(
[−, −] ◦ f) ∧ f
◦ Δ .
(2) Sewing of two open string insertions living on the same boundary. This
operation obviously increases the number of boundaries by one but leaves the
genus unchanged. It is described by
δ ◦ f ,
in the homotopy algebra.
(3) Consider a surface with more than one boundary. Take an open string insertion
of one boundary and sew it with another open string insertion on a second
boundary. This operation increases the genus by one and decreases the number
of boundaries by one. It is identified with
[−, −] ◦ f .
(4) Sewing of two closed string insertion, both lying on the same surface attaches a
handle to the surface and hence increases the genus by one, whereas the number
of boundaries does not change. We identify it with
f ◦ D(ω
−1
c ) .
(5) Take a closed string insertion of one surface and sew it with another closed
string insertion on a second surface. The genus and the number of boundaries
of the resulting surface is the sum of the genera and the sum of the number of
boundaries, respectively, of the input surfaces. The sewing in the case where
both surfaces have open and closed insertions is identified with
f ◦ D(e i ) ∧ f ◦ D(e
i )
◦ Δ ,
whereas the sewing involving a surface with closed string insertions only and
another surface with open and closed string insertions is identified with
f ◦ L q .
The above analysis provides the geometric interpretation of all individual terms
in (5.6).
Let us now focus on the vertices with open string insertions only. These vertices
are also comprised in the IBL ∞ -morphism F and defined by setting the closed string
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genera of the individual surfaces, whereas the number of boundaries decreases
by one. This operation is identified with
1
2
[−, −] ◦
f ∧ f
◦ Δ −
(
[−, −] ◦ f) ∧ f
◦ Δ .
(2) Sewing of two open string insertions living on the same boundary. This
operation obviously increases the number of boundaries by one but leaves the
genus unchanged. It is described by
δ ◦ f ,
in the homotopy algebra.
(3) Consider a surface with more than one boundary. Take an open string insertion
of one boundary and sew it with another open string insertion on a second
boundary. This operation increases the genus by one and decreases the number
of boundaries by one. It is identified with
[−, −] ◦ f .
(4) Sewing of two closed string insertion, both lying on the same surface attaches a
handle to the surface and hence increases the genus by one, whereas the number
of boundaries does not change. We identify it with
f ◦ D(ω
−1
c ) .
(5) Take a closed string insertion of one surface and sew it with another closed
string insertion on a second surface. The genus and the number of boundaries
of the resulting surface is the sum of the genera and the sum of the number of
boundaries, respectively, of the input surfaces. The sewing in the case where
both surfaces have open and closed insertions is identified with
f ◦ D(e i ) ∧ f ◦ D(e
i )
◦ Δ ,
whereas the sewing involving a surface with closed string insertions only and
another surface with open and closed string insertions is identified with
f ◦ L q .
The above analysis provides the geometric interpretation of all individual terms
in (5.6).
Let us now focus on the vertices with open string insertions only. These vertices
are also comprised in the IBL ∞ -morphism F and defined by setting the closed string