72
5 Open-Closed BV Equation
inputs to zero. More precisely, let m = f| be the restriction of f onto the subspace
without closed strings. The weighted sum of open string vertices is then given by
m =
∞
b=1
∞
g=0
¯
h
g+b−1 m
b,g ,
m
b,g
∈ A
∧b
o ,
where m b,g = f b,g |. The complement of m—the vertices with at least one closed
string input—is denoted by g, so that
f = m + g .
(5.7)
Remark 5.1 In the classical limit ¯
h → 0, we expect to recover the OCHA
defined by Kajiura and Stasheff. Indeed, the IBL ∞ -morphism F reduces to an
L ∞ -morphism, the loop algebra L c of closed strings reduces to the L ∞ -algebra
L cl := L 0 and the IBL-algebra on the space of cyclic coderivations becomes an
ordinary Lie algebra. The defining Eq. (5.6) of the QOCHA simplifies to
f cl ◦ L cl =
1
2
[f cl , f cl ] ◦ Δ ,
(5.8)
where f cl := f 1,0 is the component of f with one boundary and genus zero and
the corresponding L ∞ -morphism is given by
n
1
n! f cl
∧n
◦ Δ n (see Sect. A.2).
Separating the purely open string vertices m cl from f cl , we see that those have to
satisfy the axioms of an A ∞ -algebra (since L cl | = 0), i.e., they define a classical
open string field theory. Thus, the space A o turns into a dgla with differential
d H = [m cl , −] and Eq. (5.8) finally reads
n cl ◦ L cl = d H ◦ n cl +
1
2
[n cl , n cl ] ◦ Δ ,
(5.9)
where n cl = f cl − m cl : T V c → A o denotes the vertices with at least one closed
string input and one non-empty boundary.
Similarly, we define n = f − m and the QOCHA in terms of n reads
N ◦ L c = L
o ◦ N ,
(5.10)
where N =
∞
n=0
1
n! n ∧n ◦ Δ n and L
o =
d H + L o .
Equation (5.9) is precisely the OCHA. The physical interpretation of n cl is that
it describes the deformation of open string field theory by turning on a closed string
background. The vanishing of the right-hand side is the condition for a consistent
classical field theory of open strings, while the left-hand side vanishes if the closed
string background solves the classical closed string field theory equations of motion.
Equation (5.9) then implies that the open-closed vertices define a consistent classical
5 Open-Closed BV Equation
inputs to zero. More precisely, let m = f| be the restriction of f onto the subspace
without closed strings. The weighted sum of open string vertices is then given by
m =
∞
b=1
∞
g=0
¯
h
g+b−1 m
b,g ,
m
b,g
∈ A
∧b
o ,
where m b,g = f b,g |. The complement of m—the vertices with at least one closed
string input—is denoted by g, so that
f = m + g .
(5.7)
Remark 5.1 In the classical limit ¯
h → 0, we expect to recover the OCHA
defined by Kajiura and Stasheff. Indeed, the IBL ∞ -morphism F reduces to an
L ∞ -morphism, the loop algebra L c of closed strings reduces to the L ∞ -algebra
L cl := L 0 and the IBL-algebra on the space of cyclic coderivations becomes an
ordinary Lie algebra. The defining Eq. (5.6) of the QOCHA simplifies to
f cl ◦ L cl =
1
2
[f cl , f cl ] ◦ Δ ,
(5.8)
where f cl := f 1,0 is the component of f with one boundary and genus zero and
the corresponding L ∞ -morphism is given by
n
1
n! f cl
∧n
◦ Δ n (see Sect. A.2).
Separating the purely open string vertices m cl from f cl , we see that those have to
satisfy the axioms of an A ∞ -algebra (since L cl | = 0), i.e., they define a classical
open string field theory. Thus, the space A o turns into a dgla with differential
d H = [m cl , −] and Eq. (5.8) finally reads
n cl ◦ L cl = d H ◦ n cl +
1
2
[n cl , n cl ] ◦ Δ ,
(5.9)
where n cl = f cl − m cl : T V c → A o denotes the vertices with at least one closed
string input and one non-empty boundary.
Similarly, we define n = f − m and the QOCHA in terms of n reads
N ◦ L c = L
o ◦ N ,
(5.10)
where N =
∞
n=0
1
n! n ∧n ◦ Δ n and L
o =
d H + L o .
Equation (5.9) is precisely the OCHA. The physical interpretation of n cl is that
it describes the deformation of open string field theory by turning on a closed string
background. The vanishing of the right-hand side is the condition for a consistent
classical field theory of open strings, while the left-hand side vanishes if the closed
string background solves the classical closed string field theory equations of motion.
Equation (5.9) then implies that the open-closed vertices define a consistent classical