66
5 Open-Closed BV Equation
and m j are the numbers of open string punctures on the respective boundaries. The
BV equation then reads
∂ν
b,g
n,m +
1
2
n 1 ≤n 2 ;g 1 ≤g 2
b 1 ≤b 2 ;m 1 ≤m 2
{ν
b 1 ,g 1
n 1 +1,m 1
, ν
b 2 ,g 2
n 2 +1,m 2
} c + ¯
hΔ c ν
b,g−1
n+2,m
+
1
2
n 1 ≤n 2 ;g 1 ≤g 2
b 1 ≤b 2 ;m 1 ≤m 2
{ν
b 1 ,g 1
n 1 ,m 1 +1 , ν
b 2 ,g 2
n 2 ,m 2 +2 } o + ¯
hΔ o ν
b−1,g
n,m+2 + ¯
hΔ o ν
b+1,g−1
n,m+2
= 0 ,
where n = n 1 + n 2 , g = g 1 + g 2 , b = b 1 + b 2 , and m = m 1 + m 2 . Here and in
what follows, we will decorate with the subscript c objects that are associated with
closed strings (V c , Δ c , etc.) and with the subscript o objects that are associated with
open strings (V o , Δ o , etc.),
A solution to this BV equation can be obtained using the minimal area construction of Zwiebach: Given a genus g Riemann surface Σ with b boundaries,
m punctures on the boundaries and n punctures in the interior, the string vertex is
defined by the metric of minimal area under the condition that the length of any
non-trivial open curve in Σ with endpoints at the boundaries be greater or equal to
π and that the length of any non-trivial closed curve be greater or equal to 2π.
In order to transfer this BV-structure from the complex of geometric vertices to
the BV-algebra on V o ⊕ V c , one needs an extension of the path integral measure
defining F A in (3.18) to include surfaces with boundaries. This is straightforward
and we will thus not enter in the details. The resulting string field theory vertex
f
b,g
n,m 1 ,...,m b of genus g with n closed string insertions and b boundaries with m i
representing the number of insertions on the i-th boundary comes with the power
2g + b + n/2 − 1 in ¯
h. The full BV action reads
S(c, a) =
b,g
n
m 1 ,...,m b
¯
h
2g+b+n/2−1 f
b,g
n,m 1 ,...,m b (c, a) ,
(5.1)
where c ∈ V c is the closed string field and a ∈ V o is the open string field. The BV
equation (3.26) puts constraints on the collection of vertices f
b,g
n,m 1 ,...,m b and our goal
is to interpret these constraints in the language of homotopy algebras.
The idea is to split the set of all vertices into two disjoint sets. One contains
all vertices corresponding to closed Riemann surfaces and the other contains the
vertices associated with bordered Riemann surfaces. For the former, the action will
be given in (5.2). Taking all symmetries of vertices with open and closed inputs into
account, we can write the part of the action for the latter as
n
m 1 ,...,m b
¯
h
2g+b+n/2−1 f
b,g
n,m 1 ,...,m b (c, a) =
1
b!
¯
h
2g+b−1 f
b,g (e ¯
h 1/2 c
; ¯
e
a , . . . , ¯
e
a
b times
) ,
5 Open-Closed BV Equation
and m j are the numbers of open string punctures on the respective boundaries. The
BV equation then reads
∂ν
b,g
n,m +
1
2
n 1 ≤n 2 ;g 1 ≤g 2
b 1 ≤b 2 ;m 1 ≤m 2
{ν
b 1 ,g 1
n 1 +1,m 1
, ν
b 2 ,g 2
n 2 +1,m 2
} c + ¯
hΔ c ν
b,g−1
n+2,m
+
1
2
n 1 ≤n 2 ;g 1 ≤g 2
b 1 ≤b 2 ;m 1 ≤m 2
{ν
b 1 ,g 1
n 1 ,m 1 +1 , ν
b 2 ,g 2
n 2 ,m 2 +2 } o + ¯
hΔ o ν
b−1,g
n,m+2 + ¯
hΔ o ν
b+1,g−1
n,m+2
= 0 ,
where n = n 1 + n 2 , g = g 1 + g 2 , b = b 1 + b 2 , and m = m 1 + m 2 . Here and in
what follows, we will decorate with the subscript c objects that are associated with
closed strings (V c , Δ c , etc.) and with the subscript o objects that are associated with
open strings (V o , Δ o , etc.),
A solution to this BV equation can be obtained using the minimal area construction of Zwiebach: Given a genus g Riemann surface Σ with b boundaries,
m punctures on the boundaries and n punctures in the interior, the string vertex is
defined by the metric of minimal area under the condition that the length of any
non-trivial open curve in Σ with endpoints at the boundaries be greater or equal to
π and that the length of any non-trivial closed curve be greater or equal to 2π.
In order to transfer this BV-structure from the complex of geometric vertices to
the BV-algebra on V o ⊕ V c , one needs an extension of the path integral measure
defining F A in (3.18) to include surfaces with boundaries. This is straightforward
and we will thus not enter in the details. The resulting string field theory vertex
f
b,g
n,m 1 ,...,m b of genus g with n closed string insertions and b boundaries with m i
representing the number of insertions on the i-th boundary comes with the power
2g + b + n/2 − 1 in ¯
h. The full BV action reads
S(c, a) =
b,g
n
m 1 ,...,m b
¯
h
2g+b+n/2−1 f
b,g
n,m 1 ,...,m b (c, a) ,
(5.1)
where c ∈ V c is the closed string field and a ∈ V o is the open string field. The BV
equation (3.26) puts constraints on the collection of vertices f
b,g
n,m 1 ,...,m b and our goal
is to interpret these constraints in the language of homotopy algebras.
The idea is to split the set of all vertices into two disjoint sets. One contains
all vertices corresponding to closed Riemann surfaces and the other contains the
vertices associated with bordered Riemann surfaces. For the former, the action will
be given in (5.2). Taking all symmetries of vertices with open and closed inputs into
account, we can write the part of the action for the latter as
n
m 1 ,...,m b
¯
h
2g+b+n/2−1 f
b,g
n,m 1 ,...,m b (c, a) =
1
b!
¯
h
2g+b−1 f
b,g (e ¯
h 1/2 c
; ¯
e
a , . . . , ¯
e
a
b times
) ,