5.2 Quantum Open-Closed Homotopy Algebra
67
where f b,g ∈ Hom(T V c , C) ⊗ (Hom
cycl (T V o , C)) ∧b . Furthermore, ¯
e a :=
∞
n=1
1
n a ⊗n and T V o denotes the tensor algebra of V o . To summarize, the full
BV-quantum action of open-closed string field theory can be expressed as
S =
∞
g=0
¯
h
2g−1 ω c (l
g , ·)(e ¯
h 1/2 c ) +
∞
b=1
∞
g=0
1
b!
¯
h
2g+b−1 f
b,g (e ¯
h 1/2 c
; ¯
e
a , . . . , ¯
e
a
b times
) .
5.2
Quantum Open-Closed Homotopy Algebra
In order to give an algebraic interpretation of the quantum open-closed string field
theory, we first have to identify the algebraic structure on V c and Hom
cycl (T V o , C).
In the last part of this section we will connect the open and closed string part by
an IBL ∞ -morphism and finally define the quantum open-closed homotopy algebra,
that is, the algebraic structure of quantum open-closed string field theory.
As stated in Sect. A.1, the space of cyclic coderivations Coder
cycl (T A) is a Lie
algebra, with Lie bracket
[D 1 , D 2 ] = D 1 ◦ D 2 − (−1)
|D 1 ||D 2 | D 2 ◦ D 1 .
If A is in addition a cyclic A ∞ -algebra (A, M, ω), the space Coder cycl (T A)
becomes a dgla, where the differential is defined by d H = [M, −]. First we will
transfer the dgla structure from Coder cycl (T A) to the cyclic Hochschild complex
A := Hom cycl (A, C). Let f, g ∈ A , with both having at least one input. We define
associated maps in Hom cycl (T A, A) by
ω(d f , −) := f ,
ω(d g , −) := g ,
and lift them to cyclic coderivations D f , D g ∈ Coder cycl (T A). We define the
Gerstenhaber bracket on the cyclic Hochschild complex A by
[f, g] := (−1)
|f |+1 ω
π 1 ◦ [D f , D g ], −
.
(5.2)
In the case where one of the maps f, g ∈ A has no inputs, we define the commutator
to be identically zero. Note that the Gerstenhaber bracket as defined in (5.2) is
graded symmetric and has degree one. Thus, the structure induced on A is a
Lie algebra up to a shift in degree, that is, the actual Lie algebra lives on sA .
Furthermore, the map that associates a cyclic coderivation to an element of the cyclic
Hochschild complex defines a morphism of Lie algebras
[D f , D g ] = (−1)
|f |+1 D [f,g] .
67
where f b,g ∈ Hom(T V c , C) ⊗ (Hom
cycl (T V o , C)) ∧b . Furthermore, ¯
e a :=
∞
n=1
1
n a ⊗n and T V o denotes the tensor algebra of V o . To summarize, the full
BV-quantum action of open-closed string field theory can be expressed as
S =
∞
g=0
¯
h
2g−1 ω c (l
g , ·)(e ¯
h 1/2 c ) +
∞
b=1
∞
g=0
1
b!
¯
h
2g+b−1 f
b,g (e ¯
h 1/2 c
; ¯
e
a , . . . , ¯
e
a
b times
) .
5.2
Quantum Open-Closed Homotopy Algebra
In order to give an algebraic interpretation of the quantum open-closed string field
theory, we first have to identify the algebraic structure on V c and Hom
cycl (T V o , C).
In the last part of this section we will connect the open and closed string part by
an IBL ∞ -morphism and finally define the quantum open-closed homotopy algebra,
that is, the algebraic structure of quantum open-closed string field theory.
As stated in Sect. A.1, the space of cyclic coderivations Coder
cycl (T A) is a Lie
algebra, with Lie bracket
[D 1 , D 2 ] = D 1 ◦ D 2 − (−1)
|D 1 ||D 2 | D 2 ◦ D 1 .
If A is in addition a cyclic A ∞ -algebra (A, M, ω), the space Coder cycl (T A)
becomes a dgla, where the differential is defined by d H = [M, −]. First we will
transfer the dgla structure from Coder cycl (T A) to the cyclic Hochschild complex
A := Hom cycl (A, C). Let f, g ∈ A , with both having at least one input. We define
associated maps in Hom cycl (T A, A) by
ω(d f , −) := f ,
ω(d g , −) := g ,
and lift them to cyclic coderivations D f , D g ∈ Coder cycl (T A). We define the
Gerstenhaber bracket on the cyclic Hochschild complex A by
[f, g] := (−1)
|f |+1 ω
π 1 ◦ [D f , D g ], −
.
(5.2)
In the case where one of the maps f, g ∈ A has no inputs, we define the commutator
to be identically zero. Note that the Gerstenhaber bracket as defined in (5.2) is
graded symmetric and has degree one. Thus, the structure induced on A is a
Lie algebra up to a shift in degree, that is, the actual Lie algebra lives on sA .
Furthermore, the map that associates a cyclic coderivation to an element of the cyclic
Hochschild complex defines a morphism of Lie algebras
[D f , D g ] = (−1)
|f |+1 D [f,g] .