68
5 Open-Closed BV Equation
It turns out that we can endow A with the structure of a differential involutive Lie
bialgebra, i.e., that there is a map δ : A → A ∧2 such that d H , [−, −] and δ satisfy
the defining Eqs. (B.4)–(B.10) of an IBL-algebra. We then define δ : A → A ∧2 by
(δf )(a 1 , . . . , a n )(b 1 , . . . , b m )
:=(−1) |f |
n
i=1
m
j =1
(−1) f (e k , a i , . . . , a n , a 1 , . . . , a i−1 , e k , b j , . . . , b m , b 1 , . . . , b j −1 ) ,
where is the Koszul sign resulting from permutation of the entries, {e k } is a basis
of A, and {e k } denotes the corresponding dual basis with respect to the symplectic
structure ω. This definition ensures that δf has the right symmetry properties.
Furthermore, d H , [−, −] and δ satisfy all conditions (B.4)–(B.10). Now, let us
put this into the language of IBL ∞ -algebras. Lift the Hochschild differential, the
Gerstenhaber bracket and the cobracket δ to coderivations on SA , the symmetric
algebra over A ,
d H ∈ Coder(SA ) ,
[−, −] ∈ Coder(SA ) ,
δ ∈ Coder
2 (SA ) .
The statement that the maps d H , [−, −] and δ satisfy the defining relations of a
differential IBL-algebra is then equivalent to
(
d H +
[−, −] + x δ)
2
= 0 ,
where x is a formal expansion parameter which we may identify with ¯
h. If the
algebra A is not endowed with the structure of a cyclic A ∞ -algebra, the differential
d H is absent, but we still have an IBL-algebra defined by
L
2
o = 0 ,
where
L o :=
[−, −] + x δ ∈ Coder(SA , x)
and
|L o | = 1 .
(5.3)
We use Gothic characters for formal power series with values in coderivations. This
is the structure that will enter in the definition of the quantum open-closed homotopy
algebra. That means that we do not anticipate that the vertices of classical open
string field theory define an A ∞ -algebra but rather, as we will see soon, derive it
from the quantum open-closed homotopy algebra.
Now, we can put the parts together and define the quantum open-closed
homotopy algebra (QOCHA). The QOCHA is defined by an IBL ∞ -morphisms from
the IBL ∞ -algebra of closed strings to the IBL-algebra of open strings
(V c , L c )
IBL ∞ −morphism
−−−−−−−−−→ (A o , L o ) ,
(5.4)
5 Open-Closed BV Equation
It turns out that we can endow A with the structure of a differential involutive Lie
bialgebra, i.e., that there is a map δ : A → A ∧2 such that d H , [−, −] and δ satisfy
the defining Eqs. (B.4)–(B.10) of an IBL-algebra. We then define δ : A → A ∧2 by
(δf )(a 1 , . . . , a n )(b 1 , . . . , b m )
:=(−1) |f |
n
i=1
m
j =1
(−1) f (e k , a i , . . . , a n , a 1 , . . . , a i−1 , e k , b j , . . . , b m , b 1 , . . . , b j −1 ) ,
where is the Koszul sign resulting from permutation of the entries, {e k } is a basis
of A, and {e k } denotes the corresponding dual basis with respect to the symplectic
structure ω. This definition ensures that δf has the right symmetry properties.
Furthermore, d H , [−, −] and δ satisfy all conditions (B.4)–(B.10). Now, let us
put this into the language of IBL ∞ -algebras. Lift the Hochschild differential, the
Gerstenhaber bracket and the cobracket δ to coderivations on SA , the symmetric
algebra over A ,
d H ∈ Coder(SA ) ,
[−, −] ∈ Coder(SA ) ,
δ ∈ Coder
2 (SA ) .
The statement that the maps d H , [−, −] and δ satisfy the defining relations of a
differential IBL-algebra is then equivalent to
(
d H +
[−, −] + x δ)
2
= 0 ,
where x is a formal expansion parameter which we may identify with ¯
h. If the
algebra A is not endowed with the structure of a cyclic A ∞ -algebra, the differential
d H is absent, but we still have an IBL-algebra defined by
L
2
o = 0 ,
where
L o :=
[−, −] + x δ ∈ Coder(SA , x)
and
|L o | = 1 .
(5.3)
We use Gothic characters for formal power series with values in coderivations. This
is the structure that will enter in the definition of the quantum open-closed homotopy
algebra. That means that we do not anticipate that the vertices of classical open
string field theory define an A ∞ -algebra but rather, as we will see soon, derive it
from the quantum open-closed homotopy algebra.
Now, we can put the parts together and define the quantum open-closed
homotopy algebra (QOCHA). The QOCHA is defined by an IBL ∞ -morphisms from
the IBL ∞ -algebra of closed strings to the IBL-algebra of open strings
(V c , L c )
IBL ∞ −morphism
−−−−−−−−−→ (A o , L o ) ,
(5.4)