74
5 Open-Closed BV Equation
mixing up mutually closed and open punctures. Also operations sewing together
open and closed punctures are forbidden. Sewing and self-sewing operations within
the two separate sectors remain the same as described before.
The open-closed geometric vertices ν
b,g
n,m satisfying the (geometric) quantum
BV master equation result from a decomposition of the moduli space of bordered
Riemann surfaces with punctures both in the bulk as well as on the boundary
components. Again, it can be understood as an odd modular operad morphism going
from the (two-colored version of) Feynman transform of the above described openclosed modular operad to the moduli space operad, cf. discussion in Sects. 3.8
and 4.3. The latter one is the odd modular operad on singular chain complex
with operations induced from properly defined sewing/self-sewing of the Riemann
surfaces. As briefly mentioned in Sect. 5.1, the decomposition of the moduli spaces
comes as a solution to the corresponding minimal area problem. This is a rather
informal description of the horizontal morphism of Fig. 1.1 in the present situation.
The vertical arrow in Fig. 1.1 is the morphism from the odd moduli space operad
to the open-closed endomorphism operad provided by the open-closed CFT. Recall
that the CFT state space V = V o ⊕ V c is equipped with the odd symplectic form
ω o + ω c . The morphism is described similarly as in the closed case, cf. (3.22).
The resulting action (5.1) is the generating function of the quantum open-closed
operations f
b,g
n,m , these are graded symmetric functions of closed states. The graded
symmetry with respect to the open states holds for cyclic permutations within
a boundary component and for permutations of the whole boundary components.
Recall, cf. Sect. 3.8, that the main difference of the IBL ∞ interpretation of
the closed SFT algebra, as opposed to the loop homotopy algebra, was that
we considered, together with the n-ary brackets l
g
n , also the BV operator Δ as
a special cobracket with zero inputs and two outputs. The resulting structure
is naturally described in the dual description by the nilpotent full BV operator
¯
hΔ + {S, −}. Obviously, starting from a quantum A ∞ -algebra we could do the
same. However, this is not what is used for the interpretation/description of the
quantum-open homotopy algebra in Sect. 5.2. There, the starting point for the
description of the open sector is the cyclic Hochschild complex equipped with an
IBL-algebra structure. Roughly speaking, now we think of the inputs/outputs of
interactions being disc with open string insertions, not the open strings themselves.
So “sewing” two discs with open string insertions using the odd symplectic form
gives the multiplication, whereas the self-sewing gives the comultiplication. The
algebraic vertices are formally split into two groups, the ones corresponding to
closed strings exclusively and the remaining ones. The latter ones correspond to an
IBL ∞ -morphism form the IBL ∞ -algebra corresponding to closed strings to the IBLalgebra on the cyclic Hochschild complex of the open sector. The corresponding
mathematical structures are elucidated in the following appendix to Part I and in
Part II, Sect. 8.3.
5 Open-Closed BV Equation
mixing up mutually closed and open punctures. Also operations sewing together
open and closed punctures are forbidden. Sewing and self-sewing operations within
the two separate sectors remain the same as described before.
The open-closed geometric vertices ν
b,g
n,m satisfying the (geometric) quantum
BV master equation result from a decomposition of the moduli space of bordered
Riemann surfaces with punctures both in the bulk as well as on the boundary
components. Again, it can be understood as an odd modular operad morphism going
from the (two-colored version of) Feynman transform of the above described openclosed modular operad to the moduli space operad, cf. discussion in Sects. 3.8
and 4.3. The latter one is the odd modular operad on singular chain complex
with operations induced from properly defined sewing/self-sewing of the Riemann
surfaces. As briefly mentioned in Sect. 5.1, the decomposition of the moduli spaces
comes as a solution to the corresponding minimal area problem. This is a rather
informal description of the horizontal morphism of Fig. 1.1 in the present situation.
The vertical arrow in Fig. 1.1 is the morphism from the odd moduli space operad
to the open-closed endomorphism operad provided by the open-closed CFT. Recall
that the CFT state space V = V o ⊕ V c is equipped with the odd symplectic form
ω o + ω c . The morphism is described similarly as in the closed case, cf. (3.22).
The resulting action (5.1) is the generating function of the quantum open-closed
operations f
b,g
n,m , these are graded symmetric functions of closed states. The graded
symmetry with respect to the open states holds for cyclic permutations within
a boundary component and for permutations of the whole boundary components.
Recall, cf. Sect. 3.8, that the main difference of the IBL ∞ interpretation of
the closed SFT algebra, as opposed to the loop homotopy algebra, was that
we considered, together with the n-ary brackets l
g
n , also the BV operator Δ as
a special cobracket with zero inputs and two outputs. The resulting structure
is naturally described in the dual description by the nilpotent full BV operator
¯
hΔ + {S, −}. Obviously, starting from a quantum A ∞ -algebra we could do the
same. However, this is not what is used for the interpretation/description of the
quantum-open homotopy algebra in Sect. 5.2. There, the starting point for the
description of the open sector is the cyclic Hochschild complex equipped with an
IBL-algebra structure. Roughly speaking, now we think of the inputs/outputs of
interactions being disc with open string insertions, not the open strings themselves.
So “sewing” two discs with open string insertions using the odd symplectic form
gives the multiplication, whereas the self-sewing gives the comultiplication. The
algebraic vertices are formally split into two groups, the ones corresponding to
closed strings exclusively and the remaining ones. The latter ones correspond to an
IBL ∞ -morphism form the IBL ∞ -algebra corresponding to closed strings to the IBLalgebra on the cyclic Hochschild complex of the open sector. The corresponding
mathematical structures are elucidated in the following appendix to Part I and in
Part II, Sect. 8.3.