Further Reading
63
moduli space. In the present case, the relevant decomposition uses only the trivalent
geometric vertex ν 3 satisfying the classical BV master equation (4.3). The horizontal
arrow, provided by the open genus-0 CFT and landing in the endomorphism operad
of the open conformal theory state space, gives the cubic open STF action (4.4).
Despite the non-existence of a consistent quantum open string field theory,
nothing prevents us from constructing an associative analogue of a loop homotopy
algebra (quantum L ∞ -algebra). We may call it a quantum A ∞ -algebra, or quantum
open homotopy algebra, similarly as we could have coined the name quantum closed
homotopy algebra for a quantum L ∞ -algebra. Actually, quantum versions of homotopy algebras will exist for all modular operads and also for their colored versions,
e.g. quantum open-closed homotopy algebra over the two-colored combination of
the modular associative and modular commutative operad.
Hence, in the quantum associative case, we start from the modular associative
operad, i.e. the modular envelope of the cyclic associative operad. It is the
linearization of the modular operad described by Theorem 6.1 in Part II. Axioms of
quantum homotopy associative algebras are then obtained by applying Theorem 8.2
to its Feynman transform.
Also, for quantum A ∞ -algebras, in particular for their classical versions, i.e. A ∞ -
algebras, we could, using homological perturbation lemma, transfer the respective
algebra structures to their Q-cohomology, cf. the corresponding discussion in
Sect. 2.3 related to formulas (2.35) and (3.30), respectively.
Finally, let us make a comment on D-branes, a subject which we did not touch at
all. These can be included in the classical A ∞ -picture. We refer the interested reader
to work of Gaberdiel and Zwiebach cited below.
Further Reading
The open bosonic string field theory described in this chapter was formulated by
E. Witten in
• E. Witten, “Noncommutative geometry and string field theory”, Nucl. Phys.
B268 (1986) 253
The proof that the cubic action (4.4) does realize a decomposition of the moduli
space of open Riemann surfaces with punctures on the boundary can be found in
• B. Zwiebach, “A proof that Witten’s open string theory gives a single cover of
moduli space”, Commun. Math. Phys. 142 (1991) 193.
A derivation of the isomorphism (4.6) can be found in
• N. Moeller and I. Sachs, “Closed String Cohomology in Open String Field
Theory”, JHEP 1107 (2011) 022.
63
moduli space. In the present case, the relevant decomposition uses only the trivalent
geometric vertex ν 3 satisfying the classical BV master equation (4.3). The horizontal
arrow, provided by the open genus-0 CFT and landing in the endomorphism operad
of the open conformal theory state space, gives the cubic open STF action (4.4).
Despite the non-existence of a consistent quantum open string field theory,
nothing prevents us from constructing an associative analogue of a loop homotopy
algebra (quantum L ∞ -algebra). We may call it a quantum A ∞ -algebra, or quantum
open homotopy algebra, similarly as we could have coined the name quantum closed
homotopy algebra for a quantum L ∞ -algebra. Actually, quantum versions of homotopy algebras will exist for all modular operads and also for their colored versions,
e.g. quantum open-closed homotopy algebra over the two-colored combination of
the modular associative and modular commutative operad.
Hence, in the quantum associative case, we start from the modular associative
operad, i.e. the modular envelope of the cyclic associative operad. It is the
linearization of the modular operad described by Theorem 6.1 in Part II. Axioms of
quantum homotopy associative algebras are then obtained by applying Theorem 8.2
to its Feynman transform.
Also, for quantum A ∞ -algebras, in particular for their classical versions, i.e. A ∞ -
algebras, we could, using homological perturbation lemma, transfer the respective
algebra structures to their Q-cohomology, cf. the corresponding discussion in
Sect. 2.3 related to formulas (2.35) and (3.30), respectively.
Finally, let us make a comment on D-branes, a subject which we did not touch at
all. These can be included in the classical A ∞ -picture. We refer the interested reader
to work of Gaberdiel and Zwiebach cited below.
Further Reading
The open bosonic string field theory described in this chapter was formulated by
E. Witten in
• E. Witten, “Noncommutative geometry and string field theory”, Nucl. Phys.
B268 (1986) 253
The proof that the cubic action (4.4) does realize a decomposition of the moduli
space of open Riemann surfaces with punctures on the boundary can be found in
• B. Zwiebach, “A proof that Witten’s open string theory gives a single cover of
moduli space”, Commun. Math. Phys. 142 (1991) 193.
A derivation of the isomorphism (4.6) can be found in
• N. Moeller and I. Sachs, “Closed String Cohomology in Open String Field
Theory”, JHEP 1107 (2011) 022.