8.4 Comments and Remarks Related to Part I
215
contrary to the modular operads having operations of degree zero. This unexpected
degree is related to the degree of the BV bracket and BV operator Δ, which are
inherently present in the construction of an algebra over the Feynman transform of a
modular operad. Hence, in Sect. 6.5 we discussed odd modular operads. An example
directly relevant to quantum field theory and string field theory is the odd modular
version of the endomorphism operad. This is an operad based on a differential
graded vector space equipped with a degree −1 symplectic form. In Part I, we have
met examples of these spaces, namely the representation space of the first quantized
point particle and of the closed, open, and open-closed CFT theory representation
spaces, respectively. The differential was the BRST operator and the odd symplectic
form was related to the product on the respective representation spaces, e.g. BPZ
pairing for strings. Further examples we have met explicitly in Part I were the odd
modular operads of chain complexes on the moduli spaces of Riemann surfaces
associated with (open, closed, open-closed) string field theories. As also discussed
in Part I, conformal field theory provides an example of an odd modular operad
morphism going from the moduli space operad to the endomorphism operad.
In Sect. 7.2 we introduced the Feynman transform of modular cooperads which
are objects dual to modular operads, in order to obtain an important class of odd
modular operads. As for Part I, the most relevant modular cooperads come from
dualizations of particular modular operads. The construction is an analogue of the
classical cobar construction for cyclic cooperads. Although we mentioned the cobar
construction at the beginning of the present section merely to motivate the Feynman
transform, it is of its own interest and we will comment on it later. Nevertheless, we
did not give a detailed description of the cobar construction. This can be recognized
as the genus zero part of the more general case of the Feynman transform. The
important point is Proposition 7.2 describing explicitly an algebra over the Feynman
transform of a modular cooperad.
In Sect. 8.2, the relevance of operads in the description of string field theory
became transparent. Here, the Feynman transform was applied to the case, when the
modular cooperad is the component-wise linear dual of a modular operad. 11 In this
instance, a morphism from the Feynman transform into an odd modular operad can
be expressed directly using the structure of the original modular operad. It explicitly
leads immediately not only to a BV structure, see Theorem 8.1, on the space of
invariants (8.4), but also to a characterization of the corresponding morphism as a
solution to the quantum BV master equation on this space, see Theorem 8.2. The
BV bracket and Δ have their origins in the operadic operations of both the original
modular operad and the (odd modular) endomorphism operad, cf. Proposition 8.1.
The S(n, g)-part of the solution S to the master equation comes from evaluating the
morphism on the (dual) of the (n, g) dg-vector space component of the original
operad. We have met this through Part I. All decompositions of moduli spaces
mentioned there are examples. More concretely, let us consider, for instance, closed
11 In various places in Part I we would refer to this as to a Feynman transform of the modular
operad itself. We will continue doing this also in the rest of this section.
215
contrary to the modular operads having operations of degree zero. This unexpected
degree is related to the degree of the BV bracket and BV operator Δ, which are
inherently present in the construction of an algebra over the Feynman transform of a
modular operad. Hence, in Sect. 6.5 we discussed odd modular operads. An example
directly relevant to quantum field theory and string field theory is the odd modular
version of the endomorphism operad. This is an operad based on a differential
graded vector space equipped with a degree −1 symplectic form. In Part I, we have
met examples of these spaces, namely the representation space of the first quantized
point particle and of the closed, open, and open-closed CFT theory representation
spaces, respectively. The differential was the BRST operator and the odd symplectic
form was related to the product on the respective representation spaces, e.g. BPZ
pairing for strings. Further examples we have met explicitly in Part I were the odd
modular operads of chain complexes on the moduli spaces of Riemann surfaces
associated with (open, closed, open-closed) string field theories. As also discussed
in Part I, conformal field theory provides an example of an odd modular operad
morphism going from the moduli space operad to the endomorphism operad.
In Sect. 7.2 we introduced the Feynman transform of modular cooperads which
are objects dual to modular operads, in order to obtain an important class of odd
modular operads. As for Part I, the most relevant modular cooperads come from
dualizations of particular modular operads. The construction is an analogue of the
classical cobar construction for cyclic cooperads. Although we mentioned the cobar
construction at the beginning of the present section merely to motivate the Feynman
transform, it is of its own interest and we will comment on it later. Nevertheless, we
did not give a detailed description of the cobar construction. This can be recognized
as the genus zero part of the more general case of the Feynman transform. The
important point is Proposition 7.2 describing explicitly an algebra over the Feynman
transform of a modular cooperad.
In Sect. 8.2, the relevance of operads in the description of string field theory
became transparent. Here, the Feynman transform was applied to the case, when the
modular cooperad is the component-wise linear dual of a modular operad. 11 In this
instance, a morphism from the Feynman transform into an odd modular operad can
be expressed directly using the structure of the original modular operad. It explicitly
leads immediately not only to a BV structure, see Theorem 8.1, on the space of
invariants (8.4), but also to a characterization of the corresponding morphism as a
solution to the quantum BV master equation on this space, see Theorem 8.2. The
BV bracket and Δ have their origins in the operadic operations of both the original
modular operad and the (odd modular) endomorphism operad, cf. Proposition 8.1.
The S(n, g)-part of the solution S to the master equation comes from evaluating the
morphism on the (dual) of the (n, g) dg-vector space component of the original
operad. We have met this through Part I. All decompositions of moduli spaces
mentioned there are examples. More concretely, let us consider, for instance, closed
11 In various places in Part I we would refer to this as to a Feynman transform of the modular
operad itself. We will continue doing this also in the rest of this section.