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8 Structures Relevant to Physics
8.4
Comments and Remarks Related to Part I
In Sect. 6.1, we introduced cyclic operads. The simplest example is the cyclic
commutative operad of Example 6.1. Another example is the cyclic associative
operad. However, this one is described only later, cf. Sect. 6.2, Example 6.18, as
a symmetrization of the non-Σ cyclic associative operad, Example 6.17. Further
example is the cyclic endomorphism operad in Example 6.6 and its dual version,
cf. Example 6.7. An important point is that each cyclic operad is a quotient of the
free cyclic operad on some cyclic module, see Proposition 6.3. From the point of
view adopted in Part I, cf. Sects. 3.8 and 4.3, cyclic operads are relevant to the
classical string field theory, the commutative case to closed strings, the associative
to open strings.
In Sect. 6.2, we introduced the non-Σ cyclic operads. The simplest example is
the non-Σ cyclic associative operad in Example 6.17. As already mentioned, the
cyclic associative operad is its symmetrization, cf. Example 6.18.
In Sect. 6.3 (cyclic) operad algebras were introduced. Such an algebra is a
morphism from a cyclic operad to the endomorphism operad. This led us to nonunital Frobenius algebras in the associative case in Example 6.21 and to their
commutative versions in the commutative one in Example 6.22. These are the
familiar structures from the two-dimensional topological quantum field theory.
In Sect. 6.4 we discussed modular operads. Again, the simplest example is
the modular commutative operad of Example 6.24. It is the modular completion
(envelope) of the cyclic commutative operad. From the point of view of Part I,
Sect. 3.8, it is the operad relevant to the quantum closed string field theory. This
is the reason why we called it also quantum closed operad through the text. As
for the modular associative operad (quantum open operad), its explicit description
preceding Theorem 6.1 is rather complicated. Nevertheless, it is defined abstractly
as the modular completion of the cyclic associative operad. Another possible
description is as a symmetrization of the modular completion of the non-Σ cyclic
associative operad. These two operads can be combined rather straightforwardly
into the quantum open-closed operad. This is why we did not describe this
operad explicitly. 10 Modular version of the endomorphism operad is used for
defining representations. However, we don’t get new examples of algebras from
representations of modular operads. A cyclic operad and its modular envelope have
the same categories of representations, due to the universal property of the modular
completion.
To obtain homotopy algebras as representations of operads, and hence make
a direct contact with string field theory and quantum field theory in general, we need
the cobar construction and its modular analogue, the Feynman transform. Operads
resulting from this constructions are, however, not modular operads any more. They
are odd (twisted) modular operads. Their structure operations are of degree one, in
10 Neither we discussed, which is much more delicate, its description in terms of a modular
completion.
8 Structures Relevant to Physics
8.4
Comments and Remarks Related to Part I
In Sect. 6.1, we introduced cyclic operads. The simplest example is the cyclic
commutative operad of Example 6.1. Another example is the cyclic associative
operad. However, this one is described only later, cf. Sect. 6.2, Example 6.18, as
a symmetrization of the non-Σ cyclic associative operad, Example 6.17. Further
example is the cyclic endomorphism operad in Example 6.6 and its dual version,
cf. Example 6.7. An important point is that each cyclic operad is a quotient of the
free cyclic operad on some cyclic module, see Proposition 6.3. From the point of
view adopted in Part I, cf. Sects. 3.8 and 4.3, cyclic operads are relevant to the
classical string field theory, the commutative case to closed strings, the associative
to open strings.
In Sect. 6.2, we introduced the non-Σ cyclic operads. The simplest example is
the non-Σ cyclic associative operad in Example 6.17. As already mentioned, the
cyclic associative operad is its symmetrization, cf. Example 6.18.
In Sect. 6.3 (cyclic) operad algebras were introduced. Such an algebra is a
morphism from a cyclic operad to the endomorphism operad. This led us to nonunital Frobenius algebras in the associative case in Example 6.21 and to their
commutative versions in the commutative one in Example 6.22. These are the
familiar structures from the two-dimensional topological quantum field theory.
In Sect. 6.4 we discussed modular operads. Again, the simplest example is
the modular commutative operad of Example 6.24. It is the modular completion
(envelope) of the cyclic commutative operad. From the point of view of Part I,
Sect. 3.8, it is the operad relevant to the quantum closed string field theory. This
is the reason why we called it also quantum closed operad through the text. As
for the modular associative operad (quantum open operad), its explicit description
preceding Theorem 6.1 is rather complicated. Nevertheless, it is defined abstractly
as the modular completion of the cyclic associative operad. Another possible
description is as a symmetrization of the modular completion of the non-Σ cyclic
associative operad. These two operads can be combined rather straightforwardly
into the quantum open-closed operad. This is why we did not describe this
operad explicitly. 10 Modular version of the endomorphism operad is used for
defining representations. However, we don’t get new examples of algebras from
representations of modular operads. A cyclic operad and its modular envelope have
the same categories of representations, due to the universal property of the modular
completion.
To obtain homotopy algebras as representations of operads, and hence make
a direct contact with string field theory and quantum field theory in general, we need
the cobar construction and its modular analogue, the Feynman transform. Operads
resulting from this constructions are, however, not modular operads any more. They
are odd (twisted) modular operads. Their structure operations are of degree one, in
10 Neither we discussed, which is much more delicate, its description in terms of a modular
completion.