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6 Operads
Informally, cyclic operads are modular operads without the contractions and the
genus grading. One easily sees that the genus 0 part
:=
M (S; 0) | S ∈ Cor
of a modular operad M with the restricted a ◦ b -operations forms a cyclic operad.
One therefore has the forgetful functor
: ModOp → CycOp
from the category ModOp of modular operads with A = N and the step s = 1 to the
category of cyclic operads. It can be proved that it has a left adjoint
Mod : CycOp → ModOp
introduced in [8]. The functor Mod(−) preserves the stability.
Definition 6.18 The modular operad Mod(P) is the modular completion or the
modular envelope of the cyclic operad P.
Example 6.26 The terminal stable modular Set-operad ∗ Mod discussed in Example 6.24 is the modular completion of the terminal stable cyclic operad ∗ cyclic from
Example 6.2, i.e., one has the isomorphism
∗ Mod ∼ = Mod(∗ cyclic )
of modular Set-operads (with A = N and s = 1). Equivalently, the stable modular
operad M of oriented surfaces of arbitrary genus is the modular completion of the
stable cyclic operad M 0 of oriented surfaces of genus 0. The modular completion
has in this case a clear geometric meaning.
Recall that Com ∼ = Span(∗ cyclic ) by (6.12). We conclude that the modular
completion Mod(Com) of the cyclic Chain-operad Com is the linearization of the
terminal modular set-operad ∗ Mod . Explicitly,
Mod(Com)(S; g) =
k if (S, g) ∈ S, and
0 otherwise,
with all structure operations the canonical isomorphisms. The operad Mod(Com)
will play the fundamental rôle in our description of the algebraic structure of closed
field theory given in Sect. 8.2 where it will be denoted QC and called the quantumclosed operad.
Example 6.27 In this example we discuss the stable modular operad QO consisting
of homeomorphism classes of connected compact two-dimensional orientable
6 Operads
Informally, cyclic operads are modular operads without the contractions and the
genus grading. One easily sees that the genus 0 part
:=
M (S; 0) | S ∈ Cor
of a modular operad M with the restricted a ◦ b -operations forms a cyclic operad.
One therefore has the forgetful functor
: ModOp → CycOp
from the category ModOp of modular operads with A = N and the step s = 1 to the
category of cyclic operads. It can be proved that it has a left adjoint
Mod : CycOp → ModOp
introduced in [8]. The functor Mod(−) preserves the stability.
Definition 6.18 The modular operad Mod(P) is the modular completion or the
modular envelope of the cyclic operad P.
Example 6.26 The terminal stable modular Set-operad ∗ Mod discussed in Example 6.24 is the modular completion of the terminal stable cyclic operad ∗ cyclic from
Example 6.2, i.e., one has the isomorphism
∗ Mod ∼ = Mod(∗ cyclic )
of modular Set-operads (with A = N and s = 1). Equivalently, the stable modular
operad M of oriented surfaces of arbitrary genus is the modular completion of the
stable cyclic operad M 0 of oriented surfaces of genus 0. The modular completion
has in this case a clear geometric meaning.
Recall that Com ∼ = Span(∗ cyclic ) by (6.12). We conclude that the modular
completion Mod(Com) of the cyclic Chain-operad Com is the linearization of the
terminal modular set-operad ∗ Mod . Explicitly,
Mod(Com)(S; g) =
k if (S, g) ∈ S, and
0 otherwise,
with all structure operations the canonical isomorphisms. The operad Mod(Com)
will play the fundamental rôle in our description of the algebraic structure of closed
field theory given in Sect. 8.2 where it will be denoted QC and called the quantumclosed operad.
Example 6.27 In this example we discuss the stable modular operad QO consisting
of homeomorphism classes of connected compact two-dimensional orientable