186
8 Structures Relevant to Physics
For f ∈ MT(n + 2; g), let
Δ(f ) :=
◦ uv ⊗ • uv
M (θ ) ⊗ T (θ )
(f ) ∈ MT(n; g + s)
(8.2)
for an arbitrary bijection θ : [n + 2]
∼ =
− → [n] ] {u, v}. Finally, for g ∈ MT(n 1 +1; g 1 )
and h ∈ MT(n 2 + 1; g 2 ), let {g, h} ∈ MT(n 1 + n 2 ; g 1 + g 2 ) be defined as
{g, h} :=
S 1 S 2 =[n 1 +n 2 ]
( a ◦ b ⊗ a • b )τ
M (θ 1 )⊗T (θ 1 )⊗M (θ 2 )⊗T (θ 2 )
(g⊗h),
(8.3)
where θ 1 : [n 1 + 1]
∼ =
− → S 1 {a} and θ 2 : [n 2 + 1]
∼ =
− → S 2 {b} are arbitrary bijections 1 and τ exchanges the two middle factors. We complete the definition by setting
Δ(f ) := 0 for f ∈ MT(n; g), n = 0, 1, and {g, h} := 0 if g ∈ MT(0; g 1 ) or
h ∈ MT(0; g 2 ).
Lemma 8.1 The above operations are well defined and do not depend on the
choices of θ in (8.2) and θ 1 , θ 2 in (8.3).
Proof. It follows from the Σ n -equivariance of the differentials d T and d M that d(e)
in (8.1) is Σ n -stable, too, i.e. indeed d(e) ∈ MT(n; g). Let us prove that Δ(f ) as
defined in (8.2) does not depend on θ .
Assume that θ : [n + 2] → [n] ] {u, v} is another bijection. Then
M (θ
) ⊗ T (θ
)
(f ) =
M (θ ) ⊗ T (θ )
M (θ
−1 θ
) ⊗ T (θ
−1 θ
)
(f )
=
M (θ ) ⊗ T (θ )
(f ),
since θ −1 θ ∈ Σ n+2 and f ∈ M ([n + 2]; g) ⊗ T ([n + 2]; g) is Σ n+2 -invariant by
assumption.
Let us prove that Δ(f ) ∈ M ([n]; g + s) ⊗ T ([n]; g + s) is Σ n -invariant.
Invoking (6.53) resp. its obvious version for odd modular operads, we get for
σ ∈ Σ n
σ Δ(f ) =
M (σ ) ⊗ T (σ )
(◦ uv ⊗ • uv )
M (θ ) ⊗ T (θ )
(f )
= (◦ uv ⊗ • uv )
M ( ˜
σ θ) ⊗ T ( ˜
σ θ)
(f ),
where ˜
σ : [n] ] {u, v}
∼ =
− → [n] ] {u, v} fixes u, v and restricts to σ on [n]. Since ˜
σ θ
is just another isomorphism between [n + 2] and [n] ] {u, v}, σ Δ(f ) equals Δ(f )
by the previous paragraph.
The independence of {g, h} in (8.3) on θ 1 and θ 2 is proved precisely as the
independence of Δ(f ) on θ . Let us show that {g, h} is Σ n 1 +n 2 -invariant. Since
1 Notice that necessarily |S 1 | = n 1 and |S 2 | = n 2 .
8 Structures Relevant to Physics
For f ∈ MT(n + 2; g), let
Δ(f ) :=
◦ uv ⊗ • uv
M (θ ) ⊗ T (θ )
(f ) ∈ MT(n; g + s)
(8.2)
for an arbitrary bijection θ : [n + 2]
∼ =
− → [n] ] {u, v}. Finally, for g ∈ MT(n 1 +1; g 1 )
and h ∈ MT(n 2 + 1; g 2 ), let {g, h} ∈ MT(n 1 + n 2 ; g 1 + g 2 ) be defined as
{g, h} :=
S 1 S 2 =[n 1 +n 2 ]
( a ◦ b ⊗ a • b )τ
M (θ 1 )⊗T (θ 1 )⊗M (θ 2 )⊗T (θ 2 )
(g⊗h),
(8.3)
where θ 1 : [n 1 + 1]
∼ =
− → S 1 {a} and θ 2 : [n 2 + 1]
∼ =
− → S 2 {b} are arbitrary bijections 1 and τ exchanges the two middle factors. We complete the definition by setting
Δ(f ) := 0 for f ∈ MT(n; g), n = 0, 1, and {g, h} := 0 if g ∈ MT(0; g 1 ) or
h ∈ MT(0; g 2 ).
Lemma 8.1 The above operations are well defined and do not depend on the
choices of θ in (8.2) and θ 1 , θ 2 in (8.3).
Proof. It follows from the Σ n -equivariance of the differentials d T and d M that d(e)
in (8.1) is Σ n -stable, too, i.e. indeed d(e) ∈ MT(n; g). Let us prove that Δ(f ) as
defined in (8.2) does not depend on θ .
Assume that θ : [n + 2] → [n] ] {u, v} is another bijection. Then
M (θ
) ⊗ T (θ
)
(f ) =
M (θ ) ⊗ T (θ )
M (θ
−1 θ
) ⊗ T (θ
−1 θ
)
(f )
=
M (θ ) ⊗ T (θ )
(f ),
since θ −1 θ ∈ Σ n+2 and f ∈ M ([n + 2]; g) ⊗ T ([n + 2]; g) is Σ n+2 -invariant by
assumption.
Let us prove that Δ(f ) ∈ M ([n]; g + s) ⊗ T ([n]; g + s) is Σ n -invariant.
Invoking (6.53) resp. its obvious version for odd modular operads, we get for
σ ∈ Σ n
σ Δ(f ) =
M (σ ) ⊗ T (σ )
(◦ uv ⊗ • uv )
M (θ ) ⊗ T (θ )
(f )
= (◦ uv ⊗ • uv )
M ( ˜
σ θ) ⊗ T ( ˜
σ θ)
(f ),
where ˜
σ : [n] ] {u, v}
∼ =
− → [n] ] {u, v} fixes u, v and restricts to σ on [n]. Since ˜
σ θ
is just another isomorphism between [n + 2] and [n] ] {u, v}, σ Δ(f ) equals Δ(f )
by the previous paragraph.
The independence of {g, h} in (8.3) on θ 1 and θ 2 is proved precisely as the
independence of Δ(f ) on θ . Let us show that {g, h} is Σ n 1 +n 2 -invariant. Since
1 Notice that necessarily |S 1 | = n 1 and |S 2 | = n 2 .