8.1 BV Algebras and the Master Equation
195
and, likewise,
a • b (ϑ 1 ⊗ ϑ 2 ) = T (κ ij ) i • j = i • j .
Taking this in account, we see that for the choices of ϑ 1 and ϑ 2 above, (8.18) equals
the sum in the right-hand side of (8.16) as required.
Suppose that M is a finite-dimensional, in the sense of Definition 7.8, modular
operad with structure operations a ◦ b and ◦ uv . Suppose moreover that for each finite
sets S 1 , S 2 ∈ Cor and g ∈ A, there are only finitely many couples (g 1 , g 2 ) ∈ A ×2
such that g 1 + g 2 = g for which the restriction
g 1
a ◦
g 2
b : M
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M (S 1 S 2 ; g 1 + g 2 )
of the structure operation a ◦ b is non-zero. Such M clearly fulfills the assumptions
of Proposition 7.1, therefore its piece-wise linear dual M # is a modular cooperad,
with structure operations
a
S 1 ; g 1
◦ b
S 2 ;g 2
:= a ◦ b
# and ◦ uv
g := ◦ uv
# .
The following statement rephrases Proposition 7.2 for the case when C is the
piece-wise linear dual of a modular operad M as above.
Theorem 8.2 ([1]) Assume that M is a finite-dimensional modular operad as
above and M # its dual modular cooperad. A morphism
α : F (M
# ) → T
(8.19)
of odd modular dg-operads is then the same as a degree 0 element S ∈ MT satisfying
the master equation
d(S) + Δ(S) +
1
2
{S, S} = 0
(8.20)
in the desuspended bi-differential graded Lie algebra structure of Theorem 8.1.
Proof. The left-hand side of the master equation (8.20), as an element of the product
MT, vanishes if and only if each of its factors does. The factor in arity n and genus
g equals
dS(n; g) +
g +s=g
ΔS(n+2; g
) +
1
2
n 1 +n 2 =n
g 1 +g 2 =g
S(n 1 +1; g 1 ), S(n 2 +1; g 2 )
.
(8.21)
We therefore need to prove that (8.21) vanishes for each (n, g) ∈ N × A.
195
and, likewise,
a • b (ϑ 1 ⊗ ϑ 2 ) = T (κ ij ) i • j = i • j .
Taking this in account, we see that for the choices of ϑ 1 and ϑ 2 above, (8.18) equals
the sum in the right-hand side of (8.16) as required.
Suppose that M is a finite-dimensional, in the sense of Definition 7.8, modular
operad with structure operations a ◦ b and ◦ uv . Suppose moreover that for each finite
sets S 1 , S 2 ∈ Cor and g ∈ A, there are only finitely many couples (g 1 , g 2 ) ∈ A ×2
such that g 1 + g 2 = g for which the restriction
g 1
a ◦
g 2
b : M
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M (S 1 S 2 ; g 1 + g 2 )
of the structure operation a ◦ b is non-zero. Such M clearly fulfills the assumptions
of Proposition 7.1, therefore its piece-wise linear dual M # is a modular cooperad,
with structure operations
a
S 1 ; g 1
◦ b
S 2 ;g 2
:= a ◦ b
# and ◦ uv
g := ◦ uv
# .
The following statement rephrases Proposition 7.2 for the case when C is the
piece-wise linear dual of a modular operad M as above.
Theorem 8.2 ([1]) Assume that M is a finite-dimensional modular operad as
above and M # its dual modular cooperad. A morphism
α : F (M
# ) → T
(8.19)
of odd modular dg-operads is then the same as a degree 0 element S ∈ MT satisfying
the master equation
d(S) + Δ(S) +
1
2
{S, S} = 0
(8.20)
in the desuspended bi-differential graded Lie algebra structure of Theorem 8.1.
Proof. The left-hand side of the master equation (8.20), as an element of the product
MT, vanishes if and only if each of its factors does. The factor in arity n and genus
g equals
dS(n; g) +
g +s=g
ΔS(n+2; g
) +
1
2
n 1 +n 2 =n
g 1 +g 2 =g
S(n 1 +1; g 1 ), S(n 2 +1; g 2 )
.
(8.21)
We therefore need to prove that (8.21) vanishes for each (n, g) ∈ N × A.