8.1 BV Algebras and the Master Equation
199
always clear from the context. For instance, the full name of the term in the bracket
on which τ acts is
M (θ 1 ) ⊗ T (θ 1 ) ⊗ M (θ 2 ) ⊗ T (θ 2 ).
Isomorphism (8.23) with
V
= M (n 1 + 1; g 1 ) ⊗ M (n 2 + 1; g 2 ), W
= T (n 1 + 1; g 1 ) ⊗ T (n 2 + 1; g 2 )
V
= M (n; g), W
= T (n; g), h = a ◦ b , and f = a • b ,
converts the right-hand side of (8.30) into
S 1 S 2 =[n 1 +n 2 ]
a • b N
τ (θ 1 ⊗ θ 1 ⊗ θ 2 ⊗ θ 2 )(S(n 1 + 1; g 1 ) ⊗ S(n 2 + 1; g 2 ))
a ◦ b
# .
(8.31)
The commutativity of diagram (8.25), with
V 1 = M (n 1 + 1; g 1 ), V 2 = M (n 2 + 1; g 2 ),
W 1 = T (n 1 + 1; g 1 ) and W 2 = T (n 2 + 1; g 2 )
implies that
N
τ (θ 1 ⊗ θ 1 ⊗ θ 2 ⊗ θ 2 )(S(n 1 + 1; g 1 ) ⊗ S(n 2 + 1; g 2 ))
equals
Υ
N((θ 1 ⊗ θ 1 )(S(n 1 + 1; g 1 ))) ⊗ N((θ 1 ⊗ θ 1 )(S(n 2 + 1; g 2 )))
which can be rewritten using (8.23) twice as
Υ
θ 1 A(n 1 + 1; g 1 )θ
#
1 ⊗ θ 2 A(n 2 + 1; g 1 )θ
#
2
which clearly equals
(θ 1 ⊗ θ 2 )
A(n 1 +1; g 1 ) ⊗ A(n 2 +1; g 2 )
(θ
#
1 ⊗ θ
#
2 ).
Inserting this expression into (8.31) gives the right-hand side of (8.29). This finishes
the proof.