2.2 Scattering Matrix and Minimal Model
21
where i n : (H ⊗n ) sym → SH is the inclusion map. It is then not hard to see that F
is a chain map, that is,
(Q + ∗)F = F S ,
(2.36)
where S =
∞
n=0 s n . Furthermore, (Q+∗) 2 = 0 due to (2.17) and similarly S 2 = 0.
We will use the same symbol for the induced morphism of the underlying algebraic
structures, i.e., we will write F : (H, {s n }) → (V , Q, ∗). Let us note that F also
respects the cyclic structure, i.e.,
F
∗ ω = ω H ,
(2.37)
with F ∗ denoting the pullback of F . In this sense, we actually have an induced
morphism F : (H, {s n }, ω H ) → (V , Q, ∗, ω) of the underlying algebraic structures
including also the respective odd symplectic forms.
Remark 2.8 We can also express the minimal model theorem directly in terms of
the BV action. We take the BV vector field Q = {S, −} in (2.18) and expand it
in the formal neighborhood [φ 0 ] around a point φ 0 in the degree 0 subspace of the
space of fields F
Q
f orm
= Q
(0)
φ 0
+ Q
(1)
φ 0
+ Q
(2)
φ 0
+ · · · .
If φ 0 is an element of the Euler–Lagrange subspace E L (i.e., the space of classical
solutions) of F , then Q
(0)
φ 0
vanishes and the set {Q
(k)
φ 0
, k > 0} defines degree one
vector fields on T φ 0 F . In our example with φ 0 = 0,
Q
(1)
0 = ω
ij φ
k ω(e k , Qe i )∂ φ j ,
Q
(2)
0 = ω
ij φ
k φ
r ω(e k , e r ∗ e i )∂ φ j .
More generally, if we restrict the neighborhood [φ 0 ] to be in E L /Q, then the {Q
(k)
φ 0
}
define linear maps
Q
(k)
φ 0
= φ
i 1 · · · φ
i k l
i
i 1 ···i k
∂ φ i : T φ 0 F → S(T φ 0 F ) .
These maps are dual to the multilinear maps {l n } defined above, that is,
l n (e i 1 , . . . e i n ) = l
i
i 1 ···i n
e i .
Furthermore, the collection {Q
(k)
φ 0
} induces the Maurer–Cartan equation (2.34) on H .