60
4 Open and Closed Strings
can be used to write the action as
S o =
∞
n=1
1
n + 1
ω o (m n (Φ
⊗n ), Φ) .
Here, Φ = Φ i e i , where {e i } is a homogeneous basis of V o . To keep the notation
concise, we denote the degree |e i | of e i simply by i and degree of Φ i by −i. The
classical BV equation applied to an open string field theory action then gives
{S o , S o } o =
∂ l S o
∂Φ i ω
ij ∂ r S o
∂Φ j
(4.5)
=
∞
n 1 =1
∞
n 2 =1
ω o
e i , m n 1 (Φ
⊗n 1 )
ω
ij ω o
e j , m n 2 (Φ
⊗n 2 )
,
where we used that ∂ Φ i picks up a sign (−1) i when commuted through m n as well
as the equality
ω o (Φ, m n (Φ, · · · , Φ, e i , Φ, · · · , Φ)) = (−1)
i ω o (e i , m n (Φ, · · · , Φ)).
Then, using ω ij = ω(e i , e j ) and δ
i
j = ω ik ω(e k , e j ), we find
{S o , S o } o =
∞
n 1 =1
∞
n 2 =1
ω o
m n 2 (Φ
⊗n 2 ), m n 1 (Φ
⊗n 1 )
=
∞
n=1
2
n + 1
i+j +k=n
ω o
m i+k+1
Φ
⊗i
⊗ m j (Φ
⊗j ) ⊗ Φ
⊗k
, Φ
=
∞
n=1
2
n + 1
ω o
π 1 ◦ M
2 (Φ
⊗n ), Φ
= 0 .
All we had to use was the cyclicity of m n . Also, M ∈ Coder cycl (T A o ) is the
coderivation corresponding to m ∈ Hom cylc (T A o , A o ) and Eq. (4.5) is equivalent
to M 2 = 0, the well-known statement that the vertices of a classical open string
field theory define an A ∞ -algebra. We give a detailed description of A ∞ -algebras
in Appendix A. Their definition is very similar to that of L ∞ -algebras described in
the previous section.
Infinitesimal deformations of this type of algebras are governed by cyclic cohomology. To see this connection, we note that in the presence of an invariant inner
product, there is a natural isomorphism Hom(A ⊗k , A) → Hom(A ⊗k+1 , C). The