4.2 Open String
61
image of ˆ
m k ∈ Hom(V ⊗k
o , V o ) in Hom(V ⊗k+1
o
, C) is
f k+1 (Φ 1 , · · · , Φ k+1 ) = ω( ˆ
m k (Φ 1 , · · · , Φ k ), Φ k+1 ) .
The cyclic symmetry (4.1) implies that ˆ
m k preserves the inner product. For example,
ω( ˆ
m 2 (Φ 1 , Φ 2 ), Φ 3 ) = f 3 (Φ 1 , Φ 2 , Φ 3 ) = (−1)
|Φ 1 |(|Φ 2 |+|Φ 3 |) f 3 (Φ 2 , Φ 3 , Φ 1 )
= (−1)
|Φ 1 |(|Φ 2 |+|Φ 3 |) ω( ˆ
m 2 (Φ 2 , Φ 3 ), Φ 1 )
= ω(Φ 1 , ˆ
m 2 (Φ 2 , Φ 3 )) ,
where we used (4.1) in the second identity and |Φ| denoted the ghost number of the
string field Φ. In close analogy to the closed string, the collection of maps { ˆ
m k } can
be lifted to a coderivation on
T V o =
∞
n=0
V o [1]
⊗n .
Then the bracket [M, −], M =
m n , on Coder(T V o ) induces a Hochschild
coboundary operator d H on CC k ≡ Hom cycl (V k
o , V o ). For the cubic theory at hand,
ˆ
m 1 = Q and ˆ
m 2 = ∗, we have
d H = (−1)
k+1 Q + δ ,
where
(Qf k )(Φ 1 , . . . , Φ k ) =
k
i=1
(−1)
Φ 1 +...+Φ i−1 f k (Φ 1 , . . . , QΦ i , . . . , Φ k )
and
(δf k )(Φ 1 , · · · , Φ k+1 ) =
k
i=1
(−1)
i f k (Φ 1 , · · · , Φ i ∗ Φ i+1 , · · · , Φ k+1 )
+(−1)
Φ 1 (Φ 2 +···+Φ k+1 ) f k (Φ 2 , · · · , Φ k , Φ k+1 ∗ Φ 1 ) .
From the above, it is clear that d H takes cyclic elements to cyclic elements.
Furthermore, d H squares to zero since Q 2 = δ 2 = [Q, δ] = 0. The cyclic
cohomology is then the cohomology of d H in CC ∗ . It turns out that this cohomology
is isomorphic to the physical cohomology of closed strings. More precisely:
Let I [φ, c, b] be the world-sheet action defining open string world-sheet CFTmorphism between geometric and algebraic vertices, V o the corresponding module