36
3 String Theory
Similarly, for the elementary four-vertex one has
Ω(ψ 1 , ˆ
l 3 (ψ 2 , ψ 3 , ψ 3 )) := ˆ
f 4 (ψ 1 , ψ 2 , ψ 3 , ψ 4 ),
where the maps ˆ
l n : V ⊗ V ⊗ · · · ⊗ V → V are graded symmetric, e.g. for n = 2,
ˆ
l 2 (A, B) = (−1)
deg(A)deg(B) ˆ
l 2 (B, A).
The definition of ˆ
f 4 requires more care since it involves an integration over the
part of the moduli space covered by ν 4 described above in (3.11). We will come back
to this in more generality in Sect. 3.3. For now, let us just state that the world-sheet
conformal field theory provides a map from the set of geometric vertices {ν k , k ≥ 3}
to the set of multilinear maps { ˆ
l k , k ≥ 2}. In addition, ∂ → Q will be explained in
Sect. 3.3. The correct generalization of the free, closed string action (3.9) is then
given by
S[ψ] =
1
2
Ω(ψ, Qψ) +
1
3!
Ω(ψ, ˆ
l 2 (ψ, ψ)) +
1
4!
Ω(ψ, ˆ
l 3 (ψ, ψ, ψ)) + · · · .
There are two apparent obstacles in interpreting this action as a BV action.
First, Ω has ghost number −5, while in the BV formalism one usually assumes
a degree −1 odd symplectic form. Second, the physical string fields have ghost
number 2, whereas in the BV formalism the physical fields are taken to have degree
0. Both of these issues can be resolved by defining the degree to be the ghost number
minus two, so that the classical closed string field is a degree zero element in 1
V [−2] :=↓ 2 V . To simplify the notation, we will denote V [−2] by V again when
there is no risk of confusion. The odd symplectic structure of closed string field
theory ω : V ⊗ V → C is then identified as
ω := Ω ◦ (↑
2
⊗ ↑
2 ) = =−, c
−
0 −− ◦ (↑
2
⊗ ↑
2 ) .
Due to the shift and the c
−
0 -insertion, ω is graded anti-symmetric and has degree −1.
Similarly, the maps l n are redefined as l n =↓ 2 ◦ ˆ
l n ◦ ↑ ⊗2n . With this convention the
maps l n all have degree 1.
As mentioned above, an important difference against the point particle involves
the gauge invariance which was trivial for the latter. At the linearized level,
δ Λ ψ = QΛ
with Λ an arbitrary element in V of degree −1, will leave S[ψ] invariant since
Q 2 = 0. Clearly, this gauge invariance is reducible due to Λ → Λ + QΛ , where
Λ has degree −2, etc. This leads to ghosts for ghosts, etc. Luckily this complication
1 See Appendix A.1 and Part II for the definition of ↑and ↓.
3 String Theory
Similarly, for the elementary four-vertex one has
Ω(ψ 1 , ˆ
l 3 (ψ 2 , ψ 3 , ψ 3 )) := ˆ
f 4 (ψ 1 , ψ 2 , ψ 3 , ψ 4 ),
where the maps ˆ
l n : V ⊗ V ⊗ · · · ⊗ V → V are graded symmetric, e.g. for n = 2,
ˆ
l 2 (A, B) = (−1)
deg(A)deg(B) ˆ
l 2 (B, A).
The definition of ˆ
f 4 requires more care since it involves an integration over the
part of the moduli space covered by ν 4 described above in (3.11). We will come back
to this in more generality in Sect. 3.3. For now, let us just state that the world-sheet
conformal field theory provides a map from the set of geometric vertices {ν k , k ≥ 3}
to the set of multilinear maps { ˆ
l k , k ≥ 2}. In addition, ∂ → Q will be explained in
Sect. 3.3. The correct generalization of the free, closed string action (3.9) is then
given by
S[ψ] =
1
2
Ω(ψ, Qψ) +
1
3!
Ω(ψ, ˆ
l 2 (ψ, ψ)) +
1
4!
Ω(ψ, ˆ
l 3 (ψ, ψ, ψ)) + · · · .
There are two apparent obstacles in interpreting this action as a BV action.
First, Ω has ghost number −5, while in the BV formalism one usually assumes
a degree −1 odd symplectic form. Second, the physical string fields have ghost
number 2, whereas in the BV formalism the physical fields are taken to have degree
0. Both of these issues can be resolved by defining the degree to be the ghost number
minus two, so that the classical closed string field is a degree zero element in 1
V [−2] :=↓ 2 V . To simplify the notation, we will denote V [−2] by V again when
there is no risk of confusion. The odd symplectic structure of closed string field
theory ω : V ⊗ V → C is then identified as
ω := Ω ◦ (↑
2
⊗ ↑
2 ) = =−, c
−
0 −− ◦ (↑
2
⊗ ↑
2 ) .
Due to the shift and the c
−
0 -insertion, ω is graded anti-symmetric and has degree −1.
Similarly, the maps l n are redefined as l n =↓ 2 ◦ ˆ
l n ◦ ↑ ⊗2n . With this convention the
maps l n all have degree 1.
As mentioned above, an important difference against the point particle involves
the gauge invariance which was trivial for the latter. At the linearized level,
δ Λ ψ = QΛ
with Λ an arbitrary element in V of degree −1, will leave S[ψ] invariant since
Q 2 = 0. Clearly, this gauge invariance is reducible due to Λ → Λ + QΛ , where
Λ has degree −2, etc. This leads to ghosts for ghosts, etc. Luckily this complication
1 See Appendix A.1 and Part II for the definition of ↑and ↓.