3.6 Coalgebra Description
47
3.6
Coalgebra Description
In the scaling limit ¯
h → 0 with c ≡ ¯
h 1/2 Ψ finite, (3.28) reproduces the defining
equations of a strongly homotopy Lie or L ∞ -algebra, described in more detail in
Appendix A and in Part II. A convenient way to define L ∞ -algebras is the cobar
construction where one collects the various maps into a single object
L =
∞
n=1
i+j =n
σ
(l i ∧ 1
∧j ) ◦ σ ,
(3.29)
where
σ indicates the sum over all unshuffles σ ∈ Σ n with
σ 1 < · · · < σ i , σ i+1 < · · · < σ n .
The permutation σ denotes the map that sends c 1 ∧ · · · ∧ c n to (−1) c σ 1 ∧ · · · ∧
c σ n . As reviewed in Appendix A, L is a coderivation of SV =
∞
n=0 V ∧n . The
space Coder(SV ) is equipped with the canonical bracket [−, −] defined through
the composition of maps, cf. Appendix A. For ¯
h = 0, (3.28) is equivalent to
L
2
=
1
2
[L, L] = 0,
which is precisely the defining relation for an L ∞ -algebra. We recognize a theory
with vertices given by a cyclic L ∞ -algebra L as a classical closed string theory.
Remark 3.4 The construction of the minimal model map for classical closed string
theory proceeds in close analogy with the point particle. We already noticed in
Sect. 2.2 that the construction of the minimal model is equivalent to the construction
of tree-level S-matrix amplitudes via Feynman rules. First one chooses a gauge
(e.g., the Siegel gauge b
+
0 ψ = 0) so that one can define a propagator Q −1 . With
the aid of the latter, we construct all possible rooted trees with vertices labeled by
l n := π 1 ◦ L ◦ i n and internal edges labeled by the propagator. Here π 1 stands
for the projection to maps with one output. The collection of all these trees, with
inputs and the output restricted to the cohomology H , then defines the multilinear
maps ˜
l n = π 1 ◦ ˜
L ◦ i n in complete analogy with Sect. 2.2. Thus, ˜
l n represents the
n + 1-string S-matrix amplitudes. Furthermore, ˜
L 2 = 0 so that the map
F : (H, ˜
L) → (V , L)
is a quasi-isomorphism between L ∞ -algebras. The proof of these claims goes
analogically with the point particle case in Sect. 2.2, with
c = F (c
P ) = π 1 (1 − l
−1
1 δ)
−1 (e
∧c P − 1) ,
47
3.6
Coalgebra Description
In the scaling limit ¯
h → 0 with c ≡ ¯
h 1/2 Ψ finite, (3.28) reproduces the defining
equations of a strongly homotopy Lie or L ∞ -algebra, described in more detail in
Appendix A and in Part II. A convenient way to define L ∞ -algebras is the cobar
construction where one collects the various maps into a single object
L =
∞
n=1
i+j =n
σ
(l i ∧ 1
∧j ) ◦ σ ,
(3.29)
where
σ indicates the sum over all unshuffles σ ∈ Σ n with
σ 1 < · · · < σ i , σ i+1 < · · · < σ n .
The permutation σ denotes the map that sends c 1 ∧ · · · ∧ c n to (−1) c σ 1 ∧ · · · ∧
c σ n . As reviewed in Appendix A, L is a coderivation of SV =
∞
n=0 V ∧n . The
space Coder(SV ) is equipped with the canonical bracket [−, −] defined through
the composition of maps, cf. Appendix A. For ¯
h = 0, (3.28) is equivalent to
L
2
=
1
2
[L, L] = 0,
which is precisely the defining relation for an L ∞ -algebra. We recognize a theory
with vertices given by a cyclic L ∞ -algebra L as a classical closed string theory.
Remark 3.4 The construction of the minimal model map for classical closed string
theory proceeds in close analogy with the point particle. We already noticed in
Sect. 2.2 that the construction of the minimal model is equivalent to the construction
of tree-level S-matrix amplitudes via Feynman rules. First one chooses a gauge
(e.g., the Siegel gauge b
+
0 ψ = 0) so that one can define a propagator Q −1 . With
the aid of the latter, we construct all possible rooted trees with vertices labeled by
l n := π 1 ◦ L ◦ i n and internal edges labeled by the propagator. Here π 1 stands
for the projection to maps with one output. The collection of all these trees, with
inputs and the output restricted to the cohomology H , then defines the multilinear
maps ˜
l n = π 1 ◦ ˜
L ◦ i n in complete analogy with Sect. 2.2. Thus, ˜
l n represents the
n + 1-string S-matrix amplitudes. Furthermore, ˜
L 2 = 0 so that the map
F : (H, ˜
L) → (V , L)
is a quasi-isomorphism between L ∞ -algebras. The proof of these claims goes
analogically with the point particle case in Sect. 2.2, with
c = F (c
P ) = π 1 (1 − l
−1
1 δ)
−1 (e
∧c P − 1) ,