3.8 Summary, Comments, and Remarks Towards Part II
51
the permutations of punctures as described in Sect. 3.3. This is the horizontal arrow
in Fig. 1.1 providing the geometric vertices ν g,n .
The BV algebra morphism given by formula (3.22) that sends a geometric vertex
ν g,n into the algebraic vertex f
g
n , when extended from invariant cochains to all cochains, can again be interpreted as a morphism of two odd modular operads going
from the odd modular operad of cochains on the moduli space to the endomorphism
operad, i.e. as a representation of the former one. This is the vertical arrow in Fig. 1.1
providing the algebraic vertices f
g
n .
Finally, the composition of the two odd modular operad morphisms gives the
algebra over the Feynman transform of the modular commutative operad, i.e. the
diagonal arrow in our Fig. 1.1. As such, it is equivalent to a solution to the quantum
BV master equation on the space of complex-valued graded symmetric maps on
the CFT state space, described by algebraic vertices f
g
n : V ⊗n → C, due to
Theorem 8.3 of Part II. This is, of course, guaranteed by the construction that uses
the composition of the horizontal and vertical arrows in Fig. 1.1. The result is the
quantum closed SFT action in Zwiebach’s construction. Using the odd symplectic
form, the algebraic vertices f
g
n can be turned into degree 1 n-ary brackets l g,n :
V ⊗n → V . The resulting homotopy algebra is the loop homotopy algebra (aka
quantum L ∞ -algebra) described in Sect. 3.5 of Part I and again from the operadic
point of view in Sect. 8.2 of Part II. The graded symmetry of the vertices/brackets
is due to the trivial action of permutations on the commutative operad.
The same algebraic structure, i.e. the loop homotopy algebra can be reinterpreted
in terms of an IBL ∞ -algebra. The main difference is that the BV operator now,
contrary to the loop homotopy algebra formulation, becomes one of the operations.
In this formalism the quantum BV master equation ¯
hΔ +
1
2 {S, S} = 0 equivalently
translates to the nilpotency of the full BV operator ¯
hΔ + {S, −}. IBL ∞ -algebras,
as they are used here, are described in Appendix to Part I, cf. Sect. B.2. A more
rigorous alternate version is given in Part II, Sect. 8.3.
All constructions described above have their classical analogue. In STF it is the
scaling limit ¯
h → 0, c ≡ ¯
h 1/2 Ψ , cf. Sect. 3.6, leading to the classical BV action
and the corresponding L ∞ -algebra. Since the powers of ¯
h count loop contributions,
we may forget about loops.
In the operadic language of Part II, this scaling limit corresponds to taking
the forgetful functor from the modular commutative operad (the right adjoint to
the modular envelope functor (6.18), cf. Sect. 6.4), to the cyclic commutative
operad. Roughly speaking, as above, we forget about loops everywhere and consider
only genus 0 corollas. The horizontal arrow in Fig. 1.1 now goes from the cobar
construction of the latter (here we consider only the trees generated by corollas
instead of all graphs) into the odd cyclic operad of the genus 0 closed Riemannian
surfaces with punctures, where the self-sewing operation is now forbidden. The
vertical arrow is as before given by conformal field theory, however, now evaluated
only at zero genus. The composition, i.e. the diagonal arrow in Fig. 1.1, is the
classical (genus 0) closed string field theory.
51
the permutations of punctures as described in Sect. 3.3. This is the horizontal arrow
in Fig. 1.1 providing the geometric vertices ν g,n .
The BV algebra morphism given by formula (3.22) that sends a geometric vertex
ν g,n into the algebraic vertex f
g
n , when extended from invariant cochains to all cochains, can again be interpreted as a morphism of two odd modular operads going
from the odd modular operad of cochains on the moduli space to the endomorphism
operad, i.e. as a representation of the former one. This is the vertical arrow in Fig. 1.1
providing the algebraic vertices f
g
n .
Finally, the composition of the two odd modular operad morphisms gives the
algebra over the Feynman transform of the modular commutative operad, i.e. the
diagonal arrow in our Fig. 1.1. As such, it is equivalent to a solution to the quantum
BV master equation on the space of complex-valued graded symmetric maps on
the CFT state space, described by algebraic vertices f
g
n : V ⊗n → C, due to
Theorem 8.3 of Part II. This is, of course, guaranteed by the construction that uses
the composition of the horizontal and vertical arrows in Fig. 1.1. The result is the
quantum closed SFT action in Zwiebach’s construction. Using the odd symplectic
form, the algebraic vertices f
g
n can be turned into degree 1 n-ary brackets l g,n :
V ⊗n → V . The resulting homotopy algebra is the loop homotopy algebra (aka
quantum L ∞ -algebra) described in Sect. 3.5 of Part I and again from the operadic
point of view in Sect. 8.2 of Part II. The graded symmetry of the vertices/brackets
is due to the trivial action of permutations on the commutative operad.
The same algebraic structure, i.e. the loop homotopy algebra can be reinterpreted
in terms of an IBL ∞ -algebra. The main difference is that the BV operator now,
contrary to the loop homotopy algebra formulation, becomes one of the operations.
In this formalism the quantum BV master equation ¯
hΔ +
1
2 {S, S} = 0 equivalently
translates to the nilpotency of the full BV operator ¯
hΔ + {S, −}. IBL ∞ -algebras,
as they are used here, are described in Appendix to Part I, cf. Sect. B.2. A more
rigorous alternate version is given in Part II, Sect. 8.3.
All constructions described above have their classical analogue. In STF it is the
scaling limit ¯
h → 0, c ≡ ¯
h 1/2 Ψ , cf. Sect. 3.6, leading to the classical BV action
and the corresponding L ∞ -algebra. Since the powers of ¯
h count loop contributions,
we may forget about loops.
In the operadic language of Part II, this scaling limit corresponds to taking
the forgetful functor from the modular commutative operad (the right adjoint to
the modular envelope functor (6.18), cf. Sect. 6.4), to the cyclic commutative
operad. Roughly speaking, as above, we forget about loops everywhere and consider
only genus 0 corollas. The horizontal arrow in Fig. 1.1 now goes from the cobar
construction of the latter (here we consider only the trees generated by corollas
instead of all graphs) into the odd cyclic operad of the genus 0 closed Riemannian
surfaces with punctures, where the self-sewing operation is now forbidden. The
vertical arrow is as before given by conformal field theory, however, now evaluated
only at zero genus. The composition, i.e. the diagonal arrow in Fig. 1.1, is the
classical (genus 0) closed string field theory.