50
3 String Theory
Fig. 3.7 A schematic representation of the background independence. The vertical arrow on the
left represents a deformation of the background implemented by a deformation ΔI (Ψ 0 ) of the
CFT morphism
where K is an L ∞ -isomorphism continuously connected to the identity. However,
since the L ∞ equivalence classes identify all continuously connected closed
backgrounds, we cannot conclude from the above that L CF T (Ψ 0 ) and L Ψ 0 actually
describe the same background. The necessary refinement for this is then provided
by the open-closed homotopy algebra described in the next section, which implies
that K = 1. We should note, however, that generic on-shell closed string
backgrounds may not be continuously connected to each other and furthermore do
not preserve V c . This puts a limitation on applicability of the proof of background
independence given here.
3.8
Summary, Comments, and Remarks Towards Part II
It should be clear from the discussion in this chapter, that abstractly, quantum closed
string field theory can be defined as a morphism from the BV algebra of singular
chains on the moduli space of punctured closed Riemann surfaces to the BV algebra
of functions on a differential graded vector space V , endowed with a degree -
1 symplectic form ω, i.e. on the state space of the first quantized closed string.
Such a morphism realized by a world-sheet conformal field theory carries geometric
vertices ν g,n assembled into a solution of the quantum BV master equation (3.17)
on the space of chains on the moduli space into algebraic vertices f g,n defining
the string field theory action. Geometric vertices themselves are determined by a
decomposition of the moduli space.
The relation to Part II is as follows. The decomposition of the moduli space
can be interpreted as a morphism between two odd (aka twisted) modular operads,
see Definition 6.24. This morphism goes from the Feynman transform, cf. Definition 7.9, of the modular commutative operad in Definition 6.18 into the odd modular
operad of chains on the moduli space. On the latter there is an obvious action of
permutations of punctures. The operadic operations (3.15) and (3.16) are induced
by the twisted sewing and self-sewing of punctured closed Riemann surfaces.
We know from Theorem 8.2 that such a morphism is equivalent to a solution of
the quantum BV equation on the respective space of invariants under permutations.
In the case of the commutative operad, this is the complex of chains invariant under
3 String Theory
Fig. 3.7 A schematic representation of the background independence. The vertical arrow on the
left represents a deformation of the background implemented by a deformation ΔI (Ψ 0 ) of the
CFT morphism
where K is an L ∞ -isomorphism continuously connected to the identity. However,
since the L ∞ equivalence classes identify all continuously connected closed
backgrounds, we cannot conclude from the above that L CF T (Ψ 0 ) and L Ψ 0 actually
describe the same background. The necessary refinement for this is then provided
by the open-closed homotopy algebra described in the next section, which implies
that K = 1. We should note, however, that generic on-shell closed string
backgrounds may not be continuously connected to each other and furthermore do
not preserve V c . This puts a limitation on applicability of the proof of background
independence given here.
3.8
Summary, Comments, and Remarks Towards Part II
It should be clear from the discussion in this chapter, that abstractly, quantum closed
string field theory can be defined as a morphism from the BV algebra of singular
chains on the moduli space of punctured closed Riemann surfaces to the BV algebra
of functions on a differential graded vector space V , endowed with a degree -
1 symplectic form ω, i.e. on the state space of the first quantized closed string.
Such a morphism realized by a world-sheet conformal field theory carries geometric
vertices ν g,n assembled into a solution of the quantum BV master equation (3.17)
on the space of chains on the moduli space into algebraic vertices f g,n defining
the string field theory action. Geometric vertices themselves are determined by a
decomposition of the moduli space.
The relation to Part II is as follows. The decomposition of the moduli space
can be interpreted as a morphism between two odd (aka twisted) modular operads,
see Definition 6.24. This morphism goes from the Feynman transform, cf. Definition 7.9, of the modular commutative operad in Definition 6.18 into the odd modular
operad of chains on the moduli space. On the latter there is an obvious action of
permutations of punctures. The operadic operations (3.15) and (3.16) are induced
by the twisted sewing and self-sewing of punctured closed Riemann surfaces.
We know from Theorem 8.2 that such a morphism is equivalent to a solution of
the quantum BV equation on the respective space of invariants under permutations.
In the case of the commutative operad, this is the complex of chains invariant under