3.7 Uniqueness and Background Independence
49
3.7
Uniqueness and Background Independence
In this section, we will discuss the question whether the CFT morphism that
determines the algebraic string field theory vertices and, therefore, the particular
realization of an L ∞ -algebra, is unique up to an equivalence. This important
question in string theory is, however, not directly relevant to the rest of this book and
can thus be skipped by the reader who is more interested in the algebraic aspects of
string theory.
There is a definite answer to the above question as far as continuous deformations
of a given SFT are concerned. Let us denote the classical closed string vertices by
l n ≡ l 0
n . The bracket [−, −] on Coder(SA) (see Appendix A) induces the Chevalley–
Eilenberg differential d C = [L, −] on the deformation complex. On the other
hand, due to the isomorphism Hom(V ∧k
c , V ) → Hom(V ∧k+1
c
, C) this induces a
differential d c on the cyclic complex. Thus, any consistent infinitesimal deformation
Δl = {Δl n } n∈N of the L ∞ -structure {l n } n∈N (Δ not to be confused with the BV
operator) is d c -closed, d c (Δl) = 0. By carefully analyzing continuous deformations
of the CFT morphism, one arrives at the following result:
Let I [φ, c, ¯
c, b, ¯
b] be the closed string world-sheet CFT action on M defining the
morphism from the BV algebra of the geometric vertices {ν n } to the BV algebra of
algebraic vertices s n , V c the corresponding module of conformal tensors (closed
string Hilbert space) and Q c the BRST differential. Then
coh(d c ) = ∅ .
Remark 3.5 The above result has important consequences for the background independence of classical closed string field theory. It is intimately related to the nature
of equivalence classes of L ∞ -algebras. Clearly, L ∞ -field redefinitions preserve the
L ∞ -structure and can be interpreted as [·, ·]-gauge symmetry transformations if they
are continuously connected to the identity. On the other hand, field redefinitions
include shifts in the closed string background. These are easily seen to be L ∞ -
isomorphisms. For a given homotopy algebra we can then consider a non-vanishing
Maurer–Cartan element Ψ 0 with L(e Ψ 0 ) = 0, in order to construct a twisted
homotopy algebra L Ψ 0 = E(−Ψ 0 ) ◦ L ◦ E(Ψ 0 ) upon conjugation. The definition of
E(−Ψ 0 ) is given in Appendix A. The background independence then would imply
that the structure maps of the minimal model obtained from this homotopy algebra
are equivalent to the perturbative S-matrix elements of the world-sheet CFT in the
new background, see Fig. 3.7.
Since coh(d c ) = ∅, the L ∞ -algebra L CF T (Ψ 0 ) obtained from the world-sheet
theory in the new background Ψ 0 corresponding to the MC-element Ψ 0 is L ∞ -
equivalent to the L ∞ -algebra L Ψ 0 obtained from L by conjugation, i.e.
L CF T (Ψ 0 ) = K
−1
◦ L Ψ 0 ◦ K ,
49
3.7
Uniqueness and Background Independence
In this section, we will discuss the question whether the CFT morphism that
determines the algebraic string field theory vertices and, therefore, the particular
realization of an L ∞ -algebra, is unique up to an equivalence. This important
question in string theory is, however, not directly relevant to the rest of this book and
can thus be skipped by the reader who is more interested in the algebraic aspects of
string theory.
There is a definite answer to the above question as far as continuous deformations
of a given SFT are concerned. Let us denote the classical closed string vertices by
l n ≡ l 0
n . The bracket [−, −] on Coder(SA) (see Appendix A) induces the Chevalley–
Eilenberg differential d C = [L, −] on the deformation complex. On the other
hand, due to the isomorphism Hom(V ∧k
c , V ) → Hom(V ∧k+1
c
, C) this induces a
differential d c on the cyclic complex. Thus, any consistent infinitesimal deformation
Δl = {Δl n } n∈N of the L ∞ -structure {l n } n∈N (Δ not to be confused with the BV
operator) is d c -closed, d c (Δl) = 0. By carefully analyzing continuous deformations
of the CFT morphism, one arrives at the following result:
Let I [φ, c, ¯
c, b, ¯
b] be the closed string world-sheet CFT action on M defining the
morphism from the BV algebra of the geometric vertices {ν n } to the BV algebra of
algebraic vertices s n , V c the corresponding module of conformal tensors (closed
string Hilbert space) and Q c the BRST differential. Then
coh(d c ) = ∅ .
Remark 3.5 The above result has important consequences for the background independence of classical closed string field theory. It is intimately related to the nature
of equivalence classes of L ∞ -algebras. Clearly, L ∞ -field redefinitions preserve the
L ∞ -structure and can be interpreted as [·, ·]-gauge symmetry transformations if they
are continuously connected to the identity. On the other hand, field redefinitions
include shifts in the closed string background. These are easily seen to be L ∞ -
isomorphisms. For a given homotopy algebra we can then consider a non-vanishing
Maurer–Cartan element Ψ 0 with L(e Ψ 0 ) = 0, in order to construct a twisted
homotopy algebra L Ψ 0 = E(−Ψ 0 ) ◦ L ◦ E(Ψ 0 ) upon conjugation. The definition of
E(−Ψ 0 ) is given in Appendix A. The background independence then would imply
that the structure maps of the minimal model obtained from this homotopy algebra
are equivalent to the perturbative S-matrix elements of the world-sheet CFT in the
new background, see Fig. 3.7.
Since coh(d c ) = ∅, the L ∞ -algebra L CF T (Ψ 0 ) obtained from the world-sheet
theory in the new background Ψ 0 corresponding to the MC-element Ψ 0 is L ∞ -
equivalent to the L ∞ -algebra L Ψ 0 obtained from L by conjugation, i.e.
L CF T (Ψ 0 ) = K
−1
◦ L Ψ 0 ◦ K ,