62
4 Open and Closed Strings
of conformal tensors (open string Hilbert space) and Q the open string BRST differential. Then, the only nontrivial infinitesimal deformations of this CFT-morphism
preserving V o are infinitesimal deformations of the closed string background in the
relative cohomology of Q c
coh(d H ) ∼ = coh(b
− , Q c ) .
(4.6)
This can alternatively be expressed by saying that the only way to deform open
string theory is to place it in a nontrivial closed string background, e.g. a curved
metric on R 26 that solves the closed string equations of motion.
Remark 4.2 A particular class of deformations that do not preserve Q o are the
shifts in the open string background. Such transformations are, however, d H -exact
as are all field redefinitions of φ. From a physics perspective, the interesting fact
implied by the above result is that open string theory already contains the complete
information about the particle content of closed string theory.
4.3
Summary, Comments, and Remarks Towards Part II
Here, we comment only on open strings and leave the comments on the open-closed
theory to Sect. 5.3. Most of the discussion below will be parallel to the case of the
closed string in Sect. 3.8. Concerning physics, the main difference is that there is no
consistent open string filed theory beyond the classical level, i.e. the one containing
only interaction through a disc with (at least three) punctures on the boundary. A
consistent quantum field theory of open strings needs the inclusion of closed strings.
In the previous sections, we described the classically consistent Witten’s open
SFT (4.4), which contains only cubic interaction, and where the resulting algebraic
structure is that of a differential graded cyclic associative algebra. Classically
consistent deformations of this open string theory lead us to a generalization of
associative structures, i.e. to their homotopy versions, which are the cyclic A ∞ -
algebras. Obviously, the vertices are graded symmetric only with respect to cyclic
permutations of punctures on the disc boundary.
In the operadic language of Part II, A ∞ -algebras are algebras over the cobar
construction of the cyclic associative operad of Example 6.18. Here, the difference
against the cyclic commutative operad is the absence of a nontrivial action of
all permutations. The consequence is the above-mentioned graded symmetry of
the corresponding algebraic vertices f k and of the corresponding products m k , in
Sect. 4.2, under the cyclic permutations only.
Concerning the construction of Witten’s open SFT in the spirit of Fig. 1.1, in the
upper left corner we would have the cobar construction of the cyclic associative
operad. In the upper right corner we would have the odd cyclic operad of cyclic
chains (4.1) on the moduli space of discs with punctured boundaries, the operadic
operations being defined by sewing discs via punctures as in (4.2). The horizontal
arrow, a morphism of these two operads, would be given by a decomposition of the
4 Open and Closed Strings
of conformal tensors (open string Hilbert space) and Q the open string BRST differential. Then, the only nontrivial infinitesimal deformations of this CFT-morphism
preserving V o are infinitesimal deformations of the closed string background in the
relative cohomology of Q c
coh(d H ) ∼ = coh(b
− , Q c ) .
(4.6)
This can alternatively be expressed by saying that the only way to deform open
string theory is to place it in a nontrivial closed string background, e.g. a curved
metric on R 26 that solves the closed string equations of motion.
Remark 4.2 A particular class of deformations that do not preserve Q o are the
shifts in the open string background. Such transformations are, however, d H -exact
as are all field redefinitions of φ. From a physics perspective, the interesting fact
implied by the above result is that open string theory already contains the complete
information about the particle content of closed string theory.
4.3
Summary, Comments, and Remarks Towards Part II
Here, we comment only on open strings and leave the comments on the open-closed
theory to Sect. 5.3. Most of the discussion below will be parallel to the case of the
closed string in Sect. 3.8. Concerning physics, the main difference is that there is no
consistent open string filed theory beyond the classical level, i.e. the one containing
only interaction through a disc with (at least three) punctures on the boundary. A
consistent quantum field theory of open strings needs the inclusion of closed strings.
In the previous sections, we described the classically consistent Witten’s open
SFT (4.4), which contains only cubic interaction, and where the resulting algebraic
structure is that of a differential graded cyclic associative algebra. Classically
consistent deformations of this open string theory lead us to a generalization of
associative structures, i.e. to their homotopy versions, which are the cyclic A ∞ -
algebras. Obviously, the vertices are graded symmetric only with respect to cyclic
permutations of punctures on the disc boundary.
In the operadic language of Part II, A ∞ -algebras are algebras over the cobar
construction of the cyclic associative operad of Example 6.18. Here, the difference
against the cyclic commutative operad is the absence of a nontrivial action of
all permutations. The consequence is the above-mentioned graded symmetry of
the corresponding algebraic vertices f k and of the corresponding products m k , in
Sect. 4.2, under the cyclic permutations only.
Concerning the construction of Witten’s open SFT in the spirit of Fig. 1.1, in the
upper left corner we would have the cobar construction of the cyclic associative
operad. In the upper right corner we would have the odd cyclic operad of cyclic
chains (4.1) on the moduli space of discs with punctured boundaries, the operadic
operations being defined by sewing discs via punctures as in (4.2). The horizontal
arrow, a morphism of these two operads, would be given by a decomposition of the