5.3 Summary, Comment, and Remarks Towards Part II
73
open string field theory if the closed string background satisfies the classical closed
string equations of motion. The inverse assertion does not follow from (5.9).
However, it holds true for infinitesimal closed string deformations. More precisely,
upon linearizing equation (5.9) in c ∈ V c we get
n cl (L cl (c)) = d H (n cl (c)) ,
(5.11)
where L cl ∈ Coder cycl (T V c ) is determined by l cl = π 1 ◦ L cl ∈ Hom cycl (T V c ), the
closed string vertices of genus zero (see Sect. A.2). In string field theory, the vertex
with just one input (l cl ) 1 is the closed string BRST operator Q c . Thus Eq. (5.11) is
equivalent to
n cl (Q c (c)) = d H (n cl (c)) ,
that is, n cl ◦ i 1 induces a chain map from the BRST complex of closed strings to
the cyclic Hochschild complex of open strings. The cohomology of Q c (BRST
cohomology) defines the space of physical states, whereas the cohomology of d H
(cyclic cohomology) characterizes the infinitesimal deformations of the initial open
string field theory m cl as discussed in Sect. 4.2. There we have seen that the
BRST cohomology of closed strings is indeed isomorphic to the cyclic Hochschild
cohomology of open strings.
5.3
Summary, Comment, and Remarks Towards Part II
The detailed discussion of the open-closed SFT in Sect. 5.2 is taken from the point
of view of IBL ∞ -algebras and their morphisms. Nevertheless, the description of
geometric vertices ν
b,g
n,m as well of the open-closed SFT action in (5.1) can directly
be interpreted in the spirit of Fig. 1.1. As already noticed, this construction would
be a rather straightforward combination of the open and closed theories.
The starting point would be the open-closed modular operad. Since we are not
going to describe it in full detail, we at least indicate its nature here. It is a twocolored operad with colors corresponding to open and closed strings (punctures),
respectively. In terms of corollas used in Part II, we would have to consider corollas
with two kinds of legs. This can be pictured informally as follows. Think about
bordered Riemann surfaces with punctures both on the boundaries (open ones) as
well as in the bulk (closed ones). We allow also boundaries with no punctures. Now,
forget about all the structure but genus, number of boundaries, number of closed
punctures, and the distribution of punctures on the boundaries. What we have are
corollas remembering the genus and having their legs colored as open or closed,
the open ones grouped together accordingly to their respective boundaries. There is
an obvious two-fold action of permutations, we can either permute closed punctures
between themselves or open punctures between themselves. We allow for permuting
of punctures between different boundaries, but we do not allow for permutations
73
open string field theory if the closed string background satisfies the classical closed
string equations of motion. The inverse assertion does not follow from (5.9).
However, it holds true for infinitesimal closed string deformations. More precisely,
upon linearizing equation (5.9) in c ∈ V c we get
n cl (L cl (c)) = d H (n cl (c)) ,
(5.11)
where L cl ∈ Coder cycl (T V c ) is determined by l cl = π 1 ◦ L cl ∈ Hom cycl (T V c ), the
closed string vertices of genus zero (see Sect. A.2). In string field theory, the vertex
with just one input (l cl ) 1 is the closed string BRST operator Q c . Thus Eq. (5.11) is
equivalent to
n cl (Q c (c)) = d H (n cl (c)) ,
that is, n cl ◦ i 1 induces a chain map from the BRST complex of closed strings to
the cyclic Hochschild complex of open strings. The cohomology of Q c (BRST
cohomology) defines the space of physical states, whereas the cohomology of d H
(cyclic cohomology) characterizes the infinitesimal deformations of the initial open
string field theory m cl as discussed in Sect. 4.2. There we have seen that the
BRST cohomology of closed strings is indeed isomorphic to the cyclic Hochschild
cohomology of open strings.
5.3
Summary, Comment, and Remarks Towards Part II
The detailed discussion of the open-closed SFT in Sect. 5.2 is taken from the point
of view of IBL ∞ -algebras and their morphisms. Nevertheless, the description of
geometric vertices ν
b,g
n,m as well of the open-closed SFT action in (5.1) can directly
be interpreted in the spirit of Fig. 1.1. As already noticed, this construction would
be a rather straightforward combination of the open and closed theories.
The starting point would be the open-closed modular operad. Since we are not
going to describe it in full detail, we at least indicate its nature here. It is a twocolored operad with colors corresponding to open and closed strings (punctures),
respectively. In terms of corollas used in Part II, we would have to consider corollas
with two kinds of legs. This can be pictured informally as follows. Think about
bordered Riemann surfaces with punctures both on the boundaries (open ones) as
well as in the bulk (closed ones). We allow also boundaries with no punctures. Now,
forget about all the structure but genus, number of boundaries, number of closed
punctures, and the distribution of punctures on the boundaries. What we have are
corollas remembering the genus and having their legs colored as open or closed,
the open ones grouped together accordingly to their respective boundaries. There is
an obvious two-fold action of permutations, we can either permute closed punctures
between themselves or open punctures between themselves. We allow for permuting
of punctures between different boundaries, but we do not allow for permutations