5
Open-Closed BV Equation
An important conclusion at the end of the previous chapter is that the unique
consistent infinitesimal deformation of classical open string field theory is an openclosed vertex with one closed string puncture, cf. the italicized paragraph before
formula (4.6). To continue, we want to analyze the consistency of a generic openclosed vertex as in Fig. 4.1. For this, we first need to review the various sewing
operations on Riemann surfaces with labeled boundaries and labeled punctures
in the bulk as well as on the boundaries. Concerning the sewing of closed string
punctures, we have already discussed it in Sect. 3.3. The sewing of two open string
punctures on different vertices with the corresponding cyclic complex was treated
in the previous section. What remains, is the sewing of open string punctures on the
same surface. Acting with the geometric operator Δ on two open punctures on the
same boundary of a given Riemann surface Σ increases the number of boundaries
by one and decreases the number of open string punctures by two while leaving the
genus invariant. In contrast, acting on two punctures on different boundaries of the
same surface increases the genus by one, decreases the number of boundaries by one
and decreases the number of punctures by two. A detailed discussion of the various
possible sewings of open-closed surfaces can be found in the literature quoted at the
end of this chapter. Through the present chapter, we use notation and terminology
introduced in the appendix to Part I.
5.1
Open-Closed BV Action
The operations described above can again be packaged into a geometric BV
equation with a degree one bracket {−, −} and a degree one BV operator provided
we consider the singular chain complex with a factor (−1) (m i +1)(m j +1) assigned
under the exchange of boundary i with boundary j of the same surface. Here m i
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_5
65
Open-Closed BV Equation
An important conclusion at the end of the previous chapter is that the unique
consistent infinitesimal deformation of classical open string field theory is an openclosed vertex with one closed string puncture, cf. the italicized paragraph before
formula (4.6). To continue, we want to analyze the consistency of a generic openclosed vertex as in Fig. 4.1. For this, we first need to review the various sewing
operations on Riemann surfaces with labeled boundaries and labeled punctures
in the bulk as well as on the boundaries. Concerning the sewing of closed string
punctures, we have already discussed it in Sect. 3.3. The sewing of two open string
punctures on different vertices with the corresponding cyclic complex was treated
in the previous section. What remains, is the sewing of open string punctures on the
same surface. Acting with the geometric operator Δ on two open punctures on the
same boundary of a given Riemann surface Σ increases the number of boundaries
by one and decreases the number of open string punctures by two while leaving the
genus invariant. In contrast, acting on two punctures on different boundaries of the
same surface increases the genus by one, decreases the number of boundaries by one
and decreases the number of punctures by two. A detailed discussion of the various
possible sewings of open-closed surfaces can be found in the literature quoted at the
end of this chapter. Through the present chapter, we use notation and terminology
introduced in the appendix to Part I.
5.1
Open-Closed BV Action
The operations described above can again be packaged into a geometric BV
equation with a degree one bracket {−, −} and a degree one BV operator provided
we consider the singular chain complex with a factor (−1) (m i +1)(m j +1) assigned
under the exchange of boundary i with boundary j of the same surface. Here m i
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_5
65