Open-Closed BV Equation
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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 open-closed 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 open-closed 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.