부산대학교 도서관

The master equation shows that the string products form an algebra overthe Feynman transform of $\Cal{QC}$. B. Zwiebach constructed stringproducts on the Hilbert space of closed string field theory satisfyingthe master equation, which reflected the structure of the set$\Cal{QC}$ of diffeomorphism classes of Riemann surfaces of arbitrarygenera with labeled holes. The authors, M. Doubek and M. Markl, proved$\Cal{QC} \cong {\rm Mod}(\Cal{C}{\rm om})$; i.e., $\Cal{QC}$ is themodular completion of its cyclic suboperad $\Cal{C}{\rm om} \subset\Cal{QC}$ consisting of Riemann surfaces of genus 0.\par Later, Doubek and Markl proved a similar statement for open strings byidentifying the modular operad $\Cal{QO}$ of diffeomorphism classes ofRiemann surfaces with marked open boundaries with the modularcompletion of its genus 0 part $\Cal{A}{\rm ss}$ by establishing$\Cal{QO} \cong {\rm Mod}(\Cal{A}{\rm ss})$. In the paper under reviewthey argue that $\Cal{QO}$ is the symmetrization of a more elementaryobject $\Cal{Q}\underline{\Cal{O}}$ with the structure of anon-$\Sigma$ modular operad since the previous isomorphism follows theisomorphism $\Cal{Q}\underline{\Cal{O}} \cong \underline{{\rmMod}}(\underline{\Cal{A}{\rm ss}})$, where $\underline{\Cal{A}{\rmss}}$ is the non-$\Sigma$ version of the associative cyclic operad and$\underline{\rm Mod}(-)$ the non-$\Sigma$ modular completion functor.These isomorphisms establish analogs for the combined theory of openand closed strings. The central object is the set$\Cal{Q}\underline{\Cal{O}}C$ of diffeomorphism classes of Riemannsurfaces with both open and closed inputs; it behaves as anon-$\Sigma$-modular operad in the open and as an ordinary modularoperad in the closed inputs, which are called modular hybrids.\par The set $\Cal{Q}\underline{\Cal{O}}C$ is not the modular completion ofits genus 0 part $\underline{\Cal{O}}\Cal{C}$, but instead, it is thequotient $\Cal{Q}\underline{\Cal{O}}C \cong \underline{\rm M}{\rmod}(\underline{\Cal{O}}\Cal{C})/{\rm Cardy}$ of this completion by theCardy conditions, where Cardy conditions involve both the open andclosed interactions. Isomorphisms restricted to closed and open partsgive $\Cal{QC} \cong{\rm Mod}(\Cal{C}\text{om})$ and$\Cal{Q}\underline{\Cal{O}} \cong \underline{\rmMod}(\underline{\Cal{A}{\rm ss}})$, respectively. The authors also givea purely combinatorial proof characterizing algebras over a version of$\Cal{Q}\underline{\Cal{O}}C$ in terms of Frobenius algebra morphisms.\par Doubek and Markl establish three different versions of$\Cal{Q}\underline{\Cal{O}}C \cong \underline{\rm M}{\rmod}(\underline{\Cal{O}}\Cal{C})/{\rm Cardy}$. Furthermore, the cyclichybrid $\underline{\Cal{O}}\Cal{C}$ contains the stable and ${\rm KP}$subhybrids $\underline{\Cal{O}}\Cal{C}\supset\underline{\Cal{O}}\Cal{C}_{\rm st}\supset\underline{\Cal{O}}\Cal{C}_{\rm KP}$. A substantial part of thispaper is devoted to proving that the inclusions $\underline{\rm M}{\rmod}(\underline{\Cal{O}}\Cal{C}) \supseteq \underline{\rm M}{\rmod}(\underline{\Cal{O}}\Cal{C}_{\rm st}) \supseteq \underline{\rmM}{\rm od}(\underline{\Cal{O}}\Cal{C}_{\rm KP})$ hold. Thus, the stableand ${\rm KP}$ cases are treated as the restricted versions of theordinary one.