부산대학교 도서관

The master equation shows that the string products form an algebra over the Feynman transform of $\Cal{QC}$. B. Zwiebach constructed string products on the Hilbert space of closed string field theory satisfying the master equation, which reflected the structure of the set $\Cal{QC}$ of diffeomorphism classes of Riemann surfaces of arbitrary genera 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 the modular 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 by identifying the modular operad $\Cal{QO}$ of diffeomorphism classes of Riemann surfaces with marked open boundaries with the modular completion of its genus 0 part $\Cal{A}{\rm ss}$ by establishing $\Cal{QO} \cong {\rm Mod}(\Cal{A}{\rm ss})$. In the paper under review they argue that $\Cal{QO}$ is the symmetrization of a more elementary object $\Cal{Q}\underline{\Cal{O}}$ with the structure of a non-$\Sigma$ modular operad since the previous isomorphism follows the isomorphism $\Cal{Q}\underline{\Cal{O}} \cong \underline{{\rm Mod}}(\underline{\Cal{A}{\rm ss}})$, where $\underline{\Cal{A}{\rm ss}}$ 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 open and closed strings. The central object is the set $\Cal{Q}\underline{\Cal{O}}C$ of diffeomorphism classes of Riemann surfaces with both open and closed inputs; it behaves as a non-$\Sigma$-modular operad in the open and as an ordinary modular operad in the closed inputs, which are called modular hybrids. \par The set $\Cal{Q}\underline{\Cal{O}}C$ is not the modular completion of its genus 0 part $\underline{\Cal{O}}\Cal{C}$, but instead, it is the quotient $\Cal{Q}\underline{\Cal{O}}C \cong \underline{\rm M}{\rm od}(\underline{\Cal{O}}\Cal{C})/{\rm Cardy}$ of this completion by the Cardy conditions, where Cardy conditions involve both the open and closed interactions. Isomorphisms restricted to closed and open parts give $\Cal{QC} \cong{\rm Mod}(\Cal{C}\text{om})$ and $\Cal{Q}\underline{\Cal{O}} \cong \underline{\rm Mod}(\underline{\Cal{A}{\rm ss}})$, respectively. The authors also give a 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 $\Cal{Q}\underline{\Cal{O}}C \cong \underline{\rm M}{\rm od}(\underline{\Cal{O}}\Cal{C})/{\rm Cardy}$. Furthermore, the cyclic hybrid $\underline{\Cal{O}}\Cal{C}$ 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 this paper is devoted to proving that the inclusions $\underline{\rm M}{\rm od}(\underline{\Cal{O}}\Cal{C}) \supseteq \underline{\rm M}{\rm od}(\underline{\Cal{O}}\Cal{C}_{\rm st}) \supseteq \underline{\rm M}{\rm od}(\underline{\Cal{O}}\Cal{C}_{\rm KP})$ hold. Thus, the stable and ${\rm KP}$ cases are treated as the restricted versions of the ordinary one.