The Worldsheet Conformal Field Theory of the Fractional Superstring

Two of the important unresolved issues concerning fractional superstrings have been the appearance of new massive sectors whose spacetime statistics properties are unclear, and the appearance of new types of ``internal projections'' which alter or deform the worldsheet conformal field theory in a highly non-trivial manner. In this paper we provide a systematic analysis of these two connected issues, and explicitly map out the effective post-projection worldsheet theories for each of the fractional-superstring sectors. In this way we determine their central charges, highest weights, fusion rules, and characters, and find that these theories are isomorphic to those of free worldsheet bosons and fermions. We also analyze the recently-discovered parafermionic ``twist current'' which has been shown to play an important role in reorganizing the fractional-superstring Fock space, and find that this current can be expressed directly in terms of the primary fields of the post-projection theory. This thereby enables us to deduce some of the spacetime statistics properties of the surviving states.Comment: 56 pages (LaTeX), McGill/93-01. (discussion clarified in places, but results unchanged

Birational geometry of moduli space of del Pezzo pairs

In this paper, we investigate the geometry of moduli space $P_d$ of degree $d$ del Pezzo pair, that is, a del Pezzo surface $X$ of degree $d$ with a curve $C \sim -2K_X$. More precisely, we study compactifications for $P_d$ from both Hodge's theoretical and geometric invariant theoretical (GIT) perspective. We compute the Picard numbers of these compact moduli spaces which is an important step to set up the Hassett-Keel-Looijenga models for $P_d$. For $d=8$ case, we propose the Hassett-Keel-Looijenga program \cF_8(s)=\Proj(R(\cF_8,\Delta(s) ) as the section rings of certain \bQ-line bundle $\Delta_8(s)$ on locally symmetric variety \cF_8, which is birational to $P_8$. Moreover, we give an arithmetic stratification on \cF_8. After using the arithmetic computation of pullback $\Delta(s)$ on these arithmetic strata, we give the arithmetic predictions for the wall-crossing behavior of \cF_8(s) when $s\in [0,1]$ varies. The relation of \cF_8(s) with the K-moduli spaces of degree $8$ del Pezzo pairs is also proposed.Comment: 43 pages, comments are very welcome