Relations Between Closed String Amplitudes at Higher-order Tree Level and Open String Amplitudes

KLT relations almost factorize closed string amplitudes on $S_2$ by two open string tree amplitudes which correspond to the left- and the right- moving sectors. In this paper, we investigate string amplitudes on $D_2$ and $RP_2$. We find that KLT factorization relations do not hold in these two cases. The relations between closed and open string amplitudes have new forms. On $D_2$ and $RP_2$, the left- and the right- moving sectors are connected into a single sector. Then an amplitude with closed strings on $D_2$ or $RP_2$ can be given by one open string tree amplitude except for a phase factor. The relations depends on the topologies of the world-sheets.Under T-duality, the relations on $D_2$ and $RP_2$ give the amplitudes between closed strings scattering from D-brane and O-plane respectively by open string partial amplitudes.In the low energy limits of these two cases, the factorization relations for graviton amplitudes do not hold. The amplitudes for gravitons must be given by the new relations instead.Comment: 19 page

The system of sets of lengths in Krull monoids under set addition

Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \in H$ has a factorization into irreducible elements, and the set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. We consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\mathcal L (H)$ is additively closed under set addition.Comment: Revista Matem{\'a}tica Iberoamericana, to appea

Holomorphic Factorization and Renormalization Group in Closed String Theory

The prescription of Kawai, Lewellen and Tye for writing the closed string tree amplitudes as sums of products of open string tree amplitudes, is applied to the world sheet renormalization group equation. The main point is that regularization of the Minkowski (rather than Euclidean) world sheet theory allows factorization into left-moving and right-moving sectors to be maintained. Explicit calculations are done for the tachyon and the (gauge-fixed) graviton.Comment: 19 pages, Latex File, 9 figure