The low energy expansion of the one-loop type II superstring amplitude

The one-loop four-graviton amplitude in either of the type II superstring theories is expanded in powers of the external momenta up to and including terms of order s^4 log s R^4, where R^4 denotes a specific contraction of four linearized Weyl tensors and s is a Mandelstam invariant. Terms in this series are obtained by integrating powers of the two-dimensional scalar field theory propagator over the toroidal world-sheet as well as the moduli of the torus. The values of these coefficients match expectations based on duality relations between string theory and eleven-dimensional supergravity.Comment: harvmac (b), 25 pages, 3 eps figures. v2: Factors of 2 corrected. Conclusion unchange

Homology representations arising from the half cube, II

In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${\Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of \sym_n agree (over ${\Bbb C}$) with the representations of \sym_n on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.Comment: 19 pages AMSTeX. One figure. The Conjecture in the previous version is now a Theorem. This research was supported by NSF grant DMS-090576

On rank functions for heaps

Motivated by work of Stembridge, we study rank functions for Viennot's heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups.Comment: 18 pages AMSTeX, 3 figure