A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d − 1 that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is d-collapsible is NP-complete for d ≥ 4 and polynomial time solvable for d ≤ 2. As an intermediate step, we prove that d-collaps...

In this paper we study half-supersymmetric (D6,D8) bound state brane configuration of massive type IIA supergravity. We show this bound state can also be generated by using massive T-duality rules of type II strings in D = 9 and starting from D7-branes. We write down corresponding Killing spinors and find that these backgrounds indeed preserve 16 supersymmetries like any other Dp-brane bound st...

We study T 2 compactification of massive type IIA supergravity in presence of possible Ramond-Ramond (RR) background fluxes. The resulting theory in D = 8 is shown to possess full SL(2, R) × SL(2, R) T-duality symmetry similar to the massless case. It is shown that elements of duality symmetry interpolate between massive type IIA compactified on T 2 and ordinary type IIA compactified on T 2 wit...

Using the boundary state formalism, we perform a microscopic string analysis of the interaction between two D-branes and provide a local interpretation for the R-R force in the D0-D8 brane system. To do so, we construct BRST invariant vertex operators for the massless R-R states in the asymmetric picture that are proportional to potentials rather than field strengths. The Hilbert space of such ...

We completely solve the problem of enumerating combinatorially inequivalent d-dimensional polytopes with d + 3 vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum’s book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on...