Discrete Morse theory for the collapsibility of supremum sections

The Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman

Long String Scattering in c = 1 String Theory

We study the scattering of long strings in c = 1 string theory, both in the worldsheet description and in the non-singlet sector of the dual matrix quantum mechanics. From the worldsheet perspective, the scattering amplitudes of long strings are obtained from a decoupling limit of open strings amplitudes on FZZT branes, which we compute by integrating Virasoro conformal blocks along with structure constants of boundary Liouville theory. In particular, we study the tree level amplitudes of (1) a long string decaying by emitting a closed string, and (2) the scattering of a pair of long strings. We show that they are indeed well defined as limits of open string amplitudes, and that our results are in striking numerical agreement with computations in the adjoint and bi-adjoint sectors of the dual matrix model (based on proposals of Maldacena and solutions due to Fidkowski), thereby providing strong evidence of the duality.Comment: 42 pages, 18 figure

The c=1 String Theory S-Matrix Revisited

We revisit the perturbative S-matrix of c=1 string theory from the worldsheet perspective. We clarify the origin of the leg pole factors, the non-analyticity of the string amplitudes, and the validity as well as limitations of earlier computations based on resonance momenta. We compute the tree level 4-point amplitude and the genus one 2-point reflection amplitude by numerically integrating Virasoro conformal blocks with DOZZ structure constants on the sphere and on the torus, with sufficiently generic complex Liouville momenta, and find agreement with known answers from the c=1 matrix model.Comment: 32 pages, 7 figures; footnote and references added, typos correcte

Les relations canado-américaines : Nationalisme et continentalisme

This paper's objective is to bring forth some elements which confirm the following hypothesis : Canada is consigned to continentalism, namely to economic and cultural integration with the United States though this fact is shrouded in a Canadian nationalism of sorts. The continentalist mentality is rooted in the history of British North America, inhabited mostly by refugees from America who have remained inherently "Yankees" in spite of their anti-americanism. The Confederation itself is based on a sort of complicity with the United States. More recently there were talks of a "North American nationality", and continentalism both cultural and economic has come to be seen as a 'force of nature" which the governments, at the most, put into a chanelling process. Still, it is possible for Canadian nationalism to exist provided it does not go beyond the threshold whence it would run headlong into the continental mentality. Canada has defined itself through an international or non-national perspective far too long for today's nationalism not to remain weak and poorly established. But the Americans whose "manifest destiny" has succeeded in spreading over Canada without even their having tried to hoist their flag there find it to their advantage to maintain some form of Canadian sovereignty. Canada as a "friendly nation" can be of use to Washington. That is why there are almost as many advocates for Canada's independence in the United States as there are north of the border. Canadian nationalism can thus further the interests of some Canadian elites without seriously prejudicing continental integration which can very well afford not to be set up into formalized structures