Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.Comment: 27 pages, 6 figures, revised versio

Entanglement Entropy of the Low-Lying Excited States and Critical Properties of an Exactly Solvable Two-Leg Spin Ladder with Three-Spin Interactions

In this work, we investigate an exactly solvable two-leg spin ladder with three-spin interactions. We obtain analytically the finite-size corrections of the low-lying energies and determine the central charge as well as the scaling dimensions. The model considered in this work has the same universality class of critical behavior of the XX chain with central charge c=1. By using the correlation matrix method, we also study the finite-size corrections of the Renyi entropy of the ground state and of the excited states. Our results are in agreement with the predictions of the conformal field theory.Comment: 10 pages, 6 figures, 2 table

Minimum Length from First Principles

We show that no device or gedanken experiment is capable of measuring a distance less than the Planck length. By "measuring a distance less than the Planck length" we mean, technically, resolve the eigenvalues of the position operator to within that accuracy. The only assumptions in our argument are causality, the uncertainty principle from quantum mechanics and a dynamical criteria for gravitational collapse from classical general relativity called the hoop conjecture. The inability of any gedanken experiment to measure a sub-Planckian distance suggests the existence of a minimal length.Comment: 8 pages, Honorable Mention in the 2005 Gravity Research Foundation Essay Competitio

Grand unification and enhanced quantum gravitational effects

In grand unified theories with large numbers of fields, renormalization effects significantly modify the scale at which quantum gravity becomes strong. This in turn can modify the boundary conditions for coupling constant unification, if higher dimensional operators induced by gravity are taken into consideration. We show that the generic size of these effects from gravity can be larger than the two-loop corrections typically considered in renormalization group analyses of unification. In some cases, gravitational effects of modest size can render unification impossible.Comment: 4 pages, 1 figure, revtex; minor changes in v2 (version published in Phys. Rev. Lett.