Six-dimensional nilpotent Lie algebras

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent Lie algebras correspond to orbits of subspaces under this action. In some cases, these orbits are determined using geometric invariants, such as the Gram determinant or the Arf invariant. As a byproduct, we completely determine, for a 4-dimensional vector space $V$, the orbits of \GL(V) on the set of 2-dimensional subspaces of $V\wedge V$.Comment: Corrected a small error in Theorem 4.

Matrix factorizations and link homology II

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.Comment: 37 pages, 20 figures; version 2 corrects an inaccuracy in the proof of Proposition

Integrability vs non-integrability: Hard hexagons and hard squares compared

In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free-free boundary conditions up to sizes $40\times40$ and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point $z=-1$ of hard squares where all eigenvalues have unit modulus, and we give several conjectures for the value at $z=-1$ of the partition functions.Comment: 46 page

Toward the M(F)--Theory Embedding of Realistic Free-Fermion Models

We construct a Landau-Ginzburg model with the same data and symmetries as a $Z_2\times Z_2$ orbifold that corresponds to a class of realistic free-fermion models. Within the class of interest, we show that this orbifolding connects between different $Z_2\times Z_2$ orbifold models and commutes with the mirror symmetry. Our work suggests that duality symmetries previously discussed in the context of specific $M$ and $F$ theory compactifications may be extended to the special $Z_2\times Z_2$ orbifold that characterizes realistic free-fermion models.Comment: 15 pages. Standard Late